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+CHAPTER 2
+Approaches to Valuation
+Analysts use a wide range of models in practice, ranging from the simple to the sophisticated. These models often make very different assumptions, but they do share some common characteristics and can be classified in broader terms. There are several advantages to such a classification: It makes it easier to understand where individual models fit into the big picture, why they provide different results, and when they have fundamental errors in logic.
+In general terms, there are three approaches to valuation. The first, discounted cash flow (DCF) valuation, relates the value of an asset to the present value (PV) of expected future cash flows on that asset. The second, relative valuation, estimates the value of an asset by looking at the pricing of comparable assets relative to a common variable such as earnings, cash flows, book value, or sales. The third, contingent claim valuation, uses option pricing models to measure the value of assets that share option characteristics. Some of these assets are traded financial assets like warrants, and some of these options are not traded and are based on real assets, (projects, patents, and oil reserves are examples). The latter are often called real options. There can be significant differences in outcomes, depending on which approach is used. One of the objectives in this book is to explain the reasons for such differences in value across different models, and to help in choosing the right model to use for a specific task.
+DISCOUNTED CASH FLOW VALUATION
+While discounted cash flow valuation is only one of the three ways of approaching valuation and most valuations done in the real world are relative valuations, it is the foundation on which all other valuation approaches are built. To do relative valuation correctly, we need to understand the fundamentals of discounted cash flow valuation. To apply option pricing models to value assets, we often have to begin with a discounted cash flow valuation. This is why so much of this book focuses on discounted cash flow valuation. Anyone who understands its fundamentals will be able to analyze and use the other approaches. This section considers the basis of this approach, a philosophical rationale for discounted cash flow valuation, and an examination of the different subapproaches to discounted cash flow valuation.
+Basis for Discounted Cash Flow Valuation
+This approach has its foundation in the present value rule, where the value of any asset is the present value of expected future cash flows on it. where n = Life of the asset
+CFt = Cash flow in period t
+r = Discount rate reflecting the riskiness of the estimated cash flows
+The cash flows will vary from asset to asset—dividends for stocks, coupons (interest) and the face value for bonds, and after-tax cash flows for a real project. The discount rate will be a function of the riskiness of the estimated cash flows, with higher rates for riskier assets and lower rates for safer projects.
+You can in fact think of discounted cash flow valuation on a continuum. At one end of the spectrum you have the default-free zero coupon bond, with a guaranteed cash flow in the future. Discounting this cash flow at the riskless rate should yield the value of the bond. A little further up the risk spectrum are corporate bonds where the cash flows take the form of coupons and there is default risk. These bonds can be valued by discounting the cash flows at an interest rate that reflects the default risk. Moving up the risk ladder, we get to equities, where there are expected cash flows with substantial uncertainty around the expectations. The value here should be the present value of the expected cash flows at a discount rate that reflects the uncertainty.
+Underpinnings of Discounted Cash Flow Valuation
+In discounted cash flow valuation, we try to estimate the intrinsic value of an asset based on its fundamentals. What is intrinsic value? For lack of a better definition, consider it the value that would be attached to the firm by an unbiased analyst, who not only estimates the expected cash flows for the firm correctly, given the information available at the time, but also attaches the right discount rate to value these cash flows. Hopeless though the task of estimating intrinsic value may seem to be, especially when valuing young companies with substantial uncertainty about the future, making the best estimates that you can and persevering to estimate value can still pay off because markets make mistakes. While market prices can deviate from intrinsic value (estimated based on fundamentals), you are hoping that the two will converge sooner rather than later.
+Categorizing Discounted Cash Flow Models
+There are literally thousands of discounted cash flow models in existence. Investment banks or consulting firms often claim that their valuation models are better or more sophisticated than those used by their contemporaries. Ultimately, however, discounted cash flow models can vary only a couple of dimensions.
+Equity Valuation and Firm Valuation
+There are two paths to valuation in a business: The first is to value just the equity stake in the business, while the second is to value the entire business, which includes, besides equity, the other claimholders in the firm (bondholders, preferred stockholders). While both approaches discount expected cash flows, the relevant cash flows and discount rates are different under each. Figure 2.1 captures the essence of the two approaches. Figure 2.1 Equity versus Firm Valuation The value of equity is obtained by discounting expected cash flows to equity (i.e., the residual cash flows after meeting all expenses, reinvestment needs, tax obligations, and interest and principal payments) at the cost of equity (i.e., the rate of return required by equity investors in the firm). where n = Life of the asset
+CF to equityt = Expected cash flow to equity in period t
+ke = Cost of equity
+The dividend discount model is a special case of equity valuation, where the value of equity is the present value of expected future dividends.
+The value of the firm is obtained by discounting expected cash flows to the firm (i.e., the residual cash flows after meeting all operating expenses, reinvestment needs, and taxes, but prior to any payments to either debt or equity holders) at the weighted average cost of capital (WACC), which is the cost of the different components of financing used by the firm, weighted by their market value proportions. where n = Life of the asset
+CF to firmt = Expected cash flow to firm in period t
+WACC = Weighted average cost of capital
+While these approaches use different definitions of cash flow and discount rates, they will yield consistent estimates of value for equity as long as you are consistent in your assumptions in valuation. The key error to avoid is mismatching cash flows and discount rates, since discounting cash flows to equity at the cost of capital will lead to an upwardly biased estimate of the value of equity, while discounting cash flows to the firm at the cost of equity will yield a downwardly biased estimate of the value of the firm. Illustration 2.1 shows the equivalence of equity and firm valuation. ILLUSTRATION 2.1: Effects of Mismatching Cash Flows and Discount Rates
+Assume that you are analyzing a company with the following cash flows for the next five years. Assume also that the cost of equity is 13.625% and the firm can borrow long term at 10%. (The tax rate for the firm is 50%.) The current market value of equity is $1,073, and the value of debt outstanding is $800. The cost of equity is given as an input and is 13.625%, and the after-tax cost of debt is 5%. Given the market values of equity and debt, we can estimate the cost of capital. METHOD 1: DISCOUNT CASH FLOWS TO EQUITY AT COST OF EQUITY TO GET VALUE OF EQUITY
+We discount cash flows to equity at the cost of equity: METHOD 2: DISCOUNT CASH FLOWS TO FIRM AT COST OF CAPITAL TO GET VALUE OF FIRM Note that the value of equity is $1,073 under both approaches. It is easy to make the mistake of discounting cash flows to equity at the cost of capital or the cash flows to the firm at the cost of equity.
+ERROR 1: DISCOUNT CASH FLOWS TO EQUITY AT COST OF CAPITAL TO GET TOO HIGH A VALUE FOR EQUITY ERROR 2: DISCOUNT CASH FLOWS TO FIRM AT COST OF EQUITY TO GET TOO LOW A VALUE FOR THE FIRM The effects of using the wrong discount rate are clearly visible in the last two calculations (Error 1 and Error 2). When the cost of capital is mistakenly used to discount the cash flows to equity, the value of equity increases by $175 over its true value ($1,073). When the cash flows to the firm are erroneously discounted at the cost of equity, the value of the firm is understated by $260. It must be pointed out, though, that getting the values of equity to agree with the firm and equity valuation approaches can be much more difficult in practice than in this example. We return to this subject in Chapters 14 and 15 and consider the assumptions that we need to make to arrive at this result. Cost of Capital versus APV Approaches In Figure 2.1, we noted that a firm can finance its assets, using either equity or debt. What are the effects of using debt on value? On the plus side, the tax deductibility of interest expenses provides a tax subsidy or benefit to the firm, which increases with the tax rate faced by the firm on its income. On the minus side, debt does increase the likelihood that the firm will default on its commitments and be forced into bankruptcy. The net effect can be positive, neutral or negative. In the cost of capital approach, we capture the effects of debt in the discount rate: Cost of capital =
+Cost of equity(Proportion of equity used to fund business) + Pretax cost of debt (1 – Tax rate) (Proportion of debt used to fund business) The cash flows discounted are predebt cash flows and do not include any of the tax benefits of debt (since that would be double counting).
+In a variation, called the adjusted present value (APV) approach, we separate the effects on value of debt financing from the value of the assets of a business. Thus, we start by valuing the business as if it were all equity funded and assess the effect of debt separately, by first valuing the tax benefits from the debt and then subtracting out the expected bankruptcy costs. Value of business =
+Value of business with 100% equity financing
++ Present value of expected tax benefits of debt
+- Expected bankruptcy costs While the two approaches take different tacks to evaluating the value added or destroyed by debt, they will provide the same estimate of value, if we are consistent in our assumptions about cash flows and risk. In chapter 15, we will return to examine these approaches in more detail.
+Total Cash Flow versus Excess Cash Flow Models
+The conventional discounted cash flow model values an asset by estimating the present value of all cash flows generated by that asset at the appropriate discount rate. In excess return (and excess cash flow) models, only cash flows earned in excess of the required return are viewed as value creating, and the present value of these excess cash flows can be added to the amount invested in the asset to estimate its value. To illustrate, assume that you have an asset in which you invested $100 million and that you expect to generate $12 million in after-tax cash flows in perpetuity. Assume further that the cost of capital on this investment is 10 percent. With a total cash flow model, the value of this asset can be estimated as follows:
+Value of asset = $12 million/.1 = $120 million
+With an excess return model, we would first compute the excess return made on this asset: Excess return
+= Cash flow earned – Cost of capital × Capital invested in asset
+= $12 million – .10 × $100 million = $2 million A SIMPLE TEST OF CASH FLOWS
+There is a simple test that can be employed to determine whether the cash flows being used in a valuation are cash flows to equity or cash flows to the firm. If the cash flows that are being discounted are after interest expenses (and principal payments), they are cash flows to equity and the discount rate used should be the cost of equity. If the cash flows that are discounted are before interest expenses and principal payments, they are usually cash flows to the firm. Needless to say, there are other items that need to be considered when estimating these cash flows, and they are considered in extensive detail in the coming chapters. We then add the present value of these excess returns to the investment in the asset: Value of asset
+= Present value of excess return + Investment in the asset
+= $2 million/.1 + $100 million = $120 million Note that the answers in the two approaches are equivalent. Why, then, would we want to use an excess return model? By focusing on excess returns, this model brings home the point that it is not earnings per se that create value, but earnings in excess of a required return. Chapter 32 considers special versions of these excess return models. As in this simple example, with consistent assumptions, total cash flow and excess return models are equivalent.
+Applicability and Limitations of Discounted Cash Flow Valuation
+Discounted cash flow valuation is based on expected future cash flows and discount rates. Given these estimation requirements, this approach is easiest to use for assets (firms) whose cash flows are currently positive and can be estimated with some reliability for future periods, and where a proxy for risk that can be used to obtain discount rates is available. The further we get from this idealized setting, the more difficult (and more useful) discounted cash flow valuation becomes. Here are some scenarios where discounted cash flow valuation might run into trouble and need to be adapted.
+Firms in Trouble
+A distressed firm generally has negative earnings and cash flows, and expects to lose money for some time in the future. For these firms, estimating future cash flows is difficult to do, since there is a strong probability of bankruptcy. For firms that are expected to fail, discounted cash flow valuation does not work very well, since the method values the firm as a going concern providing positive cash flows to its investors. Even for firms that are expected to survive, cash flows will have to be estimated until they turn positive, since obtaining a present value of negative cash flows will yield a negative value for equity1 or for the firm. We will examine these firms in more detail in chapters 22 and 30.
+Cyclical Firms
+The earnings and cash flows of cyclical firms tend to follow the economy—rising during economic booms and falling during recessions. If discounted cash flow valuation is used on these firms, expected future cash flows are usually smoothed out, unless the analyst wants to undertake the onerous task of predicting the timing and duration of economic recessions and recoveries. In the depths of a recession many cyclical firms look like troubled firms, with negative earnings and cash flows. Estimating future cash flows then becomes entangled with analyst predictions about when the economy will turn and how strong the upturn will be, with more optimistic analysts arriving at higher estimates of value. This is unavoidable, but the economic biases of the analysts have to be taken into account before using these valuations.
+Firms with Unutilized Assets
+Discounted cash flow valuation reflects the value of all assets that produce cash flows. If a firm has assets that are unutilized (and hence do not produce any cash flows), the value of these assets will not be reflected in the value obtained from discounting expected future cash flows. The same caveat applies, in lesser degree, to underutilized assets, since their value will be understated in discounted cash flow valuation. While this is a problem, it is not insurmountable. The value of these assets can always be obtained externally2 and added to the value obtained from discounted cash flow valuation. Alternatively, the assets can be valued as though they are used optimally.
+Firms with Patents or Product Options
+Firms sometimes have unutilized patents or licenses that do not produce any current cash flows and are not expected to produce cash flows in the near future, but are valuable nevertheless. If this is the case, the value obtained from discounting expected cash flows to the firm will understate the true value of the firm. Again, the problem can be overcome, by valuing these assets in the open market or by using option pricing models, and then adding the value obtained from discounted cash flow valuation. Chapter 28 examines the use of option pricing models to value patents.
+Firms in the Process of Restructuring
+Firms in the process of restructuring often sell some of their assets, acquire other assets, and change their capital structure and dividend policy. Some of them also change their ownership structure (going from publicly traded to private status and vice versa) and management compensation schemes. Each of these changes makes estimating future cash flows more difficult and affects the riskiness of the firm. Using historical data for such firms can give a misleading picture of the firm's value. However, these firms can be valued, even in the light of the major changes in investment and financing policy, if future cash flows reflect the expected effects of these changes and the discount rate is adjusted to reflect the new business and financial risk in the firm. Chapter 31 takes a closer look at how value can be altered by changing the way a business is run.
+Firms Involved in Acquisitions
+There are at least two specific issues relating to acquisitions that need to be taken into account when using discounted cash flow valuation models to value target firms. The first is the thorny one of whether there is synergy in the merger and how its value can be estimated. To do so will require assumptions about the form the synergy will take and its effect on cash flows. The second, especially in hostile takeovers, is the effect of changing management on cash flows and risk. Again, the effect of the change can and should be incorporated into the estimates of future cash flows and discount rates and hence into value. Chapter 25 looks at the value of synergy and control in acquisitions.
+Private Firms
+The biggest problem in using discounted cash flow valuation models to value private firms is the measurement of risk (to use in estimating discount rates), since most risk/return models require that risk parameters be estimated from historical prices on the asset being analyzed and make assumptions about the profiles of investors in the firm that may not fit private businesses. One solution is to look at the riskiness of comparable firms that are publicly traded. The other is to relate the measure of risk to accounting variables, which are available for the private firm. Chapter 24 looks at adaptations to valuation models that are needed to value private businesses.
+The point is not that discounted cash flow valuation cannot be done in these cases, but that we have to be flexible enough to adapt our models. The fact is that valuation is simple for firms with well-defined assets that generate cash flows that can be easily forecasted. The real challenge in valuation is to extend the valuation framework to cover firms that vary to some extent or the other from this idealized framework. Much of this book is spent considering how to value such firms.
+RELATIVE VALUATION
+While we tend to focus most on discounted cash flow valuation when discussing valuation, the reality is that most valuations are relative valuations. The values of most assets, from the house you buy to the stocks you invest in, are based on how similar assets are priced in the marketplace. This section begins with a basis for relative valuation, moves on to consider the underpinnings of the model, and then considers common variants within relative valuation.
+Basis for Relative Valuation
+In relative valuation, the value of an asset is derived from the pricing of comparable assets, standardized using a common variable such as earnings, cash flows, book value, or revenues. One illustration of this approach is the use of an industry-average price-earnings ratio to value a firm, the assumption being that the other firms in the industry are comparable to the firm being valued and that the market, on average, prices these firms correctly. Another multiple in wide use is the price–book value ratio, with firms selling at a discount on book value relative to comparable firms being considered undervalued. Revenue multiple are also used to value firms, with the average price-sales ratios of firms with similar characteristics being used for comparison. While these three multiples are among the most widely used, there are others that also play a role in analysis—EV to EBITDA, EV to invested capital, and market value to replacement value (Tobin's Q), to name a few.
+Underpinnings of Relative Valuation
+Unlike discounted cash flow valuation, which is a search for intrinsic value, relative valuation relies much more on the market being right. In other words, we assume that the market is correct in the way it prices stocks on average, but that it makes errors on the pricing of individual stocks. We also assume that a comparison of multiples will allow us to identify these errors, and that these errors will be corrected over time.
+The assumption that markets correct their mistakes over time is common to both discounted cash flow and relative valuation, but those who use multiples and comparables to pick stocks argue, with some basis, that errors made in pricing individual stocks in a sector are more noticeable and more likely to be corrected quickly. For instance, they would argue that a software firm that trades at a price-earnings ratio of 10 when the rest of the sector trades at 25 times earnings is clearly undervalued and that the correction toward the sector average should occur sooner rather than later. Proponents of discounted cash flow valuation would counter that this is small consolation if the entire sector is overpriced by 50 percent.
+Categorizing Relative Valuation Models
+Analysts and investors are endlessly inventive when it comes to using relative valuation. Some compare multiples across companies, while other compare the multiple of a company to the multiples it used to trade at in the past. While most relative valuations are based on the pricing of comparable assets at the same time, there are some relative valuations that are based on fundamentals.
+Fundamentals versus Comparables
+In discounted cash flow valuation, the value of a firm is determined by its expected cash flows. Other things remaining equal, higher cash flows, lower risk, and higher growth should yield higher value. Some analysts who use multiples go back to these discounted cash flow models to extract multiples. Other analysts compare multiples across firms or time and make explicit or implicit assumptions about how firms are similar or vary on fundamentals.
+Using Fundamentals
+The first approach relates multiples to fundamentals about the firm being valued—growth rates in earnings and cash flows, reinvestment and risk. This approach to estimating multiples is equivalent to using discounted cash flow models, requiring the same information and yielding the same results. Its primary advantage is that it shows the relationship between multiples and firm characteristics, and allows us to explore how multiples change as these characteristics change. For instance, what will be the effect of changing profit margins on the price-sales ratio? What will happen to price-earnings ratios as growth rates decrease? What is the relationship between price–book value ratios and return on equity?
+Using Comparables
+The more common approach to using multiples is to compare how a firm is valued with how similar firms are priced by the market or, in some cases, with how the firm was valued in prior periods. As we see in the later chapters, finding similar and comparable firms is often a challenge, and frequently we have to accept firms that are different from the firm being valued on one dimension or the other. When this is the case, we have to either explicitly or implicitly control for differences across firms on growth, risk, and cash flow measures. In practice, controlling for these variables can range from the naive (using industry averages) to the sophisticated (multivariate regression models where the relevant variables are identified and controlled for).
+Cross-Sectional versus Time Series Comparisons
+In most cases, analysts price stocks on a relative basis, by comparing the multiples they are trading at to the multiples at which other firms in the same business are trading at contemporaneously. In some cases, however, especially for mature firms with long histories, the comparison is done across time.
+Cross-Sectional Comparisons
+When we compare the price-earnings ratio of a software firm to the average price-earnings ratio of other software firms, we are doing relative valuation and we are making cross-sectional comparisons. The conclusions can vary depending on our assumptions about the firm being valued and the comparable firms. For instance, if we assume that the firm we are valuing is similar to the average firm in the industry, we would conclude that it is cheap if it trades at a multiple that is lower than the average multiple. If, however, we assume that the firm being valued is riskier than the average firm in the industry, we might conclude that the firm should trade at a lower multiple than other firms in the business. In short, you cannot compare firms without making assumptions about their fundamentals.
+Comparisons across Time
+If you have a mature firm with a long history, you can compare the multiple it trades at today to the multiple it used to trade at in the past. Thus, Ford Motor Company may be viewed as cheap because it trades at six times earnings, if it has historically traded at 10 times earnings. To make this comparison, however, you have to assume that your firm's fundamentals have not changed over time. For instance, you would expect a high-growth firm's price-earnings ratio to drop over time and its expected growth rate to decrease as it becomes larger. Comparing multiples across time can also be complicated by changes in interest rates and the behavior of the overall market. For instance, as interest rates fall below historical norms and the overall market increases in value, you would expect most companies to trade at much higher multiples of earnings and book value than they have historically.
+Applicability and Limitations of Multiples
+The allure of multiples is that they are simple and easy to relate to. They can be used to obtain estimates of value quickly for firms and assets, and are particularly useful when a large number of comparable firms are traded on financial markets, and the market is, on average, pricing these firms correctly. They tend to be more difficult to use to value unique firms with no obvious comparables, with little or no revenues, and with negative earnings.
+By the same token, multiples are also easy to misuse and manipulate, especially when comparable firms are used. Given that no two firms are exactly alike in terms of risk and growth, the definition of comparable firms is a subjective one. Consequently, a biased analyst can choose a group of comparable firms to confirm his or her biases about a firm's value. Illustration 2.2 shows an example. While this potential for bias exists with discounted cash flow valuation as well, the analyst in DCF valuation is forced to be much more explicit about the assumptions that determine the final value. With multiples, these assumptions are often left unstated. ASSET-BASED VALUATION MODELS
+There are some analysts who add a fourth approach to valuation to the three described in this chapter. They argue that you can value the individual assets owned by a firm and aggregate them to arrive at a firm value—asset-based valuation models. In fact, there are several variants on asset-based valuation models. The first is liquidation value, which is obtained by aggregating the estimated sale proceeds of the assets owned by a firm. The second is replacement cost, where you estimate what it would cost you to replace all of the assets that a firm has today. The third is the simplest: use accounting book value as the measure of the value of the assets, with adjustments to the book value made where necessary. While analysts may use asset-based valuation approaches to estimate value, they are not alternatives to discounted cash flow, relative, or option pricing models since both replacement and liquidation values have to be obtained using one or another of these approaches. Ultimately, all valuation models attempt to value assets; the differences arise in how we identify the assets and how we attach value to each asset. In liquidation valuation, we look only at assets in place and estimate their value based on what similar assets are priced at in the market. In traditional discounted cash flow valuation, we consider all assets and include expected growth potential to arrive at value. The two approaches may, in fact, yield the same values if you have a firm that has no growth potential and the market assessments of value reflect expected cash flows. ILLUSTRATION 2.2: The Potential for Misuse with Comparable Firms
+Assume that an analyst is valuing an initial public offering (IPO) of a firm that manufactures computer software. At the same time,3 the price-earnings multiples of other publicly traded firms manufacturing software are: Firm
+Multiple Adobe Systems
+23.2 Autodesk
+20.4 Broderbund
+32.8 Computer Associates
+18.0 Lotus Development
+24.1 Microsoft
+27.4 Novell
+30.0 Oracle
+37.8 Software Publishing
+10.6 System Software
+15.7 Average PE ratio
+24.0 While the average PE ratio using the entire sample is 24, it can be changed markedly by removing a couple of firms from the group. For instance, if the two firms with the lowest PE ratios in the group (Software Publishing and System Software) are eliminated from the sample, the average PE ratio increases to 27. If the two firms with the highest PE ratios in the group (Broderbund and Oracle) are removed from the group, the average PE ratio drops to 21. The other problem with using multiples based on comparable firms is that it builds in errors (overvaluation or undervaluation) that the market might be making in valuing these firms. In Illustration 2.2, for instance, if the market has overvalued all computer software firms, using the average PE ratio of these firms to value an initial public offering will lead to an overvaluation of the IPO stock. In contrast, discounted cash flow valuation is based on firm-specific growth rates and cash flows, so it is less likely to be influenced by market errors in valuation.
+CONTINGENT CLAIM VALUATION
+Perhaps the most revolutionary development in valuation is the acceptance, at least in some cases, that the value of an asset may be greater than the present value of expected cash flows if the cash flows are contingent on the occurrence or nonoccurrence of an event. This acceptance has largely come about because of the development of option pricing models. While these models were initially used to value traded options, there has been an attempt in recent years to extend the reach of these models into more traditional valuation. There are many who argue that assets such as patents or undeveloped reserves are really options and should be valued as such, rather than with traditional discounted cash flow models.
+Basis for Approach
+A contingent claim or option is a claim that pays off only under certain contingencies—if the value of the underlying asset exceeds a prespecified value for a call option or is less than a prespecified value for a put option. Much work has been done in the past 20 years in developing models that value options, and these option pricing models can be used to value any assets that have optionlike features.
+Figure 2.2 illustrates the payoffs on call and put options as a function of the value of the underlying asset. An option can be valued as a function of the following variables: the current value and the variance in value of the underlying asset, the strike price and the time to expiration of the option, and the riskless interest rate. This was first established by Fischer Black and Myron Scholes in 1972 and has been extended and refined subsequently in numerous variants. While the Black-Scholes option pricing model ignore dividends and assumes that options will not be exercised early, it can be modified to allow for both. A discrete-time variant, the binomial option pricing model, has also been developed to price options. Figure 2.2 Payoff Diagram on Call and Put Options An asset can be valued as an option if the payoffs are a function of the value of an underlying asset. It can be valued as a call option if when that value exceeds a prespecified level the asset is worth the difference. It can be valued as a put option if it gains value as the value of the underlying asset drops below a prespecified level, and if it is worth nothing when the underlying asset's value exceeds that specified level.
+Underpinnings of Contingent Claim Valuation
+The fundamental premise behind the use of option pricing models is that discounted cash flow models tend to understate the value of assets that provide payoffs that are contingent on the occurrence of an event. As a simple example, consider an undeveloped oil reserve belonging to Petrobras. You could value this reserve based on expectations of oil prices in the future, but this estimate would miss the fact that the oil company will develop this reserve only if oil prices go up and will not if oil prices decline. An option pricing model would yield a value that incorporates this right.
+When we use option pricing models to value assets such as patents and undeveloped natural resource reserves, we are assuming that markets are sophisticated enough to recognize such options and incorporate them into the market price. If the markets do not do so right now, we assume that they will eventually; the payoff to using such models comes about when this correction occurs.
+Categorizing Option Pricing Models
+The first categorization of options is based on whether the underlying asset is a financial asset or a real asset. Most listed options, whether they be options listed on the Chicago Board Options Exchange or callable fixed income securities, are on financial assets such as stocks and bonds. In contrast, options can be on real assets such as commodities, real estate, or even investment projects; such options are often called real options.
+A second and overlapping categorization is based on whether the underlying asset is traded. The overlap occurs because most financial assets are traded, whereas relatively few real assets are traded. Options on traded assets are generally easier to value, and the inputs to the option pricing models can be obtained from financial markets. Options on nontraded assets are much more difficult to value, since there are no market inputs available on the underlying assets.
+Applicability and Limitations of Option Pricing Models
+There are several direct examples of securities that are options—LEAPS, which are long-term equity options on traded stocks; contingent value rights, which provide protection to stockholders in companies against stock price declines; and warrants, which are long-term call options issued by firms.
+There are other assets that generally are not viewed as options but still share several option characteristics. Equity, for instance, can be viewed as a call option on the value of the underlying firm, with the face value of debt representing the strike price and the term of the debt measuring the life of the option. A patent can be analyzed as a call option on a product, with the investment outlay needed to get the project going considered the strike price and the patent life becoming the time to expiration of the option.
+There are limitations in using option pricing models to value long-term options on nontraded assets. The assumptions made about constant variance and dividend yields, which are not seriously contested for short-term options, are much more difficult to defend when options have long lifetimes. When the underlying asset is not traded, the inputs for the value of the underlying asset and the variance in that value cannot be extracted from financial markets and have to be estimated. Thus the final values obtained from these applications of option pricing models have much more estimation error associated with them than the values obtained in their more standard applications (to value short-term traded options).
+CONCLUSION
+There are three basic, though not mutually exclusive, approaches to valuation. The first is discounted cash flow valuation, where cash flows are discounted at a risk-adjusted discount rate to arrive at an estimate of value. The analysis can be done purely from the perspective of equity investors by discounting expected cash flows to equity at the cost of equity, or it can be done from the viewpoint of all claimholders in the firm, by discounting expected cash flows to the firm at the weighted average cost of capital. The second is relative valuation, where the value of an asset is based on the pricing of similar assets. The third is contingent claim valuation, where an asset with the characteristics of an option is valued using an option pricing model. There should be a place for each among the tools available to any analyst interested in valuation.
+QUESTIONS AND SHORT PROBLEMS
+In the problems following, use an equity risk premium of 5.5 percent if none is specified. 1. Discounted cash flow valuation is based on the notion that the value of an asset is the present value of the expected cash flows on that asset, discounted at a rate that reflects the riskiness of those cash flows. Specify whether the following statements about discounted cash flow valuation are true or false, assuming that all variables are constant except for the one mentioned: a. As the discount rate increases, the value of an asset increases.
+True____False____
+b. As the expected growth rate in cash flows increases, the value of an asset increases.
+True____False____
+c. As the life of an asset is lengthened, the value of that asset increases.
+True____False____
+d. As the uncertainty about the expected cash flow increases, the value of an asset increases.
+True____False____
+e. An asset with an infinite life (i.e., it is expected to last forever) will have an infinite value.
+True____False____ 2. Why might discounted cash flow valuation be difficult to do for the following types of firms? a. A private firm, where the owner is planning to sell the firm.
+b. A biotechnology firm with no current products or sales, but with several promising product patents in the pipeline.
+c. A cyclical firm during a recession.
+d. A troubled firm that has made significant losses and is not expected to get out of trouble for a few years.
+e. A firm that is in the process of restructuring, where it is selling some of its assets and changing its financial mix.
+f. A firm that owns a lot of valuable land that is currently unutilized. 3. The following are the projected cash flows to equity and to the firm over the next five years: The firm has a cost of equity of 12% and a cost of capital of 9.94%. Answer the following questions: a. What is the value of the equity in this firm?
+b. What is the value of the firm? 4. You are estimating the price-earnings multiple to use to value Paramount Corporation by looking at the average price-earnings multiple of comparable firms. The following are the price-earnings ratios of firms in the entertainment business. Firm
+PE Ratio Disney (Walt)
+22.09 Time Warner
+36.00 King World Productions
+14.10 New Line Cinema
+26.70 a. What is the average PE ratio?
+b. Would you use all the comparable firms in calculating the average? Why or why not?
+c. What assumptions are you making when you use the industry-average PE ratio to value Paramount Corporation? 1 The protection of limited liability should ensure that no stock will sell for less than zero. The price of such a stock can never be negative.
+2 If these assets are traded on external markets, the market prices of these assets can be used in the valuation. If not, the cash flows can be projected, assuming full utilization of assets, and the value can be estimated.
+3 These were the PE ratios for these firms at the end of 1992.
+
+HAPTER 3Understanding Financial Statements
+Financial statements provide the fundamental information that we use to analyze and answer valuation questions. It is important, therefore, that we understand the principles governing these statements by looking at four questions: 1. How valuable are the assets of a firm? The assets of a firm can come in several forms—assets with long lives such as land and buildings, assets with shorter lives such as inventory, and intangible assets that nevertheless produce revenues for the firm such as patents and trademarks.
+2. How did the firm raise the funds to finance these assets? In acquiring assets, firms can use the funds of the owners (equity) or borrowed money (debt), and the mix is likely to change as the assets age.
+3. How profitable are these assets? A good investment is one that makes a return greater than the cost of funding it. To evaluate whether the investments that a firm has already made are good investments, we need to estimate what returns these investments are producing.
+4. How much uncertainty (or risk) is embedded in these assets? While we have not yet directly confronted the issue of risk, estimating how much uncertainty there is in existing investments, and the implications for a firm, is clearly a first step. This chapter looks at the way accountants would answer these questions, and why the answers might be different when doing valuation. Some of these differences can be traced to the differences in objectives: Accountants try to measure the current standing and immediate past performance of a firm, whereas valuation is much more forward-looking.
+THE BASIC ACCOUNTING STATEMENTS
+There are three basic accounting statements that summarize information about a firm. The first is the balance sheet, shown in Figure 3.1, which summarizes the assets owned by a firm, the value of these assets, and the mix of financing (debt and equity) used to finance these assets at a point in time. Figure 3.1 The Balance Sheet The next is the income statement, shown in Figure 3.2, which provides information on the revenues and expenses of the firm, and the resulting income made by the firm, during a period. The period can be a quarter (if it is a quarterly income statement) or a year (if it is an annual report). Figure 3.2 Income Statement Finally, there is the statement of cash flows, shown in Figure 3.3, which specifies the sources and uses of cash to the firm from operating, investing, and financing activities during a period. The statement of cash flows can be viewed as an attempt to explain what the cash flows during a period were, and why the cash balance changed during the period. Figure 3.3 Statement of Cash Flows ASSET MEASUREMENT AND VALUATION
+When analyzing any firm, we want to know the types of assets that it owns, the value of these assets, and the degree of uncertainty about this value. Accounting statements do a reasonably good job of categorizing the assets owned by a firm, a partial job of assessing the value of these assets, and a poor job of reporting uncertainty about asset value. This section begins by looking at the accounting principles underlying asset categorization and measurement, and the limitations of financial statements in providing relevant information about assets.
+Accounting Principles Underlying Asset Measurement
+An asset is any resource that has the potential either to generate future cash inflows or to reduce future cash outflows. While that is a general definition broad enough to cover almost any kind of asset, accountants add a caveat that for a resource to be an asset a firm has to have acquired it in a prior transaction and be able to quantify future benefits with reasonable precision. The accounting view of asset value is to a great extent grounded in the notion of historical cost, which is the original cost of the asset, adjusted upward for improvements made to the asset since purchase and downward for the loss in value associated with the aging of the asset. This historical cost is called the book value. While the generally accepted accounting principles (GAAP) for valuing an asset vary across different kinds of assets, three principles underlie the way assets are valued in accounting statements: 1. An abiding belief in book value as the best estimate of value. Accounting estimates of asset value begin with the book value, and unless a substantial reason is given to do otherwise, accountants view the historical cost as the best estimate of the value of an asset.
+2. A distrust of market or estimated value. When a current market value exists for an asset that is different from the book value, accounting convention seems to view this market value with suspicion. The market price of an asset is often viewed as both much too volatile and too easily manipulated to be used as an estimate of value for an asset. This suspicion runs even deeper when a value is estimated for an asset based on expected future cash flows.
+3. A preference for underestimating value rather than overestimating it. When there is more than one approach to valuing an asset, accounting convention takes the view that the more conservative (lower) estimate of value should be used rather than the less conservative (higher) estimate of value. Thus, when both market and book value are available for an asset, accounting rules often require that you use the lesser of the two numbers. Measuring Asset Value
+The financial statement in which accountants summarize and report asset value is the balance sheet. To examine how asset value is measured, let us begin with the way assets are categorized in the balance sheet. First there are the fixed assets, which include the long-term assets of the firm, such as plant, equipment, land, and buildings. Next, we have the short-term assets of the firm, including inventory (raw materials, work in progress, and finished goods, receivables (summarizing moneys owed to the firm), and cash; these are categorized as current assets. We then have investments in the assets and securities of other firms, which are generally categorized as financial investments. Finally, we have what is loosely categorized as intangible assets. These include not only assets such as patents and trademarks that presumably will create future earnings and cash flows, but also uniquely accounting assets such as goodwill that arise because of acquisitions made by the firm.
+Fixed Assets
+Generally accepted accounting principles (GAAP) in the United States require the valuation of fixed assets at historical cost, adjusted for any estimated loss in value from the aging of these assets. While in theory the adjustments for aging should reflect the loss of earning power of the asset as it ages, in practice they are much more a product of accounting rules and convention, and these adjustments are called depreciation. Depreciation methods can very broadly be categorized into straight line (where the loss in asset value is assumed to be the same every year over its lifetime) and accelerated (where the asset loses more value in the earlier years and less in the later years). While tax rules, at least in the United States, have restricted the freedom that firms have on their choices of asset life and depreciation methods, firms continue to have a significant amount of flexibility on these decisions for reporting purposes. Thus, the depreciation that is reported in the annual reports may not be, and generally is not, the same depreciation that is used in the tax statements.
+Since fixed assets are valued at book value and are adjusted for depreciation provisions, the value of a fixed asset is strongly influenced by both its depreciable life and the depreciation method used. Many firms in the United States use straight-line depreciation for financial reporting while using accelerated depreciation for tax purposes, since firms can report better earnings with the former, at least in the years right after the asset is acquired.1 In contrast, firms in other countries often use accelerated depreciation for both tax and financial reporting purposes, leading to reported income that is understated relative to that of their U.S. counterparts.
+Current Assets
+Current assets include inventory, cash, and accounts receivable. It is in this category that accountants are most amenable to the use of market value, especially in valuing marketable securities.
+Accounts Receivable
+Accounts receivable represent money owed by entities to the firm on the sale of products on credit. When the Home Depot sells products to building contractors and gives them a few weeks to make their payments, it is creating accounts receivable. The accounting convention is for accounts receivable to be recorded as the amount owed to the firm based on the billing at the time of the credit sale. The only major valuation and accounting issue is when the firm has to recognize accounts receivable that are not collectible. Firms can set aside a portion of their income to cover expected bad debts from credit sales, and accounts receivable will be reduced by this reserve. Alternatively, the bad debts can be recognized as they occur, and the firm can reduce the accounts receivable accordingly. There is the danger, however, that absent a decisive declaration of a bad debt, firms may continue to show as accounts receivable amounts that they know are unlikely ever to be collected.
+Cash
+Cash is one of the few assets for which accountants and financial analysts should agree on value. The value of a cash balance should not be open to estimation error. Having said this, we note that fewer and fewer companies actually hold cash in the conventional sense (as currency or as demand deposits in banks). Firms often invest the cash in interest-bearing accounts, commercial paper or in Treasuries so as to earn a return on their investments. In either case, market value can sometimes deviate from book value. While there is minimal default risk in either of these investments, interest rate movements can affect their value. The valuation of marketable securities is examined later in this section.
+Inventory
+Three basic approaches to valuing inventory are allowed by GAAP: first in, first out (FIFO); last in, first out (LIFO); and weighted average. 1. First in, first out (FIFO). Under FIFO, the cost of goods sold is based on the cost of material bought earliest in the period, while the cost of inventory is based on the cost of material bought later in the year. This results in inventory being valued close to current replacement cost. During periods of inflation, the use of FIFO will result in the lowest estimate of cost of goods sold among the three valuation approaches, and the highest net income.
+2. Last in, first out (LIFO). Under LIFO, the cost of goods sold is based on the cost of material bought toward the end of the period, resulting in costs that closely approximate current costs. The inventory, however, is valued on the basis of the cost of materials bought earlier in the year. During periods of inflation, the use of LIFO will result in the highest estimate of cost of goods sold among the three approaches, and the lowest net income.
+3. Weighted average. Under the weighted average approach, both inventory and the cost of goods sold are based on the average cost of all material bought during the period. When inventory turns over rapidly, this approach will more closely resemble FIFO than LIFO. Firms often adopt the LIFO approach for its tax benefits during periods of high inflation. The cost of goods sold is then higher because it is based on prices paid toward to the end of the accounting period. This, in turn, will reduce the reported taxable income and net income while increasing cash flows. Studies indicate that larger firms with rising prices for raw materials and labor, more variable inventory growth, and an absence of other tax loss carryforwards are much more likely to adopt the LIFO approach.
+Given the income and cash flow effects of inventory valuation methods, it is often difficult to compare the profitability of firms that use different methods. There is, however, one way of adjusting for these differences. Firms that choose the LIFO approach to value inventories have to specify in a footnote the difference in inventory valuation between FIFO and LIFO, and this difference is termed the LIFO reserve. It can be used to adjust the beginning and ending inventories, and consequently the cost of goods sold, and to restate income based on FIFO valuation.
+Investments (Financial) and Marketable Securities
+In the category of investments and marketable securities, accountants consider investments made by firms in the securities or assets of other firms, as well as other marketable securities, including Treasury bills or bonds. The way in which these assets are valued depends on the way the investment is categorized and the motive behind the investment. In general, an investment in the securities of another firm can be categorized as a minority passive investment, a minority active investment, or a majority active investment, and the accounting rules vary depending on the categorization.
+Minority Passive Investments
+If the securities or assets owned in another firm represent less than 20 percent of the overall ownership of that firm, an investment is treated as a minority passive investment. These investments have an acquisition value, which represents what the firm originally paid for the securities, and often a market value. Accounting principles require that these assets be subcategorized into one of three groups—investments that will be held to maturity, investments that are available for sale, and trading investments. The valuation principles vary for each. For an investment that will be held to maturity, the valuation is at historical cost or book value, and interest or dividends from this investment are shown in the income statement.
+For an investment that is available for sale, the valuation is at market value, but the unrealized gains or losses are shown as part of the equity in the balance sheet and not in the income statement. Thus, unrealized losses reduce the book value of the equity in the firm, and unrealized gains increase the book value of equity.
+For a trading investment, the valuation is at market value, and the unrealized gains and losses are shown in the income statement. Firms are allowed an element of discretion in the way they classify investments and, subsequently, in the way they value these assets. This classification ensures that firms such as investment banks, whose assets are primarily securities held in other firms for purposes of trading, revalue the bulk of these assets at market levels each period. This is called marking to market, and provides one of the few instances in which market value trumps book value in accounting statements. Note, however, that this mark-to-market ethos did not provide any advance warning in 2008 to investors in financial service firms of the overvaluation of subprime and mortgage-backed securities.
+Minority Active Investments
+If the securities or assets owned in another firm represent between 20 percent and 50 percent of the overall ownership of that firm, an investment is treated as a minority active investment. While these investments have an initial acquisition value, a proportional share (based on ownership proportion) of the net income and losses made by the firm in which the investment was made is used to adjust the acquisition cost. In addition, the dividends received from the investment reduce the acquisition cost. This approach to valuing investments is called the equity approach.
+The market value of these investments is not considered until the investment is liquidated, at which point the gain or loss from the sale relative to the adjusted acquisition cost is shown as part of the earnings in that period.
+Majority Active Investments
+If the securities or assets owned in another firm represent more than 50 percent of the overall ownership of that firm, an investment is treated as a majority active investment. In this case, the investment is no longer shown as a financial investment but is instead replaced by the assets and liabilities of the firm in which the investment was made. This approach leads to a consolidation of the balance sheets of the two firms, where the assets and liabilities of the two firms are merged and presented as one balance sheet.2 The share of the equity in the subsidiary that is owned by other investors is shown as a minority interest on the liability side of the balance sheet. To provide an illustration, assume that Firm A owns 60% of Firm B. Firm A will be required to consolidate 100% of Fir m B's revenues, earnings, and assets into its own financial statements and then show a liability (minority interest) reflecting the accounting estimate of value of the 40% of Firm B's equity that does not belong to it. A similar consolidation occurs in the other financial statements of the firm as well, with the statement of cash flows reflecting the cumulated cash inflows and outflows of the combined firm. This is in contrast to the equity approach used for minority active investments, in which only the dividends received on the investment are shown as a cash inflow in the cash flow statement.
+Here again, the market value of this investment is not considered until the ownership stake is liquidated. At that point, the difference between the market price and the net value of the equity stake in the firm is treated as a gain or loss for the period.
+Intangible Assets
+Intangible assets include a wide array of assets, ranging from patents and trademarks to goodwill. The accounting standards vary across intangible assets.
+Patents and Trademarks
+Patents and trademarks are valued differently depending on whether they are generated internally or acquired. When patents and trademarks are generated from internal research, the costs incurred in developing the asset are expensed in that period, even though the asset might have a life of several accounting periods. Thus, the intangible asset is not valued in the balance sheet of the firm. In contrast, when an intangible asset is acquired from an external party, it is treated as an asset.
+Intangible assets have to be amortized over their expected lives, with a maximum amortization period of 40 years. The standard practice is to use straight-line amortization. For tax purposes, however, firms are generally not allowed to amortize goodwill or other intangible assets with no specific lifetime, though recent changes in the tax law allow for some flexibility in this regard.
+Goodwill
+Goodwill is the by-product of acquisitions. When a firm acquires another firm, the purchase price is first allocated to tangible assets, and the excess price is then allocated to any intangible assets such as patents or trade names. Any residual becomes goodwill. While accounting principles suggest that goodwill captures the value of any intangibles that are not specifically identifiable, it is really a reflection of the difference between the book value of assets of the acquired firm and the market value paid in the acquisition. This approach is called purchase accounting, and goodwill is amortized over time. Until 2000, firms that did not want to see this charge against their earnings often used an alternative approach called pooling accounting, in which the purchase price never shows up in the balance sheet. Instead, the book values of the two companies involved in the merger were aggregated to create the consolidated balance of the combined firm. The rules on acquisition accounting have changed substantially in the past decade both in the United States and internationally. Not only is purchase accounting required on all acquisitions, but firms are no longer allowed to automatically amortize goodwill over long periods (as they were used to doing). Instead, acquiring firms are required to reassess the values of the acquired entities every year; if the values have dropped since the acquisition, the value of goodwill must be reduced (impaired) to reflect the decline in value. If the acquired firm's values have gone up, though, the goodwill cannot be increased to reflect this change.3 ILLUSTRATION 3.1: Asset Values for Boeing and the Home Depot in 1998
+The following table summarizes asset values, as measured in the balance sheets of Boeing, the aerospace giant, and the Home Depot, a building supplies retailer, at the end of the 1998 financial year (in millions of dollars): Boeing
+Home Depot Net fixed assets
+$ 8,589
+$ 8,160 Goodwill
+$ 2,312
+$ 140 Investments and notes receivable
+$ 41
+$ 0 Deferred income taxes
+$ 411
+$ 0 Prepaid pension expense
+$ 3,513
+$ 0 Customer financing
+$ 4,930
+$ 0 Other assets
+$ 542
+$ 191 Current Assets Cash
+$ 2,183
+$ 62 Short-term marketable investments
+$ 279
+$ 0 Accounts receivables
+$ 3,288
+$ 469 Current portion of customer financing
+$ 781
+$ 0 Deferred income taxes
+$ 1,495
+$ 0 Inventories
+$ 8,349
+$ 4,293 Other current assets
+$ 0
+$ 109 Total current assets
+$16,375
+$ 4,933 Total assets
+$36,672
+$13,465 There are five points worth noting about these asset values: 1. Goodwill. Boeing, which acquired Rockwell in 1996 and McDonnell Douglas in 1997, used purchase accounting for the Rockwell acquisition and pooling for McDonnell Douglas. The goodwill on the balance sheet reflects the excess of acquisition value over book value for Rockwell and is being amortized over 30 years (which Boeing would not be able to do under current rules). With McDonnell Douglas, there is no recording of the premium paid on the acquisition among the assets, suggesting that the acquisition was structured to qualify for pooling, which would also not be allowed under current rules.
+2. Customer financing and accounts receivable. Boeing often either provides financing to its customers to acquire its planes or acts as the lessor on the planes. Since these contracts tend to run over several years, the present value of the payments due in future years on the financing and the lease payments is shown as customer financing. The current portion of these payments is shown as accounts receivable. The Home Depot provides credit to its customers as well, but all these payments due are shown as accounts receivable, since they are all short-term.
+3. Inventories. Boeing values inventories using the weighted average cost method, while the Home Depot uses the FIFO approach for valuing inventories.
+4. Marketable securities. Boeing classifies its short-term investments as trading investments and records them at market value. The Home Depot has a mix of trading, available-for-sale, and held-to-maturity investments and therefore uses a mix of book and market value to value these investments.
+5. Prepaid pension expense. Boeing records the excess of its pension fund assets over its expected pension fund liabilities as an asset on the balance sheet. Finally, the balance sheet for Boeing fails to report the value of a very significant asset, which is the effect of past research and development (R&D) expenses. Since accounting convention requires that these be expensed in the year that they occur and not be capitalized, the research asset does not show up in the balance sheet. Chapter 9 considers how to capitalize research and development expenses and the effects on balance sheets. MEASURING FINANCING MIX
+The second set of questions that we would like to answer, and would like accounting statements to shed some light on, relate to the mix of debt and equity used by the firm, and the current values of each. The bulk of the information about these questions is provided on the liabilities side of the balance sheet and the footnotes to it.
+Accounting Principles Underlying Liability and Equity Measurement
+Just as with the measurement of asset value, the accounting categorization of liabilities and equity is governed by a set of fairly rigid principles. The first is a strict categorization of financing into either a liability or an equity based on the nature of the obligation. For an obligation to be recognized as a liability, it must meet three requirements: 1. The obligation must be expected to lead to a future cash outflow or the loss of a future cash inflow at some specified or determinable date.
+2. The firm cannot avoid the obligation.
+3. The transaction giving rise to the obligation has to have already happened. In keeping with the earlier principle of conservatism in estimating asset value, accountants recognize as liabilities only cash flow obligations that cannot be avoided.
+The second principle is that the values of both liabilities and equity in a firm are better estimated using historical costs with accounting adjustments, rather than with expected future cash flows or market value. The process by which accountants measure the value of liabilities and equities is inextricably linked to the way they value assets. Since assets are primarily valued at historical cost or at book value, both debt and equity also get measured primarily at book value. The next section examines the accounting measurement of both liabilities and equity.
+Measuring the Value of Liabilities and Equities
+Accountants categorize liabilities into current liabilities, long-term debt, and long-term liabilities that are not debt or equity. Next, we will examine the way they measure each of these.
+Current Liabilities
+Under current liabilities are categorized all obligations that the firm has coming due in the next year. These generally include: Accounts payable, representing credit received from suppliers and other vendors to the firm. The value of accounts payable represents the amounts due to these creditors. For this item, book and market values should be similar.
+Short-term borrowing, representing short-term loans (due in less than a year) taken to finance the operations or current asset needs of the business. Here again, the value shown represents the amounts due on such loans, and the book and market values should be similar, unless the default risk of the firm has changed dramatically since it borrowed the money.
+Short-term portion of long-term borrowing, representing the portion of the long-term debt or bonds that is coming due in the next year. Here again, the value shown is the actual amount due on these loans, and market and book values should converge as the due date approaches.
+Other short-term liabilities, which is a catchall component for any other short-term liabilities that the firm might have, including wages due to its employees and taxes due to the government. Of all the items in the balance sheet, absent outright fraud, current liabilities should be the one for which the accounting estimates of book value and financial estimates of market value are closest.
+Long-Term Debt
+Long-term debt for firms can take one of two forms. It can be a long-term loan from a bank or other financial institution, or it can be a long-term bond issued to financial markets, in which case the creditors are the investors in the bond. Accountants measure the value of long-term debt by looking at the present value of payments due on the loan or bond at the time of the borrowing. For bank loans, this will be equal to the nominal value of the loan. With bonds, however, there are three possibilities: When bonds are issued at par value, for instance, the value of the long-term debt is generally measured in terms of the nominal obligation created (i.e., principal due on the borrowing). When bonds are issued at a premium or a discount on par value, the bonds are recorded at the issue price, but the premium or discount is amortized over the life of the bond. As an extreme example, companies that issue zero coupon debt have to record the debt at the issue price, which will be significantly below the principal (face value) due at maturity. The difference between the issue price and the face value is amortized each period and is treated as a noncash interest expense that is tax deductible.
+In all these cases, the value of debt is unaffected by changes in interest rates during the life of the loan or bond. Note that as market interest rates rise or fall, the present value of the loan obligations should decrease or increase. This updated market value for debt is not shown on the balance sheet. If debt is retired prior to maturity, the difference between book value and the amount paid at retirement is treated as an extraordinary gain or loss in the income statement.
+Finally, companies that have long-term debt denominated in nondomestic currencies have to adjust the book value of debt for changes in exchange rates. Since exchange rate changes reflect underlying changes in interest rates, it does imply that this debt is likely to be valued much nearer to market value than is debt in the domestic currency.
+Other Long-Term Liabilities
+Firms often have long-term obligations that are not captured in the long-term debt item. These include obligations to lessors on assets that firms have leased, to employees in the form of pension fund and health care benefits yet to be paid, and to the government in the form of taxes deferred. In the past two decades accountants have increasingly moved toward quantifying these liabilities and showing them as long-term liabilities.
+Leases
+Firms often choose to lease long-term assets rather than buy them. Lease payments create the same kind of obligation that interest payments on debt create, and they must be viewed in a similar light. If a firm is allowed to lease a significant portion of its assets and keep it off its financial statements, a perusal of the statements will give a very misleading view of the company's financial strength. Consequently, accounting rules have been devised to force firms to reveal the extent of their lease obligations on their books.
+There are two ways of accounting for leases. In an operating lease, the lessor (or owner) transfers only the right to use the property to the lessee. At the end of the lease period, the lessee returns the property to the lessor. Since the lessee does not assume the risk of ownership, the lease expense is treated as an operating expense in the income statement and the lease does not affect the balance sheet. In a capital lease, the lessee assumes some of the risks of ownership and enjoys some of the benefits. Consequently, the lease, when signed, is recognized both as an asset and as a liability (for the lease payments) on the balance sheet. The firm gets to claim depreciation each year on the asset and also deducts the interest expense component of the lease payment each year. In general, capital leases recognize expenses sooner than equivalent operating leases.
+Since firms prefer to keep leases off the books and sometimes to defer expenses, they have a strong incentive to report all leases as operating leases. Consequently the Financial Accounting Standards Board has ruled that a lease should be treated as a capital lease if it meets any one of the following four conditions: 1. The lease life exceeds 75 percent of the life of the asset.
+2. There is a transfer of ownership to the lessee at the end of the lease term.
+3. There is an option to purchase the asset at a bargain price at the end of the lease term.
+4. The present value of the lease payments, discounted at an appropriate discount rate, exceeds 90 percent of the fair market value of the asset. The lessor uses the same criteria for determining whether the lease is a capital or operating lease and accounts for it accordingly. If it is a capital lease, the lessor records the present value of future cash flows as revenue and recognizes expenses. The lease receivable is also shown as an asset on the balance sheet, and the interest revenue is recognized over the term of the lease as paid.
+From a tax standpoint, the lessor can claim the tax benefits of the leased asset only if it is an operating lease, though the tax code uses slightly different criteria for determining whether the lease is an operating lease.4
+Employee Benefits
+Employers can provide pension and health care benefits to their employees. In many cases, the obligations created by these benefits are extensive, and a failure by the firm to adequately fund these obligations needs to be revealed in financial statements.
+Pension Plans
+In a pension plan, the firm agrees to provide certain benefits to its employees, either by specifying a defined contribution (wherein a fixed contribution is made to the plan each year by the employer, without any promises as to the benefits that will be delivered in the plan) or a defined benefit (wherein the employer promises to pay a certain benefit to the employee). In the latter case, the employer has to put sufficient money into the plan each period to meet the defined benefits.
+Under a defined contribution plan, the firm meets its obligation once it has made the prespecified contribution to the plan. Under a defined benefit plan, the firm's obligations are much more difficult to estimate, since they will be determined by a number of variables, including the benefits that employees are entitled to, the prior contributions made by the employer and the returns they have earned, and the rate of return that the employer expects to make on current contributions. As these variables change, the value of the pension fund assets can be greater than, less than, or equal to pension fund liabilities (which include the present value of promised benefits). A pension fund whose assets exceed its liabilities is an overfunded plan, whereas one whose assets are less than its liabilities is an underfunded plan, and disclosures to that effect have to be included in financial statements, generally in the footnotes.
+When a pension fund is overfunded, the firm has several options. It can withdraw the excess assets from the fund, it can discontinue contributions to the plan, or it can continue to make contributions on the assumption that the overfunding is a transitory phenomenon that could well disappear by the next period. When a fund is underfunded, the firm has a liability, though accounting standards require that firms reveal only the excess of accumulated pension fund liability5 over pension fund assets on the balance sheet.
+Health Care Benefits
+A firm can provide health care benefits in either of two ways—by making a fixed contribution to a health care plan without promising specific benefits (analogous to a defined contribution plan) or by promising specific health benefits and setting aside the funds to provide these benefits (analogous to a defined benefit plan). The accounting for health care benefits is very similar to the accounting for pension obligations.
+Deferred Taxes
+Firms often use different methods of accounting for tax and financial reporting purposes, leading to a question of how tax liabilities should be reported. Since accelerated depreciation and favorable inventory valuation methods for tax accounting purposes lead to a deferral of taxes, the taxes on the income reported in the financial statements will generally be much greater than the actual tax paid. The same principles of matching expenses to income that underlie accrual accounting suggest that the deferred income tax be recognized in the financial statements. Thus a company that pays taxes of $55,000 on its taxable income based on its tax accounting, and that would have paid taxes of $75,000 on the income reported in its financial statements, will be forced to recognize the difference ($20,000) as deferred taxes. Since the deferred taxes will be paid in later years, they will be recognized when paid.
+It is worth noting that companies that actually pay more in taxes than the taxes they report in the financial statements create an asset called a deferred tax asset. This reflects the fact that the firm's earnings in future periods will be greater as the firm is given credit for the deferred taxes.
+The question of whether the deferred tax liability is really a liability is an interesting one. On one hand, the firm does not owe the amount categorized as deferred taxes to any entity, and treating it as a liability makes the firm look more risky than it really is. On the other hand, the firm will eventually have to pay its deferred taxes, and treating the amount as a liability seems to be the conservative thing to do.
+Preferred Stock
+When a company issues preferred stock, it generally creates an obligation to pay a fixed dividend on the stock. Accounting rules have conventionally not viewed preferred stock as debt because the failure to meet preferred dividends does not result in bankruptcy. At the same time, the fact the preferred dividends are cumulative makes them more onerous than common equity. Thus, preferred stock is a hybrid security, sharing some characteristics with equity and some with debt.
+Preferred stock is valued on the balance sheet at its original issue price, with any cumulated unpaid dividends added on. Convertible preferred stock is treated similarly, but it is treated as equity on conversion.
+Equity
+The accounting measure of equity is a historical cost measure. The value of equity shown on the balance sheet reflects the original proceeds received by the firm when it issued the equity, augmented by any earnings made since (or reduced by losses, if any) and reduced by any dividends paid out during the period. While these three items go into what we can call the book value of equity, three other points need to be made about this estimate: 1. When companies buy back stock for short periods, with the intent of reissuing the stock or using it to cover option exercises, they are allowed to show the repurchased stock as treasury stock, which reduces the book value of equity. Firms are not allowed to keep treasury stock on the books for extended periods, and have to reduce their book value of equity by the value of repurchased stock in the case of stock buybacks. Since these buybacks occur at the current market price, they can result in significant reductions in the book value of equity.
+2. Firms that have significant losses over extended periods or carry out massive stock buybacks can end up with negative book values of equity.
+3. Relating back to the discussion of marketable securities, any unrealized gain or loss in marketable securities that are classified as available for sale is shown as an increase or a decrease in the book value of equity in the balance sheet. As part of their financial statements, firms provide a summary of changes in shareholders' equity during the period, where all the changes that occurred to the accounting measure of equity value are summarized.
+As a final point on equity, accounting rules still seem to consider preferred stock, with its fixed dividend, as equity or near-equity, largely because of the fact that preferred dividends can be deferred or cumulated without the risk of default. To the extent that there can still be a loss of control in the firm (as opposed to bankruptcy), we have already argued that preferred stock shares almost as many characteristics with unsecured debt as it does with equity. ILLUSTRATION 3.2: Measuring Liabilities and Equity in Boeing and the Home Depot in 1998
+The following table summarizes the accounting estimates of liabilities and equity at Boeing and the Home Depot for the 1998 financial year in millions of dollars: Boeing
+Home Depot Accounts payable and other liabilities
+$10,733
+$ 1,586 Accrued salaries and expenses
+0
+$ 1,010 Advances in excess of costs
+$ 1,251
+$ 0 Taxes payable
+$ 569
+$ 247 Short-term debt and current long-term debt
+$ 869
+$ 14 Total current liabilities
+$13,422
+$ 2,857 Accrued health care benefits
+$ 4,831
+0 Other long-term liabilities
+0
+$ 210 Deferred income taxes
+0
+$ 83 Long-term debt
+$ 6,103
+$ 1,566 Minority interests
+$ 9
+$ 0 Shareholders' Equity Par value
+$ 5,059
+$ 37 Additional paid-in capital
+$ 0
+$ 2,891 Retained earnings
+$ 7,257
+$ 5,812 Total shareholders' equity
+$12,316
+$ 8,740 Total liabilities
+$36,672
+$13,465 The most significant difference between the companies is the accrued health care liability shown by Boeing, representing the present value of expected health care obligations promised to employees in excess of health care assets. The shareholders' equity for both firms represents the book value of equity and is significantly different from the market value of equity. The follwing table summarizes the difference at the end of 1998 (in millions of dollars): Boeing
+Home Depot Book value of equity
+$12,316
+$ 8,740 Market value of equity
+$32,595
+$85,668 One final point needs to be made about the Home Depot's liabilities. The Home Depot has substantial operating leases. Because these leases are treated as operating expenses, they do not show up in the balance sheet. Since they represent commitments to make payments in the future, we would argue that operating leases should be capitalized and treated as part of the liabilities of the firm. How best to do this is considered in Chapter 9. MEASURING EARNINGS AND PROFITABILITY
+How profitable is a firm? What did it earn on the assets that it invested in? These are fundamental questions we would like financial statements to answer. Accountants use the income statement to provide information about a firm's operating activities over a specific time period. The income statement is designed to measure the earnings from assets in place. This section examines the principles underlying earnings and return measurement in accounting, and the way they are put into practice.
+Accounting Principles Underlying Measurement of Earnings and Profitability
+Two primary principles underlie the measurement of accounting earnings and profitability. The first is the principle of accrual accounting. In accrual accounting, the revenue from selling a good or service is recognized in the period in which the good is sold or the service is performed (in whole or substantially). A corresponding effort is made on the expense side to match expenses to revenues.6 This is in contrast to a cash-based system of accounting, where revenues are recognized when payment is received and expenses are recorded when paid.
+The second principle is the categorization of expenses into operating, financing, and capital expenses. Operating expenses are expenses that, at least in theory, provide benefits only for the current period; the cost of labor and materials expended to create products that are sold in the current period is a good example. Financing expenses are expenses arising from the nonequity financing used to raise capital for the business; the most common example is interest expenses. Capital expenses are expenses that are expected to generate benefits over multiple periods; for instance, the cost of buying land and buildings is treated as a capital expense.
+Operating expenses are subtracted from revenues in the current period to arrive at a measure of operating earnings of the firm. Financing expenses are subtracted from operating earnings to estimate earnings to equity investors or net income. Capital expenses are written off over their useful lives (in terms of generating benefits) as depreciation or amortization.
+Measuring Accounting Earnings and Profitability
+Since income can be generated from a number of different sources, generally accepted accounting principles (GAAP) require that income statements be classified into four sections—income from continuing operations, income from discontinued operations, extraordinary gains or losses, and adjustments for changes in accounting principles.
+Generally accepted accounting principles require the recognition of revenues when the service for which the firm is getting paid has been performed in full or substantially, and the firm has received in return either cash or a receivable that is both observable and measurable. Expenses linked directly to the production of revenues (like labor and materials) are recognized in the same period in which revenues are recognized. Any expenses that are not directly linked to the production of revenues are recognized in the period in which the firm consumes the services. Accounting has resolved one inconsistency that bedeviled it for years, with a change in the way it treats employee options. Unlike the old rules, these option grants were not treated as expenses when granted but only when exercised, the new rules require that employee options be valued and expensed, when granted (with allowances for amortization over periods). Since employee options are part of compensation, which is an operating expense, the new rules make more sense.
+While accrual accounting is straightforward in firms that produce goods and sell them, there are special cases where accrual accounting can be complicated by the nature of the product or service being offered. For instance, firms that enter into long-term contracts with their customers are allowed to recognize revenue on the basis of the percentage of the contract that is completed. As the revenue is recognized on a percentage-of-completion basis, a corresponding proportion of the expense is also recognized. When there is considerable uncertainty about the capacity of the buyer of a good or service to pay for it, the firm providing the good or service may recognize the income only when it collects portions of the selling price under the installment method.
+Reverting back to the discussion of the difference between capital and operating expenses, operating expenses should reflect only those expenses that create revenues in the current period. In practice, however, a number of expenses are classified as operating expenses that do not seem to meet this test. The first is depreciation and amortization. While the notion that capital expenditures should be written off over multiple periods is reasonable, the accounting depreciation that is computed on the original historical cost often bears little resemblance to the actual economic depreciation. The second expense is research and development expenses, which accounting standards classify as operating expenses, but which clearly provide benefits over multiple periods. The rationale used for this classification is that the benefits cannot be counted on or easily quantified.
+Much of financial analysis is built around the expected future earnings of a firm, and many of these forecasts start with the current earnings. It is therefore important to know how much of these earnings comes from the ongoing operations of the firm and how much can be attributed to unusual or extraordinary events that are unlikely to recur on a regular basis. From that standpoint, it is useful that firms categorize expenses into operating and nonrecurring expenses, since it is the earnings prior to extraordinary items that should be used in forecasting. Nonrecurring items include: Unusual or infrequent items, such as gains or losses from the divestiture of an asset or division, and write-offs or restructuring costs. Companies sometimes include such items as part of operating expenses. As an example, Boeing in 1997 took a write-off of $1,400 million to adjust the value of assets it acquired in its acquisition of McDonnell Douglas, and it showed this as part of operating expenses.
+Extraordinary items, which are defined as events that are unusual in nature, infrequent in occurrence, and material in impact. Examples include the accounting gain associated with refinancing high-coupon debt with lower-coupon debt, and gains or losses from marketable securities that are held by the firm.
+Losses associated with discontinued operations, which measure both the loss from the phaseout period and any estimated loss on sale of the operations. To qualify, however, the operations have to be separable from the firm.
+Gains or losses associated with accounting changes, which measure earnings changes created by both accounting changes made voluntarily by the firm (such as a change in inventory valuation) and accounting changes mandated by new accounting standards. ILLUSTRATION 3.3: Measures of Earnings—Boeing and the Home Depot in 1998
+The following table summarizes the income statements of Boeing and the Home Depot for the 1998 financial year: Boeing (in $ millions)
+Home Depot (in $ millons) Sales and other operating revenues
+$56,154
+$30,219 – Operating costs and expenses
+$51,022
+$27,185 – Depreciation
+$ 1,517
+$ 373 – Research and development expenses
+$ 1,895
+$ 0 Operating income
+$ 1,720
+$ 2,661 + Other income (includes interest income)
+$ 130
+$ 30 – Interest expenses
+$ 453
+$ 37 Earnings before taxes
+$ 1,397
+$ 2,654 – Income taxes
+$ 277
+$ 1,040 Net earnings (Loss)
+$ 1,120
+$ 1,614 Boeing's operating income is reduced by the research and development expense, which is treated as an operating expense by accountants. The Home Depot's operating expenses include operating leases. As noted earlier, the treatment of both these items skews earnings, and how best to adjust earnings when such expenses exist is considered in Chapter 9. Measures of Profitability While the income statement allows us to estimate how profitable a firm is in absolute terms, it is just as important that we gauge the profitability of the firm in terms of percentage returns. Two basic ratios measure profitability. One examines the profitability relative to the capital employed to get a rate of return on investment. This can be done either from the viewpoint of just the equity investors or by looking at the entire firm. Another examines profitability relative to sales, by estimating a profit margin.
+Return on Assets and Return on Capital
+The return on assets (ROA) of a firm measures its operating efficiency in generating profits from its assets, prior to the effects of financing.
+Return on assets = Earnings before interest and taxes(1 – Tax rate)/Total assets
+Earnings before interest and taxes (EBIT) is the accounting measure of operating income from the income statement, and total assets refers to the assets as measured using accounting rules—that is, using book value (BV) for most assets. Alternatively, return on assets can be written as:
+Return on assets = [Net income + Interest expenses(1 – Tax rate)]/Total assets
+By separating the financing effects from the operating effects, the return on assets provides a cleaner measure of the true return on these assets. By dividing by total assets, the return on assets does understate the profitability of firms that have substantial current assets.
+ROA can also be computed on a pretax basis with no loss of generality, by using the earnings before interest and taxes and not adjusting for taxes:
+Pretax ROA = Earnings before interest and taxes/Total assets
+This measure is useful if the firm or division is being evaluated for purchase by an acquirer with a different tax rate.
+A more useful measure of return relates the operating income to the capital invested in the firm, where capital is defined as the sum of the book value of debt and equity, net of cash. This is the return on invested capital (ROC or ROIC), and provides not only a truer measure of return but one that can be compared to the cost of capital, to measure the quality of a firm's investments. The denominator is generally termed invested capital and measures the book value of operating assets. For both measures, the book value can be measured at the beginning of the period or as an average of beginning and ending values. ILLUSTRATION 3.4: Estimating Return on Capital—Boeing and the Home Depot in 1998
+The following table summarizes the after-tax return on assets and return on capital estimates for Boeing and the Home Depot, using both average and beginning measures of capital in 1998: Boeing
+Home Depot (in $millions)
+(in $millions) After-tax operating income
+$ 1,118
+$ 1,730 Book value of capital—beginning
+$19,807
+$ 8,525 Book value of capital—ending
+$19,288
+$10,320 Book value of capital—average
+$19,548
+$ 9,423 Return on capital (based on average)
+5.72%
+18.36% Return on capital (based on beginning)
+5.64%
+20.29% Boeing had a terrible year, in 1998, in terms of after-tax returns. The Home Depot had a much better year in terms of those same returns. Decomposing Return on Capital
+The return on capital of a firm can be written as a function of the operating profit margin it has on its sales, and its capital turnover ratio. Thus, a firm can arrive at a high ROC by either increasing its profit margin or utilizing its capital more efficiently to increase sales. There are likely to be competitive constraints and technological constraints on both variables, but a firm still has some freedom within these constraints to choose the mix of profit margin and capital turnover that maximizes its ROC. The return on capital varies widely across firms in different businesses, largely as a consequence of differences in profit margins and capital turnover ratios. mgnroc.xls: This is a dataset on the Web that summarizes the operating margins, turnover ratios, and returns on capital of firms in the United States, classified by industry. Return on Equity
+While the return on capital measures the profitability of the overall firm, the return on equity (ROE) examines profitability from the perspective of the equity investor, by relating the equity investor's profits (net profit after taxes and interest expenses) to the book value of the equity investment. Since preferred stockholders have a different type of claim on the firm than do common stockholders, the net income should be estimated after preferred dividends, and the book value should be that of only common equity.
+Determinants of Noncash ROE
+Since the ROE is based on earnings after interest payments, it is affected by the financing mix the firm uses to fund its projects. In general, a firm that borrows money to finance projects and that earns a ROC on those projects that exceeds the after-tax interest rate it pays on its debt will be able to increase its ROE by borrowing. The return on equity, not including cash, can be written as follows:7 where ROC = EBIT(1 – t)/(BV of debt + BV of equity – Cash)
+D/E = BV of debt/BV of equity
+i = Interest expense on debt/BV of debt
+t = Tax rate on ordinary income
+The second term captures the benefit of financial leverage. ILLUSTRATION 3.5: Return on Equity Computations: Boeing and the Home Depot in 1998
+The following table summarizes the return on equity for Boeing and the Home Depot in 1998: Boeing
+Home Depot Return Ratios
+(in $millions)
+(in $millions) Net income
+$ 1,120
+$1,614 Book value of equity—beginning
+$12,953
+$7,214 Book value of equity—ending
+$12,316
+$8,740 Book value of equity—average
+$12,635
+$7,977 Return on equity (based on average)
+8.86%
+20.23% Return on equity (based on beginning)
+8.65%
+22.37% The results again indicate that Boeing had a substandard year in 1998, while the Home Depot reported healthier returns on equity. The returns on equity can also be estimated by decomposing into the components just specified (using the adjusted beginning-of-the-year numbers): Boeing
+Home Depot (in $millions)
+(in $millions) After-tax return on capital
+5.82%
+16.37% Debt-equity ratio
+35.18%
+48.37% Book interest rate(1 – Tax rate)
+4.22%
+4.06% Return on equity
+6.38%
+22.33% Note that a tax rate of 35% is used on both the return on capital and the book interest rate. This approach results in a return on equity that is different from the one estimated using the net income and the book value of equity. rocroe.xls: This is a dataset on the Web that summarizes the return on capital, debt equity ratios, book interest rates, and returns on equity of firms in the United States, classified by industry. MEASURING RISK
+How risky are the investments the firm has made over time? How much risk do equity investors in a firm face? These are two more questions that we would like to find the answers to in the course of an investment analysis. Accounting statements do not really claim to measure or quantify risk in a systematic way, other than to provide footnotes and disclosures where there might be risk embedded in the firm. This section examines some of the ways in which accountants try to assess risk.
+Accounting Principles Underlying Risk Measurement
+To the extent that accounting statements and ratios do attempt to measure risk, there seem to be two common themes.
+The first is that the risk being measured is the risk of default—that is, the risk that a fixed obligation, such as interest or principal due on outstanding debt, will not be met. The broader equity notion of risk, which measures the variance of actual returns around expected returns, does not seem to receive much attention. Thus, an all-equity-financed firm with positive earnings and few or no fixed obligations will generally emerge as a low-risk firm from an accounting standpoint, in spite of the fact that its earnings are unpredictable.
+The second theme is that accounting risk measures generally take a static view of risk, by looking at the capacity of a firm at a point in time to meet its obligations. For instance, when ratios are used to assess a firm's risk, the ratios are almost always based on one period's income statement and balance sheet.
+Accounting Measures of Risk
+Accounting measures of risk can be broadly categorized into two groups. The first is disclosures about potential obligations or losses in values that show up as footnotes on balance sheets, which are designed to alert potential or current investors to the possibility of significant losses. The second measure is ratios that are designed to measure both liquidity and default risk.
+Disclosures in Financial Statements
+In recent years, the disclosures that firms have to make about future obligations have proliferated. Consider, for instance, the case of contingent liabilities. These refer to potential liabilities that will be incurred under certain contingencies, as is the case, for instance, when a firm is the defendant in a lawsuit. The general rule that has been followed is to ignore contingent liabilities that hedge against risk, since the obligations on the contingent claim will be offset by benefits elsewhere.8 In recent periods, however, significant losses borne by firms from supposedly hedged derivatives positions (such as options and futures) have led to FASB requirements that these derivatives be disclosed as part of a financial statement. In fact, pension fund and health care obligations have moved from mere footnotes to actual liabilities for firms.
+Financial Ratios
+Financial statements have long been used as the basis for estimating financial ratios that measure profitability, risk, and leverage. Earlier, the section on earnings looked at two of the profitability ratios—return on equity and return on capital. This section looks at some of the financial ratios that are often used to measure the financial risk in a firm.
+Short-Term Liquidity Risk
+Short-term liquidity risk arises primarily from the need to finance current operations. To the extent that the firm has to make payments to its suppliers before it gets paid for the goods and services it provides, there is a cash shortfall that has to be met, usually through short-term borrowing. Though this financing of working capital needs is done routinely in most firms, financial ratios have been devised to keep track of the extent of the firm's exposure to the risk that it will not be able to meet its short-term obligations. The two ratios most frequently used to measure short-term liquidity risk are the current ratio and the quick ratio.
+Current Ratios
+The current ratio is the ratio of the firm's current assets (cash, inventory, accounts receivable) to its current liabilities (obligations coming due within the next period). A current ratio below 1, for instance, would indicate that the firm has more obligations coming due in the next year than assets it can expect to turn into cash. That would be an indication of liquidity risk.
+While traditional analysis suggests that firms maintain a current ratio of 2 or greater, there is a trade-off here between minimizing liquidity risk and tying up more and more cash in net working capital (Net working capital = Current assets – Current liabilities). In fact, it can be reasonably argued that a very high current ratio is indicative of an unhealthy firm that is having problems reducing its inventory. In recent years firms have worked at reducing their current ratios and managing their net working capital better.
+Reliance on current ratios has to be tempered by a few concerns. First, the ratio can be easily manipulated by firms around the time of financial reporting dates to give the illusion of safety; second, current assets and current liabilities can change by an equal amount, but the effect on the current ratio will depend on its level before the change.9
+Quick or Acid Test Ratios
+The quick or acid test ratio is a variant of the current ratio. It distinguishes current assets that can be converted quickly into cash (cash, marketable securities) from those that cannot (inventory, accounts receivable). The exclusion of accounts receivable and inventory is not a hard-and-fast rule. If there is evidence that either can be converted into cash quickly, it can, in fact, be included as part of the quick ratio.
+Turnover Ratios
+Turnover ratios measure the efficiency of working capital management by looking at the relationship of accounts receivable and inventory to sales and to the cost of goods sold: These statistics can be interpreted as measuring the speed with which the firm turns accounts receivable into cash or inventory into sales. These ratios are often expressed in terms of the number of days outstanding: A similar pair of statistics can be computed for accounts payable, relative to puchases: Since accounts receivable and inventory are assets, and accounts payable is a liability, these three statistics (standardized in terms of days outstanding) can be combined to get an estimate of how much financing the firm needs to raise to fund working capital needs. The greater the financing period for a firm, the greater is its short-term liquidity risk. wcdata.xls: This is a dataset on the Web that summarizes working capital ratios for firms in the United States, classified by industry. finratio.xls: This spreadsheet allows you to compute the working capital ratios for a firm, based upon financial statement data. Long-Term Solvency and Default Risk
+Measures of long-term solvency attempt to examine a firm's capacity to meet interest and principal payments in the long term. Clearly, the profitability ratios discussed earlier in the section are a critical component of this analysis. The ratios specifically designed to measure long-term solvency try to relate profitability to the level of debt payments in order to identify the degree of comfort with which the firm can meet these payments.
+Interest Coverage Ratios
+The interest coverage ratio measures the capacity of the firm to meet interest payments from predebt, pretax earnings. The higher the interest coverage ratio, the more secure is the firm's capacity to make interest payments from earnings. This argument, however, has to be tempered by the recognition that the amount of earnings before interest and taxes is volatile and can drop significantly if the economy enters a recession. Consequently, two firms can have the same interest coverage ratio but be viewed very differently in terms of risk.
+The denominator in the interest coverage ratio can be easily extended to cover other fixed obligations such as lease payments. If this is done, the ratio is called a fixed charges coverage ratio: Finally, this ratio, while stated in terms of earnings, can be restated in terms of cash flows by using earnings before interest, taxes, depreciation, and amortization (EBITDA) in the numerator and cash fixed charges in the denominator. Both interest coverage and fixed charges coverage ratios are open to the criticism that they do not consider capital expenditures, a cash flow that may be discretionary in the very short term, but not in the long term if the firm wants to maintain growth. One way of capturing the extent of this cash flow, relative to operating cash flows, is to compute a ratio of the two: While there are a number of different definitions of cash flows from operations, the most reasonable way of defining it is to measure the cash flows from continuing operations, before interest but after taxes and after meeting working capital needs. covratio.xls: This is a dataset on the Web that summarizes the interest coverage and fixed charges coverage ratios for firms in the United States, classified by industry. ILLUSTRATION 3.6: Interest and Fixed Charges Coverage Ratios: Boeing and the Home Depot in 1998
+The following table summarizes interest and fixed charges coverage ratios for Boeing and the Home Depot in 1998: Boeing
+Home Depot EBIT
+$1,720
+$2,661 Interest expense
+$ 453
+$ 37 Interest coverage ratio
+3.80
+71.92 EBIT
+$1,720
+$2,661 Operating lease expenses
+$ 215
+$ 290 Interest expenses
+$ 453
+$ 37 Fixed charges coverage ratio
+2.90
+9.02 EBITDA
+$3,341
+$3,034 Cash fixed charges
+$ 668
+$ 327 Cash fixed charges coverage ratio
+5.00
+9.28 Cash flows from operations
+$2,161
+$1,662 Capital expenditures
+$1,584
+$2,059 Cash flows/Capital expenditures
+1.36
+0.81 Boeing, based on its operating income in 1998, looks riskier than the Home Depot on both the interest coverage ratio basis and fixed charges coverage ratio basis. On a cash flow basis, however, Boeing does look much better. In fact, when capital expenditures are considered, the Home Depot has a lower ratio. For Boeing, the other consideration is the fact that operating income in 1998 was depressed relative to income in earlier years, and this does have an impact on the ratios across the board. It might make more sense when computing these ratios to look at the average operating income over time. finratio.xls: This spreadsheet allows you to compute the interest coverage and fixed charges coverage ratios for a firm based on financial statement data. Debt Ratios
+Interest coverage ratios measure the capacity of the firm to meet interest payments, but do not examine whether it can pay back the principal on outstanding debt. Debt ratios attempt to do this, by relating debt to total capital or to equity: The first ratio measures debt as a proportion of the total capital of the firm and cannot exceed 100 percent. The second measures debt as a proportion of the book value of equity in the firm and can be easily derived from the first, since: While these ratios presume that capital is raised from only debt and equity, they can be easily adapted to include other sources of financing, such as preferred stock. Although preferred stock is sometimes combined with common stock under the equity label, it is better to keep the two sources of financing separate and to compute the ratio of preferred stock to capital (which will include debt, equity, and preferred stock).
+There are two close variants of debt ratios. In the first, only long-term debt is used rather than total debt, with the rationale that short-term debt is transitory and will not affect the long-term solvency of the firm. Given the ease with which some firms can roll over short-term debt and the willingness of many firms to use short-term financing to fund long-term projects, these variants can provide a misleading picture of the firm's financial leverage risk.
+The second variant of debt ratios uses market value (MV) instead of book value, primarily to reflect the fact that some firms have a significantly greater capacity to borrow than their book values indicate. Many analysts disavow the use of market value in their calculations, contending that market values, in addition to being difficult to get for debt, are volatile and hence unreliable. These contentions are open to debate. It is true that the market value of debt is difficult to get for firms that do not have publicly traded bonds, but the market value of equity not only is easy to obtain, but it also is constantly updated to reflect marketwide and firm-specific changes. Furthermore, using the book value of debt as a proxy for market value in those cases where bonds are not traded does not significantly shift most market value-based debt ratios.10 ILLUSTRATION 3.7: Book Value Debt Ratios and Variants—Boeing and the Home Depot
+The following table summarizes different estimates of the debt ratio for Boeing and the Home Depot, in 2008, using book values of debt and equity for both firms: Boeing
+Home Depot (in $millions)
+(in $millions) Long-term debt
+$ 6,103
+$1,566 Short-term debt
+$ 869
+$ 14 Book value of equity
+$12,316
+$8,740 Long-term debt/Equity
+49.55%
+17.92% Long-term debt/(Long-term debt + Equity)
+33.13%
+15.20% Debt/Equity
+56.61%
+18.08% Debt/(Debt + Equity)
+36.15%
+15.31% In 2008, Boeing has a much higher book value debt ratio, considering either long-term or total debt, than the Home Depot. dbtfund.xls: This is a dataset on the Web that summarizes the book value debt ratios and market value debt ratios for firms in the United States, classified by industry. OTHER ISSUES IN ANALYZING FINANCIAL STATEMENTS
+There are significant differences in accounting standards and practices across countries and these differences may color comparisons across companies.
+Differences in Accounting Standards and Practices
+Differences in accounting standards across countries affect the measurement of earnings. These differences, however, are not so great as they are made out to be by some analysts, and they cannot explain away radical departures from fundamental principles of valuation. Choi and Levich, in a 1990 survey of accounting standards across developed markets, note that most countries subscribe to basic accounting notions of consistency, realization, and historical cost principles in preparing accounting statements. As countries increasingly move toward international financial reporting standards (IFRS), it is worth noting that IFRS and U.S. GAAP are more similar than dissimilar on many issues. It is true that there are areas of differences that still remain, and we note some of them in Table 3.1.
+Table 3.1 Key Differences between IFRS and GAAP Most of these differences can be accounted and adjusted for when comparisons are made between companies in the United States and companies in other financial markets. Statistics such as price-earnings ratios, which use stated and unadjusted earnings, can be misleading when accounting standards vary widely across the companies being compared.
+CONCLUSION
+Financial statements remain the primary source of information for most investors and analysts. There are differences, however, between how accounting and financial analysts approach answering a number of key questions about the firm.
+The first question relates to the nature and the value of the assets owned by a firm. Assets can be categorized into investments already made (assets in place) and investments yet to be made (growth assets); accounting statements provide a substantial amount of historical information about the former and very little about the latter. The focus on the original price of assets in place (book value) in accounting statements can lead to significant differences between the stated value of these assets and their market value. With growth assets, accounting rules result in low or no values for assets generated by internal research.
+The second issue is the measurement of profitability. The two principles that govern how profits are measured are accrual accounting—in which revenues and expenses are shown in the period in which transactions occur rather than when the cash is received or paid—and the categorization of expenses into operating, financing, and capital expenses. While operating and financing expenses are shown in income statements, capital expenditures are spread over several time periods and take the form of depreciation and amortization. Accounting standards miscategorize operating leases and research and development expenses as operating expenses (when the former should be categorized as financing expenses and the latter as capital expenses).
+Financial statements also deal with short-term liquidity risk and long-term default risk. While the emphasis in accounting statements is on examining the risk that firms may be unable to make payments that they have committed to make, there is very little focus on risk to equity investors.
+QUESTIONS AND SHORT PROBLEMS
+In the problems following, use an equity risk premium of 5.5 percent if none is specified.
+Coca-Cola's balance sheet for December 1998 is summarized (in millions of dollars) for problems 1 through 9: 1. Consider the assets on Coca-Cola's balance sheet and answer the following questions: a. Which assets are likely to be assessed closest to market value? Explain.
+b. Coca-Cola has net fixed assets of $3,669 million. Can you estimate how much Coca-Cola paid for these assets? Is there any way to know the age of these assets?
+c. Coca-Cola seems to have far more invested in current assets than in fixed assets. Is this significant? Explain.
+d. In the early 1980s, Coca-Cola sold off its bottling operations, and the bottlers became independent companies. How would this action have impacted the assets on Coca-Cola's balance sheet? (The manufacturing plants are most likely to be part of the bottling operations.) 2. Examine the liabilities on Coca-Cola's balance sheet. a. How much interest-bearing debt does Coca-Cola have outstanding? (You can assume that other short-term liabilities represent sundry payables, and other long-term liabilities represent health care and pension obligations.)
+b. How much did Coca-Cola obtain in equity capital when it issued stock originally to the financial markets?
+c. Is there any significance to the fact that the retained earnings amount is much larger than the original paid-in capital?
+d. The market value of Coca-Cola's equity is $140 billion. What is the book value of equity in Coca-Cola? Why is there such a large difference between the market value of equity and the book value of equity? 3. Coca-Cola's most valuable asset is its brand name. Where in the balance sheet do you see its value? Is there any way to adjust the balance sheet to reflect the value of this asset?
+4. Assume that you have been asked to analyze Coca-Cola's working capital management. a. Estimate the net working capital and noncash working capital for Coca-Cola.
+b. Estimate the firm's current ratio.
+c. Estimate the firm's quick ratio.
+d. Would you draw any conclusions about the riskiness of Coca-Cola as a firm by looking at these numbers? Why or why not? Coca-Cola's income statements for 1997 and 1998 are summarized (in millions of dollars) for problems 5 through 9: 1997
+1998 Net revenues
+$18,868
+$18,813 Cost of goods sold
+6,015
+5,562 Selling, general, and administrative expenses
+7,852
+8,284 Earnings before interest and taxes
+5,001
+4,967 Interest expenses
+258
+277 Nonoperating gains
+1,312
+508 Income tax expenses
+1,926
+1,665 Net income
+4,129
+3,533 Dividends
+1,387
+1,480 The following questions relate to Coca-Cola's income statements. 5. How much operating income did Coca-Cola earn, before taxes, in 1998? How does this compare to how much Coca-Cola earned in 1997? What are the reasons for the difference?
+6. The biggest expense for Coca-Cola is advertising, which is part of the selling, generals and administrative (G&A) expenses. A large portion of these expenses is designed to build up Coca-Cola's brand name. Should advertising expenses be treated as operating expenses or are they really capital expenses? If they are to be treated as capital expenses, how would you capitalize them? (Use the capitalization of R&D as a guide.)
+7. What effective tax rate did Coca-Cola have in 1998? How does it compare with what the company paid in 1997 as an effective tax rate? What might account for the difference?
+8. You have been asked to assess the profitability of Coca-Cola as a firm. To that end, estimate the pretax operating and net margins in 1997 and 1998 for the firm. Are there any conclusions you would draw from the comparisons across the two years?
+9. The book value of equity at Coca-Cola in 1997 was $7,274 million. The book value of interest-bearing debt was $3,875 million. Estimate: a. The return on equity (beginning of the year) in 1998.
+b. The pretax return on capital (beginning of the year) in 1998.
+c. The after-tax return on capital (beginning of the year) in 1998, using the effective tax rate in 1998. 10. SeeSaw Toys reported that it had a book value of equity of $1.5 billion at the end of 1998 and 100 million shares outstanding. During 1999, it bought back 10 million shares at a market price of $40 per share. The firm also reported a net income of $150 million for 1999, and paid dividends of $50 million. Estimate: a. The book value of equity at the end of 1999.
+b. The return on equity, using beginning book value of equity.
+c. The return on equity, using the average book value of equity. 1 Depreciation is treated as an accounting expense. Hence, the use of straight-line depreciation (which is lower than accelerated depreciation in the first few years after an asset is acquired) will result in lower expenses and higher income.
+2 Firms have evaded the requirements of consolidation by keeping their share of ownership in other firms below 50 percent.
+3 Once an acquisition is complete, the difference between market value and book value for the target firm does not automatically become goodwill. Existing assets can be reappraised first to fair value and the difference becomes goodwill.
+4 The requirements for an operating lease in the tax code are: (1) The property can be used by someone other than the lessee at the end of the lease term, (2) the lessee cannot buy the asset using a bargain purchase option, (3) the lessor has at least 20 percent of its capital at risk, (4) the lessor has a positive cash flow from the lease independent of tax benefits, and (5) the lessee does not have an investment in the lease.
+5 The accumulated pension fund liability does not take into account the projected benefit obligation, where actuarial estimates of future benefits are made. Consequently, it is much smaller than the total pension liabilities.
+6 If a cost (such as an administrative cost) cannot easily be linked with particular revenues, it is usually recognized as an expense in the period in which it is consumed.
+7 8 This assumes that the hedge is set up competently. It is entirely possible that a hedge, if sloppily set up, can end up costing the firm money.
+9 If the current assets and current liabilities increase by an equal amount, the current ratio will go down if it was greater than 1 before the increase, and go up if it was less than 1.
+10 Deviations in the market value of equity from book value are likely to be much larger than deviations for debt, and are likely to dominate in most debt ratio calculations.
+HAPTER 4The Basics of Risk
+When valuing assets and firms, we need to use discount rates that reflect the riskiness of the cash flows. In particular, the cost of debt has to incorporate a default spread for the default risk in the debt, and the cost of equity has to include a risk premium for equity risk. But how do we measure default and equity risk? More importantly, how do we come up with the default and equity risk premiums?
+This chapter lays the foundations for analyzing risk in valuation. It presents alternative models for measuring risk and converting these risk measures into acceptable hurdle rates. It begins with a discussion of equity risk and presents the analysis in three steps. In the first step, risk is defined in statistical terms to be the variance in actual returns around an expected return. The greater this variance, the more risky an investment is perceived to be. The next step, the central one, is to decompose this risk into risk that can be diversified away by investors and risk that cannot. The third step looks at how different risk and return models in finance attempt to measure this nondiversifiable risk. It compares the most widely used model, the capital asset pricing model (CAPM), with other models and explains how and why they diverge in their measures of risk and the implications for the equity risk premium.
+The final part of this chapter considers default risk and how it is measured by ratings agencies. By the end of the chapter, we should have a way of estimating the equity risk and default risk for any firm.
+WHAT IS RISK?
+Risk, for most of us, refers to the likelihood that in life’s games of chance we will receive an outcome that we will not like. For instance, the risk of driving a car too fast is getting a speeding ticket or, worse still, getting into an accident. Merriam-Webster’s Collegiate Dictionary, in fact, defines the verb to risk as “to expose to hazard or danger.” Thus risk is perceived almost entirely in negative terms.
+In finance, our definition of risk is both different and broader. Risk, as we see it, refers to the likelihood that we will receive a return on an investment that is different from the return we expect to make. Thus, risk includes not only the bad outcomes (returns that are lower than expected), but also good outcomes (returns that are higher than expected). In fact, we can refer to the former as downside risk and the latter as upside risk, but we consider both when measuring risk. In fact, the spirit of our definition of risk in finance is captured best by the Chinese symbols for risk: Loosely defined, the first symbol is the symbol for “danger,” while the second is the symbol for “opportunity,” making risk a mix of danger and opportunity. It illustrates very clearly the trade-off that every investor and business has to make—between the higher rewards that come with the opportunity and the higher risk that has to be borne as a consequence of the danger.
+Much of this chapter can be viewed as an attempt to come up with a model that best measures the danger in any investment, and then attempts to convert this into the opportunity that we would need to compensate for the danger. In finance terms, we term the danger to be “risk” and the opportunity to be “expected return.”
+What makes the measurement of risk and expected return so challenging is that it can vary depending on whose perspective we adopt. When analyzing the risk of a firm, for instance, we can measure it from the viewpoint of the firm’s managers. Alternatively, we can argue that the firm’s equity is owned by its stockholders, and that it is their perspective on risk that should matter. A firm’s stockholders, many of whom hold the stock as one investment in a larger portfolio, might perceive the risk in the firm very differently from the firm’s managers, who might have the bulk of their capital, human and financial, invested in the firm.
+We argue that risk in an investment has to be perceived through the eyes of investors in the firm. Since firms often have thousands of investors, often with very different perspectives, it can be asserted that risk has to be measured from the perspective of not just any investor in the stock, but of the marginal investor, defined to be the investor most likely to be trading on the stock at any given point in time. The objective in valuation is to measure the value of an asset to those who will be pricing it. If we want to stay true to this objective, we have to consider the viewpoint of those who set the stock prices, and they are the marginal investors.
+EQUITY RISK AND EXPECTED RETURN
+To demonstrate how risk is viewed in finance, risk analysis is presented here in three steps: first, defining risk in terms of the distribution of actual returns around an expected return; second, differentiating between risk that is specific to one or a few investments and risk that affects a much wider cross section of investments (in a market where the marginal investor is well diversified, it is only the latter risk, called market risk, that will be rewarded); and third, alternative models for measuring this market risk and the expected returns that go with it.
+Defining Risk
+Investors who buy an asset expect to earn returns over the time horizon that they hold the asset. Their actual returns over this holding period may be very different from the expected returns, and it is this difference between actual and expected returns that is a source of risk. For example, assume that you are an investor with a one-year time horizon buying a one-year Treasury bill (or any other default-free one-year bond) with a 5 percent expected return. At the end of the one-year holding period, the actual return on this investment will be 5 percent, which is equal to the expected return. The return distribution for this investment is shown in Figure 4.1. This is a riskless investment. Figure 4.1 Probability Distribution of Returns on a Risk-Free Investment To provide a contrast to the riskless investment, consider an investor who buys stock in a firm, say Boeing. This investor, having done her research, may conclude that she can make an expected return of 30 percent on Boeing over her one-year holding period. The actual return over this period will almost certainly not be equal to 30 percent; it might be much greater or much lower. The distribution of returns on this investment is illustrated in Figure 4.2. Figure 4.2 Return Distribution for Risky Investment In addition to the expected return, an investor now has to consider the following. First, note that the actual returns, in this case, are different from the expected return. The spread of the actual returns around the expected return is measured by the variance or standard deviation of the distribution; the greater the deviation of the actual returns from the expected return, the greater the variance. Second, the bias toward positive or negative returns is represented by the skewness of the distribution. The distribution in Figure 4.2 is positively skewed, since there is a higher probability of large positive returns than large negative returns. Third, the shape of the tails of the distribution is measured by the kurtosis of the distribution; fatter tails lead to higher kurtosis. In investment terms, this represents the tendency of the price of this investment to jump (up or down from current levels) in either direction.
+In the special case where the distribution of returns is normal, investors do not have to worry about skewness and kurtosis, since there is no skewness (normal distributions are symmetric) and a normal distribution is defined to have a kurtosis of zero. Figure 4.3 illustrates the return distributions on two investments with symmetric returns. Figure 4.3 Return Distribution Comparisons When return distributions are normal, the characteristics of any investment can be measured with two variables—the expected return, which represents the opportunity in the investment, and the standard deviation or variance, which represents the danger. In this scenario, a rational investor, faced with a choice between two investments with the same standard deviation but different expected returns, will always pick the one with the higher expected return.
+In the more general case, where distributions are neither symmetric nor normal, it is still conceivable that investors will choose between investments on the basis of only the expected return and the variance, if they possess utility functions that allow them to do so.1 It is far more likely, however, that they prefer positive skewed distributions to negatively skewed ones, and distributions with a lower likelihood of jumps (lower kurtosis) over those with a higher likelihood of jumps (higher kurtosis). In this world, investors will trade off the good (higher expected returns and more positive skewness) against the bad (higher variance and kurtosis) in making investments.
+In closing, it should be noted that the expected returns and variances that we run into in practice are almost always estimated using past returns rather than future returns. The assumption made when using historical variances is that past return distributions are good indicators of future return distributions. When this assumption is violated, as is the case when the asset’s characteristics have changed significantly over time, the historical estimates may not be good measures of risk. optvar.xls: This is a dataset on the Web that summarizes standard deviations and variances of stocks in various sectors in the United States. Diversifiable and Nondiversifiable Risk
+Although there are many reasons why actual returns may differ from expected returns, we can group the reasons into two categories: firm-specific and marketwide. The risks that arise from firm-specific actions affect one or a few investments, while the risks arising from marketwide reasons affect many or all investments. This distinction is critical to the way we assess risk in finance.
+Components of Risk
+When an investor buys stock or takes an equity position in a firm, he or she is exposed to many risks. Some risk may affect only one or a few firms, and this risk is categorized as firm-specific risk. Within this category, we would consider a wide range of risks, starting with the risk that a firm may have misjudged the demand for a product from its customers; we call this project risk. For instance, consider Boeing’s investment in a Super Jumbo jet. This investment is based on the assumption that airlines want a larger airplane and are willing to pay a high price for it. If Boeing has misjudged this demand, it will clearly have an impact on Boeing’s earnings and value, but it should not have a significant effect on other firms in the market. The risk could also arise from competitors proving to be stronger or weaker than anticipated, called competitive risk. For instance, assume that Boeing and Airbus are competing for an order from Qantas, the Australian airline. The possibility that Airbus may win the bid is a potential source of risk to Boeing and perhaps some of its suppliers, but again, few other firms will be affected by it. Similarly, Disney recently launched magazines aimed at teenage girls, hoping to capitalize on the success of its TV shows. Whether it succeeds is clearly important to Disney and its competitors, but it is unlikely to have an impact on the rest of the market. In fact, risk measures can be extended to include risks that may affect an entire sector but are restricted to that sector; we call this sector risk. For instance, a cut in the defense budget in the United States will adversely affect all firms in the defense business, including Boeing, but there should be no significant impact on other sectors. What is common across the three risks described—project, competitive, and sector risk—is that they affect only a small subset of firms.
+There is another group of risks that is much more pervasive and affects many if not all investments. For instance, when interest rates increase, all investments are negatively affected, albeit to different degrees. Similarly, when the economy weakens, all firms feel the effects, though cyclical firms (such as automobiles, steel, and housing) may feel it more. We term this risk market risk.
+Finally, there are risks that fall in a gray area, depending on how many assets they affect. For instance, when the dollar strengthens against other currencies, it has a significant impact on the earnings and values of firms with international operations. If most firms in the market have significant international operations, it could well be categorized as market risk. If only a few do, it would be closer to firm-specific risk. Figure 4.4 summarizes the spectrum of firm-specific and market risks. Figure 4.4 Breakdown of Risk Why Diversification Reduces or Eliminates Firm-Specific Risk: An Intuitive Explanation
+As an investor, you could invest all your portfolio in one asset. If you do so, you are exposed to both firm-specific and market risk. If, however, you expand your portfolio to include other assets or stocks, you are diversifying, and by doing so you can reduce your exposure to firm-specific risk. There are two reasons why diversification reduces or, at the limit, eliminates firm-specific risk. The first is that each investment in a diversified portfolio is a much smaller percentage of that portfolio than would be the case if you were not diversified. Any action that increases or decreases the value of only that investment or a small group of investments will have only a small impact on your overall portfolio, whereas undiversified investors are much more exposed to changes in the values of the investments in their portfolios. The second reason is that the effects of firm-specific actions on the prices of individual assets in a portfolio can be either positive or negative for each asset for any period. Thus, in very large portfolios this risk will average out to zero and will not affect the overall value of the portfolio.
+In contrast, the effects of marketwide movements are likely to be in the same direction for most or all investments in a portfolio, though some assets may be affected more than others. For instance, other things being equal, an increase in interest rates will lower the values of most assets in a portfolio. Being more diversified does not eliminate this risk.
+A Statistical Analysis of Diversification-Reducing Risk
+The effects of diversification on risk can be illustrated fairly dramatically by examining the effects of increasing the number of assets in a portfolio on portfolio variance. The variance in a portfolio is partially determined by the variances of the individual assets in the portfolio and partially by how they move together; the latter is measured statistically with a correlation coefficient or the covariance across investments in the portfolio. It is the covariance term that provides an insight into why diversification will reduce risk and by how much.
+Consider a portfolio of two assets. Asset A has an expected return of μA and a variance in returns of σ2A, while asset B has an expected return of μΒ and a variance in returns of σ2B. The correlation in returns between the two assets, which measures how the assets move together, is ρAB. The expected returns and variances of a two-asset portfolio can be written as a function of these inputs and the proportion of the portfolio going to each asset. where wA = Proportion of the portfolio in asset A
+The last term in the variance formulation is sometimes written in terms of the covariance in returns between the two assets, which is: The savings that accrue from diversification are a function of the correlation coefficient. Other things remaining equal, the higher the correlation in returns between the two assets, the smaller are the potential benefits from diversification. It is worth adding, though, that the benefits of correlation exist even for positively correlated assets and are non-existent only when the correlation is equal to one.
+Mean-Variance Models Measuring Market Risk
+While most risk and return models in use in finance agree on the first two steps of the risk analysis process (i.e., that risk comes from the distribution of actual returns around the expected return and that risk should be measured from the perspective of a marginal investor who is well diversified), they part ways when it comes to measuring nondiversifiable or market risk. This section will discuss the different models that exist in finance for measuring market risk and why they differ. It begins with what still is the most widely used model for measuring market risk in finance—the capital asset pricing model (CAPM)—and then discusses the alternatives to this model that have developed over the past two decades. While the discussion will emphasize the differences, it will also look at what the models have in common.
+Capital Asset Pricing Model
+The risk and return model that has been in use the longest and is still the standard for most practitioners is the capital asset pricing model (CAPM). This section will examine the assumptions on which the model is based and the measures of market risk that emerge from these assumptions. WHY IS THE MARGINAL INVESTOR ASSUMED TO BE DIVERSIFIED?
+The argument that diversification reduces an investor’s exposure to risk is clear both intuitively and statistically, but risk and return models in finance go further. These models look at risk through the eyes of the investor most likely to be trading on the investment at any point in time—the marginal investor. They argue that this investor, who sets prices for investments, is well diversified; thus, the only risk that he or she cares about is the risk added to a diversified portfolio or market risk. This argument can be justified simply. The risk in an investment will always be perceived to be higher for an undiversified investor than for a diversified one, since the latter does not shoulder any firm-specific risk and the former does. If both investors have the same expectations about future earnings and cash flows on an asset, the diversified investor will be willing to pay a higher price for that asset because of his or her perception of lower risk. Consequently, the asset, over time, will end up being held by diversified investors.
+This argument is powerful, especially in markets where assets can be traded easily and at low cost. Thus, it works well for a stock traded in developed markets, since investors can become diversified at fairly low cost. In addition, a significant proportion of the trading in developed market stocks is done by institutional investors, who tend to be well diversified. It becomes a more difficult argument to sustain when assets cannot be easily traded or the costs of trading are high. In these markets, the marginal investor may well be undiversified, and firm-specific risk may therefore continue to matter when looking at individual investments. For instance, real estate in most countries is still held by investors who are undiversified and have the bulk of their wealth tied up in these investments. Assumptions
+While diversification reduces the exposure of investors to firm-specific risk, most investors limit their diversification to holding only a few assets. Even large mutual funds rarely hold more than a few hundred stocks, and many of them hold as few as 10 to 20. There are two reasons why investors stop diversifying. One is that an investor or mutual fund manager can obtain most of the benefits of diversification from a relatively small portfolio, because the marginal benefits of diversification become smaller as the portfolio gets more diversified. Consequently, these benefits may not cover the marginal costs of diversification, which include transactions and monitoring costs. Another reason for limiting diversification is that many investors (and funds) believe they can find undervalued assets and thus choose not to hold those assets that they believe to be fairly valued or overvalued.
+The capital asset pricing model assumes that there are no transaction costs, all assets are traded, and investments are infinitely divisible (i.e., you can buy any fraction of a unit of the asset). It also assumes that everyone has access to the same information and that investors therefore cannot find under- or overvalued assets in the marketplace. By making these assumptions, it allows investors to keep diversifying without additional cost. At the limit, their portfolios will not only include every traded asset in the market but these assets will be held in proportion to their market value.
+The fact that this portfolio includes all traded assets in the market is the reason it is called the market portfolio, which should not be a surprising result, given the benefits of diversification and the absence of transaction costs in the capital asset pricing model. If diversification reduces exposure to firm-specific risk and there are no costs associated with adding more assets to the portfolio, the logical limit to diversification is to hold a small proportion of every traded asset in the economy. If this seems abstract, consider the market portfolio to be an extremely well diversified mutual fund that holds stocks and real assets. In the CAPM, all investors will hold combinations of the riskier asset and that supremely diversified mutual fund.2
+Investor Portfolios in the CAPM
+If every investor in the market holds the identical market portfolio, how exactly do investors reflect their risk aversion in their investments? In the capital asset pricing model, investors adjust for their risk preferences in their allocation decision, where they decide how much to invest in a riskless asset and how much in the market portfolio. Investors who are risk averse might choose to put much or even all of their wealth in the riskless asset. Investors who want to take more risk will invest the bulk or even all of their wealth in the market portfolio. Investors who invest all their wealth in the market portfolio and are desirous of taking on still more risk would do so by borrowing at the riskless rate and investing in the same market portfolio as everyone else.
+These results are predicated on two additional assumptions. First, there exists a riskless asset, where the expected returns are known with certainty. Second, investors can lend and borrow at the riskless rate to arrive at their optimal allocations. While lending at the riskless rate can be accomplished fairly simply by buying Treasury bills or bonds, borrowing at the riskless rate might be more difficult for individuals to do. There are variations of the CAPM that allow these assumptions to be relaxed and still arrive at conclusions that are consistent with the model.
+Measuring the Market Risk of an Individual Asset
+The risk of any asset to an investor is the risk added by that asset to the investor’s overall portfolio. In the CAPM world, where all investors hold the market portfolio, the risk to an investor of an individual asset will be the risk that this asset adds to the market portfolio. Intuitively, if an asset moves independently of the market portfolio, it will not add much risk to the market portfolio. In other words, most of the risk in this asset is firm-specific and can be diversified away. In contrast, if an asset tends to move up when the market portfolio moves up and down when it moves down, it will add risk to the market portfolio. This asset has more market risk and less firm-specific risk. Statistically, this added risk is measured by the covariance of the asset with the market portfolio.
+Measuring the Nondiversifiable Risk
+In a world in which investors hold a combination of only two assets—the riskless asset and the market portfolio—the risk of any individual asset will be measured relative to the market portfolio. In particular, the risk of any asset will be the risk it adds to the market portfolio. To arrive at the appropriate measure of this added risk, assume that σ2m is the variance of the market portfolio prior to the addition of the new asset and that the variance of the individual asset being added to this portfolio is σ2i. The market value portfolio weight on this asset is wi, and the covariance in returns between the individual asset and the market portfolio is σim. The variance of the market portfolio prior to and after the addition of the individual asset can then be written as: The market value weight on any individual asset in the market portfolio should be small, since the market portfolio includes all traded assets in the economy. Consequently, the first term in the equation should approach zero, and the second term should approach σ2m′ leaving the third term (σim′ the covariance) as the measure of the risk added by asset i.
+Standardizing Covariances
+The covariance is a percentage value, and it is difficult to pass judgment on the relative risk of an investment by looking at this value. In other words, knowing that the covariance of Boeing with the market portfolio is 55 percent does not provide us a clue as to whether Boeing is riskier or safer than the average asset. We therefore standardize the risk measure by dividing the covariance of each asset with the market portfolio by the variance of the market portfolio. This yields a risk measure called the beta of the asset: Since the covariance of the market portfolio with itself is its variance, the beta of the market portfolio (and, by extension, the average asset in it) is 1. Assets that are riskier than average (using this measure of risk) will have betas that exceed 1, and assets that are safer than average will have betas that are lower than 1. The riskless asset will have a beta of zero.
+Getting Expected Returns
+The fact that every investor holds some combination of the riskless asset and the market portfolio leads to the next conclusion, which is that the expected return on an asset is linearly related to the beta of the asset. In particular, the expected return on an asset can be written as a function of the risk-free rate and the beta of that asset: where E(Ri) = Expected return on asset i
+Rf = Risk-free rate
+E(Rm) = Expected return on market portfolio
+βi = Beta of asset i
+To use the capital asset pricing model, we need three inputs. While the next chapter looks at the estimation process in far more detail, each of these inputs is estimated as follows: The riskless asset is defined to be an asset for which the investor knows the expected return with certainty for the time horizon of the analysis.
+The risk premium is the premium demanded by investors for investing in the market portfolio, which includes all risky assets in the market, instead of investing in a riskless asset.
+The beta, defined as the covariance of the asset divided by the market portfolio, measures the risk added by an investment to the market portfolio. In summary, in the capital asset pricing model all the market risk is captured in one beta measured relative to a market portfolio, which at least in theory should include all traded assets in the marketplace held in proportion to their market value.
+Arbitrage Pricing Model
+The restrictive assumptions on transaction costs and private information in the capital asset pricing model, and the model’s dependence on the market portfolio, have long been viewed with skepticism by both academics and practitioners. Ross (1976) suggested an alternative model for measuring risk called the arbitrage pricing model (APM).
+Assumptions
+If investors can invest risklessly and earn more than the riskless rate, they have found an arbitrage opportunity. The premise of the arbitrage pricing model is that investors take advantage of such arbitrage opportunities, and in the process eliminate them. If two portfolios have the same exposure to risk but offer different expected returns, investors will buy the portfolio that has the higher expected returns and sell the portfolio with the lower expected returns, and earn the difference as a riskless profit. To prevent this arbitrage from occurring, the two portfolios have to earn the same expected return.
+Like the capital asset pricing model, the arbitrage pricing model begins by breaking risk down into firm-specific and market risk components. As in the capital asset pricing model, firm-specific risk covers information that affects primarily the firm. Market risk affects many or all firms and would include unanticipated changes in a number of economic variables, including gross national product, inflation, and interest rates. Incorporating both types of risk into a return model, we get: where R is the actual return, E(R) is the expected return, m is the marketwide component of unanticipated risk, and ε is the firm-specific component. Thus, the actual return can be different from the expected return, because of either market risk or firm-specific actions.
+Sources of Marketwide Risk
+While both the capital asset pricing model and the arbitrage pricing model make a distinction between firm-specific and marketwide risk, they measure market risk differently. The CAPM assumes that market risk is captured in the market portfolio, whereas the arbitrage pricing model allows for multiple sources of marketwide risk and measures the sensitivity of investments to changes in each source. In general, the market component of unanticipated returns can be decomposed into economic factors: where βj = Sensitivity of investment to unanticipated changes in market risk factor j
+Fj = Unanticipated changes in market risk factor j
+Note that the measure of an investment’s sensitivity to any macroeconomic (or market) factor takes the form of a beta, called a factor beta. In fact, this beta has many of the same properties as the market beta in the CAPM.
+Effects of Diversification
+The benefits of diversification were discussed earlier, in the context of the breakdown of risk into market and firm-specific risk. The primary point of that discussion was that diversification eliminates firm-specific risk. The arbitrage pricing model uses the same argument and concludes that the return on a portfolio will not have a firm-specific component of unanticipated returns. The return on a portfolio can be written as the sum of two weighted averages—that of the anticipated returns in the portfolio and that of the market factors: where wj = Portfolio weight on asset j (where there are n assets)
+Rj = Expected return on asset j
+βi,j = Beta on factor i for asset j
+Expected Returns and Betas
+The final step in this process is estimating an expected return as a function of the betas just specified. To do this, we should first note that the beta of a portfolio is the weighted average of the betas of the assets in the portfolio. This property, in conjunction with the absence of arbitrage, leads to the conclusion that expected returns should be linearly related to betas. To see why, assume that there is only one factor and three portfolios. Portfolio A has a beta of 2.0 and an expected return of 20 percent; portfolio B has a beta of 1.0 and an expected return of 12 percent; and portfolio C has a beta of 1.5 and an expected return of 14 percent. Note that investors can put half of their wealth in portfolio A and half in portfolio B and end up with portfolios with a beta of 1.5 and an expected return of 16 percent. Consequently no investor will choose to hold portfolio C until the prices of assets in that portfolio drop and the expected return increases to 16 percent. By the same rationale, the expected returns of every portfolio should be a linear function of the beta. If they were not, we could combine two other portfolios, one with a higher beta and one with a lower beta, to earn a higher return than the portfolio in question, creating an opportunity for arbitrage. This argument can be extended to multiple factors with the same results. Therefore, the expected return on an asset can be written as: where Rf = Expected return on a zero-beta portfolio
+E(Rj) = Expected return on a portfolio with a factor beta of 1 for factor j, and zero for all other factors (where j = 1, 2, ... , K factors)
+The terms in the brackets can be considered to be risk premiums for each of the factors in the model.
+The capital asset pricing model can be considered to be a special case of the arbitrage pricing model, where there is only one economic factor driving marketwide returns, and the market portfolio is the factor. The APM in Practice
+The arbitrage pricing model requires estimates of each of the factor betas and factor risk premiums in addition to the riskless rate. In practice, these are usually estimated using historical data on asset returns and a factor analysis. Intuitively, in a factor analysis, we examine the historical data looking for common patterns that affect broad groups of assets (rather than just one sector or a few assets). A factor analysis provides two output measures: 1. It specifies the number of common factors that affected the historical return data.
+2. It measures the beta of each investment relative to each of the common factors and provides an estimate of the actual risk premium earned by each factor. The factor analysis does not, however, identify the factors in economic terms. In summary, in the arbitrage pricing model the market risk is measured relative to multiple unspecified macroeconomic variables, with the sensitivity of the investment relative to each factor being measured by a beta. The number of factors, the factor betas, and the factor risk premiums can all be estimated using the factor analysis.
+Multifactor Models for Risk and Return
+The arbitrage pricing model’s failure to identify the factors specifically in the model may be a statistical strength, but it is an intuitive weakness. The solution seems simple: Replace the unidentified statistical factors with specific economic factors, and the resultant model should have an economic basis while still retaining much of the strength of the arbitrage pricing model. That is precisely what multifactor models try to do.
+Deriving a Multifactor Model
+Multifactor models generally are determined by historical data rather than by economic modeling. Once the number of factors has been identified in the arbitrage pricing model, their behavior over time can be extracted from the data. The behavior of the unnamed factors over time can then be compared to the behavior of macroeconomic variables over that same period, to see whether any of the variables is correlated, over time, with the identified factors.
+For instance, Chen, Roll, and Ross (1986) suggest that the following macroeconomic variables are highly correlated with the factors that come out of factor analysis: industrial production, changes in default premium, shifts in the term structure, unanticipated inflation, and changes in the real rate of return. These variables can then be correlated with returns to come up with a model of expected returns, with firm-specific betas calculated relative to each variable. where βGNP = Beta relative to changes in industrial production
+E(RGNP) = Expected return on a portfolio with a beta of one on the industrial production factor and zero on all other factors
+βI = Beta relative to changes in inflation
+E(RI) = Expected return on a portfolio with a beta of one on the inflation factor and zero on all other factors
+The costs of going from the arbitrage pricing model to a macroeconomic multifactor model can be traced directly to the errors that can be made in identifying the factors. The economic factors in the model can change over time, as will the risk premium associated with each one. For instance, oil price changes were a significant economic factor driving expected returns in the 1970s but are not as significant in other time periods. Using the wrong factor or missing a significant factor in a multifactor model can lead to inferior estimates of expected return.
+ALTERNATIVE MODELS FOR EQUITY RISK
+The CAPM, arbitrage pricing model, and multifactor model represent attempts by financial economists to build risk and return models from the mean-variance base established by Harry Markowitz (1991). There are many, though, who believe the basis for the model is flawed and that we should be looking at alternatives, and in this section, we will look at some of them.
+Different Return Distributions
+From its very beginnings, the mean-variance framework has been controversial. While there have been many who have challenged its applicability, we will consider these challenges in three groups. The first group argues that stock prices, in particular, and investment returns, in general, exhibit too many large values to be drawn from a normal distribution. They argue that the fat tails on stock price distributions lend themselves better to a class of distributions, called power law distributions, which exhibit infinite variance and long periods of price dependence. The second group takes issue with the symmetry of the normal distribution and argues for measures that incorporate the asymmetry observed in actual return distributions into risk measures. The third group posits that distributions that allow for price jumps are more realistic and that risk measures should consider the likelihood and magnitude of price jumps.
+Fat Tails and Power Law Distributions
+Benoit Mandelbrot (1961; Mandelbrot and Hudson, 2004), a mathematician who also did pioneering work on the behavior of stock prices, was one of those who took issue with the use of normal and lognormal distributions. He argued, based on his observation of stock and real asset prices, that a power law distribution characterized them better. In a power-law distribution, the relationship between two variables, Y and X, can be written as follows: In this equation, α is a constant (constant of proportionality), and k is the power law exponent. Mandelbrot’s key point was that the normal and log normal distributions were best suited for series that exhibited mild and well-behaved randomness, whereas power law distributions were more suited for series that exhibited large movements and what he termed wild randomness. Wild randomness occurs when a single observation can affect the population in a disproportionate way; stock and commodity prices exhibit wild randomness. Stock and commodity prices, with their long periods of relatively small movements, punctuated by wild swings in both directions, seem to fit better into the wild randomness group.
+What are the consequences for risk measures? If asset prices follow power law distributions, the standard deviation or volatility ceases to be a good risk measure and a good basis for computing probabilities. Assume, for instance, that the standard deviation in annual stock returns is 15 percent and that the average return is 10 percent. Using the normal distribution as the base for probability predictions, this will imply that the stock returns will exceed 40 percent (average plus two standard deviations) only once every 44 years and 55 percent only (average plus three standard deviations) only once every 740 years. In fact, stock returns will be greater than 85 percent (average plus five standard deviations) only once every 3.5 million years. In reality, stock returns exceed these values far more frequently, a finding consistent with power law distributions, where the probability of larger values declines linearly as a function of the power law exponent. As the value gets doubled, the probability of its occurrence drops by the square of the exponent. Thus, if the exponent in the distribution is 2, the likelihood of returns of 25 percent, 50 percent, and 100 percent can be computed as follows: Returns will exceed 25 percent: once every 6 years.
+Returns will exceed 50 percent: once every 24 years.
+Returns will exceed 100 percent: once every 96 years. Note that as the returns get doubled, the likelihood increases four-fold (the square of the exponent). As the exponent decreases, the likelihood of larger values increases; an exponent between 0 and 2 will yield extreme values more often than a normal distribution. An exponent between 1 and 2 yields power law distributions called stable Paretian distributions, which have infinite variance. In an early study, Fama (1965) estimated the exponent for stocks to be between 1.7 and 1.9, but subsequent studies have found that the exponent is higher in both equity and currency markets.3
+In practical terms, the power law proponents argue that using measures such as volatility (and its derivatives such as beta) underestimate the risk of large movements. The power law exponents for assets, in their view, provide investors with more realistic risk measures for these assets. Assets with higher exponents are less risky (since extreme values become less common) than asset with lower exponents.
+Mandelbrot’s challenge to the normal distribution was more than a procedural one. Mandelbrot’s world, in contrast to the Gaussian mean-variance one, is a world where prices move jaggedly over time and look as though they have no pattern at a distance, but where patterns repeat themselves, when observed closely. In the 1970s, Mandelbrot created a branch of mathematics called fractal geometry where processes are not described by conventional statistical or mathematical measures but by fractals; a fractal is a geometric shape that when broken down into smaller parts replicates that shape. To illustrate the concept, he uses the example of the coastline that, from a distance, looks irregular and up close looks roughly the same—fractal patterns repeat themselves. In fractal geometry, higher fractal dimensions translate into more jagged shapes; the rugged Cornish coastline has a fractal dimension of 1.25 whereas the much smoother South African coastline has a fractal dimension of 1.02. Using the same reasoning, stock prices that look random, when observed at longer time intervals, start revealing self-repeating patterns, when observed over shorter time periods. More volatile stocks score higher on measures of fractal dimension, thus making it a measure of risk. With fractal geometry, Mandelbrot was able to explain not only the higher frequency of price jumps (relative to the normal distribution) but also long periods where prices move in the same direction and the resulting price bubbles.
+Asymmetric Distributions
+Intuitively, it should be downside risk that concerns us and not upside risk. In other words, it is not investments that go up significantly that create heartburn and unease but investments that go down significantly. The mean-variance framework, by weighting both upside volatility and downside movements equally, does not distinguish between the two. With a normal or any other symmetric distribution, the distinction between upside and downside risk is irrelevant because the risks are equivalent. With asymmetric distributions, though, there can be a difference between upside and downside risk. Studies of risk aversion in humans conclude that (1) they are loss averse; that is, they weigh the pain of a loss more than the joy of an equivalent gain and (2) they value very large positive payoffs—long shots—far more than they should given the likelihood of these payoffs.
+In practice, return distributions for stocks and most other assets are not symmetric. Instead, asset returns exhibit fat tails (i.e, more jumps) and are more likely to have extreme positive values than extreme negative values (simply because returns are constrained to be no less than –100 percent). As a consequence, the distribution of stock returns has a higher incidence of extreme returns (fat tails or kurtosis) and a tilt toward very large positive returns (positive skewness). Critics of the mean-variance approach argue that it takes too narrow a view of both rewards and risk. In their view, a fuller return measure should consider not just the magnitude of expected returns but also the likelihood of very large positive returns or skewness, and a more complete risk measure should incorporate both variance and the possibility of big jumps (co-kurtosis). Note that even as these approaches deviate from the mean-variance approach in terms of how they define risk, they stay true to the portfolio measure of risk. In other words, it is not the possibility of large positive payoffs (skewness) or big jumps (kurtosis) that they argue should be considered, but only that portion of the skewness (co-skewness) and kurtosis (co-kurtosis) that is market-related and not diversifiable.
+Jump Process Models
+The normal, power law, and asymmetric distributions that form the basis for the models we have discussed in this section are all continuous distributions. Observing the reality that stock prices do jump, there are some who have argued for the use of jump process distributions to derive risk measures.
+Press (1967), in one of the earliest papers that attempted to model stock price jumps, argued that stock prices follow a combination of a continuous price distribution and a Poisson distribution, where prices jump at irregular intervals. The key parameters of the Poisson distribution are the expected size of the price jump (μ), the variance in this value (δ2), and the likelihood of a price jump in any specified time period (λ), and Press estimated these values for 10 stocks. In subsequent papers, Beckers (1981) and Ball and Torous (1983) suggest ways of refining these estimates. In an attempt to bridge the gap between the CAPM and jump process models, Jarrow and Rosenfeld (1984) derive a version of the capital asset pricing model that includes a jump component that captures the likelihood of market jumps and an individual asset’s correlation with these jumps.
+While jump process models have gained some traction in option pricing, they have had limited success in equity markets, largely because the parameters of jump process models are difficult to estimate with any degree of precision. Thus, while everyone agrees that stock prices jump, there is little consensus on the best way to measure how often this happens, whether these jumps are diversifiable, and how best to incorporate their effect into risk measures.
+Regression or Proxy Models
+The conventional models for risk and return in finance (CAPM, arbitrage pricing model, and even multifactor models) start by making assumptions about how investors behave and how markets work to derive models that measure risk and link those measures to expected returns. While these models have the advantage of a foundation in economic theory, they seem to fall short in explaining differences in returns across investments. The reasons for the failure of these models run the gamut: The assumptions made about markets are unrealistic (no transactions costs, perfect information) and investors don’t behave rationally (and behavioral finance research provides ample evidence of this).
+With proxy models, we essentially give up on building risk and return models from economic theory. Instead, we start with how investments are priced by markets and relate returns earned to observable variables. Rather than talk in abstractions, consider the work done by Fama and French in the early 1990s. Examining returns earned by individual stocks from 1962 to 1990, they concluded that CAPM betas did not explain much of the variation in these returns. They then took a different tack and looked for company-specific variables that did a better job of explaining return differences they pinpointed two variables—the market capitalization of a firm and its price-to-book ratio (the ratio of market cap to accounting book value for equity). Specifically, they concluded that small market cap stocks earned much higher annual returns than large market cap stocks and that low price to book ratio stocks earned much higher annual returns than stocks that traded at high price-to-book ratios. Rather than view this as evidence of market inefficiency (which is what prior studies that had found the same phenomena had done), they argued if these stocks earned higher returns over long time periods, they must be riskier than stocks that earned lower returns. In effect, market capitalization and price-to-book ratios were better proxies for risk, according to their reasoning, than betas. In fact, they regressed returns on stocks against the market capitalization of a company and its price-to-book ratio to arrive at the following regression for U.S. stocks; In a pure proxy model, you could plug the market capitalization and book-to-market ratio for any company into this regression to get expected monthly returns.
+In the two decades since the Fama-French paper brought proxy models to the fore, researchers have probed the data (which has become more detailed and voluminous over time) to find better and additional proxies for risk. Some of the proxies are highlighted here: Earnings momentum. Equity research analysts will find vindication in research that seems to indicate that companies that have reported stronger than expected earnings growth in the past earn higher returns than the rest of the market.
+Price momentum. Chartists will smile when they read this, but researchers have concluded that price momentum carries over into future periods. Thus, the expected returns will be higher for stocks that have outperformed markets in recent time periods and lower for stocks that have lagged.
+Liquidity. In a nod to real-world costs, there seems to be clear evidence that stocks that are less liquid (lower trading volume, higher bid-ask spreads) earn higher returns than more liquid stocks. While the use of pure proxy models by practitioners is rare, they have adapted the findings for these models into their day-to-day use. Many analysts have melded the CAPM with proxy models to create composite or melded models. For instance, many analysts who value small companies derive expected returns for these companies by adding a small cap premium to the CAPM expected return: The threshold for small capitalization varies across time but is generally set at the bottom decile of publicly traded companies, and the small cap premium itself is estimated by looking at the historical premium earned by small cap stocks over the market. Using the Fama-French findings, the CAPM has been expanded to include market capitalization and price-to-book ratios as additional variables, with the expected return stated as: The size and the book-to-market betas are estimated by regressing a stock’s returns against the size premium and book-to-market premiums over time; this is analogous to the way we get the market beta, by regressing stock returns against overall market returns.
+While the use of proxy and melded models offers a way of adjusting expected returns to reflect market reality, there are three dangers in using these models. 1. Data mining. As the amount of data that we have on companies increases and becomes more accessible, it is inevitable that we will find more variables that are related to returns. It is also likely that most of these variables are not proxies for risk and that the correlation is a function of the time period that we look at. In effect, proxy models are statistical models and not economic models. Thus, there is no easy way to separate the variables that matter from those that do not.
+2. Standard error. Since proxy models come from looking at historical data, they carry all of the burden of the noise in the data. Stock returns are extremely volatile over time, and any historical premia that we compute (for market capitalization or any other variable) are going to have significant standard errors. The standard errors on the size and book-to-market betas in the three-factor Fama-French model may be so large that using them in practice creates almost as much noise as it adds in precision.
+3. Pricing error or risk proxy. For decades, value investors have argued that you should invest in stocks with low PE ratios that trade at low multiples of book value and have high dividend yields, pointing to the fact that you will earn higher returns by doing so. (In fact, a scan of Benjamin Graham’s screens from security analysis4 for cheap companies unearths most of the proxies that you see in use today.) Proxy models incorporate all of these variables into the expected return and thus render these assets to be fairly priced. Using the circular logic of these models, markets are always efficient because any inefficiency that exists is just another risk proxy that needs to get built into the model. A COMPARATIVE ANALYSIS OF EQUITY RISK MODELS
+When faced with the choice of estimating expected returns on equity or cost of equity, we are therefore faced with several choices, ranging from the CAPM to proxy models. Table 4.1 summarizes the different models and presents their pluses and minuses.
+Table 4.1 Alternative Models for Cost of Equity The decision has to be based as much on theoretical considerations as it will be on pragmatic considerations. The CAPM is the simplest of the models, insofar as it requires only one firm-specific input (the beta), and that input can be estimated readily from public information. To replace the CAPM with an alternative model, whether it be from the mean variance family (arbitrage pricing model or multifactor models), alternative return process families (power, asymmetric, and jump distribution models), or proxy models, we need evidence of substantial improvement in accuracy in future forecasts (and not just in explaining past returns).
+Ultimately, the survival of the capital asset pricing model as the default model for risk in real-world applications is a testament to both its intuitive appeal and the failure of more complex models to deliver significant improvement in terms of estimating expected returns. We would argue that a judicious use of the capital asset pricing model, without an over reliance on historical data, is still the most effective way of dealing with risk in valuation in most cases. In some sectors (commodities) and segments (closely held companies, illiquid stocks), using other, more complete models will be justified. We will return to the question of how improvements in estimating the inputs to the CAPM can generate far more payoff than switching to more complicated models for cost of equity.
+MODELS OF DEFAULT RISK
+The risk discussed so far in this chapter relates to cash flows on investments being different from expected cash flows. There are some investments, however, in which the cash flows are promised when the investment is made. This is the case, for instance, when you lend to a business or buy a corporate bond; the borrower may default on interest and principal payments on the borrowing. Generally speaking, borrowers with higher default risk should pay higher interest rates on their borrowing than those with lower default risk. This section examines the measurement of default risk and the relationship of default risk to interest rates on borrowing.
+In contrast to the general risk and return models for equity, which evaluate the effects of market risk on expected returns, models of default risk measure the consequences of firm-specific default risk on promised returns. While diversification can be used to explain why firm-specific risk will not be priced into expected returns for equities, the same rationale cannot be applied to securities that have limited upside potential and much greater downside potential from firm-specific events. To see what is meant by limited upside potential, consider investing in the bond issued by a company. The coupons are fixed at the time of the issue, and these coupons represent the promised cash flow on the bond. The best-case scenario for you as an investor is that you receive the promised cash flows; you are not entitled to more than these cash flows even if the company is wildly successful. All other scenarios contain only bad news, though in varying degrees, with the delivered cash flows being less than the promised cash flows. Consequently, the expected return on a corporate bond is likely to reflect the firm-specific default risk of the firm issuing the bond.
+Determinants of Default Risk
+The default risk of a firm is a function of two variables. The first is the firm’s capacity to generate cash flows from operations, and the second is its financial obligations—including interest and principal payments.5 Firms that generate high cash flows relative to their financial obligations should have lower default risk than do firms that generate low cash flows relative to obligations. Thus, firms with significant existing investments that generate high cash flows will have lower default risk than will firms that do not have such investments.
+In addition to the magnitude of a firm’s cash flows, the default risk is also affected by the volatility in these cash flows. The more stability there is in cash flows, the lower is the default risk in the firm. Firms that operate in predictable and stable businesses will have lower default risk than will otherwise similar firms that operate in cyclical or volatile businesses.
+Most models of default risk use financial ratios to measure the cash flow coverage (i.e., the magnitude of cash flows relative to obligations) and control for industry effects in order to evaluate the variability in cash flows.
+Bond Ratings and Interest Rates
+The most widely used measure of a firm’s default risk is its bond rating, which is generally assigned by an independent ratings agency. The two best known are Standard & Poor’s (S&P) and Moody’s. Thousands of companies are rated by these two agencies, and their views carry significant weight with financial markets.
+The Ratings Process
+The process of rating a bond starts when the issuing company requests a rating from a bond ratings agency. The ratings agency then collects information from both publicly available sources, such as financial statements, and the company itself and makes a decision on the rating. If the company disagrees with the rating, it is given the opportunity to present additional information. This process is presented schematically for one ratings agency, Standard & Poor’s, in Figure 4.5. Figure 4.5 The Ratings Process The ratings assigned by these agencies are letter ratings. A rating of AAA from Standard & Poor’s and Aaa from Moody’s represents the highest rating, granted to firms that are viewed as having the lowest default risk. As the default risk increases, the ratings decline toward D for firms in default (Standard & Poor’s). A rating at or above BBB by Standard & Poor’s (or Baa by Moody’s) is categorized as above investment grade, reflecting the view of the ratings agency that there is relatively little default risk in investing in bonds issued by these firms.
+Determinants of Bond Ratings
+The bond ratings assigned by ratings agencies are primarily based on publicly available information, though private information conveyed by the firm to the ratings agency does play a role. The rating assigned to a company’s bonds will depend in large part on financial ratios that measure the capacity of the company to meet debt payments and generate stable and predictable cash flows. While a multitude of financial ratios exist, Table 4.2 summarizes some of the key ratios used to measure default risk.
+Table 4.2 Definition of Financial Ratios: S&P Financial Ratio
+Definition EBITDA/Revenues
+EBITDA/Revenues ROIC
+ROIC = EBIT/(BV of debt + BV of equity – Cash) EBIT/Interest expenses
+Interest coverage ratio EBITDA/Interest
+EBITDA/Interest expenses FFO/debt
+(Net Income + Depreciation)/Debt Free operating CF/Debt
+Funds from operations/Debt Discounted CF/Debt
+Discounted cash flows/Debt Debt/EBITDA
+BV of Debt/EBITDA D/(D+E)
+BV of Debt/(BV of Debt + BV of equity) There is a strong relationship between the bond rating a company receives and its performance on these financial ratios. Table 4.3 provides a summary of the median ratios6 from 2007 to 2009 for different S&P ratings classes for manufacturing firms.
+Table 4.3 Financial Ratios and S&P Ratings Not surprisingly, firms that generate income and cash flows significantly higher than debt payments, that are profitable, and that have low debt ratios are more likely to be highly rated than are firms that do not have these characteristics. There will be individual firms whose ratings are not consistent with their financial ratios, however, because the ratings agency does add subjective judgments into the final mix. Thus a firm that performs poorly on financial ratios but is expected to improve its performance dramatically over the next period may receive a higher rating than is justified by its current financials. For most firms, however, the financial ratios should provide a reasonable basis for estimating the bond rating.
+Bond Ratings and Interest Rates
+The interest rate on a corporate bond should be a function of its default risk, which is measured by its rating. If the rating is a good measure of the default risk, higher-rated bonds should be priced to yield lower interest rates than those of lower-rated bonds. In fact, the difference between the interest rate on a bond with default risk and a default-free government bond is the default spread. This default spread will vary by maturity of the bond and can also change from period to period, depending on economic conditions. Chapter 7 considers how best to estimate these default spreads and how they might vary over time.
+CONCLUSION
+Risk, as defined in finance, is measured based on deviations of actual returns on an investment from its expected returns. There are two types of risk. The first, called equity risk, arises in investments where there are no promised cash flows, but there are expected cash flows. The second, default risk, arises on investments with promised cash flows.
+On investments with equity risk, the risk is best measured by looking at the variance of actual returns around the expected returns, with greater variance indicating greater risk. This risk can be broken down into risk that affects one or a few investments, called firm-specific risk, and risk that affects many investments, refered to as market risk. When investors diversify, they can reduce their exposure to firm-specific risk. If we assume that the investors who trade at the margin are well diversified, the risk we should be looking at with equity investments is the nondiversifiable or market risk. The different models of equity risk introduced in this chapter share this objective of measuring market risk, but they differ in the way they do it. In the capital asset pricing model, exposure to market risk is measured by a market beta, which estimates how much risk an individual investment will add to a portfolio that includes all traded assets. The arbitrage pricing model and the multifactor model allow for multiple sources of market risk and estimate betas for an investment relative to each source. Regression or proxy models for risk look for firm characteristics, such as size, that have been correlated with high returns in the past and use these to measure market risk. In all these models, the risk measures are used to estimate the expected return on an equity investment. This expected return can be considered the cost of equity for a company.
+On investments with default risk, risk is measured by the likelihood that the promised cash flows might not be delivered. Investments with higher default risk should have higher interest rates, and the premium that we demand over a riskless rate is the default spread. For many U.S. companies, default risk is measured by rating agencies in the form of a bond rating; these ratings determine, in large part, the interest rates at which these firms can borrow. Even in the absence of ratings, interest rates will include a default spread that reflects the lenders’ assessments of default risk. These default-risk-adjusted interest rates represent the cost of borrowing or debt for a business.
+QUESTIONS AND SHORT PROBLEMS
+In the problems following, use an equity risk premium of 5.5 percent if none is specified. 1. The following table lists the stock prices for Microsoft from 1989 to 1998. The company did not pay any dividends during the period. Year
+Price 1989
+$ 1.20 1990
+$ 2.09 1991
+$ 4.64 1992
+$ 5.34 1993
+$ 5.05 1994
+$ 7.64 1995
+$10.97 1996
+$20.66 1997
+$32.31 1998
+$69.34 a. Estimate the average annual return you would have made on your investment.
+b. Estimate the standard deviation and variance in annual returns.
+c. If you were investing in Microsoft today, would you expect the historical standard deviations and variances to continue to hold? Why or why not? 2. Unicom is a regulated utility serving northern Illinois. The following table lists the stock prices and dividends on Unicom from 1989 to 1998. Year
+Price
+Dividends 1989
+$36.10
+$3.00 1990
+$33.60
+$3.00 1991
+$37.80
+$3.00 1992
+$30.90
+$2.30 1993
+$26.80
+$1.60 1994
+$24.80
+$1.60 1995
+$31.60
+$1.60 1996
+$28.50
+$1.60 1997
+$24.25
+$1.60 1998
+$35.60
+$1.60 a. Estimate the average annual return you would have made on your investment.
+b. Estimate the standard deviation and variance in annual returns.
+c. If you were investing in Unicom today, would you expect the historical standard deviations and variances to continue to hold? Why or why not? 3. The following table summarizes the annual returns you would have made on two companies—Scientific Atlanta, a satellite and data equipment manufacturer, and AT&T, the telecommunications giant—from 1989 to 1998. Year
+Scientific Atlanta
+AT&T 1989
+80.95%
+58.26% 1990
+–47.37%
+–33.79% 1991
+31.00%
+29.88% 1992
+132.44%
+30.35% 1993
+32.02%
+2.94% 1994
+25.37%
+–4.29% 1995
+–28.57%
+28.86% 1996
+0.00%
+–6.36% 1997
+11.67%
+48.64% 1998
+36.19%
+23.55% a. Estimate the average annual return and standard deviation in annual returns in each company.
+b. Estimate the covariance and correlation in returns between the two companies.
+c. Estimate the variance of a portfolio composed, in equal parts, of the two investments. 4. You are in a world where there are only two assets, gold and stocks. You are interested in investing your money in one, the other, or both assets. Consequently you collect the following data on the returns on the two assets over the past six years. Gold
+Stock Market Average return
+8%
+20% Standard deviation
+25%
+22% Correlation
+–0.4 a. If you were constrained to pick just one, which one would you choose?
+b. A friend argues that this is wrong. He says that you are ignoring the big payoffs that you can get on the other asset. How would you go about alleviating his concern?
+c. How would a portfolio composed of equal proportions in gold and stocks do in terms of mean and variance?
+d. You now learn that GPEC (a cartel of gold-producing countries) is going to vary the amount of gold it produces in relation to stock prices in the United States. (GPEC will produce less gold when stock markets are up and more when they are down.) What effect will this have on your portfolio? Explain. 5. You are interested in creating a portfolio of two stocks—Coca-Cola and Texas Utilities. Over the past decade, an investment in Coca-Cola stock would have earned an average annual return of 25%, with a standard deviation in returns of 36%. An investment in Texas Utilities stock would have earned an average annual return of 12%, with a standard deviation of 22%. The correlation in returns across the two stocks is 0.28. a. Assuming that the average return and standard deviation, estimated using past returns, will continue to hold in the future, estimate the future average returns and standard deviation of a portfolio composed 60% of Coca-Cola and 40% of Texas Utilities stock.
+b. Now assume that Coca-Cola’s international diversification will reduce the correlation to 0.20, while increasing Coca-Cola’s standard deviation in returns to 45%. Assuming all of the other numbers remain unchanged, estimate one standard deviation of the portfolio in (a). 6. Assume that you have half your money invested in Times Mirror, the media company, and the other half invested in Unilever, the consumer product company. The expected returns and standard deviations on the two investments are: Times Mirror
+Unilever Expected return
+14%
+18% Standard deviation
+25%
+40% Estimate the variance of the portfolio as a function of the correlation coefficient (start with –1 and increase the correlation to +1 in 0.2 increments).
+7. You have been asked to analyze the standard deviation of a portfolio composed of the following three assets: Expected Return
+Standard Deviation Sony Corporation
+11%
+23% Tesoro Petroleum 9%
+27% Storage Technology
+16%
+50% You have also been provided with the correlations across these three investments: Estimate the variance of a portfolio, equally weighted across all three assets.
+8. Assume that the average variance of return for an individual security is 50 and that the average covariance is 10. What is the expected variance of a portfolio of 5, 10, 20, 50, and 100 securities? How many securities need to be held before the risk of a portfolio is only 10% more than the minimum?
+9. Assume you have all your wealth (a million dollars) invested in the Vanguard 500 index fund, and that you expect to earn an annual return of 12%, with a standard deviation in returns of 25%. Since you have become more risk averse, you decide to shift $200,000 from the Vanguard 500 index fund to Treasury bills. The T-bill rate is 5%. Estimate the expected return and standard deviation of your new portfolio.
+10. Every investor in the capital asset pricing model owns a combination of the market portfolio and a riskless asset. Assume that the standard deviation of the market portfolio is 30% and that the expected return on the portfolio is 15%. What proportion of the following investors’ wealth would you suggest investing in the market portfolio and what proportion in the riskless asset? (The riskless asset has an expected return of 5%.) a. An investor who desires a portfolio with no standard deviation.
+b. An investor who desires a portfolio with a standard deviation of 15%.
+c. An investor who desires a portfolio with a standard deviation of 30%.
+d. An investor who desires a portfolio with a standard deviation of 45%.
+e. An investor who desires a portfolio with an expected return of 12%. 11. The following table lists returns on the market portfolio and on Scientific Atlanta, each year from 1989 to 1998. Year
+Scientific Atlanta
+Market Portfolio 1989
+80.95%
+31.49% 1990
+–47.37%
+–3.17% 1991
+31.00%
+30.57% 1992
+132.44%
+7.58% 1993
+32.02%
+10.36% 1994
+25.37%
+2.55% 1995
+–28.57%
+37.57% 1996
+0.00%
+22.68% 1997
+11.67%
+33.10% 1998
+36.19%
+28.32% a. Estimate the covariance in returns between Scientific Atlanta and the market portfolio.
+b. Estimate the variances in returns on both investments.
+c. Estimate the beta for Scientific Atlanta. 12. United Airlines has a beta of 1.5. The standard deviation in the market portfolio is 22%, and United Airlines has a standard deviation of 66%. a. Estimate the correlation between United Airlines and the market portfolio.
+b. What proportion of United Airlines’ risk is market risk? 13. You are using the arbitrage pricing model to estimate the expected return on Bethlehem Steel, and have derived the following estimates for the factor betas and risk premium: Factor
+Beta
+Risk Premium 1
+1.2
+2.5% 2
+0.6
+1.5% 3
+1.5
+1.0% 4
+2.2
+0.8% 5
+0.5
+1.2% a. Which risk factor is Bethlehem Steel most exposed to? Is there any way, within the arbitrage pricing model, to identify the risk factor?
+b. If the risk-free rate is 5%, estimate the expected return on Bethlehem Steel.
+c. Now assume that the beta in the capital asset pricing model for Bethlehem Steel is 1.1, and that the risk premium for the market portfolio is 5%. Estimate the expected return using the CAPM.
+d. Why are the expected returns different using the two models? 14. You are using the multifactor model to estimate the expected return on Emerson Electric, and have derived the following estimates for the factor betas and risk premiums: With a riskless rate of 6%, estimate the expected return on Emerson Electric.
+15. The following equation is reproduced from the study by Fama and French of returns between 1963 and 1990. where MV is the market value of equity in hundreds of millions of dollars and BV is the book value of equity in hundreds of millions of dollars. The return is a monthly return. a. Estimate the expected annual return on Lucent Technologies if the market value of its equity is $180 billion and the book value of its equity is $73.5 billion.
+b. Lucent Technologies has a beta of 1.55. If the riskless rate is 6% and the risk premium for the market portfolio is 5.5%, estimate the expected return.
+c. Why are the expected returns different under the two approaches? 1 A utility function is a way of summarizing investor preferences into a generic term called “utility” on the basis of some choice variables. In this case, for instance, the investors’ utility or satisfaction is stated as a function of wealth. By doing so, we effectively can answer questions such as, Will investors be twice as happy if they have twice as much wealth? Does each marginal increase in wealth lead to less additional utility than the prior marginal increase? In one specific form of this function, the quadratic utility function, the entire utility of an investor can be compressed into the expected wealth measure and the standard deviation in that wealth.
+2 The significance of introducing the riskless asset into the choice mix and the implications for portfolio choice were first noted in Sharpe (1964) and Lintner (1965). Hence, the model is sometimes called the Sharpe-Lintner model.
+3 In a paper in Nature (Gabaix, X., Gopikrishnan, P., Plerou, V., and Stanley, H.E., 2003, A theory of power law distributions in financial market fluctuations, Nature 423, 267–70), researchers looked at stock prices on 500 stocks between 1929 and 1987 and concluded that the exponent for stock returns is roughly 3.
+4 Graham, B., 1949, The Intelligent Investor (New York: HarperBusiness, reprinted in 2005).
+5 Financial obligation refers to any payment that the firm has legally obligated itself to make, such as interest and principal payments. It does not include discretionary cash flows, such as dividend payments or new capital expenditures, which can be deferred or delayed without legal consequences, though there may be economic consequences.
+6 See the Standard & Poor’s online site (www.standardandpoors.com/ratings/criteria/index.htm).
+
+CHAPTER 5
+Option Pricing Theory and Models
+In general, the value of any asset is the present value of the expected cash flows on that asset. This chapter considers an exception to that rule when it looks at assets with two specific characteristics: 1. The assets derive their value from the values of other assets.
+2. The cash flows on the assets are contingent on the occurrence of specific events. These assets are called options, and the present value of the expected cash flows on these assets will understate their true value. This chapter describes the cash flow characteristics of options, considers the factors that determine their value, and examines how best to value them.
+BASICS OF OPTION PRICING
+An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called a strike price or an exercise price) at or before the expiration date of the option. Since it is a right and not an obligation, the holder can choose not to exercise the right and can allow the option to expire. There are two types of options—call options and put options.
+Call and Put Options: Description and Payoff Diagrams
+A call option gives the buyer of the option the right to buy the underlying asset at the strike price or the exercise price at any time prior to the expiration date of the option. The buyer pays a price for this right. If at expiration the value of the asset is less than the strike price, the option is not exercised and expires worthless. If, however, the value of the asset is greater than the strike price, the option is exercised—the buyer of the option buys the stock at the exercise price, and the difference between the asset value and the exercise price comprises the gross profit on the investment. The net profit on the investment is the difference between the gross profit and the price paid for the call initially.
+A payoff diagram illustrates the cash payoff on an option at expiration. For a call, the net payoff is negative (and equal to the price paid for the call) if the value of the underlying asset is less than the strike price. If the price of the underlying asset exceeds the strike price, the gross payoff is the difference between the value of the underlying asset and the strike price, and the net payoff is the difference between the gross payoff and the price of the call. This is illustrated in Figure 5.1. Figure 5.1 Payoff on Call Option A put option gives the buyer of the option the right to sell the underlying asset at a fixed price, again called the strike or exercise price, at any time prior to the expiration date of the option. The buyer pays a price for this right. If the price of the underlying asset is greater than the strike price, the option will not be exercised and will expire worthless. But if the price of the underlying asset is less than the strike price, the owner of the put option will exercise the option and sell the stock at the strike price, claiming the difference between the strike price and the market value of the asset as the gross profit. Again, netting out the initial cost paid for the put yields the net profit from the transaction.
+A put has a negative net payoff if the value of the underlying asset exceeds the strike price, and has a gross payoff equal to the difference between the strike price and the value of the underlying asset if the asset value is less than the strike price. This is summarized in Figure 5.2. Figure 5.2 Payoff on Put Option DETERMINANTS OF OPTION VALUE
+The value of an option is determined by six variables relating to the underlying asset and financial markets. 1. Current value of the underlying asset. Options are assets that derive value from an underlying asset. Consequently, changes in the value of the underlying asset affect the value of the options on that asset. Since calls provide the right to buy the underlying asset at a fixed price, an increase in the value of the asset will increase the value of the calls. Puts, on the other hand, become less valuable as the value of the asset increases.
+2. Variance in value of the underlying asset. The buyer of an option acquires the right to buy or sell the underlying asset at a fixed price. The higher the variance in the value of the underlying asset, the greater the value of the option.1 This is true for both calls and puts. While it may seem counterintuitive that an increase in a risk measure (variance) should increase value, options are different from other securities since buyers of options can never lose more than the price they pay for them; in fact, they have the potential to earn significant returns from large price movements.
+3. Dividends paid on the underlying asset. The value of the underlying asset can be expected to decrease if dividend payments are made on the asset during the life of the option. Consequently, the value of a call on the asset is a decreasing function of the size of expected dividend payments, and the value of a put is an increasing function of expected dividend payments. A more intuitive way of thinking about dividend payments, for call options, is as a cost of delaying exercise on in-the-money options. To see why, consider an option on a traded stock. Once a call option is in-the-money (i.e., the holder of the option will make a gross payoff by exercising the option), exercising the call option will provide the holder with the stock and entitle him or her to the dividends on the stock in subsequent periods. Failing to exercise the option will mean that these dividends are forgone.
+4. Strike price of the option. A key characteristic used to describe an option is the strike price. In the case of calls, where the holder acquires the right to buy at a fixed price, the value of the call will decline as the strike price increases. In the case of puts, where the holder has the right to sell at a fixed price, the value will increase as the strike price increases.
+5. Time to expiration on the option. Both calls and puts are more valuable the greater the time to expiration. This is because the longer time to expiration provides more time for the value of the underlying asset to move, increasing the value of both types of options. Additionally, in the case of a call, where the buyer has the right to pay a fixed price at expiration, the present value of this fixed price decreases as the life of the option increases, increasing the value of the call.
+6. Riskless interest rate corresponding to life of the option. Since the buyer of an option pays the price of the option up front, an opportunity cost is involved. This cost will depend on the level of interest rates and the time to expiration of the option. The riskless interest rate also enters into the valuation of options when the present value of the exercise price is calculated, since the exercise price does not have to be paid (received) until expiration on calls (puts). Increases in the interest rate will increase the value of calls and reduce the value of puts. Table 5.1 summarizes the variables and their predicted effects on call and put prices.
+Table 5.1 Summary of Variables Affecting Call and Put Prices Effect On Factor
+Call Value
+Put Value Increase in underlying asset’s value
+Increases
+Decreases Increase in variance of underlying asset
+Increases
+Increases Increase in strike price
+Decreases
+Increases Increase in dividends paid
+Decreases
+Increases Increase in time to expiration
+Increases
+Increases Increase in interest rates
+Increases
+Decreases American versus European Options: Variables Relating to Early Exercise
+A primary distinction between American and European options is that an American option can be exercised at any time prior to its expiration, while European options can be exercised only at expiration. The possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. There is one compensating factor that enables the former to be valued using models designed for the latter. In most cases, the time premium associated with the remaining life of an option and transaction costs make early exercise suboptimal. In other words, the holders of in-the-money options generally get much more by selling the options to someone else than by exercising the options.
+OPTION PRICING MODELS
+Option pricing theory has made vast strides since 1972, when Fischer Black and Myron Scholes published their pathbreaking paper that provided a model for valuing dividend-protected European options. Black and Scholes used a “replicating portfolio”—a portfolio composed of the underlying asset and the risk-free asset that had the same cash flows as the option being valued—and the notion of arbitrage to come up with their final formulation. Although their derivation is mathematically complicated, there is a simpler binomial model for valuing options that draws on the same logic.
+Binomial Model
+The binomial option pricing model is based on a simple formulation for the asset price process in which the asset, in any time period, can move to one of two possible prices. The general formulation of a stock price process that follows the binomial path is shown in Figure 5.3. In this figure, S is the current stock price; the price moves up to Su with probability p and down to Sd with probability 1 – p in any time period. Figure 5.3 General Formulation for Binomial Price Path Creating a Replicating Portfolio
+The objective in creating a replicating portfolio is to use a combination of risk-free borrowing/lending and the underlying asset to create the same cash flows as the option being valued. The principles of arbitrage apply then, and the value of the option must be equal to the value of the replicating portfolio. In the case of the general formulation shown in Figure 5.3, where stock prices can move either up to Su or down to Sd in any time period, the replicating portfolio for a call with strike price K will involve borrowing $B and acquiring Δ of the underlying asset, where: where Cu = Value of the call if the stock price is Su
+Cd = Value of the call if the stock price is Sd
+In a multiperiod binomial process, the valuation has to proceed iteratively (i.e., starting with the final time period and moving backward in time until the current point in time). The portfolios replicating the option are created at each step and valued, providing the values for the option in that time period. The final output from the binomial option pricing model is a statement of the value of the option in terms of the replicating portfolio, composed of Δ shares (option delta) of the underlying asset and risk-free borrowing/lending. ILLUSTRATION 5.1: Binomial Option Valuation
+Assume that the objective is to value a call with a strike price of $50, which is expected to expire in two time periods, on an underlying asset whose price currently is $50 and is expected to follow a binomial process: Now assume that the interest rate is 11%. In addition, define: The objective is to combined Δ shares of stock and B dollars of borrowing to replicate the cash flows from the call with a strike price of $50. This can be done iteratively, starting with the last period and working back through the binomial tree.
+Step 1: Start with the end nodes and work backward: Thus, if the stock price is $70 at t = 1, borrowing $45 and buying one share of the stock will give the same cash flows as buying the call. The value of the call at t = 1, if the stock price is $70, is therefore: Considering the other leg of the binomial tree at t = 1, If the stock price is $35 at t = 1, then the call is worth nothing.
+Step 2: Move backward to the earlier time period and create a replicating portfolio that will provide the cash flows the option will provide. In other words, borrowing $22.50 and buying five-sevenths of a share will provide the same cash flows as a call with a strike price of $50 over the call’s lifetime. The value of the call therefore has to be the same as the cost of creating this position. The Determinants of Value
+The binomial model provides insight into the determinants of option value. The value of an option is not determined by the expected price of the asset but by its current price, which, of course, reflects expectations about the future. This is a direct consequence of arbitrage. If the option value deviates from the value of the replicating portfolio, investors can create an arbitrage position (i.e., one that requires no investment, involves no risk, and delivers positive returns). To illustrate, if the portfolio that replicates the call costs more than the call does in the market, an investor could buy the call, sell the replicating portfolio, and be guaranteed the difference as a profit. The cash flows on the two positions will offset each other, leading to no cash flows in subsequent periods. The call option value also increases as the time to expiration is extended, as the price movements (u and d) increase, and with increases in the interest rate.
+While the binomial model provides an intuitive feel for the determinants of option value, it requires a large number of inputs, in terms of expected future prices at each node. As time periods are made shorter in the binomial model, you can make one of two assumptions about asset prices.You can assume that price changes become smaller as periods get shorter; this leads to price changes becoming infinitesimally small as time periods approach zero, leading to a continuous price process. Alternatively, you can assume that price changes stay large even as the period gets shorter; this leads to a jump price process, where prices can jump in any period. This section considers the option pricing models that emerge with each of these assumptions.
+Black-Scholes Model
+When the price process is continuous (i.e., price changes become smaller as time periods get shorter), the binomial model for pricing options converges on the Black-Scholes model. The model, named after its cocreators, Fischer Black and Myron Scholes, allows us to estimate the value of any option using a small number of inputs, and has been shown to be robust in valuing many listed options.
+The Model
+While the derivation of the Black-Scholes model is far too complicated to present here, it is based on the idea of creating a portfolio of the underlying asset and the riskless asset with the same cash flows, and hence the same cost, as the option being valued. The value of a call option in the Black-Scholes model can be written as a function of the five variables: S = Current value of the underlying asset
+K = Strike price of the option
+t = Life to expiration of the option
+r = Riskless interest rate corresponding to the life of the option
+σ2 = Variance in the ln(value) of the underlying asset The value of a call is then: Note that e–rt is the present value factor, and reflects the fact that the exercise price on the call option does not have to be paid until expiration, since the model values European options. N(d1) and N(d2) are probabilities, estimated by using a cumulative standardized normal distribution, and the values of d1 and d2 obtained for an option. The cumulative distribution is shown in Figure 5.4. Figure 5.4 Cumulative Normal Distribution In approximate terms, N(d2) yields the likelihood that an option will generate positive cash flows for its owner at exercise (i.e., that S > K in the case of a call option and that K > S in the case of a put option). The portfolio that replicates the call option is created by buying N(d1) units of the underlying asset, and borrowing Ke–rt N(d2). The portfolio will have the same cash flows as the call option, and thus the same value as the option. N(d1), which is the number of units of the underlying asset that are needed to create the replicating portfolio, is called the option delta. A NOTE ON ESTIMATING THE INPUTS TO THE BLACK-SCHOLES MODEL
+The Black-Scholes model requires inputs that are consistent on time measurement. There are two places where this affects estimates. The first relates to the fact that the model works in continuous time, rather than discrete time. That is why we use the continuous time version of present value (exp–rt) rather than the discrete version, (1 + r)–t. It also means that the inputs such as the riskless rate have to be modified to make them continuous time inputs. For instance, if the one-year Treasury bond rate is 6.2 percent, the risk-free rate that is used in the Black-Scholes model should be: The second relates to the period over which the inputs are estimated. For instance, the preceding rate is an annual rate. The variance that is entered into the model also has to be an annualized variance. The variance, estimated from ln(asset prices), can be annualized easily because variances are linear in time if the serial correlation is zero. Thus, if monthly or weekly prices are used to estimate variance, the variance is annualized by multiplying by 12 or 52, respectively. ILLUSTRATION 5.2: Valuing an Option Using the Black-Scholes Model
+On March 6, 2001, Cisco Systems was trading at $13.62. We will attempt to value a July 2001 call option with a strike price of $15, trading on the CBOE on the same day for $2. The following are the other parameters of the options: The annualized standard deviation in Cisco Systems stock price over the previous year was 81%. This standard deviation is estimated using weekly stock prices over the year, and the resulting number was annualized as follows: The option expiration date is Friday, July 20, 2001. There are 103 days to expiration, and the annualized Treasury bill rate corresponding to this option life is 4.63%. The inputs for the Black-Scholes model are as follows: Current stock price (S) = $13.62
+Strike price on the option = $15
+Option life = 103/365 = 0.2822
+Standard deviation in ln(stock prices) = 81%
+Riskless rate = 4.63% Inputting these numbers into the model, we get: Using the normal distribution, we can estimate the N(d1) and N(d2): The value of the call can now be estimated: Since the call is trading at $2, it is slightly overvalued, assuming that the estimate of standard deviation used is correct. IMPLIED VOLATILITY
+The only input in the Black Scholes on which there can be significant disagreement among investors is the variance. While the variance is often estimated by looking at historical data, the values for options that emerge from using the historical variance can be different from the market prices. For any option, there is some variance at which the estimated value will be equal to the market price. This variance is called an implied variance.
+Consider the Cisco option valued in Illustration 5.2. With a standard deviation of 81 percent, the value of the call option with a strike price of $15 was estimated to be $1.87. Since the market price is higher than the calculated value, we tried higher standard deviations, and at a standard deviation 85.40 percent the value of the option is $2 (which is the market price). This is the implied standard deviation or implied volatility. Model Limitations and Fixes
+The Black-Scholes model was designed to value European options that can be exercised only at maturity and whose underlying assets do not pay dividends. In addition, options are valued based on the assumption that option exercise does not affect the value of the underlying asset. In practice, assets do pay dividends, options sometimes get exercised early, and exercising an option can affect the value of the underlying asset. Adjustments exist that, while not perfect, provide partial corrections to the Black-Scholes model.
+Dividends
+The payment of a dividend reduces the stock price; note that on the ex-dividend day, the stock price generally declines. Consequently, call options become less valuable and put options more valuable as expected dividend payments increase. There are two ways of dealing with dividends in the Black-Scholes model: 1. Short-term options. One approach to dealing with dividends is to estimate the present value of expected dividends that will be paid by the underlying asset during the option life and subtract it from the current value of the asset to use as S in the model. 2. Long-term options. Since it becomes less practical to estimate the present value of dividends the longer the option life, an alternate approach can be used. If the dividend yield (y = Dividends/Current value of the asset) on the underlying asset is expected to remain unchanged during the life of the option, the Black-Scholes model can be modified to take dividends into account. From an intuitive standpoint, the adjustments have two effects. First, the value of the asset is discounted back to the present at the dividend yield to take into account the expected drop in asset value resulting from dividend payments. Second, the interest rate is offset by the dividend yield to reflect the lower carrying cost from holding the asset (in the replicating portfolio). The net effect will be a reduction in the value of calls estimated using this model. ILLUSTRATION 5.3: Valuing a Short-Term Option with Dividend Adjustments—The Black-Scholes Correction
+Assume that it is March 6, 2001, and that AT&T is trading at $20.50 a share. Consider a call option on the stock with a strike price of $20, expiring on July 20, 2001. Using past stock prices, the annualized standard deviation in the log of stock prices for AT&T is estimated at 60%. There is one dividend, amounting to $0.15, and it will be paid in 23 days. The riskless rate is 4.63%. Present value of expected dividend = $0.15/1.046323/365 = $0.15
+Dividend-adjusted stock price = $20.50 – $0.15 = $20.35
+Time to expiration = 103/365 = 0.2822
+Variance in ln(stock prices) = 0.62 = 0.36
+Riskless rate = 4.63% The value from the Black-Scholes model is: The call option was trading at $2.60 on that day. ILLUSTRATION 5.4: Valuing a Long-Term Option with Dividend Adjustments—Primes and Scores
+The CBOE offers longer-term call and put options on some stocks. On March 6, 2001, for instance, you could have purchased an AT&T call expiring on January 17, 2003. The stock price for AT&T is $20.50 (as in the previous example). The following is the valuation of a call option with a strike price of $20. Instead of estimating the present value of dividends over the next two years, assume that AT&T’s dividend yield will remain 2.51% over this period and that the risk-free rate for a two-year Treasury bond is 4.85%. The inputs to the Black-Scholes model are: The value from the Black-Scholes model is: The call was trading at $5.80 on March 8, 2001. optst.xls: This spreadsheet allows you to estimate the value of a short-term option when the expected dividends during the option life can be estimated. optlt.xls: This spreadsheet allows you to estimate the value of an option when the underlying asset has a constant dividend yield. Early Exercise
+The Black-Scholes model was designed to value European options that can be exercised only at expiration. In contrast, most options that we encounter in practice are American options and can be exercised at any time until expiration. As mentioned earlier, the possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. In general, though, with traded options, it is almost always better to sell the option to someone else rather than exercise early, since options have a time premium (i.e., they sell for more than their exercise value). There are two exceptions. One occurs when the underlying asset pays large dividends, thus reducing the expected value of the asset. In this case, call options may be exercised just before an ex-dividend date, if the time premium on the options is less than the expected decline in asset value as a consequence of the dividend payment. The other exception arises when an investor holds both the underlying asset and deep in-the-money puts (i.e., puts with strike prices well above the current price of the underlying asset) on that asset at a time when interest rates are high. In this case, the time premium on the put may be less than the potential gain from exercising the put early and earning interest on the exercise price.
+There are two basic ways of dealing with the possibility of early exercise. One is to continue to use the unadjusted Black-Scholes model and to regard the resulting value as a floor or conservative estimate of the true value. The other is to try to adjust the value of the option for the possibility of early exercise. There are two approaches for doing so. One uses the Black-Scholes model to value the option to each potential exercise date. With options on stocks, this basically requires that the investor values options to each ex-dividend day and chooses the maximum of the estimated call values. The second approach is to use a modified version of the binomial model to consider the possibility of early exercise. In this version, the up and the down movements for asset prices in each period can be estimated from the variance. 2
+Approach 1: Pseudo-American Valuation
+Step 1: Define when dividends will be paid and how much the dividends will be.
+Step 2: Value the call option to each ex-dividend date using the dividend-adjusted approach described earlier, where the stock price is reduced by the present value of expected dividends.
+Step 3: Choose the maximum of the call values estimated for each ex-dividend day. ILLUSTRATION 5.5: Using Pseudo-American Option Valuation to Adjust for Early Exercise
+Consider an option with a strike price of $35 on a stock trading at $40. The variance in the ln(stock prices) is 0.05, and the riskless rate is 4%. The option has a remaining life of eight months, and there are three dividends expected during this period: Expected Dividend
+Ex-Dividend Day $0.80
+In 1 month $0.80
+In 4 months $0.80
+In 7 months The call option is first valued to just before the first ex-dividend date: The value from the Black-Scholes model is: The call option is then valued to before the second ex-dividend date: The value of the call based on these parameters is: The call option is then valued to before the third ex-dividend date: The value of the call based on these parameters is: The call option is then valued to expiration: The value of the call based on these parameters is: Approach 2: Using the Binomial Model
+The binomial model is much more capable of handling early exercise because it considers the cash flows at each time period, rather than just at expiration. The biggest limitation of the binomial model is determining what stock prices will be at the end of each period, but this can be overcome by using a variant that allows us to estimate the up and the down movements in stock prices from the estimated variance. There are four steps involved:
+Step 1: If the variance in ln(stock prices) has been estimated for the Black-Scholes valuation, convert these into inputs for the binomial model: where u and d are the up and the down movements per unit time for the binomial, and dt is the number of periods within each year (or unit time).
+Step 2: Specify the period in which the dividends will be paid and make the assumption that the price will drop by the amount of the dividend in that period.
+Step 3: Value the call at each node of the tree, allowing for the possibility of early exercise just before ex-dividend dates. There will be early exercise if the remaining time premium on the option is less than the expected drop in option value as a consequence of the dividend payment.
+Step 4: Value the call at time 0, using the standard binomial approach. bstobin.xls: This spreadsheet allows you to estimate the parameters for a binomial model from the inputs to a Black-Scholes model. Impact of Exercise on Underlying Asset Value
+The Black-Scholes model is based on the assumption that exercising an option does not affect the value of the underlying asset. This may be true for listed options on stocks, but it is not true for some types of options. For instance, the exercise of warrants increases the number of shares outstanding and brings fresh cash into the firm, both of which will affect the stock price.3 The expected negative impact (dilution) of exercise will decrease the value of warrants, compared to otherwise similar call options. The adjustment for dilution to the stock price is fairly simple in the Black-Scholes valuation. The stock price is adjusted for the expected dilution from the exercise of the options. In the case of warrants, for instance: where S = Current value of the stock
+nw = Number of warrants outstanding
+W = Value of warrants outstanding
+ns = Number of shares outstanding
+When the warrants are exercised, the number of shares outstanding will increase, reducing the stock price. The numerator reflects the market value of equity, including both stocks and warrants outstanding. The reduction in S will reduce the value of the call option.
+There is an element of circularity in this analysis, since the value of the warrant is needed to estimate the dilution-adjusted S and the dilution-adjusted S is needed to estimate the value of the warrant. This problem can be resolved by starting the process off with an assumed value for the warrant (e.g., the exercise value or the current market price of the warrant). This will yield a value for the warrant, and this estimated value can then be used as an input to reestimate the warrant’s value until there is convergence. FROM BLACK-SCHOLES TO BINOMIAL
+The process of converting the continuous variance in a Black-Scholes model to a binomial tree is a fairly simple one. Assume, for instance, that you have an asset that is trading at $30 currently and that you estimate the annualized standard deviation in the asset value to be 40 percent; the annualized riskless rate is 5 percent. For simplicity, let us assume that the option that you are valuing has a four-year life and that each period is a year. To estimate the prices at the end of each of the four years, we begin by first estimating the up and down movements in the binomial: Based on these estimates, we can obtain the prices at the end of the first node of the tree (the end of the first year): Progressing through the rest of the tree, we obtain the following numbers: ILLUSTRATION 5.6: Valuing a Warrant on Avatek Corporation
+Avatek Corporation is a real estate firm with 19.637 million shares outstanding, trading at $0.38 a share. In March 2001 the company had 1.8 million options outstanding, with four years to expiration and with an exercise price of $2.25. The stock paid no dividends, and the standard deviation in ln(stock prices) was 93%. The four-year Treasury bond rate was 4.9%. (The options were trading at $0.12 apiece at the time of this analysis.)
+The inputs to the warrant valuation model are as follows: The results of the Black-Scholes valuation of this option are: The options were trading at $0.12 in March 2001. Since the value was equal to the price, there was no need for further iterations. If there had been a difference, we would have reestimated the adjusted stock price and option value. If the options had been non-traded (as is the case with management options), this calculation would have required an iterative process, where the option value is used to get the adjusted value per share and the value per share to get the option value. warrant.xls: This spreadsheet allows you to estimate the value of an option when there is a potential dilution from exercise. The Black-Scholes Model for Valuing Puts
+The value of a put can be derived from the value of a call with the same strike price and the same expiration date: where C is the value of the call and P is the value of the put. This relationship between the call and put values is called put-call parity, and any deviations from parity can be used by investors to make riskless profits. To see why put-call parity holds, consider selling a call and buying a put with exercise price K and expiration date t, and simultaneously buying the underlying asset at the current price S. The payoff from this position is riskless and always yields K at expiration (t). To see this, assume that the stock price at expiration is S*. The payoff on each of the positions in the portfolio can be written as follows: Position
+Payoffs at t if S* > K
+Payoffs at t if S* < K Sell call
+–(S* – K)
+0 Buy put
+0
+K – S* Buy stock
+S*
+S* Total
+K
+K Since this position yields K with certainty, the cost of creating this position must be equal to the present value of K at the riskless rate (K e–rt). Substituting the Black-Scholes equation for the value of an equivalent call into this equation, we get: Thus, the replicating portfolio for a put is created by selling short [1 – N(d1)] shares of stock and investing K e–rt[1 – N(d2)] in the riskless asset. ILLUSTRATION 5.7: Valuing a Put Using Put-Call Parity: Cisco Systems and AT&T
+Consider the call on Cisco Systems that we valued in Illustration 5.2. The call had a strike price of $15 on the stock, had 103 days left to expiration, and was valued at $1.87. The stock was trading at $13.62, and the riskless rate was 4.63%. The put can be valued as follows: The put was trading at $3.38.
+Also, a long-term call on AT&T was valued in Illustration 5.4. The call had a strike price of $20, 1.8333 years left to expiration, and a value of $6.63. The stock was trading at $20.50 and was expected to maintain a dividend yield of 2.51% over the period. The riskless rate was 4.85%. The put value can be estimated as follows: The put was trading at $3.80. Both the call and the put were trading at different prices from our estimates, which may indicate that we have not correctly estimated the stock’s volatility. Jump Process Option Pricing Models
+If price changes remain larger as the time periods in the binomial model are shortened, it can no longer be assumed that prices change continuously. When price changes remain large, a price process that allows for price jumps is much more realistic. Cox and Ross (1976) valued options when prices follow a pure jump process, where the jumps can only be positive. Thus, in the next interval, the stock price will either have a large positive jump with a specified probability or drift downward at a given rate.
+Merton (1976) considered a distribution where there are price jumps superimposed on a continuous price process. He specified the rate at which jumps occur (λ) and the average jump size (k), measured as a percentage of the stock price. The model derived to value options with this process is called a jump diffusion model. In this model, the value of an option is determined by the five variables specified in the Black-Scholes model, and the parameters of the jump process (λ, k). Unfortunately, the estimates of the jump process parameters are so difficult to make for most firms that they overwhelm any advantages that accrue from using a more realistic model. These models, therefore, have seen limited use in practice.
+EXTENSIONS OF OPTION PRICING
+All the option pricing models described so far—the binomial, the Black-Scholes, and the jump process models—are designed to value options with clearly defined exercise prices and maturities on underlying assets that are traded. However, the options we encounter in investment analysis or valuation are often on real assets rather than financial assets. Categorized as real options, they can take much more complicated forms. This section considers some of these variations.
+Capped and Barrier Options
+With a simple call option, there is no specified upper limit on the profits that can be made by the buyer of the call. Asset prices, at least in theory, can keep going up, and the payoffs increase proportionately. In some call options, though, the buyer is entitled to profits up to a specified price but not above it. For instance, consider a call option with a strike price of K1 on an asset. In an unrestricted call option, the payoff on this option will increase as the underlying asset’s price increases above K1. Assume, however, that if the price reaches K2, the payoff is capped at (K2 – K1). The payoff diagram on this option is shown in Figure 5.5. Figure 5.5 Payoff on Capped Call This option is called a capped call. Notice, also, that once the price reaches K2, there is no longer any time premium associated with the option, and the option will therefore be exercised. Capped calls are part of a family of options called barrier options, where the payoff on and the life of the option are functions of whether the underlying asset price reaches a certain level during a specified period.
+The value of a capped call is always lower than the value of the same call without the payoff limit. A simple approximation of this value can be obtained by valuing the call twice, once with the given exercise price and once with the cap, and taking the difference in the two values. In the preceding example, then, the value of the call with an exercise price of K1 and a cap at K2 can be written as: Barrier options can take many forms. In a knockout option, an option ceases to exist if the underlying asset reaches a certain price. In the case of a call option, this knockout price is usually set below the strike price, and this option is called a down-and-out option. In the case of a put option, the knockout price will be set above the exercise price, and this option is called an up-and-out option. Like the capped call, these options are worth less than their unrestricted counterparts. Many real options have limits on potential upside, or knockout provisions, and ignoring these limits can result in the overstatement of the value of these options.
+Compound Options
+Some options derive their value not from an underlying asset, but from other options. These options are called compound options. Compound options can take any of four forms—a call on a call, a put on a put, a call on a put, or a put on a call. Geske (1979) developed the analytical formulation for valuing compound options by replacing the standard normal distribution used in a simple option model with a bivariate normal distribution in the calculation.
+Consider, for instance, the option to expand a project that is discussed in Chapter 30. While we will value this option using a simple option pricing model, in reality there could be multiple stages in expansion, with each stage representing an option for the following stage. In this case, we will undervalue the option by considering it as a simple rather than a compound option.
+Notwithstanding this discussion, the valuation of compound options becomes progressively more difficult as more options are added to the chain. In this case, rather than wreck the valuation on the shoals of estimation error, it may be better to accept the conservative estimate that is provided with a simple valuation model as a floor on the value.
+Rainbow Options
+In a simple option, the uncertainty is about the price of the underlying asset. Some options are exposed to two or more sources of uncertainty, and these options are rainbow options. Using the simple option pricing model to value such options can lead to biased estimates of value. As an example, consider an undeveloped oil reserve as an option, where the firm that owns the reserve has the right to develop the reserve. Here there are two sources of uncertainty. The first is obviously the price of oil, and the second is the quantity of oil that is in the reserve. To value this undeveloped reserve, we can make the simplifying assumption that we know the quantity of oil in the reserve with certainty. In reality, however, uncertainty about the quantity will affect the value of this option and make the decision to exercise more difficult.4
+CONCLUSION
+An option is an asset with payoffs that are contingent on the value of an underlying asset. A call option provides its holder with the right to buy the underlying asset at a fixed price, whereas a put option provides its holder with the right to sell at a fixed price, at any time before the expiration of the option. The value of an option is determined by six variables—the current value of the underlying asset, the variance in this value, the expected dividends on the asset, the strike price and life of the option, and the riskless interest rate. This is illustrated in both the binomial and the Black-Scholes models, which value options by creating replicating portfolios composed of the underlying asset and riskless lending or borrowing. These models can be used to value assets that have option like characteristics.
+QUESTIONS AND SHORT PROBLEMS
+In the problems following, use an equity risk premium of 5.5 percent if none is specified. 1. The following are prices of options traded on Microsoft Corporation, which pays no dividends. The stock is trading at $83, and the annualized riskless rate is 3.8%. The standard deviation in ln(stock prices) (based on historical data) is 30%. a. Estimate the value of a three-month call with a strike price of $85.
+b. Using the inputs from the Black-Scholes model, specify how you would replicate this call.
+c. What is the implied standard deviation in this call?
+d. Assume now that you buy a call with a strike price of $85 and sell a call with a strike price of $90. Draw the payoff diagram on this position.
+e. Using put-call parity, estimate the value of a three-month put with a strike price of $85. 2. You are trying to value three-month call and put options on Merck with a strike price of $30. The stock is trading at $28.75, and the company expects to pay a quarterly dividend per share of $0.28 in two months. The annualized riskless interest rate is 3.6%, and the standard deviation in log stock prices is 20%. a. Estimate the value of the call and put options, using the Black-Scholes model.
+b. What effect does the expected dividend payment have on call values? On put values? Why? 3. There is the possibility that the options on Merck described in the preceding problem could be exercised early. a. Use the pseudo-American call option technique to determine whether this will affect the value of the call.
+b. Why does the possibility of early exercise exist? What types of options are most likely to be exercised early? 4. You have been provided the following information on a three-month call: a. If you wanted to replicate buying this call, how much money would you need to borrow?
+b. If you wanted to replicate buying this call, how many shares of stock would you need to buy? 5. Go Video, a manufacturer of video recorders, was trading at $4 per share in May 1994. There were 11 million shares outstanding. At the same time, it had 550,000 one-year warrants outstanding, with a strike price of $4.25. The stock has had a standard deviation of 60%. The stock does not pay a dividend. The riskless rate is 5%. a. Estimate the value of the warrants, ignoring dilution.
+b. Estimate the value of the warrants, allowing for dilution.
+c. Why does dilution reduce the value of the warrants? 6. You are trying to value a long-term call option on the NYSE Composite index, expiring in five years, with a strike price of 275. The index is currently at 250, and the annualized standard deviation in stock prices is 15%. The average dividend yield on the index is 3% and is expected to remain unchanged over the next five years. The five-year Treasury bond rate is 5%. a. Estimate the value of the long-term call option.
+b. Estimate the value of a put option with the same parameters.
+c. What are the implicit assumptions you are making when you use the Black-Scholes model to value this option? Which of these assumptions are likely to be violated? What are the consequences for your valuation? 7. A new security on AT&T will entitle the investor to all dividends on AT&T over the next three years, limiting upside potential to 20% but also providing downside protection below 10%. AT&T stock is trading at $50, and three-year call and put options are traded on the exchange at the following prices: How much would you be willing to pay for this security? 1 Note, though, that higher variance can reduce the value of the underlying asset. As a call option becomes more in-the-money, the more it resembles the underlying asset. For very deep in-the-money call options, higher variance can reduce the value of the option.
+2 To illustrate, if σ2 is the variance in ln(stock prices), the up and the down movements in the binomial can be estimated as follows: where u and d are the up and down movements per unit time for the binomial, T is the life of the option, and m is the number of periods within that lifetime.
+3 Warrants are call options issued by firms, either as part of management compensation contracts or to raise equity.
+4 The analogy to a listed option on a stock is the case where you do not know with certainty what the stock price is when you exercise the option. The more uncertain you are about the stock price, the more margin for error you have to give yourself when you exercise the option, to ensure that you are in fact earning a profit.
+CHAPTER 5
+Option Pricing Theory and Models
+In general, the value of any asset is the present value of the expected cash flows on that asset. This chapter considers an exception to that rule when it looks at assets with two specific characteristics: 1. The assets derive their value from the values of other assets.
+2. The cash flows on the assets are contingent on the occurrence of specific events. These assets are called options, and the present value of the expected cash flows on these assets will understate their true value. This chapter describes the cash flow characteristics of options, considers the factors that determine their value, and examines how best to value them.
+BASICS OF OPTION PRICING
+An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called a strike price or an exercise price) at or before the expiration date of the option. Since it is a right and not an obligation, the holder can choose not to exercise the right and can allow the option to expire. There are two types of options—call options and put options.
+Call and Put Options: Description and Payoff Diagrams
+A call option gives the buyer of the option the right to buy the underlying asset at the strike price or the exercise price at any time prior to the expiration date of the option. The buyer pays a price for this right. If at expiration the value of the asset is less than the strike price, the option is not exercised and expires worthless. If, however, the value of the asset is greater than the strike price, the option is exercised—the buyer of the option buys the stock at the exercise price, and the difference between the asset value and the exercise price comprises the gross profit on the investment. The net profit on the investment is the difference between the gross profit and the price paid for the call initially.
+A payoff diagram illustrates the cash payoff on an option at expiration. For a call, the net payoff is negative (and equal to the price paid for the call) if the value of the underlying asset is less than the strike price. If the price of the underlying asset exceeds the strike price, the gross payoff is the difference between the value of the underlying asset and the strike price, and the net payoff is the difference between the gross payoff and the price of the call. This is illustrated in Figure 5.1. Figure 5.1 Payoff on Call Option A put option gives the buyer of the option the right to sell the underlying asset at a fixed price, again called the strike or exercise price, at any time prior to the expiration date of the option. The buyer pays a price for this right. If the price of the underlying asset is greater than the strike price, the option will not be exercised and will expire worthless. But if the price of the underlying asset is less than the strike price, the owner of the put option will exercise the option and sell the stock at the strike price, claiming the difference between the strike price and the market value of the asset as the gross profit. Again, netting out the initial cost paid for the put yields the net profit from the transaction.
+A put has a negative net payoff if the value of the underlying asset exceeds the strike price, and has a gross payoff equal to the difference between the strike price and the value of the underlying asset if the asset value is less than the strike price. This is summarized in Figure 5.2. Figure 5.2 Payoff on Put Option DETERMINANTS OF OPTION VALUE
+The value of an option is determined by six variables relating to the underlying asset and financial markets. 1. Current value of the underlying asset. Options are assets that derive value from an underlying asset. Consequently, changes in the value of the underlying asset affect the value of the options on that asset. Since calls provide the right to buy the underlying asset at a fixed price, an increase in the value of the asset will increase the value of the calls. Puts, on the other hand, become less valuable as the value of the asset increases.
+2. Variance in value of the underlying asset. The buyer of an option acquires the right to buy or sell the underlying asset at a fixed price. The higher the variance in the value of the underlying asset, the greater the value of the option.1 This is true for both calls and puts. While it may seem counterintuitive that an increase in a risk measure (variance) should increase value, options are different from other securities since buyers of options can never lose more than the price they pay for them; in fact, they have the potential to earn significant returns from large price movements.
+3. Dividends paid on the underlying asset. The value of the underlying asset can be expected to decrease if dividend payments are made on the asset during the life of the option. Consequently, the value of a call on the asset is a decreasing function of the size of expected dividend payments, and the value of a put is an increasing function of expected dividend payments. A more intuitive way of thinking about dividend payments, for call options, is as a cost of delaying exercise on in-the-money options. To see why, consider an option on a traded stock. Once a call option is in-the-money (i.e., the holder of the option will make a gross payoff by exercising the option), exercising the call option will provide the holder with the stock and entitle him or her to the dividends on the stock in subsequent periods. Failing to exercise the option will mean that these dividends are forgone.
+4. Strike price of the option. A key characteristic used to describe an option is the strike price. In the case of calls, where the holder acquires the right to buy at a fixed price, the value of the call will decline as the strike price increases. In the case of puts, where the holder has the right to sell at a fixed price, the value will increase as the strike price increases.
+5. Time to expiration on the option. Both calls and puts are more valuable the greater the time to expiration. This is because the longer time to expiration provides more time for the value of the underlying asset to move, increasing the value of both types of options. Additionally, in the case of a call, where the buyer has the right to pay a fixed price at expiration, the present value of this fixed price decreases as the life of the option increases, increasing the value of the call.
+6. Riskless interest rate corresponding to life of the option. Since the buyer of an option pays the price of the option up front, an opportunity cost is involved. This cost will depend on the level of interest rates and the time to expiration of the option. The riskless interest rate also enters into the valuation of options when the present value of the exercise price is calculated, since the exercise price does not have to be paid (received) until expiration on calls (puts). Increases in the interest rate will increase the value of calls and reduce the value of puts. Table 5.1 summarizes the variables and their predicted effects on call and put prices.
+Table 5.1 Summary of Variables Affecting Call and Put Prices Effect On Factor
+Call Value
+Put Value Increase in underlying asset’s value
+Increases
+Decreases Increase in variance of underlying asset
+Increases
+Increases Increase in strike price
+Decreases
+Increases Increase in dividends paid
+Decreases
+Increases Increase in time to expiration
+Increases
+Increases Increase in interest rates
+Increases
+Decreases American versus European Options: Variables Relating to Early Exercise
+A primary distinction between American and European options is that an American option can be exercised at any time prior to its expiration, while European options can be exercised only at expiration. The possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. There is one compensating factor that enables the former to be valued using models designed for the latter. In most cases, the time premium associated with the remaining life of an option and transaction costs make early exercise suboptimal. In other words, the holders of in-the-money options generally get much more by selling the options to someone else than by exercising the options.
+OPTION PRICING MODELS
+Option pricing theory has made vast strides since 1972, when Fischer Black and Myron Scholes published their pathbreaking paper that provided a model for valuing dividend-protected European options. Black and Scholes used a “replicating portfolio”—a portfolio composed of the underlying asset and the risk-free asset that had the same cash flows as the option being valued—and the notion of arbitrage to come up with their final formulation. Although their derivation is mathematically complicated, there is a simpler binomial model for valuing options that draws on the same logic.
+Binomial Model
+The binomial option pricing model is based on a simple formulation for the asset price process in which the asset, in any time period, can move to one of two possible prices. The general formulation of a stock price process that follows the binomial path is shown in Figure 5.3. In this figure, S is the current stock price; the price moves up to Su with probability p and down to Sd with probability 1 – p in any time period. Figure 5.3 General Formulation for Binomial Price Path Creating a Replicating Portfolio
+The objective in creating a replicating portfolio is to use a combination of risk-free borrowing/lending and the underlying asset to create the same cash flows as the option being valued. The principles of arbitrage apply then, and the value of the option must be equal to the value of the replicating portfolio. In the case of the general formulation shown in Figure 5.3, where stock prices can move either up to Su or down to Sd in any time period, the replicating portfolio for a call with strike price K will involve borrowing $B and acquiring Δ of the underlying asset, where: where Cu = Value of the call if the stock price is Su
+Cd = Value of the call if the stock price is Sd
+In a multiperiod binomial process, the valuation has to proceed iteratively (i.e., starting with the final time period and moving backward in time until the current point in time). The portfolios replicating the option are created at each step and valued, providing the values for the option in that time period. The final output from the binomial option pricing model is a statement of the value of the option in terms of the replicating portfolio, composed of Δ shares (option delta) of the underlying asset and risk-free borrowing/lending. ILLUSTRATION 5.1: Binomial Option Valuation
+Assume that the objective is to value a call with a strike price of $50, which is expected to expire in two time periods, on an underlying asset whose price currently is $50 and is expected to follow a binomial process: Now assume that the interest rate is 11%. In addition, define: The objective is to combined Δ shares of stock and B dollars of borrowing to replicate the cash flows from the call with a strike price of $50. This can be done iteratively, starting with the last period and working back through the binomial tree.
+Step 1: Start with the end nodes and work backward: Thus, if the stock price is $70 at t = 1, borrowing $45 and buying one share of the stock will give the same cash flows as buying the call. The value of the call at t = 1, if the stock price is $70, is therefore: Considering the other leg of the binomial tree at t = 1, If the stock price is $35 at t = 1, then the call is worth nothing.
+Step 2: Move backward to the earlier time period and create a replicating portfolio that will provide the cash flows the option will provide. In other words, borrowing $22.50 and buying five-sevenths of a share will provide the same cash flows as a call with a strike price of $50 over the call’s lifetime. The value of the call therefore has to be the same as the cost of creating this position. The Determinants of Value
+The binomial model provides insight into the determinants of option value. The value of an option is not determined by the expected price of the asset but by its current price, which, of course, reflects expectations about the future. This is a direct consequence of arbitrage. If the option value deviates from the value of the replicating portfolio, investors can create an arbitrage position (i.e., one that requires no investment, involves no risk, and delivers positive returns). To illustrate, if the portfolio that replicates the call costs more than the call does in the market, an investor could buy the call, sell the replicating portfolio, and be guaranteed the difference as a profit. The cash flows on the two positions will offset each other, leading to no cash flows in subsequent periods. The call option value also increases as the time to expiration is extended, as the price movements (u and d) increase, and with increases in the interest rate.
+While the binomial model provides an intuitive feel for the determinants of option value, it requires a large number of inputs, in terms of expected future prices at each node. As time periods are made shorter in the binomial model, you can make one of two assumptions about asset prices.You can assume that price changes become smaller as periods get shorter; this leads to price changes becoming infinitesimally small as time periods approach zero, leading to a continuous price process. Alternatively, you can assume that price changes stay large even as the period gets shorter; this leads to a jump price process, where prices can jump in any period. This section considers the option pricing models that emerge with each of these assumptions.
+Black-Scholes Model
+When the price process is continuous (i.e., price changes become smaller as time periods get shorter), the binomial model for pricing options converges on the Black-Scholes model. The model, named after its cocreators, Fischer Black and Myron Scholes, allows us to estimate the value of any option using a small number of inputs, and has been shown to be robust in valuing many listed options.
+The Model
+While the derivation of the Black-Scholes model is far too complicated to present here, it is based on the idea of creating a portfolio of the underlying asset and the riskless asset with the same cash flows, and hence the same cost, as the option being valued. The value of a call option in the Black-Scholes model can be written as a function of the five variables: S = Current value of the underlying asset
+K = Strike price of the option
+t = Life to expiration of the option
+r = Riskless interest rate corresponding to the life of the option
+σ2 = Variance in the ln(value) of the underlying asset The value of a call is then: Note that e–rt is the present value factor, and reflects the fact that the exercise price on the call option does not have to be paid until expiration, since the model values European options. N(d1) and N(d2) are probabilities, estimated by using a cumulative standardized normal distribution, and the values of d1 and d2 obtained for an option. The cumulative distribution is shown in Figure 5.4. Figure 5.4 Cumulative Normal Distribution In approximate terms, N(d2) yields the likelihood that an option will generate positive cash flows for its owner at exercise (i.e., that S > K in the case of a call option and that K > S in the case of a put option). The portfolio that replicates the call option is created by buying N(d1) units of the underlying asset, and borrowing Ke–rt N(d2). The portfolio will have the same cash flows as the call option, and thus the same value as the option. N(d1), which is the number of units of the underlying asset that are needed to create the replicating portfolio, is called the option delta. A NOTE ON ESTIMATING THE INPUTS TO THE BLACK-SCHOLES MODEL
+The Black-Scholes model requires inputs that are consistent on time measurement. There are two places where this affects estimates. The first relates to the fact that the model works in continuous time, rather than discrete time. That is why we use the continuous time version of present value (exp–rt) rather than the discrete version, (1 + r)–t. It also means that the inputs such as the riskless rate have to be modified to make them continuous time inputs. For instance, if the one-year Treasury bond rate is 6.2 percent, the risk-free rate that is used in the Black-Scholes model should be: The second relates to the period over which the inputs are estimated. For instance, the preceding rate is an annual rate. The variance that is entered into the model also has to be an annualized variance. The variance, estimated from ln(asset prices), can be annualized easily because variances are linear in time if the serial correlation is zero. Thus, if monthly or weekly prices are used to estimate variance, the variance is annualized by multiplying by 12 or 52, respectively. ILLUSTRATION 5.2: Valuing an Option Using the Black-Scholes Model
+On March 6, 2001, Cisco Systems was trading at $13.62. We will attempt to value a July 2001 call option with a strike price of $15, trading on the CBOE on the same day for $2. The following are the other parameters of the options: The annualized standard deviation in Cisco Systems stock price over the previous year was 81%. This standard deviation is estimated using weekly stock prices over the year, and the resulting number was annualized as follows: The option expiration date is Friday, July 20, 2001. There are 103 days to expiration, and the annualized Treasury bill rate corresponding to this option life is 4.63%. The inputs for the Black-Scholes model are as follows: Current stock price (S) = $13.62
+Strike price on the option = $15
+Option life = 103/365 = 0.2822
+Standard deviation in ln(stock prices) = 81%
+Riskless rate = 4.63% Inputting these numbers into the model, we get: Using the normal distribution, we can estimate the N(d1) and N(d2): The value of the call can now be estimated: Since the call is trading at $2, it is slightly overvalued, assuming that the estimate of standard deviation used is correct. IMPLIED VOLATILITY
+The only input in the Black Scholes on which there can be significant disagreement among investors is the variance. While the variance is often estimated by looking at historical data, the values for options that emerge from using the historical variance can be different from the market prices. For any option, there is some variance at which the estimated value will be equal to the market price. This variance is called an implied variance.
+Consider the Cisco option valued in Illustration 5.2. With a standard deviation of 81 percent, the value of the call option with a strike price of $15 was estimated to be $1.87. Since the market price is higher than the calculated value, we tried higher standard deviations, and at a standard deviation 85.40 percent the value of the option is $2 (which is the market price). This is the implied standard deviation or implied volatility. Model Limitations and Fixes
+The Black-Scholes model was designed to value European options that can be exercised only at maturity and whose underlying assets do not pay dividends. In addition, options are valued based on the assumption that option exercise does not affect the value of the underlying asset. In practice, assets do pay dividends, options sometimes get exercised early, and exercising an option can affect the value of the underlying asset. Adjustments exist that, while not perfect, provide partial corrections to the Black-Scholes model.
+Dividends
+The payment of a dividend reduces the stock price; note that on the ex-dividend day, the stock price generally declines. Consequently, call options become less valuable and put options more valuable as expected dividend payments increase. There are two ways of dealing with dividends in the Black-Scholes model: 1. Short-term options. One approach to dealing with dividends is to estimate the present value of expected dividends that will be paid by the underlying asset during the option life and subtract it from the current value of the asset to use as S in the model. 2. Long-term options. Since it becomes less practical to estimate the present value of dividends the longer the option life, an alternate approach can be used. If the dividend yield (y = Dividends/Current value of the asset) on the underlying asset is expected to remain unchanged during the life of the option, the Black-Scholes model can be modified to take dividends into account. From an intuitive standpoint, the adjustments have two effects. First, the value of the asset is discounted back to the present at the dividend yield to take into account the expected drop in asset value resulting from dividend payments. Second, the interest rate is offset by the dividend yield to reflect the lower carrying cost from holding the asset (in the replicating portfolio). The net effect will be a reduction in the value of calls estimated using this model. ILLUSTRATION 5.3: Valuing a Short-Term Option with Dividend Adjustments—The Black-Scholes Correction
+Assume that it is March 6, 2001, and that AT&T is trading at $20.50 a share. Consider a call option on the stock with a strike price of $20, expiring on July 20, 2001. Using past stock prices, the annualized standard deviation in the log of stock prices for AT&T is estimated at 60%. There is one dividend, amounting to $0.15, and it will be paid in 23 days. The riskless rate is 4.63%. Present value of expected dividend = $0.15/1.046323/365 = $0.15
+Dividend-adjusted stock price = $20.50 – $0.15 = $20.35
+Time to expiration = 103/365 = 0.2822
+Variance in ln(stock prices) = 0.62 = 0.36
+Riskless rate = 4.63% The value from the Black-Scholes model is: The call option was trading at $2.60 on that day. ILLUSTRATION 5.4: Valuing a Long-Term Option with Dividend Adjustments—Primes and Scores
+The CBOE offers longer-term call and put options on some stocks. On March 6, 2001, for instance, you could have purchased an AT&T call expiring on January 17, 2003. The stock price for AT&T is $20.50 (as in the previous example). The following is the valuation of a call option with a strike price of $20. Instead of estimating the present value of dividends over the next two years, assume that AT&T’s dividend yield will remain 2.51% over this period and that the risk-free rate for a two-year Treasury bond is 4.85%. The inputs to the Black-Scholes model are: The value from the Black-Scholes model is: The call was trading at $5.80 on March 8, 2001. optst.xls: This spreadsheet allows you to estimate the value of a short-term option when the expected dividends during the option life can be estimated. optlt.xls: This spreadsheet allows you to estimate the value of an option when the underlying asset has a constant dividend yield. Early Exercise
+The Black-Scholes model was designed to value European options that can be exercised only at expiration. In contrast, most options that we encounter in practice are American options and can be exercised at any time until expiration. As mentioned earlier, the possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. In general, though, with traded options, it is almost always better to sell the option to someone else rather than exercise early, since options have a time premium (i.e., they sell for more than their exercise value). There are two exceptions. One occurs when the underlying asset pays large dividends, thus reducing the expected value of the asset. In this case, call options may be exercised just before an ex-dividend date, if the time premium on the options is less than the expected decline in asset value as a consequence of the dividend payment. The other exception arises when an investor holds both the underlying asset and deep in-the-money puts (i.e., puts with strike prices well above the current price of the underlying asset) on that asset at a time when interest rates are high. In this case, the time premium on the put may be less than the potential gain from exercising the put early and earning interest on the exercise price.
+There are two basic ways of dealing with the possibility of early exercise. One is to continue to use the unadjusted Black-Scholes model and to regard the resulting value as a floor or conservative estimate of the true value. The other is to try to adjust the value of the option for the possibility of early exercise. There are two approaches for doing so. One uses the Black-Scholes model to value the option to each potential exercise date. With options on stocks, this basically requires that the investor values options to each ex-dividend day and chooses the maximum of the estimated call values. The second approach is to use a modified version of the binomial model to consider the possibility of early exercise. In this version, the up and the down movements for asset prices in each period can be estimated from the variance. 2
+Approach 1: Pseudo-American Valuation
+Step 1: Define when dividends will be paid and how much the dividends will be.
+Step 2: Value the call option to each ex-dividend date using the dividend-adjusted approach described earlier, where the stock price is reduced by the present value of expected dividends.
+Step 3: Choose the maximum of the call values estimated for each ex-dividend day. ILLUSTRATION 5.5: Using Pseudo-American Option Valuation to Adjust for Early Exercise
+Consider an option with a strike price of $35 on a stock trading at $40. The variance in the ln(stock prices) is 0.05, and the riskless rate is 4%. The option has a remaining life of eight months, and there are three dividends expected during this period: Expected Dividend
+Ex-Dividend Day $0.80
+In 1 month $0.80
+In 4 months $0.80
+In 7 months The call option is first valued to just before the first ex-dividend date: The value from the Black-Scholes model is: The call option is then valued to before the second ex-dividend date: The value of the call based on these parameters is: The call option is then valued to before the third ex-dividend date: The value of the call based on these parameters is: The call option is then valued to expiration: The value of the call based on these parameters is: Approach 2: Using the Binomial Model
+The binomial model is much more capable of handling early exercise because it considers the cash flows at each time period, rather than just at expiration. The biggest limitation of the binomial model is determining what stock prices will be at the end of each period, but this can be overcome by using a variant that allows us to estimate the up and the down movements in stock prices from the estimated variance. There are four steps involved:
+Step 1: If the variance in ln(stock prices) has been estimated for the Black-Scholes valuation, convert these into inputs for the binomial model: where u and d are the up and the down movements per unit time for the binomial, and dt is the number of periods within each year (or unit time).
+Step 2: Specify the period in which the dividends will be paid and make the assumption that the price will drop by the amount of the dividend in that period.
+Step 3: Value the call at each node of the tree, allowing for the possibility of early exercise just before ex-dividend dates. There will be early exercise if the remaining time premium on the option is less than the expected drop in option value as a consequence of the dividend payment.
+Step 4: Value the call at time 0, using the standard binomial approach. bstobin.xls: This spreadsheet allows you to estimate the parameters for a binomial model from the inputs to a Black-Scholes model. Impact of Exercise on Underlying Asset Value
+The Black-Scholes model is based on the assumption that exercising an option does not affect the value of the underlying asset. This may be true for listed options on stocks, but it is not true for some types of options. For instance, the exercise of warrants increases the number of shares outstanding and brings fresh cash into the firm, both of which will affect the stock price.3 The expected negative impact (dilution) of exercise will decrease the value of warrants, compared to otherwise similar call options. The adjustment for dilution to the stock price is fairly simple in the Black-Scholes valuation. The stock price is adjusted for the expected dilution from the exercise of the options. In the case of warrants, for instance: where S = Current value of the stock
+nw = Number of warrants outstanding
+W = Value of warrants outstanding
+ns = Number of shares outstanding
+When the warrants are exercised, the number of shares outstanding will increase, reducing the stock price. The numerator reflects the market value of equity, including both stocks and warrants outstanding. The reduction in S will reduce the value of the call option.
+There is an element of circularity in this analysis, since the value of the warrant is needed to estimate the dilution-adjusted S and the dilution-adjusted S is needed to estimate the value of the warrant. This problem can be resolved by starting the process off with an assumed value for the warrant (e.g., the exercise value or the current market price of the warrant). This will yield a value for the warrant, and this estimated value can then be used as an input to reestimate the warrant’s value until there is convergence. FROM BLACK-SCHOLES TO BINOMIAL
+The process of converting the continuous variance in a Black-Scholes model to a binomial tree is a fairly simple one. Assume, for instance, that you have an asset that is trading at $30 currently and that you estimate the annualized standard deviation in the asset value to be 40 percent; the annualized riskless rate is 5 percent. For simplicity, let us assume that the option that you are valuing has a four-year life and that each period is a year. To estimate the prices at the end of each of the four years, we begin by first estimating the up and down movements in the binomial: Based on these estimates, we can obtain the prices at the end of the first node of the tree (the end of the first year): Progressing through the rest of the tree, we obtain the following numbers: ILLUSTRATION 5.6: Valuing a Warrant on Avatek Corporation
+Avatek Corporation is a real estate firm with 19.637 million shares outstanding, trading at $0.38 a share. In March 2001 the company had 1.8 million options outstanding, with four years to expiration and with an exercise price of $2.25. The stock paid no dividends, and the standard deviation in ln(stock prices) was 93%. The four-year Treasury bond rate was 4.9%. (The options were trading at $0.12 apiece at the time of this analysis.)
+The inputs to the warrant valuation model are as follows: The results of the Black-Scholes valuation of this option are: The options were trading at $0.12 in March 2001. Since the value was equal to the price, there was no need for further iterations. If there had been a difference, we would have reestimated the adjusted stock price and option value. If the options had been non-traded (as is the case with management options), this calculation would have required an iterative process, where the option value is used to get the adjusted value per share and the value per share to get the option value. warrant.xls: This spreadsheet allows you to estimate the value of an option when there is a potential dilution from exercise. The Black-Scholes Model for Valuing Puts
+The value of a put can be derived from the value of a call with the same strike price and the same expiration date: where C is the value of the call and P is the value of the put. This relationship between the call and put values is called put-call parity, and any deviations from parity can be used by investors to make riskless profits. To see why put-call parity holds, consider selling a call and buying a put with exercise price K and expiration date t, and simultaneously buying the underlying asset at the current price S. The payoff from this position is riskless and always yields K at expiration (t). To see this, assume that the stock price at expiration is S*. The payoff on each of the positions in the portfolio can be written as follows: Position
+Payoffs at t if S* > K
+Payoffs at t if S* < K Sell call
+–(S* – K)
+0 Buy put
+0
+K – S* Buy stock
+S*
+S* Total
+K
+K Since this position yields K with certainty, the cost of creating this position must be equal to the present value of K at the riskless rate (K e–rt). Substituting the Black-Scholes equation for the value of an equivalent call into this equation, we get: Thus, the replicating portfolio for a put is created by selling short [1 – N(d1)] shares of stock and investing K e–rt[1 – N(d2)] in the riskless asset. ILLUSTRATION 5.7: Valuing a Put Using Put-Call Parity: Cisco Systems and AT&T
+Consider the call on Cisco Systems that we valued in Illustration 5.2. The call had a strike price of $15 on the stock, had 103 days left to expiration, and was valued at $1.87. The stock was trading at $13.62, and the riskless rate was 4.63%. The put can be valued as follows: The put was trading at $3.38.
+Also, a long-term call on AT&T was valued in Illustration 5.4. The call had a strike price of $20, 1.8333 years left to expiration, and a value of $6.63. The stock was trading at $20.50 and was expected to maintain a dividend yield of 2.51% over the period. The riskless rate was 4.85%. The put value can be estimated as follows: The put was trading at $3.80. Both the call and the put were trading at different prices from our estimates, which may indicate that we have not correctly estimated the stock’s volatility. Jump Process Option Pricing Models
+If price changes remain larger as the time periods in the binomial model are shortened, it can no longer be assumed that prices change continuously. When price changes remain large, a price process that allows for price jumps is much more realistic. Cox and Ross (1976) valued options when prices follow a pure jump process, where the jumps can only be positive. Thus, in the next interval, the stock price will either have a large positive jump with a specified probability or drift downward at a given rate.
+Merton (1976) considered a distribution where there are price jumps superimposed on a continuous price process. He specified the rate at which jumps occur (λ) and the average jump size (k), measured as a percentage of the stock price. The model derived to value options with this process is called a jump diffusion model. In this model, the value of an option is determined by the five variables specified in the Black-Scholes model, and the parameters of the jump process (λ, k). Unfortunately, the estimates of the jump process parameters are so difficult to make for most firms that they overwhelm any advantages that accrue from using a more realistic model. These models, therefore, have seen limited use in practice.
+EXTENSIONS OF OPTION PRICING
+All the option pricing models described so far—the binomial, the Black-Scholes, and the jump process models—are designed to value options with clearly defined exercise prices and maturities on underlying assets that are traded. However, the options we encounter in investment analysis or valuation are often on real assets rather than financial assets. Categorized as real options, they can take much more complicated forms. This section considers some of these variations.
+Capped and Barrier Options
+With a simple call option, there is no specified upper limit on the profits that can be made by the buyer of the call. Asset prices, at least in theory, can keep going up, and the payoffs increase proportionately. In some call options, though, the buyer is entitled to profits up to a specified price but not above it. For instance, consider a call option with a strike price of K1 on an asset. In an unrestricted call option, the payoff on this option will increase as the underlying asset’s price increases above K1. Assume, however, that if the price reaches K2, the payoff is capped at (K2 – K1). The payoff diagram on this option is shown in Figure 5.5. Figure 5.5 Payoff on Capped Call This option is called a capped call. Notice, also, that once the price reaches K2, there is no longer any time premium associated with the option, and the option will therefore be exercised. Capped calls are part of a family of options called barrier options, where the payoff on and the life of the option are functions of whether the underlying asset price reaches a certain level during a specified period.
+The value of a capped call is always lower than the value of the same call without the payoff limit. A simple approximation of this value can be obtained by valuing the call twice, once with the given exercise price and once with the cap, and taking the difference in the two values. In the preceding example, then, the value of the call with an exercise price of K1 and a cap at K2 can be written as: Barrier options can take many forms. In a knockout option, an option ceases to exist if the underlying asset reaches a certain price. In the case of a call option, this knockout price is usually set below the strike price, and this option is called a down-and-out option. In the case of a put option, the knockout price will be set above the exercise price, and this option is called an up-and-out option. Like the capped call, these options are worth less than their unrestricted counterparts. Many real options have limits on potential upside, or knockout provisions, and ignoring these limits can result in the overstatement of the value of these options.
+Compound Options
+Some options derive their value not from an underlying asset, but from other options. These options are called compound options. Compound options can take any of four forms—a call on a call, a put on a put, a call on a put, or a put on a call. Geske (1979) developed the analytical formulation for valuing compound options by replacing the standard normal distribution used in a simple option model with a bivariate normal distribution in the calculation.
+Consider, for instance, the option to expand a project that is discussed in Chapter 30. While we will value this option using a simple option pricing model, in reality there could be multiple stages in expansion, with each stage representing an option for the following stage. In this case, we will undervalue the option by considering it as a simple rather than a compound option.
+Notwithstanding this discussion, the valuation of compound options becomes progressively more difficult as more options are added to the chain. In this case, rather than wreck the valuation on the shoals of estimation error, it may be better to accept the conservative estimate that is provided with a simple valuation model as a floor on the value.
+Rainbow Options
+In a simple option, the uncertainty is about the price of the underlying asset. Some options are exposed to two or more sources of uncertainty, and these options are rainbow options. Using the simple option pricing model to value such options can lead to biased estimates of value. As an example, consider an undeveloped oil reserve as an option, where the firm that owns the reserve has the right to develop the reserve. Here there are two sources of uncertainty. The first is obviously the price of oil, and the second is the quantity of oil that is in the reserve. To value this undeveloped reserve, we can make the simplifying assumption that we know the quantity of oil in the reserve with certainty. In reality, however, uncertainty about the quantity will affect the value of this option and make the decision to exercise more difficult.4
+CONCLUSION
+An option is an asset with payoffs that are contingent on the value of an underlying asset. A call option provides its holder with the right to buy the underlying asset at a fixed price, whereas a put option provides its holder with the right to sell at a fixed price, at any time before the expiration of the option. The value of an option is determined by six variables—the current value of the underlying asset, the variance in this value, the expected dividends on the asset, the strike price and life of the option, and the riskless interest rate. This is illustrated in both the binomial and the Black-Scholes models, which value options by creating replicating portfolios composed of the underlying asset and riskless lending or borrowing. These models can be used to value assets that have option like characteristics.
+QUESTIONS AND SHORT PROBLEMS
+In the problems following, use an equity risk premium of 5.5 percent if none is specified. 1. The following are prices of options traded on Microsoft Corporation, which pays no dividends. The stock is trading at $83, and the annualized riskless rate is 3.8%. The standard deviation in ln(stock prices) (based on historical data) is 30%. a. Estimate the value of a three-month call with a strike price of $85.
+b. Using the inputs from the Black-Scholes model, specify how you would replicate this call.
+c. What is the implied standard deviation in this call?
+d. Assume now that you buy a call with a strike price of $85 and sell a call with a strike price of $90. Draw the payoff diagram on this position.
+e. Using put-call parity, estimate the value of a three-month put with a strike price of $85. 2. You are trying to value three-month call and put options on Merck with a strike price of $30. The stock is trading at $28.75, and the company expects to pay a quarterly dividend per share of $0.28 in two months. The annualized riskless interest rate is 3.6%, and the standard deviation in log stock prices is 20%. a. Estimate the value of the call and put options, using the Black-Scholes model.
+b. What effect does the expected dividend payment have on call values? On put values? Why? 3. There is the possibility that the options on Merck described in the preceding problem could be exercised early. a. Use the pseudo-American call option technique to determine whether this will affect the value of the call.
+b. Why does the possibility of early exercise exist? What types of options are most likely to be exercised early? 4. You have been provided the following information on a three-month call: a. If you wanted to replicate buying this call, how much money would you need to borrow?
+b. If you wanted to replicate buying this call, how many shares of stock would you need to buy? 5. Go Video, a manufacturer of video recorders, was trading at $4 per share in May 1994. There were 11 million shares outstanding. At the same time, it had 550,000 one-year warrants outstanding, with a strike price of $4.25. The stock has had a standard deviation of 60%. The stock does not pay a dividend. The riskless rate is 5%. a. Estimate the value of the warrants, ignoring dilution.
+b. Estimate the value of the warrants, allowing for dilution.
+c. Why does dilution reduce the value of the warrants? 6. You are trying to value a long-term call option on the NYSE Composite index, expiring in five years, with a strike price of 275. The index is currently at 250, and the annualized standard deviation in stock prices is 15%. The average dividend yield on the index is 3% and is expected to remain unchanged over the next five years. The five-year Treasury bond rate is 5%. a. Estimate the value of the long-term call option.
+b. Estimate the value of a put option with the same parameters.
+c. What are the implicit assumptions you are making when you use the Black-Scholes model to value this option? Which of these assumptions are likely to be violated? What are the consequences for your valuation? 7. A new security on AT&T will entitle the investor to all dividends on AT&T over the next three years, limiting upside potential to 20% but also providing downside protection below 10%. AT&T stock is trading at $50, and three-year call and put options are traded on the exchange at the following prices: How much would you be willing to pay for this security? 1 Note, though, that higher variance can reduce the value of the underlying asset. As a call option becomes more in-the-money, the more it resembles the underlying asset. For very deep in-the-money call options, higher variance can reduce the value of the option.
+2 To illustrate, if σ2 is the variance in ln(stock prices), the up and the down movements in the binomial can be estimated as follows: where u and d are the up and down movements per unit time for the binomial, T is the life of the option, and m is the number of periods within that lifetime.
+3 Warrants are call options issued by firms, either as part of management compensation contracts or to raise equity.
+4 The analogy to a listed option on a stock is the case where you do not know with certainty what the stock price is when you exercise the option. The more uncertain you are about the stock price, the more margin for error you have to give yourself when you exercise the option, to ensure that you are in fact earning a profit.
+CHAPTER 5
+Option Pricing Theory and Models
+In general, the value of any asset is the present value of the expected cash flows on that asset. This chapter considers an exception to that rule when it looks at assets with two specific characteristics: 1. The assets derive their value from the values of other assets.
+2. The cash flows on the assets are contingent on the occurrence of specific events. These assets are called options, and the present value of the expected cash flows on these assets will understate their true value. This chapter describes the cash flow characteristics of options, considers the factors that determine their value, and examines how best to value them.
+BASICS OF OPTION PRICING
+An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called a strike price or an exercise price) at or before the expiration date of the option. Since it is a right and not an obligation, the holder can choose not to exercise the right and can allow the option to expire. There are two types of options—call options and put options.
+Call and Put Options: Description and Payoff Diagrams
+A call option gives the buyer of the option the right to buy the underlying asset at the strike price or the exercise price at any time prior to the expiration date of the option. The buyer pays a price for this right. If at expiration the value of the asset is less than the strike price, the option is not exercised and expires worthless. If, however, the value of the asset is greater than the strike price, the option is exercised—the buyer of the option buys the stock at the exercise price, and the difference between the asset value and the exercise price comprises the gross profit on the investment. The net profit on the investment is the difference between the gross profit and the price paid for the call initially.
+A payoff diagram illustrates the cash payoff on an option at expiration. For a call, the net payoff is negative (and equal to the price paid for the call) if the value of the underlying asset is less than the strike price. If the price of the underlying asset exceeds the strike price, the gross payoff is the difference between the value of the underlying asset and the strike price, and the net payoff is the difference between the gross payoff and the price of the call. This is illustrated in Figure 5.1. Figure 5.1 Payoff on Call Option A put option gives the buyer of the option the right to sell the underlying asset at a fixed price, again called the strike or exercise price, at any time prior to the expiration date of the option. The buyer pays a price for this right. If the price of the underlying asset is greater than the strike price, the option will not be exercised and will expire worthless. But if the price of the underlying asset is less than the strike price, the owner of the put option will exercise the option and sell the stock at the strike price, claiming the difference between the strike price and the market value of the asset as the gross profit. Again, netting out the initial cost paid for the put yields the net profit from the transaction.
+A put has a negative net payoff if the value of the underlying asset exceeds the strike price, and has a gross payoff equal to the difference between the strike price and the value of the underlying asset if the asset value is less than the strike price. This is summarized in Figure 5.2. Figure 5.2 Payoff on Put Option DETERMINANTS OF OPTION VALUE
+The value of an option is determined by six variables relating to the underlying asset and financial markets. 1. Current value of the underlying asset. Options are assets that derive value from an underlying asset. Consequently, changes in the value of the underlying asset affect the value of the options on that asset. Since calls provide the right to buy the underlying asset at a fixed price, an increase in the value of the asset will increase the value of the calls. Puts, on the other hand, become less valuable as the value of the asset increases.
+2. Variance in value of the underlying asset. The buyer of an option acquires the right to buy or sell the underlying asset at a fixed price. The higher the variance in the value of the underlying asset, the greater the value of the option.1 This is true for both calls and puts. While it may seem counterintuitive that an increase in a risk measure (variance) should increase value, options are different from other securities since buyers of options can never lose more than the price they pay for them; in fact, they have the potential to earn significant returns from large price movements.
+3. Dividends paid on the underlying asset. The value of the underlying asset can be expected to decrease if dividend payments are made on the asset during the life of the option. Consequently, the value of a call on the asset is a decreasing function of the size of expected dividend payments, and the value of a put is an increasing function of expected dividend payments. A more intuitive way of thinking about dividend payments, for call options, is as a cost of delaying exercise on in-the-money options. To see why, consider an option on a traded stock. Once a call option is in-the-money (i.e., the holder of the option will make a gross payoff by exercising the option), exercising the call option will provide the holder with the stock and entitle him or her to the dividends on the stock in subsequent periods. Failing to exercise the option will mean that these dividends are forgone.
+4. Strike price of the option. A key characteristic used to describe an option is the strike price. In the case of calls, where the holder acquires the right to buy at a fixed price, the value of the call will decline as the strike price increases. In the case of puts, where the holder has the right to sell at a fixed price, the value will increase as the strike price increases.
+5. Time to expiration on the option. Both calls and puts are more valuable the greater the time to expiration. This is because the longer time to expiration provides more time for the value of the underlying asset to move, increasing the value of both types of options. Additionally, in the case of a call, where the buyer has the right to pay a fixed price at expiration, the present value of this fixed price decreases as the life of the option increases, increasing the value of the call.
+6. Riskless interest rate corresponding to life of the option. Since the buyer of an option pays the price of the option up front, an opportunity cost is involved. This cost will depend on the level of interest rates and the time to expiration of the option. The riskless interest rate also enters into the valuation of options when the present value of the exercise price is calculated, since the exercise price does not have to be paid (received) until expiration on calls (puts). Increases in the interest rate will increase the value of calls and reduce the value of puts. Table 5.1 summarizes the variables and their predicted effects on call and put prices.
+Table 5.1 Summary of Variables Affecting Call and Put Prices Effect On Factor
+Call Value
+Put Value Increase in underlying asset’s value
+Increases
+Decreases Increase in variance of underlying asset
+Increases
+Increases Increase in strike price
+Decreases
+Increases Increase in dividends paid
+Decreases
+Increases Increase in time to expiration
+Increases
+Increases Increase in interest rates
+Increases
+Decreases American versus European Options: Variables Relating to Early Exercise
+A primary distinction between American and European options is that an American option can be exercised at any time prior to its expiration, while European options can be exercised only at expiration. The possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. There is one compensating factor that enables the former to be valued using models designed for the latter. In most cases, the time premium associated with the remaining life of an option and transaction costs make early exercise suboptimal. In other words, the holders of in-the-money options generally get much more by selling the options to someone else than by exercising the options.
+OPTION PRICING MODELS
+Option pricing theory has made vast strides since 1972, when Fischer Black and Myron Scholes published their pathbreaking paper that provided a model for valuing dividend-protected European options. Black and Scholes used a “replicating portfolio”—a portfolio composed of the underlying asset and the risk-free asset that had the same cash flows as the option being valued—and the notion of arbitrage to come up with their final formulation. Although their derivation is mathematically complicated, there is a simpler binomial model for valuing options that draws on the same logic.
+Binomial Model
+The binomial option pricing model is based on a simple formulation for the asset price process in which the asset, in any time period, can move to one of two possible prices. The general formulation of a stock price process that follows the binomial path is shown in Figure 5.3. In this figure, S is the current stock price; the price moves up to Su with probability p and down to Sd with probability 1 – p in any time period. Figure 5.3 General Formulation for Binomial Price Path Creating a Replicating Portfolio
+The objective in creating a replicating portfolio is to use a combination of risk-free borrowing/lending and the underlying asset to create the same cash flows as the option being valued. The principles of arbitrage apply then, and the value of the option must be equal to the value of the replicating portfolio. In the case of the general formulation shown in Figure 5.3, where stock prices can move either up to Su or down to Sd in any time period, the replicating portfolio for a call with strike price K will involve borrowing $B and acquiring Δ of the underlying asset, where: where Cu = Value of the call if the stock price is Su
+Cd = Value of the call if the stock price is Sd
+In a multiperiod binomial process, the valuation has to proceed iteratively (i.e., starting with the final time period and moving backward in time until the current point in time). The portfolios replicating the option are created at each step and valued, providing the values for the option in that time period. The final output from the binomial option pricing model is a statement of the value of the option in terms of the replicating portfolio, composed of Δ shares (option delta) of the underlying asset and risk-free borrowing/lending. ILLUSTRATION 5.1: Binomial Option Valuation
+Assume that the objective is to value a call with a strike price of $50, which is expected to expire in two time periods, on an underlying asset whose price currently is $50 and is expected to follow a binomial process: Now assume that the interest rate is 11%. In addition, define: The objective is to combined Δ shares of stock and B dollars of borrowing to replicate the cash flows from the call with a strike price of $50. This can be done iteratively, starting with the last period and working back through the binomial tree.
+Step 1: Start with the end nodes and work backward: Thus, if the stock price is $70 at t = 1, borrowing $45 and buying one share of the stock will give the same cash flows as buying the call. The value of the call at t = 1, if the stock price is $70, is therefore: Considering the other leg of the binomial tree at t = 1, If the stock price is $35 at t = 1, then the call is worth nothing.
+Step 2: Move backward to the earlier time period and create a replicating portfolio that will provide the cash flows the option will provide. In other words, borrowing $22.50 and buying five-sevenths of a share will provide the same cash flows as a call with a strike price of $50 over the call’s lifetime. The value of the call therefore has to be the same as the cost of creating this position. The Determinants of Value
+The binomial model provides insight into the determinants of option value. The value of an option is not determined by the expected price of the asset but by its current price, which, of course, reflects expectations about the future. This is a direct consequence of arbitrage. If the option value deviates from the value of the replicating portfolio, investors can create an arbitrage position (i.e., one that requires no investment, involves no risk, and delivers positive returns). To illustrate, if the portfolio that replicates the call costs more than the call does in the market, an investor could buy the call, sell the replicating portfolio, and be guaranteed the difference as a profit. The cash flows on the two positions will offset each other, leading to no cash flows in subsequent periods. The call option value also increases as the time to expiration is extended, as the price movements (u and d) increase, and with increases in the interest rate.
+While the binomial model provides an intuitive feel for the determinants of option value, it requires a large number of inputs, in terms of expected future prices at each node. As time periods are made shorter in the binomial model, you can make one of two assumptions about asset prices.You can assume that price changes become smaller as periods get shorter; this leads to price changes becoming infinitesimally small as time periods approach zero, leading to a continuous price process. Alternatively, you can assume that price changes stay large even as the period gets shorter; this leads to a jump price process, where prices can jump in any period. This section considers the option pricing models that emerge with each of these assumptions.
+Black-Scholes Model
+When the price process is continuous (i.e., price changes become smaller as time periods get shorter), the binomial model for pricing options converges on the Black-Scholes model. The model, named after its cocreators, Fischer Black and Myron Scholes, allows us to estimate the value of any option using a small number of inputs, and has been shown to be robust in valuing many listed options.
+The Model
+While the derivation of the Black-Scholes model is far too complicated to present here, it is based on the idea of creating a portfolio of the underlying asset and the riskless asset with the same cash flows, and hence the same cost, as the option being valued. The value of a call option in the Black-Scholes model can be written as a function of the five variables: S = Current value of the underlying asset
+K = Strike price of the option
+t = Life to expiration of the option
+r = Riskless interest rate corresponding to the life of the option
+σ2 = Variance in the ln(value) of the underlying asset The value of a call is then: Note that e–rt is the present value factor, and reflects the fact that the exercise price on the call option does not have to be paid until expiration, since the model values European options. N(d1) and N(d2) are probabilities, estimated by using a cumulative standardized normal distribution, and the values of d1 and d2 obtained for an option. The cumulative distribution is shown in Figure 5.4. Figure 5.4 Cumulative Normal Distribution In approximate terms, N(d2) yields the likelihood that an option will generate positive cash flows for its owner at exercise (i.e., that S > K in the case of a call option and that K > S in the case of a put option). The portfolio that replicates the call option is created by buying N(d1) units of the underlying asset, and borrowing Ke–rt N(d2). The portfolio will have the same cash flows as the call option, and thus the same value as the option. N(d1), which is the number of units of the underlying asset that are needed to create the replicating portfolio, is called the option delta. A NOTE ON ESTIMATING THE INPUTS TO THE BLACK-SCHOLES MODEL
+The Black-Scholes model requires inputs that are consistent on time measurement. There are two places where this affects estimates. The first relates to the fact that the model works in continuous time, rather than discrete time. That is why we use the continuous time version of present value (exp–rt) rather than the discrete version, (1 + r)–t. It also means that the inputs such as the riskless rate have to be modified to make them continuous time inputs. For instance, if the one-year Treasury bond rate is 6.2 percent, the risk-free rate that is used in the Black-Scholes model should be: The second relates to the period over which the inputs are estimated. For instance, the preceding rate is an annual rate. The variance that is entered into the model also has to be an annualized variance. The variance, estimated from ln(asset prices), can be annualized easily because variances are linear in time if the serial correlation is zero. Thus, if monthly or weekly prices are used to estimate variance, the variance is annualized by multiplying by 12 or 52, respectively. ILLUSTRATION 5.2: Valuing an Option Using the Black-Scholes Model
+On March 6, 2001, Cisco Systems was trading at $13.62. We will attempt to value a July 2001 call option with a strike price of $15, trading on the CBOE on the same day for $2. The following are the other parameters of the options: The annualized standard deviation in Cisco Systems stock price over the previous year was 81%. This standard deviation is estimated using weekly stock prices over the year, and the resulting number was annualized as follows: The option expiration date is Friday, July 20, 2001. There are 103 days to expiration, and the annualized Treasury bill rate corresponding to this option life is 4.63%. The inputs for the Black-Scholes model are as follows: Current stock price (S) = $13.62
+Strike price on the option = $15
+Option life = 103/365 = 0.2822
+Standard deviation in ln(stock prices) = 81%
+Riskless rate = 4.63% Inputting these numbers into the model, we get: Using the normal distribution, we can estimate the N(d1) and N(d2): The value of the call can now be estimated: Since the call is trading at $2, it is slightly overvalued, assuming that the estimate of standard deviation used is correct. IMPLIED VOLATILITY
+The only input in the Black Scholes on which there can be significant disagreement among investors is the variance. While the variance is often estimated by looking at historical data, the values for options that emerge from using the historical variance can be different from the market prices. For any option, there is some variance at which the estimated value will be equal to the market price. This variance is called an implied variance.
+Consider the Cisco option valued in Illustration 5.2. With a standard deviation of 81 percent, the value of the call option with a strike price of $15 was estimated to be $1.87. Since the market price is higher than the calculated value, we tried higher standard deviations, and at a standard deviation 85.40 percent the value of the option is $2 (which is the market price). This is the implied standard deviation or implied volatility. Model Limitations and Fixes
+The Black-Scholes model was designed to value European options that can be exercised only at maturity and whose underlying assets do not pay dividends. In addition, options are valued based on the assumption that option exercise does not affect the value of the underlying asset. In practice, assets do pay dividends, options sometimes get exercised early, and exercising an option can affect the value of the underlying asset. Adjustments exist that, while not perfect, provide partial corrections to the Black-Scholes model.
+Dividends
+The payment of a dividend reduces the stock price; note that on the ex-dividend day, the stock price generally declines. Consequently, call options become less valuable and put options more valuable as expected dividend payments increase. There are two ways of dealing with dividends in the Black-Scholes model: 1. Short-term options. One approach to dealing with dividends is to estimate the present value of expected dividends that will be paid by the underlying asset during the option life and subtract it from the current value of the asset to use as S in the model. 2. Long-term options. Since it becomes less practical to estimate the present value of dividends the longer the option life, an alternate approach can be used. If the dividend yield (y = Dividends/Current value of the asset) on the underlying asset is expected to remain unchanged during the life of the option, the Black-Scholes model can be modified to take dividends into account. From an intuitive standpoint, the adjustments have two effects. First, the value of the asset is discounted back to the present at the dividend yield to take into account the expected drop in asset value resulting from dividend payments. Second, the interest rate is offset by the dividend yield to reflect the lower carrying cost from holding the asset (in the replicating portfolio). The net effect will be a reduction in the value of calls estimated using this model. ILLUSTRATION 5.3: Valuing a Short-Term Option with Dividend Adjustments—The Black-Scholes Correction
+Assume that it is March 6, 2001, and that AT&T is trading at $20.50 a share. Consider a call option on the stock with a strike price of $20, expiring on July 20, 2001. Using past stock prices, the annualized standard deviation in the log of stock prices for AT&T is estimated at 60%. There is one dividend, amounting to $0.15, and it will be paid in 23 days. The riskless rate is 4.63%. Present value of expected dividend = $0.15/1.046323/365 = $0.15
+Dividend-adjusted stock price = $20.50 – $0.15 = $20.35
+Time to expiration = 103/365 = 0.2822
+Variance in ln(stock prices) = 0.62 = 0.36
+Riskless rate = 4.63% The value from the Black-Scholes model is: The call option was trading at $2.60 on that day. ILLUSTRATION 5.4: Valuing a Long-Term Option with Dividend Adjustments—Primes and Scores
+The CBOE offers longer-term call and put options on some stocks. On March 6, 2001, for instance, you could have purchased an AT&T call expiring on January 17, 2003. The stock price for AT&T is $20.50 (as in the previous example). The following is the valuation of a call option with a strike price of $20. Instead of estimating the present value of dividends over the next two years, assume that AT&T’s dividend yield will remain 2.51% over this period and that the risk-free rate for a two-year Treasury bond is 4.85%. The inputs to the Black-Scholes model are: The value from the Black-Scholes model is: The call was trading at $5.80 on March 8, 2001. optst.xls: This spreadsheet allows you to estimate the value of a short-term option when the expected dividends during the option life can be estimated. optlt.xls: This spreadsheet allows you to estimate the value of an option when the underlying asset has a constant dividend yield. Early Exercise
+The Black-Scholes model was designed to value European options that can be exercised only at expiration. In contrast, most options that we encounter in practice are American options and can be exercised at any time until expiration. As mentioned earlier, the possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. In general, though, with traded options, it is almost always better to sell the option to someone else rather than exercise early, since options have a time premium (i.e., they sell for more than their exercise value). There are two exceptions. One occurs when the underlying asset pays large dividends, thus reducing the expected value of the asset. In this case, call options may be exercised just before an ex-dividend date, if the time premium on the options is less than the expected decline in asset value as a consequence of the dividend payment. The other exception arises when an investor holds both the underlying asset and deep in-the-money puts (i.e., puts with strike prices well above the current price of the underlying asset) on that asset at a time when interest rates are high. In this case, the time premium on the put may be less than the potential gain from exercising the put early and earning interest on the exercise price.
+There are two basic ways of dealing with the possibility of early exercise. One is to continue to use the unadjusted Black-Scholes model and to regard the resulting value as a floor or conservative estimate of the true value. The other is to try to adjust the value of the option for the possibility of early exercise. There are two approaches for doing so. One uses the Black-Scholes model to value the option to each potential exercise date. With options on stocks, this basically requires that the investor values options to each ex-dividend day and chooses the maximum of the estimated call values. The second approach is to use a modified version of the binomial model to consider the possibility of early exercise. In this version, the up and the down movements for asset prices in each period can be estimated from the variance. 2
+Approach 1: Pseudo-American Valuation
+Step 1: Define when dividends will be paid and how much the dividends will be.
+Step 2: Value the call option to each ex-dividend date using the dividend-adjusted approach described earlier, where the stock price is reduced by the present value of expected dividends.
+Step 3: Choose the maximum of the call values estimated for each ex-dividend day. ILLUSTRATION 5.5: Using Pseudo-American Option Valuation to Adjust for Early Exercise
+Consider an option with a strike price of $35 on a stock trading at $40. The variance in the ln(stock prices) is 0.05, and the riskless rate is 4%. The option has a remaining life of eight months, and there are three dividends expected during this period: Expected Dividend
+Ex-Dividend Day $0.80
+In 1 month $0.80
+In 4 months $0.80
+In 7 months The call option is first valued to just before the first ex-dividend date: The value from the Black-Scholes model is: The call option is then valued to before the second ex-dividend date: The value of the call based on these parameters is: The call option is then valued to before the third ex-dividend date: The value of the call based on these parameters is: The call option is then valued to expiration: The value of the call based on these parameters is: Approach 2: Using the Binomial Model
+The binomial model is much more capable of handling early exercise because it considers the cash flows at each time period, rather than just at expiration. The biggest limitation of the binomial model is determining what stock prices will be at the end of each period, but this can be overcome by using a variant that allows us to estimate the up and the down movements in stock prices from the estimated variance. There are four steps involved:
+Step 1: If the variance in ln(stock prices) has been estimated for the Black-Scholes valuation, convert these into inputs for the binomial model: where u and d are the up and the down movements per unit time for the binomial, and dt is the number of periods within each year (or unit time).
+Step 2: Specify the period in which the dividends will be paid and make the assumption that the price will drop by the amount of the dividend in that period.
+Step 3: Value the call at each node of the tree, allowing for the possibility of early exercise just before ex-dividend dates. There will be early exercise if the remaining time premium on the option is less than the expected drop in option value as a consequence of the dividend payment.
+Step 4: Value the call at time 0, using the standard binomial approach. bstobin.xls: This spreadsheet allows you to estimate the parameters for a binomial model from the inputs to a Black-Scholes model. Impact of Exercise on Underlying Asset Value
+The Black-Scholes model is based on the assumption that exercising an option does not affect the value of the underlying asset. This may be true for listed options on stocks, but it is not true for some types of options. For instance, the exercise of warrants increases the number of shares outstanding and brings fresh cash into the firm, both of which will affect the stock price.3 The expected negative impact (dilution) of exercise will decrease the value of warrants, compared to otherwise similar call options. The adjustment for dilution to the stock price is fairly simple in the Black-Scholes valuation. The stock price is adjusted for the expected dilution from the exercise of the options. In the case of warrants, for instance: where S = Current value of the stock
+nw = Number of warrants outstanding
+W = Value of warrants outstanding
+ns = Number of shares outstanding
+When the warrants are exercised, the number of shares outstanding will increase, reducing the stock price. The numerator reflects the market value of equity, including both stocks and warrants outstanding. The reduction in S will reduce the value of the call option.
+There is an element of circularity in this analysis, since the value of the warrant is needed to estimate the dilution-adjusted S and the dilution-adjusted S is needed to estimate the value of the warrant. This problem can be resolved by starting the process off with an assumed value for the warrant (e.g., the exercise value or the current market price of the warrant). This will yield a value for the warrant, and this estimated value can then be used as an input to reestimate the warrant’s value until there is convergence. FROM BLACK-SCHOLES TO BINOMIAL
+The process of converting the continuous variance in a Black-Scholes model to a binomial tree is a fairly simple one. Assume, for instance, that you have an asset that is trading at $30 currently and that you estimate the annualized standard deviation in the asset value to be 40 percent; the annualized riskless rate is 5 percent. For simplicity, let us assume that the option that you are valuing has a four-year life and that each period is a year. To estimate the prices at the end of each of the four years, we begin by first estimating the up and down movements in the binomial: Based on these estimates, we can obtain the prices at the end of the first node of the tree (the end of the first year): Progressing through the rest of the tree, we obtain the following numbers: ILLUSTRATION 5.6: Valuing a Warrant on Avatek Corporation
+Avatek Corporation is a real estate firm with 19.637 million shares outstanding, trading at $0.38 a share. In March 2001 the company had 1.8 million options outstanding, with four years to expiration and with an exercise price of $2.25. The stock paid no dividends, and the standard deviation in ln(stock prices) was 93%. The four-year Treasury bond rate was 4.9%. (The options were trading at $0.12 apiece at the time of this analysis.)
+The inputs to the warrant valuation model are as follows: The results of the Black-Scholes valuation of this option are: The options were trading at $0.12 in March 2001. Since the value was equal to the price, there was no need for further iterations. If there had been a difference, we would have reestimated the adjusted stock price and option value. If the options had been non-traded (as is the case with management options), this calculation would have required an iterative process, where the option value is used to get the adjusted value per share and the value per share to get the option value. warrant.xls: This spreadsheet allows you to estimate the value of an option when there is a potential dilution from exercise. The Black-Scholes Model for Valuing Puts
+The value of a put can be derived from the value of a call with the same strike price and the same expiration date: where C is the value of the call and P is the value of the put. This relationship between the call and put values is called put-call parity, and any deviations from parity can be used by investors to make riskless profits. To see why put-call parity holds, consider selling a call and buying a put with exercise price K and expiration date t, and simultaneously buying the underlying asset at the current price S. The payoff from this position is riskless and always yields K at expiration (t). To see this, assume that the stock price at expiration is S*. The payoff on each of the positions in the portfolio can be written as follows: Position
+Payoffs at t if S* > K
+Payoffs at t if S* < K Sell call
+–(S* – K)
+0 Buy put
+0
+K – S* Buy stock
+S*
+S* Total
+K
+K Since this position yields K with certainty, the cost of creating this position must be equal to the present value of K at the riskless rate (K e–rt). Substituting the Black-Scholes equation for the value of an equivalent call into this equation, we get: Thus, the replicating portfolio for a put is created by selling short [1 – N(d1)] shares of stock and investing K e–rt[1 – N(d2)] in the riskless asset. ILLUSTRATION 5.7: Valuing a Put Using Put-Call Parity: Cisco Systems and AT&T
+Consider the call on Cisco Systems that we valued in Illustration 5.2. The call had a strike price of $15 on the stock, had 103 days left to expiration, and was valued at $1.87. The stock was trading at $13.62, and the riskless rate was 4.63%. The put can be valued as follows: The put was trading at $3.38.
+Also, a long-term call on AT&T was valued in Illustration 5.4. The call had a strike price of $20, 1.8333 years left to expiration, and a value of $6.63. The stock was trading at $20.50 and was expected to maintain a dividend yield of 2.51% over the period. The riskless rate was 4.85%. The put value can be estimated as follows: The put was trading at $3.80. Both the call and the put were trading at different prices from our estimates, which may indicate that we have not correctly estimated the stock’s volatility. Jump Process Option Pricing Models
+If price changes remain larger as the time periods in the binomial model are shortened, it can no longer be assumed that prices change continuously. When price changes remain large, a price process that allows for price jumps is much more realistic. Cox and Ross (1976) valued options when prices follow a pure jump process, where the jumps can only be positive. Thus, in the next interval, the stock price will either have a large positive jump with a specified probability or drift downward at a given rate.
+Merton (1976) considered a distribution where there are price jumps superimposed on a continuous price process. He specified the rate at which jumps occur (λ) and the average jump size (k), measured as a percentage of the stock price. The model derived to value options with this process is called a jump diffusion model. In this model, the value of an option is determined by the five variables specified in the Black-Scholes model, and the parameters of the jump process (λ, k). Unfortunately, the estimates of the jump process parameters are so difficult to make for most firms that they overwhelm any advantages that accrue from using a more realistic model. These models, therefore, have seen limited use in practice.
+EXTENSIONS OF OPTION PRICING
+All the option pricing models described so far—the binomial, the Black-Scholes, and the jump process models—are designed to value options with clearly defined exercise prices and maturities on underlying assets that are traded. However, the options we encounter in investment analysis or valuation are often on real assets rather than financial assets. Categorized as real options, they can take much more complicated forms. This section considers some of these variations.
+Capped and Barrier Options
+With a simple call option, there is no specified upper limit on the profits that can be made by the buyer of the call. Asset prices, at least in theory, can keep going up, and the payoffs increase proportionately. In some call options, though, the buyer is entitled to profits up to a specified price but not above it. For instance, consider a call option with a strike price of K1 on an asset. In an unrestricted call option, the payoff on this option will increase as the underlying asset’s price increases above K1. Assume, however, that if the price reaches K2, the payoff is capped at (K2 – K1). The payoff diagram on this option is shown in Figure 5.5. Figure 5.5 Payoff on Capped Call This option is called a capped call. Notice, also, that once the price reaches K2, there is no longer any time premium associated with the option, and the option will therefore be exercised. Capped calls are part of a family of options called barrier options, where the payoff on and the life of the option are functions of whether the underlying asset price reaches a certain level during a specified period.
+The value of a capped call is always lower than the value of the same call without the payoff limit. A simple approximation of this value can be obtained by valuing the call twice, once with the given exercise price and once with the cap, and taking the difference in the two values. In the preceding example, then, the value of the call with an exercise price of K1 and a cap at K2 can be written as: Barrier options can take many forms. In a knockout option, an option ceases to exist if the underlying asset reaches a certain price. In the case of a call option, this knockout price is usually set below the strike price, and this option is called a down-and-out option. In the case of a put option, the knockout price will be set above the exercise price, and this option is called an up-and-out option. Like the capped call, these options are worth less than their unrestricted counterparts. Many real options have limits on potential upside, or knockout provisions, and ignoring these limits can result in the overstatement of the value of these options.
+Compound Options
+Some options derive their value not from an underlying asset, but from other options. These options are called compound options. Compound options can take any of four forms—a call on a call, a put on a put, a call on a put, or a put on a call. Geske (1979) developed the analytical formulation for valuing compound options by replacing the standard normal distribution used in a simple option model with a bivariate normal distribution in the calculation.
+Consider, for instance, the option to expand a project that is discussed in Chapter 30. While we will value this option using a simple option pricing model, in reality there could be multiple stages in expansion, with each stage representing an option for the following stage. In this case, we will undervalue the option by considering it as a simple rather than a compound option.
+Notwithstanding this discussion, the valuation of compound options becomes progressively more difficult as more options are added to the chain. In this case, rather than wreck the valuation on the shoals of estimation error, it may be better to accept the conservative estimate that is provided with a simple valuation model as a floor on the value.
+Rainbow Options
+In a simple option, the uncertainty is about the price of the underlying asset. Some options are exposed to two or more sources of uncertainty, and these options are rainbow options. Using the simple option pricing model to value such options can lead to biased estimates of value. As an example, consider an undeveloped oil reserve as an option, where the firm that owns the reserve has the right to develop the reserve. Here there are two sources of uncertainty. The first is obviously the price of oil, and the second is the quantity of oil that is in the reserve. To value this undeveloped reserve, we can make the simplifying assumption that we know the quantity of oil in the reserve with certainty. In reality, however, uncertainty about the quantity will affect the value of this option and make the decision to exercise more difficult.4
+CONCLUSION
+An option is an asset with payoffs that are contingent on the value of an underlying asset. A call option provides its holder with the right to buy the underlying asset at a fixed price, whereas a put option provides its holder with the right to sell at a fixed price, at any time before the expiration of the option. The value of an option is determined by six variables—the current value of the underlying asset, the variance in this value, the expected dividends on the asset, the strike price and life of the option, and the riskless interest rate. This is illustrated in both the binomial and the Black-Scholes models, which value options by creating replicating portfolios composed of the underlying asset and riskless lending or borrowing. These models can be used to value assets that have option like characteristics.
+QUESTIONS AND SHORT PROBLEMS
+In the problems following, use an equity risk premium of 5.5 percent if none is specified. 1. The following are prices of options traded on Microsoft Corporation, which pays no dividends. The stock is trading at $83, and the annualized riskless rate is 3.8%. The standard deviation in ln(stock prices) (based on historical data) is 30%. a. Estimate the value of a three-month call with a strike price of $85.
+b. Using the inputs from the Black-Scholes model, specify how you would replicate this call.
+c. What is the implied standard deviation in this call?
+d. Assume now that you buy a call with a strike price of $85 and sell a call with a strike price of $90. Draw the payoff diagram on this position.
+e. Using put-call parity, estimate the value of a three-month put with a strike price of $85. 2. You are trying to value three-month call and put options on Merck with a strike price of $30. The stock is trading at $28.75, and the company expects to pay a quarterly dividend per share of $0.28 in two months. The annualized riskless interest rate is 3.6%, and the standard deviation in log stock prices is 20%. a. Estimate the value of the call and put options, using the Black-Scholes model.
+b. What effect does the expected dividend payment have on call values? On put values? Why? 3. There is the possibility that the options on Merck described in the preceding problem could be exercised early. a. Use the pseudo-American call option technique to determine whether this will affect the value of the call.
+b. Why does the possibility of early exercise exist? What types of options are most likely to be exercised early? 4. You have been provided the following information on a three-month call: a. If you wanted to replicate buying this call, how much money would you need to borrow?
+b. If you wanted to replicate buying this call, how many shares of stock would you need to buy? 5. Go Video, a manufacturer of video recorders, was trading at $4 per share in May 1994. There were 11 million shares outstanding. At the same time, it had 550,000 one-year warrants outstanding, with a strike price of $4.25. The stock has had a standard deviation of 60%. The stock does not pay a dividend. The riskless rate is 5%. a. Estimate the value of the warrants, ignoring dilution.
+b. Estimate the value of the warrants, allowing for dilution.
+c. Why does dilution reduce the value of the warrants? 6. You are trying to value a long-term call option on the NYSE Composite index, expiring in five years, with a strike price of 275. The index is currently at 250, and the annualized standard deviation in stock prices is 15%. The average dividend yield on the index is 3% and is expected to remain unchanged over the next five years. The five-year Treasury bond rate is 5%. a. Estimate the value of the long-term call option.
+b. Estimate the value of a put option with the same parameters.
+c. What are the implicit assumptions you are making when you use the Black-Scholes model to value this option? Which of these assumptions are likely to be violated? What are the consequences for your valuation? 7. A new security on AT&T will entitle the investor to all dividends on AT&T over the next three years, limiting upside potential to 20% but also providing downside protection below 10%. AT&T stock is trading at $50, and three-year call and put options are traded on the exchange at the following prices: How much would you be willing to pay for this security? 1 Note, though, that higher variance can reduce the value of the underlying asset. As a call option becomes more in-the-money, the more it resembles the underlying asset. For very deep in-the-money call options, higher variance can reduce the value of the option.
+2 To illustrate, if σ2 is the variance in ln(stock prices), the up and the down movements in the binomial can be estimated as follows: where u and d are the up and down movements per unit time for the binomial, T is the life of the option, and m is the number of periods within that lifetime.
+3 Warrants are call options issued by firms, either as part of management compensation contracts or to raise equity.
+4 The analogy to a listed option on a stock is the case where you do not know with certainty what the stock price is when you exercise the option. The more uncertain you are about the stock price, the more margin for error you have to give yourself when you exercise the option, to ensure that you are in fact earning a profit.
+CHAPTER 5
+Option Pricing Theory and Models
+In general, the value of any asset is the present value of the expected cash flows on that asset. This chapter considers an exception to that rule when it looks at assets with two specific characteristics: 1. The assets derive their value from the values of other assets.
+2. The cash flows on the assets are contingent on the occurrence of specific events. These assets are called options, and the present value of the expected cash flows on these assets will understate their true value. This chapter describes the cash flow characteristics of options, considers the factors that determine their value, and examines how best to value them.
+BASICS OF OPTION PRICING
+An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called a strike price or an exercise price) at or before the expiration date of the option. Since it is a right and not an obligation, the holder can choose not to exercise the right and can allow the option to expire. There are two types of options—call options and put options.
+Call and Put Options: Description and Payoff Diagrams
+A call option gives the buyer of the option the right to buy the underlying asset at the strike price or the exercise price at any time prior to the expiration date of the option. The buyer pays a price for this right. If at expiration the value of the asset is less than the strike price, the option is not exercised and expires worthless. If, however, the value of the asset is greater than the strike price, the option is exercised—the buyer of the option buys the stock at the exercise price, and the difference between the asset value and the exercise price comprises the gross profit on the investment. The net profit on the investment is the difference between the gross profit and the price paid for the call initially.
+A payoff diagram illustrates the cash payoff on an option at expiration. For a call, the net payoff is negative (and equal to the price paid for the call) if the value of the underlying asset is less than the strike price. If the price of the underlying asset exceeds the strike price, the gross payoff is the difference between the value of the underlying asset and the strike price, and the net payoff is the difference between the gross payoff and the price of the call. This is illustrated in Figure 5.1. Figure 5.1 Payoff on Call Option A put option gives the buyer of the option the right to sell the underlying asset at a fixed price, again called the strike or exercise price, at any time prior to the expiration date of the option. The buyer pays a price for this right. If the price of the underlying asset is greater than the strike price, the option will not be exercised and will expire worthless. But if the price of the underlying asset is less than the strike price, the owner of the put option will exercise the option and sell the stock at the strike price, claiming the difference between the strike price and the market value of the asset as the gross profit. Again, netting out the initial cost paid for the put yields the net profit from the transaction.
+A put has a negative net payoff if the value of the underlying asset exceeds the strike price, and has a gross payoff equal to the difference between the strike price and the value of the underlying asset if the asset value is less than the strike price. This is summarized in Figure 5.2. Figure 5.2 Payoff on Put Option DETERMINANTS OF OPTION VALUE
+The value of an option is determined by six variables relating to the underlying asset and financial markets. 1. Current value of the underlying asset. Options are assets that derive value from an underlying asset. Consequently, changes in the value of the underlying asset affect the value of the options on that asset. Since calls provide the right to buy the underlying asset at a fixed price, an increase in the value of the asset will increase the value of the calls. Puts, on the other hand, become less valuable as the value of the asset increases.
+2. Variance in value of the underlying asset. The buyer of an option acquires the right to buy or sell the underlying asset at a fixed price. The higher the variance in the value of the underlying asset, the greater the value of the option.1 This is true for both calls and puts. While it may seem counterintuitive that an increase in a risk measure (variance) should increase value, options are different from other securities since buyers of options can never lose more than the price they pay for them; in fact, they have the potential to earn significant returns from large price movements.
+3. Dividends paid on the underlying asset. The value of the underlying asset can be expected to decrease if dividend payments are made on the asset during the life of the option. Consequently, the value of a call on the asset is a decreasing function of the size of expected dividend payments, and the value of a put is an increasing function of expected dividend payments. A more intuitive way of thinking about dividend payments, for call options, is as a cost of delaying exercise on in-the-money options. To see why, consider an option on a traded stock. Once a call option is in-the-money (i.e., the holder of the option will make a gross payoff by exercising the option), exercising the call option will provide the holder with the stock and entitle him or her to the dividends on the stock in subsequent periods. Failing to exercise the option will mean that these dividends are forgone.
+4. Strike price of the option. A key characteristic used to describe an option is the strike price. In the case of calls, where the holder acquires the right to buy at a fixed price, the value of the call will decline as the strike price increases. In the case of puts, where the holder has the right to sell at a fixed price, the value will increase as the strike price increases.
+5. Time to expiration on the option. Both calls and puts are more valuable the greater the time to expiration. This is because the longer time to expiration provides more time for the value of the underlying asset to move, increasing the value of both types of options. Additionally, in the case of a call, where the buyer has the right to pay a fixed price at expiration, the present value of this fixed price decreases as the life of the option increases, increasing the value of the call.
+6. Riskless interest rate corresponding to life of the option. Since the buyer of an option pays the price of the option up front, an opportunity cost is involved. This cost will depend on the level of interest rates and the time to expiration of the option. The riskless interest rate also enters into the valuation of options when the present value of the exercise price is calculated, since the exercise price does not have to be paid (received) until expiration on calls (puts). Increases in the interest rate will increase the value of calls and reduce the value of puts. Table 5.1 summarizes the variables and their predicted effects on call and put prices.
+Table 5.1 Summary of Variables Affecting Call and Put Prices Effect On Factor
+Call Value
+Put Value Increase in underlying asset’s value
+Increases
+Decreases Increase in variance of underlying asset
+Increases
+Increases Increase in strike price
+Decreases
+Increases Increase in dividends paid
+Decreases
+Increases Increase in time to expiration
+Increases
+Increases Increase in interest rates
+Increases
+Decreases American versus European Options: Variables Relating to Early Exercise
+A primary distinction between American and European options is that an American option can be exercised at any time prior to its expiration, while European options can be exercised only at expiration. The possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. There is one compensating factor that enables the former to be valued using models designed for the latter. In most cases, the time premium associated with the remaining life of an option and transaction costs make early exercise suboptimal. In other words, the holders of in-the-money options generally get much more by selling the options to someone else than by exercising the options.
+OPTION PRICING MODELS
+Option pricing theory has made vast strides since 1972, when Fischer Black and Myron Scholes published their pathbreaking paper that provided a model for valuing dividend-protected European options. Black and Scholes used a “replicating portfolio”—a portfolio composed of the underlying asset and the risk-free asset that had the same cash flows as the option being valued—and the notion of arbitrage to come up with their final formulation. Although their derivation is mathematically complicated, there is a simpler binomial model for valuing options that draws on the same logic.
+Binomial Model
+The binomial option pricing model is based on a simple formulation for the asset price process in which the asset, in any time period, can move to one of two possible prices. The general formulation of a stock price process that follows the binomial path is shown in Figure 5.3. In this figure, S is the current stock price; the price moves up to Su with probability p and down to Sd with probability 1 – p in any time period. Figure 5.3 General Formulation for Binomial Price Path Creating a Replicating Portfolio
+The objective in creating a replicating portfolio is to use a combination of risk-free borrowing/lending and the underlying asset to create the same cash flows as the option being valued. The principles of arbitrage apply then, and the value of the option must be equal to the value of the replicating portfolio. In the case of the general formulation shown in Figure 5.3, where stock prices can move either up to Su or down to Sd in any time period, the replicating portfolio for a call with strike price K will involve borrowing $B and acquiring Δ of the underlying asset, where: where Cu = Value of the call if the stock price is Su
+Cd = Value of the call if the stock price is Sd
+In a multiperiod binomial process, the valuation has to proceed iteratively (i.e., starting with the final time period and moving backward in time until the current point in time). The portfolios replicating the option are created at each step and valued, providing the values for the option in that time period. The final output from the binomial option pricing model is a statement of the value of the option in terms of the replicating portfolio, composed of Δ shares (option delta) of the underlying asset and risk-free borrowing/lending. ILLUSTRATION 5.1: Binomial Option Valuation
+Assume that the objective is to value a call with a strike price of $50, which is expected to expire in two time periods, on an underlying asset whose price currently is $50 and is expected to follow a binomial process: Now assume that the interest rate is 11%. In addition, define: The objective is to combined Δ shares of stock and B dollars of borrowing to replicate the cash flows from the call with a strike price of $50. This can be done iteratively, starting with the last period and working back through the binomial tree.
+Step 1: Start with the end nodes and work backward: Thus, if the stock price is $70 at t = 1, borrowing $45 and buying one share of the stock will give the same cash flows as buying the call. The value of the call at t = 1, if the stock price is $70, is therefore: Considering the other leg of the binomial tree at t = 1, If the stock price is $35 at t = 1, then the call is worth nothing.
+Step 2: Move backward to the earlier time period and create a replicating portfolio that will provide the cash flows the option will provide. In other words, borrowing $22.50 and buying five-sevenths of a share will provide the same cash flows as a call with a strike price of $50 over the call’s lifetime. The value of the call therefore has to be the same as the cost of creating this position. The Determinants of Value
+The binomial model provides insight into the determinants of option value. The value of an option is not determined by the expected price of the asset but by its current price, which, of course, reflects expectations about the future. This is a direct consequence of arbitrage. If the option value deviates from the value of the replicating portfolio, investors can create an arbitrage position (i.e., one that requires no investment, involves no risk, and delivers positive returns). To illustrate, if the portfolio that replicates the call costs more than the call does in the market, an investor could buy the call, sell the replicating portfolio, and be guaranteed the difference as a profit. The cash flows on the two positions will offset each other, leading to no cash flows in subsequent periods. The call option value also increases as the time to expiration is extended, as the price movements (u and d) increase, and with increases in the interest rate.
+While the binomial model provides an intuitive feel for the determinants of option value, it requires a large number of inputs, in terms of expected future prices at each node. As time periods are made shorter in the binomial model, you can make one of two assumptions about asset prices.You can assume that price changes become smaller as periods get shorter; this leads to price changes becoming infinitesimally small as time periods approach zero, leading to a continuous price process. Alternatively, you can assume that price changes stay large even as the period gets shorter; this leads to a jump price process, where prices can jump in any period. This section considers the option pricing models that emerge with each of these assumptions.
+Black-Scholes Model
+When the price process is continuous (i.e., price changes become smaller as time periods get shorter), the binomial model for pricing options converges on the Black-Scholes model. The model, named after its cocreators, Fischer Black and Myron Scholes, allows us to estimate the value of any option using a small number of inputs, and has been shown to be robust in valuing many listed options.
+The Model
+While the derivation of the Black-Scholes model is far too complicated to present here, it is based on the idea of creating a portfolio of the underlying asset and the riskless asset with the same cash flows, and hence the same cost, as the option being valued. The value of a call option in the Black-Scholes model can be written as a function of the five variables: S = Current value of the underlying asset
+K = Strike price of the option
+t = Life to expiration of the option
+r = Riskless interest rate corresponding to the life of the option
+σ2 = Variance in the ln(value) of the underlying asset The value of a call is then: Note that e–rt is the present value factor, and reflects the fact that the exercise price on the call option does not have to be paid until expiration, since the model values European options. N(d1) and N(d2) are probabilities, estimated by using a cumulative standardized normal distribution, and the values of d1 and d2 obtained for an option. The cumulative distribution is shown in Figure 5.4. Figure 5.4 Cumulative Normal Distribution In approximate terms, N(d2) yields the likelihood that an option will generate positive cash flows for its owner at exercise (i.e., that S > K in the case of a call option and that K > S in the case of a put option). The portfolio that replicates the call option is created by buying N(d1) units of the underlying asset, and borrowing Ke–rt N(d2). The portfolio will have the same cash flows as the call option, and thus the same value as the option. N(d1), which is the number of units of the underlying asset that are needed to create the replicating portfolio, is called the option delta. A NOTE ON ESTIMATING THE INPUTS TO THE BLACK-SCHOLES MODEL
+The Black-Scholes model requires inputs that are consistent on time measurement. There are two places where this affects estimates. The first relates to the fact that the model works in continuous time, rather than discrete time. That is why we use the continuous time version of present value (exp–rt) rather than the discrete version, (1 + r)–t. It also means that the inputs such as the riskless rate have to be modified to make them continuous time inputs. For instance, if the one-year Treasury bond rate is 6.2 percent, the risk-free rate that is used in the Black-Scholes model should be: The second relates to the period over which the inputs are estimated. For instance, the preceding rate is an annual rate. The variance that is entered into the model also has to be an annualized variance. The variance, estimated from ln(asset prices), can be annualized easily because variances are linear in time if the serial correlation is zero. Thus, if monthly or weekly prices are used to estimate variance, the variance is annualized by multiplying by 12 or 52, respectively. ILLUSTRATION 5.2: Valuing an Option Using the Black-Scholes Model
+On March 6, 2001, Cisco Systems was trading at $13.62. We will attempt to value a July 2001 call option with a strike price of $15, trading on the CBOE on the same day for $2. The following are the other parameters of the options: The annualized standard deviation in Cisco Systems stock price over the previous year was 81%. This standard deviation is estimated using weekly stock prices over the year, and the resulting number was annualized as follows: The option expiration date is Friday, July 20, 2001. There are 103 days to expiration, and the annualized Treasury bill rate corresponding to this option life is 4.63%. The inputs for the Black-Scholes model are as follows: Current stock price (S) = $13.62
+Strike price on the option = $15
+Option life = 103/365 = 0.2822
+Standard deviation in ln(stock prices) = 81%
+Riskless rate = 4.63% Inputting these numbers into the model, we get: Using the normal distribution, we can estimate the N(d1) and N(d2): The value of the call can now be estimated: Since the call is trading at $2, it is slightly overvalued, assuming that the estimate of standard deviation used is correct. IMPLIED VOLATILITY
+The only input in the Black Scholes on which there can be significant disagreement among investors is the variance. While the variance is often estimated by looking at historical data, the values for options that emerge from using the historical variance can be different from the market prices. For any option, there is some variance at which the estimated value will be equal to the market price. This variance is called an implied variance.
+Consider the Cisco option valued in Illustration 5.2. With a standard deviation of 81 percent, the value of the call option with a strike price of $15 was estimated to be $1.87. Since the market price is higher than the calculated value, we tried higher standard deviations, and at a standard deviation 85.40 percent the value of the option is $2 (which is the market price). This is the implied standard deviation or implied volatility. Model Limitations and Fixes
+The Black-Scholes model was designed to value European options that can be exercised only at maturity and whose underlying assets do not pay dividends. In addition, options are valued based on the assumption that option exercise does not affect the value of the underlying asset. In practice, assets do pay dividends, options sometimes get exercised early, and exercising an option can affect the value of the underlying asset. Adjustments exist that, while not perfect, provide partial corrections to the Black-Scholes model.
+Dividends
+The payment of a dividend reduces the stock price; note that on the ex-dividend day, the stock price generally declines. Consequently, call options become less valuable and put options more valuable as expected dividend payments increase. There are two ways of dealing with dividends in the Black-Scholes model: 1. Short-term options. One approach to dealing with dividends is to estimate the present value of expected dividends that will be paid by the underlying asset during the option life and subtract it from the current value of the asset to use as S in the model. 2. Long-term options. Since it becomes less practical to estimate the present value of dividends the longer the option life, an alternate approach can be used. If the dividend yield (y = Dividends/Current value of the asset) on the underlying asset is expected to remain unchanged during the life of the option, the Black-Scholes model can be modified to take dividends into account. From an intuitive standpoint, the adjustments have two effects. First, the value of the asset is discounted back to the present at the dividend yield to take into account the expected drop in asset value resulting from dividend payments. Second, the interest rate is offset by the dividend yield to reflect the lower carrying cost from holding the asset (in the replicating portfolio). The net effect will be a reduction in the value of calls estimated using this model. ILLUSTRATION 5.3: Valuing a Short-Term Option with Dividend Adjustments—The Black-Scholes Correction
+Assume that it is March 6, 2001, and that AT&T is trading at $20.50 a share. Consider a call option on the stock with a strike price of $20, expiring on July 20, 2001. Using past stock prices, the annualized standard deviation in the log of stock prices for AT&T is estimated at 60%. There is one dividend, amounting to $0.15, and it will be paid in 23 days. The riskless rate is 4.63%. Present value of expected dividend = $0.15/1.046323/365 = $0.15
+Dividend-adjusted stock price = $20.50 – $0.15 = $20.35
+Time to expiration = 103/365 = 0.2822
+Variance in ln(stock prices) = 0.62 = 0.36
+Riskless rate = 4.63% The value from the Black-Scholes model is: The call option was trading at $2.60 on that day. ILLUSTRATION 5.4: Valuing a Long-Term Option with Dividend Adjustments—Primes and Scores
+The CBOE offers longer-term call and put options on some stocks. On March 6, 2001, for instance, you could have purchased an AT&T call expiring on January 17, 2003. The stock price for AT&T is $20.50 (as in the previous example). The following is the valuation of a call option with a strike price of $20. Instead of estimating the present value of dividends over the next two years, assume that AT&T’s dividend yield will remain 2.51% over this period and that the risk-free rate for a two-year Treasury bond is 4.85%. The inputs to the Black-Scholes model are: The value from the Black-Scholes model is: The call was trading at $5.80 on March 8, 2001. optst.xls: This spreadsheet allows you to estimate the value of a short-term option when the expected dividends during the option life can be estimated. optlt.xls: This spreadsheet allows you to estimate the value of an option when the underlying asset has a constant dividend yield. Early Exercise
+The Black-Scholes model was designed to value European options that can be exercised only at expiration. In contrast, most options that we encounter in practice are American options and can be exercised at any time until expiration. As mentioned earlier, the possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. In general, though, with traded options, it is almost always better to sell the option to someone else rather than exercise early, since options have a time premium (i.e., they sell for more than their exercise value). There are two exceptions. One occurs when the underlying asset pays large dividends, thus reducing the expected value of the asset. In this case, call options may be exercised just before an ex-dividend date, if the time premium on the options is less than the expected decline in asset value as a consequence of the dividend payment. The other exception arises when an investor holds both the underlying asset and deep in-the-money puts (i.e., puts with strike prices well above the current price of the underlying asset) on that asset at a time when interest rates are high. In this case, the time premium on the put may be less than the potential gain from exercising the put early and earning interest on the exercise price.
+There are two basic ways of dealing with the possibility of early exercise. One is to continue to use the unadjusted Black-Scholes model and to regard the resulting value as a floor or conservative estimate of the true value. The other is to try to adjust the value of the option for the possibility of early exercise. There are two approaches for doing so. One uses the Black-Scholes model to value the option to each potential exercise date. With options on stocks, this basically requires that the investor values options to each ex-dividend day and chooses the maximum of the estimated call values. The second approach is to use a modified version of the binomial model to consider the possibility of early exercise. In this version, the up and the down movements for asset prices in each period can be estimated from the variance. 2
+Approach 1: Pseudo-American Valuation
+Step 1: Define when dividends will be paid and how much the dividends will be.
+Step 2: Value the call option to each ex-dividend date using the dividend-adjusted approach described earlier, where the stock price is reduced by the present value of expected dividends.
+Step 3: Choose the maximum of the call values estimated for each ex-dividend day. ILLUSTRATION 5.5: Using Pseudo-American Option Valuation to Adjust for Early Exercise
+Consider an option with a strike price of $35 on a stock trading at $40. The variance in the ln(stock prices) is 0.05, and the riskless rate is 4%. The option has a remaining life of eight months, and there are three dividends expected during this period: Expected Dividend
+Ex-Dividend Day $0.80
+In 1 month $0.80
+In 4 months $0.80
+In 7 months The call option is first valued to just before the first ex-dividend date: The value from the Black-Scholes model is: The call option is then valued to before the second ex-dividend date: The value of the call based on these parameters is: The call option is then valued to before the third ex-dividend date: The value of the call based on these parameters is: The call option is then valued to expiration: The value of the call based on these parameters is: Approach 2: Using the Binomial Model
+The binomial model is much more capable of handling early exercise because it considers the cash flows at each time period, rather than just at expiration. The biggest limitation of the binomial model is determining what stock prices will be at the end of each period, but this can be overcome by using a variant that allows us to estimate the up and the down movements in stock prices from the estimated variance. There are four steps involved:
+Step 1: If the variance in ln(stock prices) has been estimated for the Black-Scholes valuation, convert these into inputs for the binomial model: where u and d are the up and the down movements per unit time for the binomial, and dt is the number of periods within each year (or unit time).
+Step 2: Specify the period in which the dividends will be paid and make the assumption that the price will drop by the amount of the dividend in that period.
+Step 3: Value the call at each node of the tree, allowing for the possibility of early exercise just before ex-dividend dates. There will be early exercise if the remaining time premium on the option is less than the expected drop in option value as a consequence of the dividend payment.
+Step 4: Value the call at time 0, using the standard binomial approach. bstobin.xls: This spreadsheet allows you to estimate the parameters for a binomial model from the inputs to a Black-Scholes model. Impact of Exercise on Underlying Asset Value
+The Black-Scholes model is based on the assumption that exercising an option does not affect the value of the underlying asset. This may be true for listed options on stocks, but it is not true for some types of options. For instance, the exercise of warrants increases the number of shares outstanding and brings fresh cash into the firm, both of which will affect the stock price.3 The expected negative impact (dilution) of exercise will decrease the value of warrants, compared to otherwise similar call options. The adjustment for dilution to the stock price is fairly simple in the Black-Scholes valuation. The stock price is adjusted for the expected dilution from the exercise of the options. In the case of warrants, for instance: where S = Current value of the stock
+nw = Number of warrants outstanding
+W = Value of warrants outstanding
+ns = Number of shares outstanding
+When the warrants are exercised, the number of shares outstanding will increase, reducing the stock price. The numerator reflects the market value of equity, including both stocks and warrants outstanding. The reduction in S will reduce the value of the call option.
+There is an element of circularity in this analysis, since the value of the warrant is needed to estimate the dilution-adjusted S and the dilution-adjusted S is needed to estimate the value of the warrant. This problem can be resolved by starting the process off with an assumed value for the warrant (e.g., the exercise value or the current market price of the warrant). This will yield a value for the warrant, and this estimated value can then be used as an input to reestimate the warrant’s value until there is convergence. FROM BLACK-SCHOLES TO BINOMIAL
+The process of converting the continuous variance in a Black-Scholes model to a binomial tree is a fairly simple one. Assume, for instance, that you have an asset that is trading at $30 currently and that you estimate the annualized standard deviation in the asset value to be 40 percent; the annualized riskless rate is 5 percent. For simplicity, let us assume that the option that you are valuing has a four-year life and that each period is a year. To estimate the prices at the end of each of the four years, we begin by first estimating the up and down movements in the binomial: Based on these estimates, we can obtain the prices at the end of the first node of the tree (the end of the first year): Progressing through the rest of the tree, we obtain the following numbers: ILLUSTRATION 5.6: Valuing a Warrant on Avatek Corporation
+Avatek Corporation is a real estate firm with 19.637 million shares outstanding, trading at $0.38 a share. In March 2001 the company had 1.8 million options outstanding, with four years to expiration and with an exercise price of $2.25. The stock paid no dividends, and the standard deviation in ln(stock prices) was 93%. The four-year Treasury bond rate was 4.9%. (The options were trading at $0.12 apiece at the time of this analysis.)
+The inputs to the warrant valuation model are as follows: The results of the Black-Scholes valuation of this option are: The options were trading at $0.12 in March 2001. Since the value was equal to the price, there was no need for further iterations. If there had been a difference, we would have reestimated the adjusted stock price and option value. If the options had been non-traded (as is the case with management options), this calculation would have required an iterative process, where the option value is used to get the adjusted value per share and the value per share to get the option value. warrant.xls: This spreadsheet allows you to estimate the value of an option when there is a potential dilution from exercise. The Black-Scholes Model for Valuing Puts
+The value of a put can be derived from the value of a call with the same strike price and the same expiration date: where C is the value of the call and P is the value of the put. This relationship between the call and put values is called put-call parity, and any deviations from parity can be used by investors to make riskless profits. To see why put-call parity holds, consider selling a call and buying a put with exercise price K and expiration date t, and simultaneously buying the underlying asset at the current price S. The payoff from this position is riskless and always yields K at expiration (t). To see this, assume that the stock price at expiration is S*. The payoff on each of the positions in the portfolio can be written as follows: Position
+Payoffs at t if S* > K
+Payoffs at t if S* < K Sell call
+–(S* – K)
+0 Buy put
+0
+K – S* Buy stock
+S*
+S* Total
+K
+K Since this position yields K with certainty, the cost of creating this position must be equal to the present value of K at the riskless rate (K e–rt). Substituting the Black-Scholes equation for the value of an equivalent call into this equation, we get: Thus, the replicating portfolio for a put is created by selling short [1 – N(d1)] shares of stock and investing K e–rt[1 – N(d2)] in the riskless asset. ILLUSTRATION 5.7: Valuing a Put Using Put-Call Parity: Cisco Systems and AT&T
+Consider the call on Cisco Systems that we valued in Illustration 5.2. The call had a strike price of $15 on the stock, had 103 days left to expiration, and was valued at $1.87. The stock was trading at $13.62, and the riskless rate was 4.63%. The put can be valued as follows: The put was trading at $3.38.
+Also, a long-term call on AT&T was valued in Illustration 5.4. The call had a strike price of $20, 1.8333 years left to expiration, and a value of $6.63. The stock was trading at $20.50 and was expected to maintain a dividend yield of 2.51% over the period. The riskless rate was 4.85%. The put value can be estimated as follows: The put was trading at $3.80. Both the call and the put were trading at different prices from our estimates, which may indicate that we have not correctly estimated the stock’s volatility. Jump Process Option Pricing Models
+If price changes remain larger as the time periods in the binomial model are shortened, it can no longer be assumed that prices change continuously. When price changes remain large, a price process that allows for price jumps is much more realistic. Cox and Ross (1976) valued options when prices follow a pure jump process, where the jumps can only be positive. Thus, in the next interval, the stock price will either have a large positive jump with a specified probability or drift downward at a given rate.
+Merton (1976) considered a distribution where there are price jumps superimposed on a continuous price process. He specified the rate at which jumps occur (λ) and the average jump size (k), measured as a percentage of the stock price. The model derived to value options with this process is called a jump diffusion model. In this model, the value of an option is determined by the five variables specified in the Black-Scholes model, and the parameters of the jump process (λ, k). Unfortunately, the estimates of the jump process parameters are so difficult to make for most firms that they overwhelm any advantages that accrue from using a more realistic model. These models, therefore, have seen limited use in practice.
+EXTENSIONS OF OPTION PRICING
+All the option pricing models described so far—the binomial, the Black-Scholes, and the jump process models—are designed to value options with clearly defined exercise prices and maturities on underlying assets that are traded. However, the options we encounter in investment analysis or valuation are often on real assets rather than financial assets. Categorized as real options, they can take much more complicated forms. This section considers some of these variations.
+Capped and Barrier Options
+With a simple call option, there is no specified upper limit on the profits that can be made by the buyer of the call. Asset prices, at least in theory, can keep going up, and the payoffs increase proportionately. In some call options, though, the buyer is entitled to profits up to a specified price but not above it. For instance, consider a call option with a strike price of K1 on an asset. In an unrestricted call option, the payoff on this option will increase as the underlying asset’s price increases above K1. Assume, however, that if the price reaches K2, the payoff is capped at (K2 – K1). The payoff diagram on this option is shown in Figure 5.5. Figure 5.5 Payoff on Capped Call This option is called a capped call. Notice, also, that once the price reaches K2, there is no longer any time premium associated with the option, and the option will therefore be exercised. Capped calls are part of a family of options called barrier options, where the payoff on and the life of the option are functions of whether the underlying asset price reaches a certain level during a specified period.
+The value of a capped call is always lower than the value of the same call without the payoff limit. A simple approximation of this value can be obtained by valuing the call twice, once with the given exercise price and once with the cap, and taking the difference in the two values. In the preceding example, then, the value of the call with an exercise price of K1 and a cap at K2 can be written as: Barrier options can take many forms. In a knockout option, an option ceases to exist if the underlying asset reaches a certain price. In the case of a call option, this knockout price is usually set below the strike price, and this option is called a down-and-out option. In the case of a put option, the knockout price will be set above the exercise price, and this option is called an up-and-out option. Like the capped call, these options are worth less than their unrestricted counterparts. Many real options have limits on potential upside, or knockout provisions, and ignoring these limits can result in the overstatement of the value of these options.
+Compound Options
+Some options derive their value not from an underlying asset, but from other options. These options are called compound options. Compound options can take any of four forms—a call on a call, a put on a put, a call on a put, or a put on a call. Geske (1979) developed the analytical formulation for valuing compound options by replacing the standard normal distribution used in a simple option model with a bivariate normal distribution in the calculation.
+Consider, for instance, the option to expand a project that is discussed in Chapter 30. While we will value this option using a simple option pricing model, in reality there could be multiple stages in expansion, with each stage representing an option for the following stage. In this case, we will undervalue the option by considering it as a simple rather than a compound option.
+Notwithstanding this discussion, the valuation of compound options becomes progressively more difficult as more options are added to the chain. In this case, rather than wreck the valuation on the shoals of estimation error, it may be better to accept the conservative estimate that is provided with a simple valuation model as a floor on the value.
+Rainbow Options
+In a simple option, the uncertainty is about the price of the underlying asset. Some options are exposed to two or more sources of uncertainty, and these options are rainbow options. Using the simple option pricing model to value such options can lead to biased estimates of value. As an example, consider an undeveloped oil reserve as an option, where the firm that owns the reserve has the right to develop the reserve. Here there are two sources of uncertainty. The first is obviously the price of oil, and the second is the quantity of oil that is in the reserve. To value this undeveloped reserve, we can make the simplifying assumption that we know the quantity of oil in the reserve with certainty. In reality, however, uncertainty about the quantity will affect the value of this option and make the decision to exercise more difficult.4
+CONCLUSION
+An option is an asset with payoffs that are contingent on the value of an underlying asset. A call option provides its holder with the right to buy the underlying asset at a fixed price, whereas a put option provides its holder with the right to sell at a fixed price, at any time before the expiration of the option. The value of an option is determined by six variables—the current value of the underlying asset, the variance in this value, the expected dividends on the asset, the strike price and life of the option, and the riskless interest rate. This is illustrated in both the binomial and the Black-Scholes models, which value options by creating replicating portfolios composed of the underlying asset and riskless lending or borrowing. These models can be used to value assets that have option like characteristics.
+QUESTIONS AND SHORT PROBLEMS
+In the problems following, use an equity risk premium of 5.5 percent if none is specified. 1. The following are prices of options traded on Microsoft Corporation, which pays no dividends. The stock is trading at $83, and the annualized riskless rate is 3.8%. The standard deviation in ln(stock prices) (based on historical data) is 30%. a. Estimate the value of a three-month call with a strike price of $85.
+b. Using the inputs from the Black-Scholes model, specify how you would replicate this call.
+c. What is the implied standard deviation in this call?
+d. Assume now that you buy a call with a strike price of $85 and sell a call with a strike price of $90. Draw the payoff diagram on this position.
+e. Using put-call parity, estimate the value of a three-month put with a strike price of $85. 2. You are trying to value three-month call and put options on Merck with a strike price of $30. The stock is trading at $28.75, and the company expects to pay a quarterly dividend per share of $0.28 in two months. The annualized riskless interest rate is 3.6%, and the standard deviation in log stock prices is 20%. a. Estimate the value of the call and put options, using the Black-Scholes model.
+b. What effect does the expected dividend payment have on call values? On put values? Why? 3. There is the possibility that the options on Merck described in the preceding problem could be exercised early. a. Use the pseudo-American call option technique to determine whether this will affect the value of the call.
+b. Why does the possibility of early exercise exist? What types of options are most likely to be exercised early? 4. You have been provided the following information on a three-month call: a. If you wanted to replicate buying this call, how much money would you need to borrow?
+b. If you wanted to replicate buying this call, how many shares of stock would you need to buy? 5. Go Video, a manufacturer of video recorders, was trading at $4 per share in May 1994. There were 11 million shares outstanding. At the same time, it had 550,000 one-year warrants outstanding, with a strike price of $4.25. The stock has had a standard deviation of 60%. The stock does not pay a dividend. The riskless rate is 5%. a. Estimate the value of the warrants, ignoring dilution.
+b. Estimate the value of the warrants, allowing for dilution.
+c. Why does dilution reduce the value of the warrants? 6. You are trying to value a long-term call option on the NYSE Composite index, expiring in five years, with a strike price of 275. The index is currently at 250, and the annualized standard deviation in stock prices is 15%. The average dividend yield on the index is 3% and is expected to remain unchanged over the next five years. The five-year Treasury bond rate is 5%. a. Estimate the value of the long-term call option.
+b. Estimate the value of a put option with the same parameters.
+c. What are the implicit assumptions you are making when you use the Black-Scholes model to value this option? Which of these assumptions are likely to be violated? What are the consequences for your valuation? 7. A new security on AT&T will entitle the investor to all dividends on AT&T over the next three years, limiting upside potential to 20% but also providing downside protection below 10%. AT&T stock is trading at $50, and three-year call and put options are traded on the exchange at the following prices: How much would you be willing to pay for this security? 1 Note, though, that higher variance can reduce the value of the underlying asset. As a call option becomes more in-the-money, the more it resembles the underlying asset. For very deep in-the-money call options, higher variance can reduce the value of the option.
+2 To illustrate, if σ2 is the variance in ln(stock prices), the up and the down movements in the binomial can be estimated as follows: where u and d are the up and down movements per unit time for the binomial, T is the life of the option, and m is the number of periods within that lifetime.
+3 Warrants are call options issued by firms, either as part of management compensation contracts or to raise equity.
+4 The analogy to a listed option on a stock is the case where you do not know with certainty what the stock price is when you exercise the option. The more uncertain you are about the stock price, the more margin for error you have to give yourself when you exercise the option, to ensure that you are in fact earning a profit.
+CHAPTER 5
+Option Pricing Theory and Models
+In general, the value of any asset is the present value of the expected cash flows on that asset. This chapter considers an exception to that rule when it looks at assets with two specific characteristics: 1. The assets derive their value from the values of other assets.
+2. The cash flows on the assets are contingent on the occurrence of specific events. These assets are called options, and the present value of the expected cash flows on these assets will understate their true value. This chapter describes the cash flow characteristics of options, considers the factors that determine their value, and examines how best to value them.
+BASICS OF OPTION PRICING
+An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called a strike price or an exercise price) at or before the expiration date of the option. Since it is a right and not an obligation, the holder can choose not to exercise the right and can allow the option to expire. There are two types of options—call options and put options.
+Call and Put Options: Description and Payoff Diagrams
+A call option gives the buyer of the option the right to buy the underlying asset at the strike price or the exercise price at any time prior to the expiration date of the option. The buyer pays a price for this right. If at expiration the value of the asset is less than the strike price, the option is not exercised and expires worthless. If, however, the value of the asset is greater than the strike price, the option is exercised—the buyer of the option buys the stock at the exercise price, and the difference between the asset value and the exercise price comprises the gross profit on the investment. The net profit on the investment is the difference between the gross profit and the price paid for the call initially.
+A payoff diagram illustrates the cash payoff on an option at expiration. For a call, the net payoff is negative (and equal to the price paid for the call) if the value of the underlying asset is less than the strike price. If the price of the underlying asset exceeds the strike price, the gross payoff is the difference between the value of the underlying asset and the strike price, and the net payoff is the difference between the gross payoff and the price of the call. This is illustrated in Figure 5.1. Figure 5.1 Payoff on Call Option A put option gives the buyer of the option the right to sell the underlying asset at a fixed price, again called the strike or exercise price, at any time prior to the expiration date of the option. The buyer pays a price for this right. If the price of the underlying asset is greater than the strike price, the option will not be exercised and will expire worthless. But if the price of the underlying asset is less than the strike price, the owner of the put option will exercise the option and sell the stock at the strike price, claiming the difference between the strike price and the market value of the asset as the gross profit. Again, netting out the initial cost paid for the put yields the net profit from the transaction.
+A put has a negative net payoff if the value of the underlying asset exceeds the strike price, and has a gross payoff equal to the difference between the strike price and the value of the underlying asset if the asset value is less than the strike price. This is summarized in Figure 5.2. Figure 5.2 Payoff on Put Option DETERMINANTS OF OPTION VALUE
+The value of an option is determined by six variables relating to the underlying asset and financial markets. 1. Current value of the underlying asset. Options are assets that derive value from an underlying asset. Consequently, changes in the value of the underlying asset affect the value of the options on that asset. Since calls provide the right to buy the underlying asset at a fixed price, an increase in the value of the asset will increase the value of the calls. Puts, on the other hand, become less valuable as the value of the asset increases.
+2. Variance in value of the underlying asset. The buyer of an option acquires the right to buy or sell the underlying asset at a fixed price. The higher the variance in the value of the underlying asset, the greater the value of the option.1 This is true for both calls and puts. While it may seem counterintuitive that an increase in a risk measure (variance) should increase value, options are different from other securities since buyers of options can never lose more than the price they pay for them; in fact, they have the potential to earn significant returns from large price movements.
+3. Dividends paid on the underlying asset. The value of the underlying asset can be expected to decrease if dividend payments are made on the asset during the life of the option. Consequently, the value of a call on the asset is a decreasing function of the size of expected dividend payments, and the value of a put is an increasing function of expected dividend payments. A more intuitive way of thinking about dividend payments, for call options, is as a cost of delaying exercise on in-the-money options. To see why, consider an option on a traded stock. Once a call option is in-the-money (i.e., the holder of the option will make a gross payoff by exercising the option), exercising the call option will provide the holder with the stock and entitle him or her to the dividends on the stock in subsequent periods. Failing to exercise the option will mean that these dividends are forgone.
+4. Strike price of the option. A key characteristic used to describe an option is the strike price. In the case of calls, where the holder acquires the right to buy at a fixed price, the value of the call will decline as the strike price increases. In the case of puts, where the holder has the right to sell at a fixed price, the value will increase as the strike price increases.
+5. Time to expiration on the option. Both calls and puts are more valuable the greater the time to expiration. This is because the longer time to expiration provides more time for the value of the underlying asset to move, increasing the value of both types of options. Additionally, in the case of a call, where the buyer has the right to pay a fixed price at expiration, the present value of this fixed price decreases as the life of the option increases, increasing the value of the call.
+6. Riskless interest rate corresponding to life of the option. Since the buyer of an option pays the price of the option up front, an opportunity cost is involved. This cost will depend on the level of interest rates and the time to expiration of the option. The riskless interest rate also enters into the valuation of options when the present value of the exercise price is calculated, since the exercise price does not have to be paid (received) until expiration on calls (puts). Increases in the interest rate will increase the value of calls and reduce the value of puts. Table 5.1 summarizes the variables and their predicted effects on call and put prices.
+Table 5.1 Summary of Variables Affecting Call and Put Prices Effect On Factor
+Call Value
+Put Value Increase in underlying asset’s value
+Increases
+Decreases Increase in variance of underlying asset
+Increases
+Increases Increase in strike price
+Decreases
+Increases Increase in dividends paid
+Decreases
+Increases Increase in time to expiration
+Increases
+Increases Increase in interest rates
+Increases
+Decreases American versus European Options: Variables Relating to Early Exercise
+A primary distinction between American and European options is that an American option can be exercised at any time prior to its expiration, while European options can be exercised only at expiration. The possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. There is one compensating factor that enables the former to be valued using models designed for the latter. In most cases, the time premium associated with the remaining life of an option and transaction costs make early exercise suboptimal. In other words, the holders of in-the-money options generally get much more by selling the options to someone else than by exercising the options.
+OPTION PRICING MODELS
+Option pricing theory has made vast strides since 1972, when Fischer Black and Myron Scholes published their pathbreaking paper that provided a model for valuing dividend-protected European options. Black and Scholes used a “replicating portfolio”—a portfolio composed of the underlying asset and the risk-free asset that had the same cash flows as the option being valued—and the notion of arbitrage to come up with their final formulation. Although their derivation is mathematically complicated, there is a simpler binomial model for valuing options that draws on the same logic.
+Binomial Model
+The binomial option pricing model is based on a simple formulation for the asset price process in which the asset, in any time period, can move to one of two possible prices. The general formulation of a stock price process that follows the binomial path is shown in Figure 5.3. In this figure, S is the current stock price; the price moves up to Su with probability p and down to Sd with probability 1 – p in any time period. Figure 5.3 General Formulation for Binomial Price Path Creating a Replicating Portfolio
+The objective in creating a replicating portfolio is to use a combination of risk-free borrowing/lending and the underlying asset to create the same cash flows as the option being valued. The principles of arbitrage apply then, and the value of the option must be equal to the value of the replicating portfolio. In the case of the general formulation shown in Figure 5.3, where stock prices can move either up to Su or down to Sd in any time period, the replicating portfolio for a call with strike price K will involve borrowing $B and acquiring Δ of the underlying asset, where: where Cu = Value of the call if the stock price is Su
+Cd = Value of the call if the stock price is Sd
+In a multiperiod binomial process, the valuation has to proceed iteratively (i.e., starting with the final time period and moving backward in time until the current point in time). The portfolios replicating the option are created at each step and valued, providing the values for the option in that time period. The final output from the binomial option pricing model is a statement of the value of the option in terms of the replicating portfolio, composed of Δ shares (option delta) of the underlying asset and risk-free borrowing/lending. ILLUSTRATION 5.1: Binomial Option Valuation
+Assume that the objective is to value a call with a strike price of $50, which is expected to expire in two time periods, on an underlying asset whose price currently is $50 and is expected to follow a binomial process: Now assume that the interest rate is 11%. In addition, define: The objective is to combined Δ shares of stock and B dollars of borrowing to replicate the cash flows from the call with a strike price of $50. This can be done iteratively, starting with the last period and working back through the binomial tree.
+Step 1: Start with the end nodes and work backward: Thus, if the stock price is $70 at t = 1, borrowing $45 and buying one share of the stock will give the same cash flows as buying the call. The value of the call at t = 1, if the stock price is $70, is therefore: Considering the other leg of the binomial tree at t = 1, If the stock price is $35 at t = 1, then the call is worth nothing.
+Step 2: Move backward to the earlier time period and create a replicating portfolio that will provide the cash flows the option will provide. In other words, borrowing $22.50 and buying five-sevenths of a share will provide the same cash flows as a call with a strike price of $50 over the call’s lifetime. The value of the call therefore has to be the same as the cost of creating this position. The Determinants of Value
+The binomial model provides insight into the determinants of option value. The value of an option is not determined by the expected price of the asset but by its current price, which, of course, reflects expectations about the future. This is a direct consequence of arbitrage. If the option value deviates from the value of the replicating portfolio, investors can create an arbitrage position (i.e., one that requires no investment, involves no risk, and delivers positive returns). To illustrate, if the portfolio that replicates the call costs more than the call does in the market, an investor could buy the call, sell the replicating portfolio, and be guaranteed the difference as a profit. The cash flows on the two positions will offset each other, leading to no cash flows in subsequent periods. The call option value also increases as the time to expiration is extended, as the price movements (u and d) increase, and with increases in the interest rate.
+While the binomial model provides an intuitive feel for the determinants of option value, it requires a large number of inputs, in terms of expected future prices at each node. As time periods are made shorter in the binomial model, you can make one of two assumptions about asset prices.You can assume that price changes become smaller as periods get shorter; this leads to price changes becoming infinitesimally small as time periods approach zero, leading to a continuous price process. Alternatively, you can assume that price changes stay large even as the period gets shorter; this leads to a jump price process, where prices can jump in any period. This section considers the option pricing models that emerge with each of these assumptions.
+Black-Scholes Model
+When the price process is continuous (i.e., price changes become smaller as time periods get shorter), the binomial model for pricing options converges on the Black-Scholes model. The model, named after its cocreators, Fischer Black and Myron Scholes, allows us to estimate the value of any option using a small number of inputs, and has been shown to be robust in valuing many listed options.
+The Model
+While the derivation of the Black-Scholes model is far too complicated to present here, it is based on the idea of creating a portfolio of the underlying asset and the riskless asset with the same cash flows, and hence the same cost, as the option being valued. The value of a call option in the Black-Scholes model can be written as a function of the five variables: S = Current value of the underlying asset
+K = Strike price of the option
+t = Life to expiration of the option
+r = Riskless interest rate corresponding to the life of the option
+σ2 = Variance in the ln(value) of the underlying asset The value of a call is then: Note that e–rt is the present value factor, and reflects the fact that the exercise price on the call option does not have to be paid until expiration, since the model values European options. N(d1) and N(d2) are probabilities, estimated by using a cumulative standardized normal distribution, and the values of d1 and d2 obtained for an option. The cumulative distribution is shown in Figure 5.4. Figure 5.4 Cumulative Normal Distribution In approximate terms, N(d2) yields the likelihood that an option will generate positive cash flows for its owner at exercise (i.e., that S > K in the case of a call option and that K > S in the case of a put option). The portfolio that replicates the call option is created by buying N(d1) units of the underlying asset, and borrowing Ke–rt N(d2). The portfolio will have the same cash flows as the call option, and thus the same value as the option. N(d1), which is the number of units of the underlying asset that are needed to create the replicating portfolio, is called the option delta. A NOTE ON ESTIMATING THE INPUTS TO THE BLACK-SCHOLES MODEL
+The Black-Scholes model requires inputs that are consistent on time measurement. There are two places where this affects estimates. The first relates to the fact that the model works in continuous time, rather than discrete time. That is why we use the continuous time version of present value (exp–rt) rather than the discrete version, (1 + r)–t. It also means that the inputs such as the riskless rate have to be modified to make them continuous time inputs. For instance, if the one-year Treasury bond rate is 6.2 percent, the risk-free rate that is used in the Black-Scholes model should be: The second relates to the period over which the inputs are estimated. For instance, the preceding rate is an annual rate. The variance that is entered into the model also has to be an annualized variance. The variance, estimated from ln(asset prices), can be annualized easily because variances are linear in time if the serial correlation is zero. Thus, if monthly or weekly prices are used to estimate variance, the variance is annualized by multiplying by 12 or 52, respectively. ILLUSTRATION 5.2: Valuing an Option Using the Black-Scholes Model
+On March 6, 2001, Cisco Systems was trading at $13.62. We will attempt to value a July 2001 call option with a strike price of $15, trading on the CBOE on the same day for $2. The following are the other parameters of the options: The annualized standard deviation in Cisco Systems stock price over the previous year was 81%. This standard deviation is estimated using weekly stock prices over the year, and the resulting number was annualized as follows: The option expiration date is Friday, July 20, 2001. There are 103 days to expiration, and the annualized Treasury bill rate corresponding to this option life is 4.63%. The inputs for the Black-Scholes model are as follows: Current stock price (S) = $13.62
+Strike price on the option = $15
+Option life = 103/365 = 0.2822
+Standard deviation in ln(stock prices) = 81%
+Riskless rate = 4.63% Inputting these numbers into the model, we get: Using the normal distribution, we can estimate the N(d1) and N(d2): The value of the call can now be estimated: Since the call is trading at $2, it is slightly overvalued, assuming that the estimate of standard deviation used is correct. IMPLIED VOLATILITY
+The only input in the Black Scholes on which there can be significant disagreement among investors is the variance. While the variance is often estimated by looking at historical data, the values for options that emerge from using the historical variance can be different from the market prices. For any option, there is some variance at which the estimated value will be equal to the market price. This variance is called an implied variance.
+Consider the Cisco option valued in Illustration 5.2. With a standard deviation of 81 percent, the value of the call option with a strike price of $15 was estimated to be $1.87. Since the market price is higher than the calculated value, we tried higher standard deviations, and at a standard deviation 85.40 percent the value of the option is $2 (which is the market price). This is the implied standard deviation or implied volatility. Model Limitations and Fixes
+The Black-Scholes model was designed to value European options that can be exercised only at maturity and whose underlying assets do not pay dividends. In addition, options are valued based on the assumption that option exercise does not affect the value of the underlying asset. In practice, assets do pay dividends, options sometimes get exercised early, and exercising an option can affect the value of the underlying asset. Adjustments exist that, while not perfect, provide partial corrections to the Black-Scholes model.
+Dividends
+The payment of a dividend reduces the stock price; note that on the ex-dividend day, the stock price generally declines. Consequently, call options become less valuable and put options more valuable as expected dividend payments increase. There are two ways of dealing with dividends in the Black-Scholes model: 1. Short-term options. One approach to dealing with dividends is to estimate the present value of expected dividends that will be paid by the underlying asset during the option life and subtract it from the current value of the asset to use as S in the model. 2. Long-term options. Since it becomes less practical to estimate the present value of dividends the longer the option life, an alternate approach can be used. If the dividend yield (y = Dividends/Current value of the asset) on the underlying asset is expected to remain unchanged during the life of the option, the Black-Scholes model can be modified to take dividends into account. From an intuitive standpoint, the adjustments have two effects. First, the value of the asset is discounted back to the present at the dividend yield to take into account the expected drop in asset value resulting from dividend payments. Second, the interest rate is offset by the dividend yield to reflect the lower carrying cost from holding the asset (in the replicating portfolio). The net effect will be a reduction in the value of calls estimated using this model. ILLUSTRATION 5.3: Valuing a Short-Term Option with Dividend Adjustments—The Black-Scholes Correction
+Assume that it is March 6, 2001, and that AT&T is trading at $20.50 a share. Consider a call option on the stock with a strike price of $20, expiring on July 20, 2001. Using past stock prices, the annualized standard deviation in the log of stock prices for AT&T is estimated at 60%. There is one dividend, amounting to $0.15, and it will be paid in 23 days. The riskless rate is 4.63%. Present value of expected dividend = $0.15/1.046323/365 = $0.15
+Dividend-adjusted stock price = $20.50 – $0.15 = $20.35
+Time to expiration = 103/365 = 0.2822
+Variance in ln(stock prices) = 0.62 = 0.36
+Riskless rate = 4.63% The value from the Black-Scholes model is: The call option was trading at $2.60 on that day. ILLUSTRATION 5.4: Valuing a Long-Term Option with Dividend Adjustments—Primes and Scores
+The CBOE offers longer-term call and put options on some stocks. On March 6, 2001, for instance, you could have purchased an AT&T call expiring on January 17, 2003. The stock price for AT&T is $20.50 (as in the previous example). The following is the valuation of a call option with a strike price of $20. Instead of estimating the present value of dividends over the next two years, assume that AT&T’s dividend yield will remain 2.51% over this period and that the risk-free rate for a two-year Treasury bond is 4.85%. The inputs to the Black-Scholes model are: The value from the Black-Scholes model is: The call was trading at $5.80 on March 8, 2001. optst.xls: This spreadsheet allows you to estimate the value of a short-term option when the expected dividends during the option life can be estimated. optlt.xls: This spreadsheet allows you to estimate the value of an option when the underlying asset has a constant dividend yield. Early Exercise
+The Black-Scholes model was designed to value European options that can be exercised only at expiration. In contrast, most options that we encounter in practice are American options and can be exercised at any time until expiration. As mentioned earlier, the possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. In general, though, with traded options, it is almost always better to sell the option to someone else rather than exercise early, since options have a time premium (i.e., they sell for more than their exercise value). There are two exceptions. One occurs when the underlying asset pays large dividends, thus reducing the expected value of the asset. In this case, call options may be exercised just before an ex-dividend date, if the time premium on the options is less than the expected decline in asset value as a consequence of the dividend payment. The other exception arises when an investor holds both the underlying asset and deep in-the-money puts (i.e., puts with strike prices well above the current price of the underlying asset) on that asset at a time when interest rates are high. In this case, the time premium on the put may be less than the potential gain from exercising the put early and earning interest on the exercise price.
+There are two basic ways of dealing with the possibility of early exercise. One is to continue to use the unadjusted Black-Scholes model and to regard the resulting value as a floor or conservative estimate of the true value. The other is to try to adjust the value of the option for the possibility of early exercise. There are two approaches for doing so. One uses the Black-Scholes model to value the option to each potential exercise date. With options on stocks, this basically requires that the investor values options to each ex-dividend day and chooses the maximum of the estimated call values. The second approach is to use a modified version of the binomial model to consider the possibility of early exercise. In this version, the up and the down movements for asset prices in each period can be estimated from the variance. 2
+Approach 1: Pseudo-American Valuation
+Step 1: Define when dividends will be paid and how much the dividends will be.
+Step 2: Value the call option to each ex-dividend date using the dividend-adjusted approach described earlier, where the stock price is reduced by the present value of expected dividends.
+Step 3: Choose the maximum of the call values estimated for each ex-dividend day. ILLUSTRATION 5.5: Using Pseudo-American Option Valuation to Adjust for Early Exercise
+Consider an option with a strike price of $35 on a stock trading at $40. The variance in the ln(stock prices) is 0.05, and the riskless rate is 4%. The option has a remaining life of eight months, and there are three dividends expected during this period: Expected Dividend
+Ex-Dividend Day $0.80
+In 1 month $0.80
+In 4 months $0.80
+In 7 months The call option is first valued to just before the first ex-dividend date: The value from the Black-Scholes model is: The call option is then valued to before the second ex-dividend date: The value of the call based on these parameters is: The call option is then valued to before the third ex-dividend date: The value of the call based on these parameters is: The call option is then valued to expiration: The value of the call based on these parameters is: Approach 2: Using the Binomial Model
+The binomial model is much more capable of handling early exercise because it considers the cash flows at each time period, rather than just at expiration. The biggest limitation of the binomial model is determining what stock prices will be at the end of each period, but this can be overcome by using a variant that allows us to estimate the up and the down movements in stock prices from the estimated variance. There are four steps involved:
+Step 1: If the variance in ln(stock prices) has been estimated for the Black-Scholes valuation, convert these into inputs for the binomial model: where u and d are the up and the down movements per unit time for the binomial, and dt is the number of periods within each year (or unit time).
+Step 2: Specify the period in which the dividends will be paid and make the assumption that the price will drop by the amount of the dividend in that period.
+Step 3: Value the call at each node of the tree, allowing for the possibility of early exercise just before ex-dividend dates. There will be early exercise if the remaining time premium on the option is less than the expected drop in option value as a consequence of the dividend payment.
+Step 4: Value the call at time 0, using the standard binomial approach. bstobin.xls: This spreadsheet allows you to estimate the parameters for a binomial model from the inputs to a Black-Scholes model. Impact of Exercise on Underlying Asset Value
+The Black-Scholes model is based on the assumption that exercising an option does not affect the value of the underlying asset. This may be true for listed options on stocks, but it is not true for some types of options. For instance, the exercise of warrants increases the number of shares outstanding and brings fresh cash into the firm, both of which will affect the stock price.3 The expected negative impact (dilution) of exercise will decrease the value of warrants, compared to otherwise similar call options. The adjustment for dilution to the stock price is fairly simple in the Black-Scholes valuation. The stock price is adjusted for the expected dilution from the exercise of the options. In the case of warrants, for instance: where S = Current value of the stock
+nw = Number of warrants outstanding
+W = Value of warrants outstanding
+ns = Number of shares outstanding
+When the warrants are exercised, the number of shares outstanding will increase, reducing the stock price. The numerator reflects the market value of equity, including both stocks and warrants outstanding. The reduction in S will reduce the value of the call option.
+There is an element of circularity in this analysis, since the value of the warrant is needed to estimate the dilution-adjusted S and the dilution-adjusted S is needed to estimate the value of the warrant. This problem can be resolved by starting the process off with an assumed value for the warrant (e.g., the exercise value or the current market price of the warrant). This will yield a value for the warrant, and this estimated value can then be used as an input to reestimate the warrant’s value until there is convergence. FROM BLACK-SCHOLES TO BINOMIAL
+The process of converting the continuous variance in a Black-Scholes model to a binomial tree is a fairly simple one. Assume, for instance, that you have an asset that is trading at $30 currently and that you estimate the annualized standard deviation in the asset value to be 40 percent; the annualized riskless rate is 5 percent. For simplicity, let us assume that the option that you are valuing has a four-year life and that each period is a year. To estimate the prices at the end of each of the four years, we begin by first estimating the up and down movements in the binomial: Based on these estimates, we can obtain the prices at the end of the first node of the tree (the end of the first year): Progressing through the rest of the tree, we obtain the following numbers: ILLUSTRATION 5.6: Valuing a Warrant on Avatek Corporation
+Avatek Corporation is a real estate firm with 19.637 million shares outstanding, trading at $0.38 a share. In March 2001 the company had 1.8 million options outstanding, with four years to expiration and with an exercise price of $2.25. The stock paid no dividends, and the standard deviation in ln(stock prices) was 93%. The four-year Treasury bond rate was 4.9%. (The options were trading at $0.12 apiece at the time of this analysis.)
+The inputs to the warrant valuation model are as follows: The results of the Black-Scholes valuation of this option are: The options were trading at $0.12 in March 2001. Since the value was equal to the price, there was no need for further iterations. If there had been a difference, we would have reestimated the adjusted stock price and option value. If the options had been non-traded (as is the case with management options), this calculation would have required an iterative process, where the option value is used to get the adjusted value per share and the value per share to get the option value. warrant.xls: This spreadsheet allows you to estimate the value of an option when there is a potential dilution from exercise. The Black-Scholes Model for Valuing Puts
+The value of a put can be derived from the value of a call with the same strike price and the same expiration date: where C is the value of the call and P is the value of the put. This relationship between the call and put values is called put-call parity, and any deviations from parity can be used by investors to make riskless profits. To see why put-call parity holds, consider selling a call and buying a put with exercise price K and expiration date t, and simultaneously buying the underlying asset at the current price S. The payoff from this position is riskless and always yields K at expiration (t). To see this, assume that the stock price at expiration is S*. The payoff on each of the positions in the portfolio can be written as follows: Position
+Payoffs at t if S* > K
+Payoffs at t if S* < K Sell call
+–(S* – K)
+0 Buy put
+0
+K – S* Buy stock
+S*
+S* Total
+K
+K Since this position yields K with certainty, the cost of creating this position must be equal to the present value of K at the riskless rate (K e–rt). Substituting the Black-Scholes equation for the value of an equivalent call into this equation, we get: Thus, the replicating portfolio for a put is created by selling short [1 – N(d1)] shares of stock and investing K e–rt[1 – N(d2)] in the riskless asset. ILLUSTRATION 5.7: Valuing a Put Using Put-Call Parity: Cisco Systems and AT&T
+Consider the call on Cisco Systems that we valued in Illustration 5.2. The call had a strike price of $15 on the stock, had 103 days left to expiration, and was valued at $1.87. The stock was trading at $13.62, and the riskless rate was 4.63%. The put can be valued as follows: The put was trading at $3.38.
+Also, a long-term call on AT&T was valued in Illustration 5.4. The call had a strike price of $20, 1.8333 years left to expiration, and a value of $6.63. The stock was trading at $20.50 and was expected to maintain a dividend yield of 2.51% over the period. The riskless rate was 4.85%. The put value can be estimated as follows: The put was trading at $3.80. Both the call and the put were trading at different prices from our estimates, which may indicate that we have not correctly estimated the stock’s volatility. Jump Process Option Pricing Models
+If price changes remain larger as the time periods in the binomial model are shortened, it can no longer be assumed that prices change continuously. When price changes remain large, a price process that allows for price jumps is much more realistic. Cox and Ross (1976) valued options when prices follow a pure jump process, where the jumps can only be positive. Thus, in the next interval, the stock price will either have a large positive jump with a specified probability or drift downward at a given rate.
+Merton (1976) considered a distribution where there are price jumps superimposed on a continuous price process. He specified the rate at which jumps occur (λ) and the average jump size (k), measured as a percentage of the stock price. The model derived to value options with this process is called a jump diffusion model. In this model, the value of an option is determined by the five variables specified in the Black-Scholes model, and the parameters of the jump process (λ, k). Unfortunately, the estimates of the jump process parameters are so difficult to make for most firms that they overwhelm any advantages that accrue from using a more realistic model. These models, therefore, have seen limited use in practice.
+EXTENSIONS OF OPTION PRICING
+All the option pricing models described so far—the binomial, the Black-Scholes, and the jump process models—are designed to value options with clearly defined exercise prices and maturities on underlying assets that are traded. However, the options we encounter in investment analysis or valuation are often on real assets rather than financial assets. Categorized as real options, they can take much more complicated forms. This section considers some of these variations.
+Capped and Barrier Options
+With a simple call option, there is no specified upper limit on the profits that can be made by the buyer of the call. Asset prices, at least in theory, can keep going up, and the payoffs increase proportionately. In some call options, though, the buyer is entitled to profits up to a specified price but not above it. For instance, consider a call option with a strike price of K1 on an asset. In an unrestricted call option, the payoff on this option will increase as the underlying asset’s price increases above K1. Assume, however, that if the price reaches K2, the payoff is capped at (K2 – K1). The payoff diagram on this option is shown in Figure 5.5. Figure 5.5 Payoff on Capped Call This option is called a capped call. Notice, also, that once the price reaches K2, there is no longer any time premium associated with the option, and the option will therefore be exercised. Capped calls are part of a family of options called barrier options, where the payoff on and the life of the option are functions of whether the underlying asset price reaches a certain level during a specified period.
+The value of a capped call is always lower than the value of the same call without the payoff limit. A simple approximation of this value can be obtained by valuing the call twice, once with the given exercise price and once with the cap, and taking the difference in the two values. In the preceding example, then, the value of the call with an exercise price of K1 and a cap at K2 can be written as: Barrier options can take many forms. In a knockout option, an option ceases to exist if the underlying asset reaches a certain price. In the case of a call option, this knockout price is usually set below the strike price, and this option is called a down-and-out option. In the case of a put option, the knockout price will be set above the exercise price, and this option is called an up-and-out option. Like the capped call, these options are worth less than their unrestricted counterparts. Many real options have limits on potential upside, or knockout provisions, and ignoring these limits can result in the overstatement of the value of these options.
+Compound Options
+Some options derive their value not from an underlying asset, but from other options. These options are called compound options. Compound options can take any of four forms—a call on a call, a put on a put, a call on a put, or a put on a call. Geske (1979) developed the analytical formulation for valuing compound options by replacing the standard normal distribution used in a simple option model with a bivariate normal distribution in the calculation.
+Consider, for instance, the option to expand a project that is discussed in Chapter 30. While we will value this option using a simple option pricing model, in reality there could be multiple stages in expansion, with each stage representing an option for the following stage. In this case, we will undervalue the option by considering it as a simple rather than a compound option.
+Notwithstanding this discussion, the valuation of compound options becomes progressively more difficult as more options are added to the chain. In this case, rather than wreck the valuation on the shoals of estimation error, it may be better to accept the conservative estimate that is provided with a simple valuation model as a floor on the value.
+Rainbow Options
+In a simple option, the uncertainty is about the price of the underlying asset. Some options are exposed to two or more sources of uncertainty, and these options are rainbow options. Using the simple option pricing model to value such options can lead to biased estimates of value. As an example, consider an undeveloped oil reserve as an option, where the firm that owns the reserve has the right to develop the reserve. Here there are two sources of uncertainty. The first is obviously the price of oil, and the second is the quantity of oil that is in the reserve. To value this undeveloped reserve, we can make the simplifying assumption that we know the quantity of oil in the reserve with certainty. In reality, however, uncertainty about the quantity will affect the value of this option and make the decision to exercise more difficult.4
+CONCLUSION
+An option is an asset with payoffs that are contingent on the value of an underlying asset. A call option provides its holder with the right to buy the underlying asset at a fixed price, whereas a put option provides its holder with the right to sell at a fixed price, at any time before the expiration of the option. The value of an option is determined by six variables—the current value of the underlying asset, the variance in this value, the expected dividends on the asset, the strike price and life of the option, and the riskless interest rate. This is illustrated in both the binomial and the Black-Scholes models, which value options by creating replicating portfolios composed of the underlying asset and riskless lending or borrowing. These models can be used to value assets that have option like characteristics.
+QUESTIONS AND SHORT PROBLEMS
+In the problems following, use an equity risk premium of 5.5 percent if none is specified. 1. The following are prices of options traded on Microsoft Corporation, which pays no dividends. The stock is trading at $83, and the annualized riskless rate is 3.8%. The standard deviation in ln(stock prices) (based on historical data) is 30%. a. Estimate the value of a three-month call with a strike price of $85.
+b. Using the inputs from the Black-Scholes model, specify how you would replicate this call.
+c. What is the implied standard deviation in this call?
+d. Assume now that you buy a call with a strike price of $85 and sell a call with a strike price of $90. Draw the payoff diagram on this position.
+e. Using put-call parity, estimate the value of a three-month put with a strike price of $85. 2. You are trying to value three-month call and put options on Merck with a strike price of $30. The stock is trading at $28.75, and the company expects to pay a quarterly dividend per share of $0.28 in two months. The annualized riskless interest rate is 3.6%, and the standard deviation in log stock prices is 20%. a. Estimate the value of the call and put options, using the Black-Scholes model.
+b. What effect does the expected dividend payment have on call values? On put values? Why? 3. There is the possibility that the options on Merck described in the preceding problem could be exercised early. a. Use the pseudo-American call option technique to determine whether this will affect the value of the call.
+b. Why does the possibility of early exercise exist? What types of options are most likely to be exercised early? 4. You have been provided the following information on a three-month call: a. If you wanted to replicate buying this call, how much money would you need to borrow?
+b. If you wanted to replicate buying this call, how many shares of stock would you need to buy? 5. Go Video, a manufacturer of video recorders, was trading at $4 per share in May 1994. There were 11 million shares outstanding. At the same time, it had 550,000 one-year warrants outstanding, with a strike price of $4.25. The stock has had a standard deviation of 60%. The stock does not pay a dividend. The riskless rate is 5%. a. Estimate the value of the warrants, ignoring dilution.
+b. Estimate the value of the warrants, allowing for dilution.
+c. Why does dilution reduce the value of the warrants? 6. You are trying to value a long-term call option on the NYSE Composite index, expiring in five years, with a strike price of 275. The index is currently at 250, and the annualized standard deviation in stock prices is 15%. The average dividend yield on the index is 3% and is expected to remain unchanged over the next five years. The five-year Treasury bond rate is 5%. a. Estimate the value of the long-term call option.
+b. Estimate the value of a put option with the same parameters.
+c. What are the implicit assumptions you are making when you use the Black-Scholes model to value this option? Which of these assumptions are likely to be violated? What are the consequences for your valuation? 7. A new security on AT&T will entitle the investor to all dividends on AT&T over the next three years, limiting upside potential to 20% but also providing downside protection below 10%. AT&T stock is trading at $50, and three-year call and put options are traded on the exchange at the following prices: How much would you be willing to pay for this security? 1 Note, though, that higher variance can reduce the value of the underlying asset. As a call option becomes more in-the-money, the more it resembles the underlying asset. For very deep in-the-money call options, higher variance can reduce the value of the option.
+2 To illustrate, if σ2 is the variance in ln(stock prices), the up and the down movements in the binomial can be estimated as follows: where u and d are the up and down movements per unit time for the binomial, T is the life of the option, and m is the number of periods within that lifetime.
+3 Warrants are call options issued by firms, either as part of management compensation contracts or to raise equity.
+4 The analogy to a listed option on a stock is the case where you do not know with certainty what the stock price is when you exercise the option. The more uncertain you are about the stock price, the more margin for error you have to give yourself when you exercise the option, to ensure that you are in fact earning a profit.
+CHAPTER 5
+Option Pricing Theory and Models
+In general, the value of any asset is the present value of the expected cash flows on that asset. This chapter considers an exception to that rule when it looks at assets with two specific characteristics: 1. The assets derive their value from the values of other assets.
+2. The cash flows on the assets are contingent on the occurrence of specific events. These assets are called options, and the present value of the expected cash flows on these assets will understate their true value. This chapter describes the cash flow characteristics of options, considers the factors that determine their value, and examines how best to value them.
+BASICS OF OPTION PRICING
+An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called a strike price or an exercise price) at or before the expiration date of the option. Since it is a right and not an obligation, the holder can choose not to exercise the right and can allow the option to expire. There are two types of options—call options and put options.
+Call and Put Options: Description and Payoff Diagrams
+A call option gives the buyer of the option the right to buy the underlying asset at the strike price or the exercise price at any time prior to the expiration date of the option. The buyer pays a price for this right. If at expiration the value of the asset is less than the strike price, the option is not exercised and expires worthless. If, however, the value of the asset is greater than the strike price, the option is exercised—the buyer of the option buys the stock at the exercise price, and the difference between the asset value and the exercise price comprises the gross profit on the investment. The net profit on the investment is the difference between the gross profit and the price paid for the call initially.
+A payoff diagram illustrates the cash payoff on an option at expiration. For a call, the net payoff is negative (and equal to the price paid for the call) if the value of the underlying asset is less than the strike price. If the price of the underlying asset exceeds the strike price, the gross payoff is the difference between the value of the underlying asset and the strike price, and the net payoff is the difference between the gross payoff and the price of the call. This is illustrated in Figure 5.1. Figure 5.1 Payoff on Call Option A put option gives the buyer of the option the right to sell the underlying asset at a fixed price, again called the strike or exercise price, at any time prior to the expiration date of the option. The buyer pays a price for this right. If the price of the underlying asset is greater than the strike price, the option will not be exercised and will expire worthless. But if the price of the underlying asset is less than the strike price, the owner of the put option will exercise the option and sell the stock at the strike price, claiming the difference between the strike price and the market value of the asset as the gross profit. Again, netting out the initial cost paid for the put yields the net profit from the transaction.
+A put has a negative net payoff if the value of the underlying asset exceeds the strike price, and has a gross payoff equal to the difference between the strike price and the value of the underlying asset if the asset value is less than the strike price. This is summarized in Figure 5.2. Figure 5.2 Payoff on Put Option DETERMINANTS OF OPTION VALUE
+The value of an option is determined by six variables relating to the underlying asset and financial markets. 1. Current value of the underlying asset. Options are assets that derive value from an underlying asset. Consequently, changes in the value of the underlying asset affect the value of the options on that asset. Since calls provide the right to buy the underlying asset at a fixed price, an increase in the value of the asset will increase the value of the calls. Puts, on the other hand, become less valuable as the value of the asset increases.
+2. Variance in value of the underlying asset. The buyer of an option acquires the right to buy or sell the underlying asset at a fixed price. The higher the variance in the value of the underlying asset, the greater the value of the option.1 This is true for both calls and puts. While it may seem counterintuitive that an increase in a risk measure (variance) should increase value, options are different from other securities since buyers of options can never lose more than the price they pay for them; in fact, they have the potential to earn significant returns from large price movements.
+3. Dividends paid on the underlying asset. The value of the underlying asset can be expected to decrease if dividend payments are made on the asset during the life of the option. Consequently, the value of a call on the asset is a decreasing function of the size of expected dividend payments, and the value of a put is an increasing function of expected dividend payments. A more intuitive way of thinking about dividend payments, for call options, is as a cost of delaying exercise on in-the-money options. To see why, consider an option on a traded stock. Once a call option is in-the-money (i.e., the holder of the option will make a gross payoff by exercising the option), exercising the call option will provide the holder with the stock and entitle him or her to the dividends on the stock in subsequent periods. Failing to exercise the option will mean that these dividends are forgone.
+4. Strike price of the option. A key characteristic used to describe an option is the strike price. In the case of calls, where the holder acquires the right to buy at a fixed price, the value of the call will decline as the strike price increases. In the case of puts, where the holder has the right to sell at a fixed price, the value will increase as the strike price increases.
+5. Time to expiration on the option. Both calls and puts are more valuable the greater the time to expiration. This is because the longer time to expiration provides more time for the value of the underlying asset to move, increasing the value of both types of options. Additionally, in the case of a call, where the buyer has the right to pay a fixed price at expiration, the present value of this fixed price decreases as the life of the option increases, increasing the value of the call.
+6. Riskless interest rate corresponding to life of the option. Since the buyer of an option pays the price of the option up front, an opportunity cost is involved. This cost will depend on the level of interest rates and the time to expiration of the option. The riskless interest rate also enters into the valuation of options when the present value of the exercise price is calculated, since the exercise price does not have to be paid (received) until expiration on calls (puts). Increases in the interest rate will increase the value of calls and reduce the value of puts. Table 5.1 summarizes the variables and their predicted effects on call and put prices.
+Table 5.1 Summary of Variables Affecting Call and Put Prices Effect On Factor
+Call Value
+Put Value Increase in underlying asset’s value
+Increases
+Decreases Increase in variance of underlying asset
+Increases
+Increases Increase in strike price
+Decreases
+Increases Increase in dividends paid
+Decreases
+Increases Increase in time to expiration
+Increases
+Increases Increase in interest rates
+Increases
+Decreases American versus European Options: Variables Relating to Early Exercise
+A primary distinction between American and European options is that an American option can be exercised at any time prior to its expiration, while European options can be exercised only at expiration. The possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. There is one compensating factor that enables the former to be valued using models designed for the latter. In most cases, the time premium associated with the remaining life of an option and transaction costs make early exercise suboptimal. In other words, the holders of in-the-money options generally get much more by selling the options to someone else than by exercising the options.
+OPTION PRICING MODELS
+Option pricing theory has made vast strides since 1972, when Fischer Black and Myron Scholes published their pathbreaking paper that provided a model for valuing dividend-protected European options. Black and Scholes used a “replicating portfolio”—a portfolio composed of the underlying asset and the risk-free asset that had the same cash flows as the option being valued—and the notion of arbitrage to come up with their final formulation. Although their derivation is mathematically complicated, there is a simpler binomial model for valuing options that draws on the same logic.
+Binomial Model
+The binomial option pricing model is based on a simple formulation for the asset price process in which the asset, in any time period, can move to one of two possible prices. The general formulation of a stock price process that follows the binomial path is shown in Figure 5.3. In this figure, S is the current stock price; the price moves up to Su with probability p and down to Sd with probability 1 – p in any time period. Figure 5.3 General Formulation for Binomial Price Path Creating a Replicating Portfolio
+The objective in creating a replicating portfolio is to use a combination of risk-free borrowing/lending and the underlying asset to create the same cash flows as the option being valued. The principles of arbitrage apply then, and the value of the option must be equal to the value of the replicating portfolio. In the case of the general formulation shown in Figure 5.3, where stock prices can move either up to Su or down to Sd in any time period, the replicating portfolio for a call with strike price K will involve borrowing $B and acquiring Δ of the underlying asset, where: where Cu = Value of the call if the stock price is Su
+Cd = Value of the call if the stock price is Sd
+In a multiperiod binomial process, the valuation has to proceed iteratively (i.e., starting with the final time period and moving backward in time until the current point in time). The portfolios replicating the option are created at each step and valued, providing the values for the option in that time period. The final output from the binomial option pricing model is a statement of the value of the option in terms of the replicating portfolio, composed of Δ shares (option delta) of the underlying asset and risk-free borrowing/lending. ILLUSTRATION 5.1: Binomial Option Valuation
+Assume that the objective is to value a call with a strike price of $50, which is expected to expire in two time periods, on an underlying asset whose price currently is $50 and is expected to follow a binomial process: Now assume that the interest rate is 11%. In addition, define: The objective is to combined Δ shares of stock and B dollars of borrowing to replicate the cash flows from the call with a strike price of $50. This can be done iteratively, starting with the last period and working back through the binomial tree.
+Step 1: Start with the end nodes and work backward: Thus, if the stock price is $70 at t = 1, borrowing $45 and buying one share of the stock will give the same cash flows as buying the call. The value of the call at t = 1, if the stock price is $70, is therefore: Considering the other leg of the binomial tree at t = 1, If the stock price is $35 at t = 1, then the call is worth nothing.
+Step 2: Move backward to the earlier time period and create a replicating portfolio that will provide the cash flows the option will provide. In other words, borrowing $22.50 and buying five-sevenths of a share will provide the same cash flows as a call with a strike price of $50 over the call’s lifetime. The value of the call therefore has to be the same as the cost of creating this position. The Determinants of Value
+The binomial model provides insight into the determinants of option value. The value of an option is not determined by the expected price of the asset but by its current price, which, of course, reflects expectations about the future. This is a direct consequence of arbitrage. If the option value deviates from the value of the replicating portfolio, investors can create an arbitrage position (i.e., one that requires no investment, involves no risk, and delivers positive returns). To illustrate, if the portfolio that replicates the call costs more than the call does in the market, an investor could buy the call, sell the replicating portfolio, and be guaranteed the difference as a profit. The cash flows on the two positions will offset each other, leading to no cash flows in subsequent periods. The call option value also increases as the time to expiration is extended, as the price movements (u and d) increase, and with increases in the interest rate.
+While the binomial model provides an intuitive feel for the determinants of option value, it requires a large number of inputs, in terms of expected future prices at each node. As time periods are made shorter in the binomial model, you can make one of two assumptions about asset prices.You can assume that price changes become smaller as periods get shorter; this leads to price changes becoming infinitesimally small as time periods approach zero, leading to a continuous price process. Alternatively, you can assume that price changes stay large even as the period gets shorter; this leads to a jump price process, where prices can jump in any period. This section considers the option pricing models that emerge with each of these assumptions.
+Black-Scholes Model
+When the price process is continuous (i.e., price changes become smaller as time periods get shorter), the binomial model for pricing options converges on the Black-Scholes model. The model, named after its cocreators, Fischer Black and Myron Scholes, allows us to estimate the value of any option using a small number of inputs, and has been shown to be robust in valuing many listed options.
+The Model
+While the derivation of the Black-Scholes model is far too complicated to present here, it is based on the idea of creating a portfolio of the underlying asset and the riskless asset with the same cash flows, and hence the same cost, as the option being valued. The value of a call option in the Black-Scholes model can be written as a function of the five variables: S = Current value of the underlying asset
+K = Strike price of the option
+t = Life to expiration of the option
+r = Riskless interest rate corresponding to the life of the option
+σ2 = Variance in the ln(value) of the underlying asset The value of a call is then: Note that e–rt is the present value factor, and reflects the fact that the exercise price on the call option does not have to be paid until expiration, since the model values European options. N(d1) and N(d2) are probabilities, estimated by using a cumulative standardized normal distribution, and the values of d1 and d2 obtained for an option. The cumulative distribution is shown in Figure 5.4. Figure 5.4 Cumulative Normal Distribution In approximate terms, N(d2) yields the likelihood that an option will generate positive cash flows for its owner at exercise (i.e., that S > K in the case of a call option and that K > S in the case of a put option). The portfolio that replicates the call option is created by buying N(d1) units of the underlying asset, and borrowing Ke–rt N(d2). The portfolio will have the same cash flows as the call option, and thus the same value as the option. N(d1), which is the number of units of the underlying asset that are needed to create the replicating portfolio, is called the option delta. A NOTE ON ESTIMATING THE INPUTS TO THE BLACK-SCHOLES MODEL
+The Black-Scholes model requires inputs that are consistent on time measurement. There are two places where this affects estimates. The first relates to the fact that the model works in continuous time, rather than discrete time. That is why we use the continuous time version of present value (exp–rt) rather than the discrete version, (1 + r)–t. It also means that the inputs such as the riskless rate have to be modified to make them continuous time inputs. For instance, if the one-year Treasury bond rate is 6.2 percent, the risk-free rate that is used in the Black-Scholes model should be: The second relates to the period over which the inputs are estimated. For instance, the preceding rate is an annual rate. The variance that is entered into the model also has to be an annualized variance. The variance, estimated from ln(asset prices), can be annualized easily because variances are linear in time if the serial correlation is zero. Thus, if monthly or weekly prices are used to estimate variance, the variance is annualized by multiplying by 12 or 52, respectively. ILLUSTRATION 5.2: Valuing an Option Using the Black-Scholes Model
+On March 6, 2001, Cisco Systems was trading at $13.62. We will attempt to value a July 2001 call option with a strike price of $15, trading on the CBOE on the same day for $2. The following are the other parameters of the options: The annualized standard deviation in Cisco Systems stock price over the previous year was 81%. This standard deviation is estimated using weekly stock prices over the year, and the resulting number was annualized as follows: The option expiration date is Friday, July 20, 2001. There are 103 days to expiration, and the annualized Treasury bill rate corresponding to this option life is 4.63%. The inputs for the Black-Scholes model are as follows: Current stock price (S) = $13.62
+Strike price on the option = $15
+Option life = 103/365 = 0.2822
+Standard deviation in ln(stock prices) = 81%
+Riskless rate = 4.63% Inputting these numbers into the model, we get: Using the normal distribution, we can estimate the N(d1) and N(d2): The value of the call can now be estimated: Since the call is trading at $2, it is slightly overvalued, assuming that the estimate of standard deviation used is correct. IMPLIED VOLATILITY
+The only input in the Black Scholes on which there can be significant disagreement among investors is the variance. While the variance is often estimated by looking at historical data, the values for options that emerge from using the historical variance can be different from the market prices. For any option, there is some variance at which the estimated value will be equal to the market price. This variance is called an implied variance.
+Consider the Cisco option valued in Illustration 5.2. With a standard deviation of 81 percent, the value of the call option with a strike price of $15 was estimated to be $1.87. Since the market price is higher than the calculated value, we tried higher standard deviations, and at a standard deviation 85.40 percent the value of the option is $2 (which is the market price). This is the implied standard deviation or implied volatility. Model Limitations and Fixes
+The Black-Scholes model was designed to value European options that can be exercised only at maturity and whose underlying assets do not pay dividends. In addition, options are valued based on the assumption that option exercise does not affect the value of the underlying asset. In practice, assets do pay dividends, options sometimes get exercised early, and exercising an option can affect the value of the underlying asset. Adjustments exist that, while not perfect, provide partial corrections to the Black-Scholes model.
+Dividends
+The payment of a dividend reduces the stock price; note that on the ex-dividend day, the stock price generally declines. Consequently, call options become less valuable and put options more valuable as expected dividend payments increase. There are two ways of dealing with dividends in the Black-Scholes model: 1. Short-term options. One approach to dealing with dividends is to estimate the present value of expected dividends that will be paid by the underlying asset during the option life and subtract it from the current value of the asset to use as S in the model. 2. Long-term options. Since it becomes less practical to estimate the present value of dividends the longer the option life, an alternate approach can be used. If the dividend yield (y = Dividends/Current value of the asset) on the underlying asset is expected to remain unchanged during the life of the option, the Black-Scholes model can be modified to take dividends into account. From an intuitive standpoint, the adjustments have two effects. First, the value of the asset is discounted back to the present at the dividend yield to take into account the expected drop in asset value resulting from dividend payments. Second, the interest rate is offset by the dividend yield to reflect the lower carrying cost from holding the asset (in the replicating portfolio). The net effect will be a reduction in the value of calls estimated using this model. ILLUSTRATION 5.3: Valuing a Short-Term Option with Dividend Adjustments—The Black-Scholes Correction
+Assume that it is March 6, 2001, and that AT&T is trading at $20.50 a share. Consider a call option on the stock with a strike price of $20, expiring on July 20, 2001. Using past stock prices, the annualized standard deviation in the log of stock prices for AT&T is estimated at 60%. There is one dividend, amounting to $0.15, and it will be paid in 23 days. The riskless rate is 4.63%. Present value of expected dividend = $0.15/1.046323/365 = $0.15
+Dividend-adjusted stock price = $20.50 – $0.15 = $20.35
+Time to expiration = 103/365 = 0.2822
+Variance in ln(stock prices) = 0.62 = 0.36
+Riskless rate = 4.63% The value from the Black-Scholes model is: The call option was trading at $2.60 on that day. ILLUSTRATION 5.4: Valuing a Long-Term Option with Dividend Adjustments—Primes and Scores
+The CBOE offers longer-term call and put options on some stocks. On March 6, 2001, for instance, you could have purchased an AT&T call expiring on January 17, 2003. The stock price for AT&T is $20.50 (as in the previous example). The following is the valuation of a call option with a strike price of $20. Instead of estimating the present value of dividends over the next two years, assume that AT&T’s dividend yield will remain 2.51% over this period and that the risk-free rate for a two-year Treasury bond is 4.85%. The inputs to the Black-Scholes model are: The value from the Black-Scholes model is: The call was trading at $5.80 on March 8, 2001. optst.xls: This spreadsheet allows you to estimate the value of a short-term option when the expected dividends during the option life can be estimated. optlt.xls: This spreadsheet allows you to estimate the value of an option when the underlying asset has a constant dividend yield. Early Exercise
+The Black-Scholes model was designed to value European options that can be exercised only at expiration. In contrast, most options that we encounter in practice are American options and can be exercised at any time until expiration. As mentioned earlier, the possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. In general, though, with traded options, it is almost always better to sell the option to someone else rather than exercise early, since options have a time premium (i.e., they sell for more than their exercise value). There are two exceptions. One occurs when the underlying asset pays large dividends, thus reducing the expected value of the asset. In this case, call options may be exercised just before an ex-dividend date, if the time premium on the options is less than the expected decline in asset value as a consequence of the dividend payment. The other exception arises when an investor holds both the underlying asset and deep in-the-money puts (i.e., puts with strike prices well above the current price of the underlying asset) on that asset at a time when interest rates are high. In this case, the time premium on the put may be less than the potential gain from exercising the put early and earning interest on the exercise price.
+There are two basic ways of dealing with the possibility of early exercise. One is to continue to use the unadjusted Black-Scholes model and to regard the resulting value as a floor or conservative estimate of the true value. The other is to try to adjust the value of the option for the possibility of early exercise. There are two approaches for doing so. One uses the Black-Scholes model to value the option to each potential exercise date. With options on stocks, this basically requires that the investor values options to each ex-dividend day and chooses the maximum of the estimated call values. The second approach is to use a modified version of the binomial model to consider the possibility of early exercise. In this version, the up and the down movements for asset prices in each period can be estimated from the variance. 2
+Approach 1: Pseudo-American Valuation
+Step 1: Define when dividends will be paid and how much the dividends will be.
+Step 2: Value the call option to each ex-dividend date using the dividend-adjusted approach described earlier, where the stock price is reduced by the present value of expected dividends.
+Step 3: Choose the maximum of the call values estimated for each ex-dividend day. ILLUSTRATION 5.5: Using Pseudo-American Option Valuation to Adjust for Early Exercise
+Consider an option with a strike price of $35 on a stock trading at $40. The variance in the ln(stock prices) is 0.05, and the riskless rate is 4%. The option has a remaining life of eight months, and there are three dividends expected during this period: Expected Dividend
+Ex-Dividend Day $0.80
+In 1 month $0.80
+In 4 months $0.80
+In 7 months The call option is first valued to just before the first ex-dividend date: The value from the Black-Scholes model is: The call option is then valued to before the second ex-dividend date: The value of the call based on these parameters is: The call option is then valued to before the third ex-dividend date: The value of the call based on these parameters is: The call option is then valued to expiration: The value of the call based on these parameters is: Approach 2: Using the Binomial Model
+The binomial model is much more capable of handling early exercise because it considers the cash flows at each time period, rather than just at expiration. The biggest limitation of the binomial model is determining what stock prices will be at the end of each period, but this can be overcome by using a variant that allows us to estimate the up and the down movements in stock prices from the estimated variance. There are four steps involved:
+Step 1: If the variance in ln(stock prices) has been estimated for the Black-Scholes valuation, convert these into inputs for the binomial model: where u and d are the up and the down movements per unit time for the binomial, and dt is the number of periods within each year (or unit time).
+Step 2: Specify the period in which the dividends will be paid and make the assumption that the price will drop by the amount of the dividend in that period.
+Step 3: Value the call at each node of the tree, allowing for the possibility of early exercise just before ex-dividend dates. There will be early exercise if the remaining time premium on the option is less than the expected drop in option value as a consequence of the dividend payment.
+Step 4: Value the call at time 0, using the standard binomial approach. bstobin.xls: This spreadsheet allows you to estimate the parameters for a binomial model from the inputs to a Black-Scholes model. Impact of Exercise on Underlying Asset Value
+The Black-Scholes model is based on the assumption that exercising an option does not affect the value of the underlying asset. This may be true for listed options on stocks, but it is not true for some types of options. For instance, the exercise of warrants increases the number of shares outstanding and brings fresh cash into the firm, both of which will affect the stock price.3 The expected negative impact (dilution) of exercise will decrease the value of warrants, compared to otherwise similar call options. The adjustment for dilution to the stock price is fairly simple in the Black-Scholes valuation. The stock price is adjusted for the expected dilution from the exercise of the options. In the case of warrants, for instance: where S = Current value of the stock
+nw = Number of warrants outstanding
+W = Value of warrants outstanding
+ns = Number of shares outstanding
+When the warrants are exercised, the number of shares outstanding will increase, reducing the stock price. The numerator reflects the market value of equity, including both stocks and warrants outstanding. The reduction in S will reduce the value of the call option.
+There is an element of circularity in this analysis, since the value of the warrant is needed to estimate the dilution-adjusted S and the dilution-adjusted S is needed to estimate the value of the warrant. This problem can be resolved by starting the process off with an assumed value for the warrant (e.g., the exercise value or the current market price of the warrant). This will yield a value for the warrant, and this estimated value can then be used as an input to reestimate the warrant’s value until there is convergence. FROM BLACK-SCHOLES TO BINOMIAL
+The process of converting the continuous variance in a Black-Scholes model to a binomial tree is a fairly simple one. Assume, for instance, that you have an asset that is trading at $30 currently and that you estimate the annualized standard deviation in the asset value to be 40 percent; the annualized riskless rate is 5 percent. For simplicity, let us assume that the option that you are valuing has a four-year life and that each period is a year. To estimate the prices at the end of each of the four years, we begin by first estimating the up and down movements in the binomial: Based on these estimates, we can obtain the prices at the end of the first node of the tree (the end of the first year): Progressing through the rest of the tree, we obtain the following numbers: ILLUSTRATION 5.6: Valuing a Warrant on Avatek Corporation
+Avatek Corporation is a real estate firm with 19.637 million shares outstanding, trading at $0.38 a share. In March 2001 the company had 1.8 million options outstanding, with four years to expiration and with an exercise price of $2.25. The stock paid no dividends, and the standard deviation in ln(stock prices) was 93%. The four-year Treasury bond rate was 4.9%. (The options were trading at $0.12 apiece at the time of this analysis.)
+The inputs to the warrant valuation model are as follows: The results of the Black-Scholes valuation of this option are: The options were trading at $0.12 in March 2001. Since the value was equal to the price, there was no need for further iterations. If there had been a difference, we would have reestimated the adjusted stock price and option value. If the options had been non-traded (as is the case with management options), this calculation would have required an iterative process, where the option value is used to get the adjusted value per share and the value per share to get the option value. warrant.xls: This spreadsheet allows you to estimate the value of an option when there is a potential dilution from exercise. The Black-Scholes Model for Valuing Puts
+The value of a put can be derived from the value of a call with the same strike price and the same expiration date: where C is the value of the call and P is the value of the put. This relationship between the call and put values is called put-call parity, and any deviations from parity can be used by investors to make riskless profits. To see why put-call parity holds, consider selling a call and buying a put with exercise price K and expiration date t, and simultaneously buying the underlying asset at the current price S. The payoff from this position is riskless and always yields K at expiration (t). To see this, assume that the stock price at expiration is S*. The payoff on each of the positions in the portfolio can be written as follows: Position
+Payoffs at t if S* > K
+Payoffs at t if S* < K Sell call
+–(S* – K)
+0 Buy put
+0
+K – S* Buy stock
+S*
+S* Total
+K
+K Since this position yields K with certainty, the cost of creating this position must be equal to the present value of K at the riskless rate (K e–rt). Substituting the Black-Scholes equation for the value of an equivalent call into this equation, we get: Thus, the replicating portfolio for a put is created by selling short [1 – N(d1)] shares of stock and investing K e–rt[1 – N(d2)] in the riskless asset. ILLUSTRATION 5.7: Valuing a Put Using Put-Call Parity: Cisco Systems and AT&T
+Consider the call on Cisco Systems that we valued in Illustration 5.2. The call had a strike price of $15 on the stock, had 103 days left to expiration, and was valued at $1.87. The stock was trading at $13.62, and the riskless rate was 4.63%. The put can be valued as follows: The put was trading at $3.38.
+Also, a long-term call on AT&T was valued in Illustration 5.4. The call had a strike price of $20, 1.8333 years left to expiration, and a value of $6.63. The stock was trading at $20.50 and was expected to maintain a dividend yield of 2.51% over the period. The riskless rate was 4.85%. The put value can be estimated as follows: The put was trading at $3.80. Both the call and the put were trading at different prices from our estimates, which may indicate that we have not correctly estimated the stock’s volatility. Jump Process Option Pricing Models
+If price changes remain larger as the time periods in the binomial model are shortened, it can no longer be assumed that prices change continuously. When price changes remain large, a price process that allows for price jumps is much more realistic. Cox and Ross (1976) valued options when prices follow a pure jump process, where the jumps can only be positive. Thus, in the next interval, the stock price will either have a large positive jump with a specified probability or drift downward at a given rate.
+Merton (1976) considered a distribution where there are price jumps superimposed on a continuous price process. He specified the rate at which jumps occur (λ) and the average jump size (k), measured as a percentage of the stock price. The model derived to value options with this process is called a jump diffusion model. In this model, the value of an option is determined by the five variables specified in the Black-Scholes model, and the parameters of the jump process (λ, k). Unfortunately, the estimates of the jump process parameters are so difficult to make for most firms that they overwhelm any advantages that accrue from using a more realistic model. These models, therefore, have seen limited use in practice.
+EXTENSIONS OF OPTION PRICING
+All the option pricing models described so far—the binomial, the Black-Scholes, and the jump process models—are designed to value options with clearly defined exercise prices and maturities on underlying assets that are traded. However, the options we encounter in investment analysis or valuation are often on real assets rather than financial assets. Categorized as real options, they can take much more complicated forms. This section considers some of these variations.
+Capped and Barrier Options
+With a simple call option, there is no specified upper limit on the profits that can be made by the buyer of the call. Asset prices, at least in theory, can keep going up, and the payoffs increase proportionately. In some call options, though, the buyer is entitled to profits up to a specified price but not above it. For instance, consider a call option with a strike price of K1 on an asset. In an unrestricted call option, the payoff on this option will increase as the underlying asset’s price increases above K1. Assume, however, that if the price reaches K2, the payoff is capped at (K2 – K1). The payoff diagram on this option is shown in Figure 5.5. Figure 5.5 Payoff on Capped Call This option is called a capped call. Notice, also, that once the price reaches K2, there is no longer any time premium associated with the option, and the option will therefore be exercised. Capped calls are part of a family of options called barrier options, where the payoff on and the life of the option are functions of whether the underlying asset price reaches a certain level during a specified period.
+The value of a capped call is always lower than the value of the same call without the payoff limit. A simple approximation of this value can be obtained by valuing the call twice, once with the given exercise price and once with the cap, and taking the difference in the two values. In the preceding example, then, the value of the call with an exercise price of K1 and a cap at K2 can be written as: Barrier options can take many forms. In a knockout option, an option ceases to exist if the underlying asset reaches a certain price. In the case of a call option, this knockout price is usually set below the strike price, and this option is called a down-and-out option. In the case of a put option, the knockout price will be set above the exercise price, and this option is called an up-and-out option. Like the capped call, these options are worth less than their unrestricted counterparts. Many real options have limits on potential upside, or knockout provisions, and ignoring these limits can result in the overstatement of the value of these options.
+Compound Options
+Some options derive their value not from an underlying asset, but from other options. These options are called compound options. Compound options can take any of four forms—a call on a call, a put on a put, a call on a put, or a put on a call. Geske (1979) developed the analytical formulation for valuing compound options by replacing the standard normal distribution used in a simple option model with a bivariate normal distribution in the calculation.
+Consider, for instance, the option to expand a project that is discussed in Chapter 30. While we will value this option using a simple option pricing model, in reality there could be multiple stages in expansion, with each stage representing an option for the following stage. In this case, we will undervalue the option by considering it as a simple rather than a compound option.
+Notwithstanding this discussion, the valuation of compound options becomes progressively more difficult as more options are added to the chain. In this case, rather than wreck the valuation on the shoals of estimation error, it may be better to accept the conservative estimate that is provided with a simple valuation model as a floor on the value.
+Rainbow Options
+In a simple option, the uncertainty is about the price of the underlying asset. Some options are exposed to two or more sources of uncertainty, and these options are rainbow options. Using the simple option pricing model to value such options can lead to biased estimates of value. As an example, consider an undeveloped oil reserve as an option, where the firm that owns the reserve has the right to develop the reserve. Here there are two sources of uncertainty. The first is obviously the price of oil, and the second is the quantity of oil that is in the reserve. To value this undeveloped reserve, we can make the simplifying assumption that we know the quantity of oil in the reserve with certainty. In reality, however, uncertainty about the quantity will affect the value of this option and make the decision to exercise more difficult.4
+CONCLUSION
+An option is an asset with payoffs that are contingent on the value of an underlying asset. A call option provides its holder with the right to buy the underlying asset at a fixed price, whereas a put option provides its holder with the right to sell at a fixed price, at any time before the expiration of the option. The value of an option is determined by six variables—the current value of the underlying asset, the variance in this value, the expected dividends on the asset, the strike price and life of the option, and the riskless interest rate. This is illustrated in both the binomial and the Black-Scholes models, which value options by creating replicating portfolios composed of the underlying asset and riskless lending or borrowing. These models can be used to value assets that have option like characteristics.
+QUESTIONS AND SHORT PROBLEMS
+In the problems following, use an equity risk premium of 5.5 percent if none is specified. 1. The following are prices of options traded on Microsoft Corporation, which pays no dividends. The stock is trading at $83, and the annualized riskless rate is 3.8%. The standard deviation in ln(stock prices) (based on historical data) is 30%. a. Estimate the value of a three-month call with a strike price of $85.
+b. Using the inputs from the Black-Scholes model, specify how you would replicate this call.
+c. What is the implied standard deviation in this call?
+d. Assume now that you buy a call with a strike price of $85 and sell a call with a strike price of $90. Draw the payoff diagram on this position.
+e. Using put-call parity, estimate the value of a three-month put with a strike price of $85. 2. You are trying to value three-month call and put options on Merck with a strike price of $30. The stock is trading at $28.75, and the company expects to pay a quarterly dividend per share of $0.28 in two months. The annualized riskless interest rate is 3.6%, and the standard deviation in log stock prices is 20%. a. Estimate the value of the call and put options, using the Black-Scholes model.
+b. What effect does the expected dividend payment have on call values? On put values? Why? 3. There is the possibility that the options on Merck described in the preceding problem could be exercised early. a. Use the pseudo-American call option technique to determine whether this will affect the value of the call.
+b. Why does the possibility of early exercise exist? What types of options are most likely to be exercised early? 4. You have been provided the following information on a three-month call: a. If you wanted to replicate buying this call, how much money would you need to borrow?
+b. If you wanted to replicate buying this call, how many shares of stock would you need to buy? 5. Go Video, a manufacturer of video recorders, was trading at $4 per share in May 1994. There were 11 million shares outstanding. At the same time, it had 550,000 one-year warrants outstanding, with a strike price of $4.25. The stock has had a standard deviation of 60%. The stock does not pay a dividend. The riskless rate is 5%. a. Estimate the value of the warrants, ignoring dilution.
+b. Estimate the value of the warrants, allowing for dilution.
+c. Why does dilution reduce the value of the warrants? 6. You are trying to value a long-term call option on the NYSE Composite index, expiring in five years, with a strike price of 275. The index is currently at 250, and the annualized standard deviation in stock prices is 15%. The average dividend yield on the index is 3% and is expected to remain unchanged over the next five years. The five-year Treasury bond rate is 5%. a. Estimate the value of the long-term call option.
+b. Estimate the value of a put option with the same parameters.
+c. What are the implicit assumptions you are making when you use the Black-Scholes model to value this option? Which of these assumptions are likely to be violated? What are the consequences for your valuation? 7. A new security on AT&T will entitle the investor to all dividends on AT&T over the next three years, limiting upside potential to 20% but also providing downside protection below 10%. AT&T stock is trading at $50, and three-year call and put options are traded on the exchange at the following prices: How much would you be willing to pay for this security? 1 Note, though, that higher variance can reduce the value of the underlying asset. As a call option becomes more in-the-money, the more it resembles the underlying asset. For very deep in-the-money call options, higher variance can reduce the value of the option.
+2 To illustrate, if σ2 is the variance in ln(stock prices), the up and the down movements in the binomial can be estimated as follows: where u and d are the up and down movements per unit time for the binomial, T is the life of the option, and m is the number of periods within that lifetime.
+3 Warrants are call options issued by firms, either as part of management compensation contracts or to raise equity.
+4 The analogy to a listed option on a stock is the case where you do not know with certainty what the stock price is when you exercise the option. The more uncertain you are about the stock price, the more margin for error you have to give yourself when you exercise the option, to ensure that you are in fact earning a profit.
+
+CHAPTER 5
+Option Pricing Theory and Models
+In general, the value of any asset is the present value of the expected cash flows on that asset. This chapter considers an exception to that rule when it looks at assets with two specific characteristics: 1. The assets derive their value from the values of other assets.
+2. The cash flows on the assets are contingent on the occurrence of specific events. These assets are called options, and the present value of the expected cash flows on these assets will understate their true value. This chapter describes the cash flow characteristics of options, considers the factors that determine their value, and examines how best to value them.
+BASICS OF OPTION PRICING
+An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called a strike price or an exercise price) at or before the expiration date of the option. Since it is a right and not an obligation, the holder can choose not to exercise the right and can allow the option to expire. There are two types of options—call options and put options.
+Call and Put Options: Description and Payoff Diagrams
+A call option gives the buyer of the option the right to buy the underlying asset at the strike price or the exercise price at any time prior to the expiration date of the option. The buyer pays a price for this right. If at expiration the value of the asset is less than the strike price, the option is not exercised and expires worthless. If, however, the value of the asset is greater than the strike price, the option is exercised—the buyer of the option buys the stock at the exercise price, and the difference between the asset value and the exercise price comprises the gross profit on the investment. The net profit on the investment is the difference between the gross profit and the price paid for the call initially.
+A payoff diagram illustrates the cash payoff on an option at expiration. For a call, the net payoff is negative (and equal to the price paid for the call) if the value of the underlying asset is less than the strike price. If the price of the underlying asset exceeds the strike price, the gross payoff is the difference between the value of the underlying asset and the strike price, and the net payoff is the difference between the gross payoff and the price of the call. This is illustrated in Figure 5.1. Figure 5.1 Payoff on Call Option A put option gives the buyer of the option the right to sell the underlying asset at a fixed price, again called the strike or exercise price, at any time prior to the expiration date of the option. The buyer pays a price for this right. If the price of the underlying asset is greater than the strike price, the option will not be exercised and will expire worthless. But if the price of the underlying asset is less than the strike price, the owner of the put option will exercise the option and sell the stock at the strike price, claiming the difference between the strike price and the market value of the asset as the gross profit. Again, netting out the initial cost paid for the put yields the net profit from the transaction.
+A put has a negative net payoff if the value of the underlying asset exceeds the strike price, and has a gross payoff equal to the difference between the strike price and the value of the underlying asset if the asset value is less than the strike price. This is summarized in Figure 5.2. Figure 5.2 Payoff on Put Option DETERMINANTS OF OPTION VALUE
+The value of an option is determined by six variables relating to the underlying asset and financial markets. 1. Current value of the underlying asset. Options are assets that derive value from an underlying asset. Consequently, changes in the value of the underlying asset affect the value of the options on that asset. Since calls provide the right to buy the underlying asset at a fixed price, an increase in the value of the asset will increase the value of the calls. Puts, on the other hand, become less valuable as the value of the asset increases.
+2. Variance in value of the underlying asset. The buyer of an option acquires the right to buy or sell the underlying asset at a fixed price. The higher the variance in the value of the underlying asset, the greater the value of the option.1 This is true for both calls and puts. While it may seem counterintuitive that an increase in a risk measure (variance) should increase value, options are different from other securities since buyers of options can never lose more than the price they pay for them; in fact, they have the potential to earn significant returns from large price movements.
+3. Dividends paid on the underlying asset. The value of the underlying asset can be expected to decrease if dividend payments are made on the asset during the life of the option. Consequently, the value of a call on the asset is a decreasing function of the size of expected dividend payments, and the value of a put is an increasing function of expected dividend payments. A more intuitive way of thinking about dividend payments, for call options, is as a cost of delaying exercise on in-the-money options. To see why, consider an option on a traded stock. Once a call option is in-the-money (i.e., the holder of the option will make a gross payoff by exercising the option), exercising the call option will provide the holder with the stock and entitle him or her to the dividends on the stock in subsequent periods. Failing to exercise the option will mean that these dividends are forgone.
+4. Strike price of the option. A key characteristic used to describe an option is the strike price. In the case of calls, where the holder acquires the right to buy at a fixed price, the value of the call will decline as the strike price increases. In the case of puts, where the holder has the right to sell at a fixed price, the value will increase as the strike price increases.
+5. Time to expiration on the option. Both calls and puts are more valuable the greater the time to expiration. This is because the longer time to expiration provides more time for the value of the underlying asset to move, increasing the value of both types of options. Additionally, in the case of a call, where the buyer has the right to pay a fixed price at expiration, the present value of this fixed price decreases as the life of the option increases, increasing the value of the call.
+6. Riskless interest rate corresponding to life of the option. Since the buyer of an option pays the price of the option up front, an opportunity cost is involved. This cost will depend on the level of interest rates and the time to expiration of the option. The riskless interest rate also enters into the valuation of options when the present value of the exercise price is calculated, since the exercise price does not have to be paid (received) until expiration on calls (puts). Increases in the interest rate will increase the value of calls and reduce the value of puts. Table 5.1 summarizes the variables and their predicted effects on call and put prices.
+Table 5.1 Summary of Variables Affecting Call and Put Prices Effect On Factor
+Call Value
+Put Value Increase in underlying asset’s value
+Increases
+Decreases Increase in variance of underlying asset
+Increases
+Increases Increase in strike price
+Decreases
+Increases Increase in dividends paid
+Decreases
+Increases Increase in time to expiration
+Increases
+Increases Increase in interest rates
+Increases
+Decreases American versus European Options: Variables Relating to Early Exercise
+A primary distinction between American and European options is that an American option can be exercised at any time prior to its expiration, while European options can be exercised only at expiration. The possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. There is one compensating factor that enables the former to be valued using models designed for the latter. In most cases, the time premium associated with the remaining life of an option and transaction costs make early exercise suboptimal. In other words, the holders of in-the-money options generally get much more by selling the options to someone else than by exercising the options.
+OPTION PRICING MODELS
+Option pricing theory has made vast strides since 1972, when Fischer Black and Myron Scholes published their pathbreaking paper that provided a model for valuing dividend-protected European options. Black and Scholes used a “replicating portfolio”—a portfolio composed of the underlying asset and the risk-free asset that had the same cash flows as the option being valued—and the notion of arbitrage to come up with their final formulation. Although their derivation is mathematically complicated, there is a simpler binomial model for valuing options that draws on the same logic.
+Binomial Model
+The binomial option pricing model is based on a simple formulation for the asset price process in which the asset, in any time period, can move to one of two possible prices. The general formulation of a stock price process that follows the binomial path is shown in Figure 5.3. In this figure, S is the current stock price; the price moves up to Su with probability p and down to Sd with probability 1 – p in any time period. Figure 5.3 General Formulation for Binomial Price Path Creating a Replicating Portfolio
+The objective in creating a replicating portfolio is to use a combination of risk-free borrowing/lending and the underlying asset to create the same cash flows as the option being valued. The principles of arbitrage apply then, and the value of the option must be equal to the value of the replicating portfolio. In the case of the general formulation shown in Figure 5.3, where stock prices can move either up to Su or down to Sd in any time period, the replicating portfolio for a call with strike price K will involve borrowing $B and acquiring Δ of the underlying asset, where: where Cu = Value of the call if the stock price is Su
+Cd = Value of the call if the stock price is Sd
+In a multiperiod binomial process, the valuation has to proceed iteratively (i.e., starting with the final time period and moving backward in time until the current point in time). The portfolios replicating the option are created at each step and valued, providing the values for the option in that time period. The final output from the binomial option pricing model is a statement of the value of the option in terms of the replicating portfolio, composed of Δ shares (option delta) of the underlying asset and risk-free borrowing/lending. ILLUSTRATION 5.1: Binomial Option Valuation
+Assume that the objective is to value a call with a strike price of $50, which is expected to expire in two time periods, on an underlying asset whose price currently is $50 and is expected to follow a binomial process: Now assume that the interest rate is 11%. In addition, define: The objective is to combined Δ shares of stock and B dollars of borrowing to replicate the cash flows from the call with a strike price of $50. This can be done iteratively, starting with the last period and working back through the binomial tree.
+Step 1: Start with the end nodes and work backward: Thus, if the stock price is $70 at t = 1, borrowing $45 and buying one share of the stock will give the same cash flows as buying the call. The value of the call at t = 1, if the stock price is $70, is therefore: Considering the other leg of the binomial tree at t = 1, If the stock price is $35 at t = 1, then the call is worth nothing.
+Step 2: Move backward to the earlier time period and create a replicating portfolio that will provide the cash flows the option will provide. In other words, borrowing $22.50 and buying five-sevenths of a share will provide the same cash flows as a call with a strike price of $50 over the call’s lifetime. The value of the call therefore has to be the same as the cost of creating this position. The Determinants of Value
+The binomial model provides insight into the determinants of option value. The value of an option is not determined by the expected price of the asset but by its current price, which, of course, reflects expectations about the future. This is a direct consequence of arbitrage. If the option value deviates from the value of the replicating portfolio, investors can create an arbitrage position (i.e., one that requires no investment, involves no risk, and delivers positive returns). To illustrate, if the portfolio that replicates the call costs more than the call does in the market, an investor could buy the call, sell the replicating portfolio, and be guaranteed the difference as a profit. The cash flows on the two positions will offset each other, leading to no cash flows in subsequent periods. The call option value also increases as the time to expiration is extended, as the price movements (u and d) increase, and with increases in the interest rate.
+While the binomial model provides an intuitive feel for the determinants of option value, it requires a large number of inputs, in terms of expected future prices at each node. As time periods are made shorter in the binomial model, you can make one of two assumptions about asset prices.You can assume that price changes become smaller as periods get shorter; this leads to price changes becoming infinitesimally small as time periods approach zero, leading to a continuous price process. Alternatively, you can assume that price changes stay large even as the period gets shorter; this leads to a jump price process, where prices can jump in any period. This section considers the option pricing models that emerge with each of these assumptions.
+Black-Scholes Model
+When the price process is continuous (i.e., price changes become smaller as time periods get shorter), the binomial model for pricing options converges on the Black-Scholes model. The model, named after its cocreators, Fischer Black and Myron Scholes, allows us to estimate the value of any option using a small number of inputs, and has been shown to be robust in valuing many listed options.
+The Model
+While the derivation of the Black-Scholes model is far too complicated to present here, it is based on the idea of creating a portfolio of the underlying asset and the riskless asset with the same cash flows, and hence the same cost, as the option being valued. The value of a call option in the Black-Scholes model can be written as a function of the five variables: S = Current value of the underlying asset
+K = Strike price of the option
+t = Life to expiration of the option
+r = Riskless interest rate corresponding to the life of the option
+σ2 = Variance in the ln(value) of the underlying asset The value of a call is then: Note that e–rt is the present value factor, and reflects the fact that the exercise price on the call option does not have to be paid until expiration, since the model values European options. N(d1) and N(d2) are probabilities, estimated by using a cumulative standardized normal distribution, and the values of d1 and d2 obtained for an option. The cumulative distribution is shown in Figure 5.4. Figure 5.4 Cumulative Normal Distribution In approximate terms, N(d2) yields the likelihood that an option will generate positive cash flows for its owner at exercise (i.e., that S > K in the case of a call option and that K > S in the case of a put option). The portfolio that replicates the call option is created by buying N(d1) units of the underlying asset, and borrowing Ke–rt N(d2). The portfolio will have the same cash flows as the call option, and thus the same value as the option. N(d1), which is the number of units of the underlying asset that are needed to create the replicating portfolio, is called the option delta. A NOTE ON ESTIMATING THE INPUTS TO THE BLACK-SCHOLES MODEL
+The Black-Scholes model requires inputs that are consistent on time measurement. There are two places where this affects estimates. The first relates to the fact that the model works in continuous time, rather than discrete time. That is why we use the continuous time version of present value (exp–rt) rather than the discrete version, (1 + r)–t. It also means that the inputs such as the riskless rate have to be modified to make them continuous time inputs. For instance, if the one-year Treasury bond rate is 6.2 percent, the risk-free rate that is used in the Black-Scholes model should be: The second relates to the period over which the inputs are estimated. For instance, the preceding rate is an annual rate. The variance that is entered into the model also has to be an annualized variance. The variance, estimated from ln(asset prices), can be annualized easily because variances are linear in time if the serial correlation is zero. Thus, if monthly or weekly prices are used to estimate variance, the variance is annualized by multiplying by 12 or 52, respectively. ILLUSTRATION 5.2: Valuing an Option Using the Black-Scholes Model
+On March 6, 2001, Cisco Systems was trading at $13.62. We will attempt to value a July 2001 call option with a strike price of $15, trading on the CBOE on the same day for $2. The following are the other parameters of the options: The annualized standard deviation in Cisco Systems stock price over the previous year was 81%. This standard deviation is estimated using weekly stock prices over the year, and the resulting number was annualized as follows: The option expiration date is Friday, July 20, 2001. There are 103 days to expiration, and the annualized Treasury bill rate corresponding to this option life is 4.63%. The inputs for the Black-Scholes model are as follows: Current stock price (S) = $13.62
+Strike price on the option = $15
+Option life = 103/365 = 0.2822
+Standard deviation in ln(stock prices) = 81%
+Riskless rate = 4.63% Inputting these numbers into the model, we get: Using the normal distribution, we can estimate the N(d1) and N(d2): The value of the call can now be estimated: Since the call is trading at $2, it is slightly overvalued, assuming that the estimate of standard deviation used is correct. IMPLIED VOLATILITY
+The only input in the Black Scholes on which there can be significant disagreement among investors is the variance. While the variance is often estimated by looking at historical data, the values for options that emerge from using the historical variance can be different from the market prices. For any option, there is some variance at which the estimated value will be equal to the market price. This variance is called an implied variance.
+Consider the Cisco option valued in Illustration 5.2. With a standard deviation of 81 percent, the value of the call option with a strike price of $15 was estimated to be $1.87. Since the market price is higher than the calculated value, we tried higher standard deviations, and at a standard deviation 85.40 percent the value of the option is $2 (which is the market price). This is the implied standard deviation or implied volatility. Model Limitations and Fixes
+The Black-Scholes model was designed to value European options that can be exercised only at maturity and whose underlying assets do not pay dividends. In addition, options are valued based on the assumption that option exercise does not affect the value of the underlying asset. In practice, assets do pay dividends, options sometimes get exercised early, and exercising an option can affect the value of the underlying asset. Adjustments exist that, while not perfect, provide partial corrections to the Black-Scholes model.
+Dividends
+The payment of a dividend reduces the stock price; note that on the ex-dividend day, the stock price generally declines. Consequently, call options become less valuable and put options more valuable as expected dividend payments increase. There are two ways of dealing with dividends in the Black-Scholes model: 1. Short-term options. One approach to dealing with dividends is to estimate the present value of expected dividends that will be paid by the underlying asset during the option life and subtract it from the current value of the asset to use as S in the model. 2. Long-term options. Since it becomes less practical to estimate the present value of dividends the longer the option life, an alternate approach can be used. If the dividend yield (y = Dividends/Current value of the asset) on the underlying asset is expected to remain unchanged during the life of the option, the Black-Scholes model can be modified to take dividends into account. From an intuitive standpoint, the adjustments have two effects. First, the value of the asset is discounted back to the present at the dividend yield to take into account the expected drop in asset value resulting from dividend payments. Second, the interest rate is offset by the dividend yield to reflect the lower carrying cost from holding the asset (in the replicating portfolio). The net effect will be a reduction in the value of calls estimated using this model. ILLUSTRATION 5.3: Valuing a Short-Term Option with Dividend Adjustments—The Black-Scholes Correction
+Assume that it is March 6, 2001, and that AT&T is trading at $20.50 a share. Consider a call option on the stock with a strike price of $20, expiring on July 20, 2001. Using past stock prices, the annualized standard deviation in the log of stock prices for AT&T is estimated at 60%. There is one dividend, amounting to $0.15, and it will be paid in 23 days. The riskless rate is 4.63%. Present value of expected dividend = $0.15/1.046323/365 = $0.15
+Dividend-adjusted stock price = $20.50 – $0.15 = $20.35
+Time to expiration = 103/365 = 0.2822
+Variance in ln(stock prices) = 0.62 = 0.36
+Riskless rate = 4.63% The value from the Black-Scholes model is: The call option was trading at $2.60 on that day. ILLUSTRATION 5.4: Valuing a Long-Term Option with Dividend Adjustments—Primes and Scores
+The CBOE offers longer-term call and put options on some stocks. On March 6, 2001, for instance, you could have purchased an AT&T call expiring on January 17, 2003. The stock price for AT&T is $20.50 (as in the previous example). The following is the valuation of a call option with a strike price of $20. Instead of estimating the present value of dividends over the next two years, assume that AT&T’s dividend yield will remain 2.51% over this period and that the risk-free rate for a two-year Treasury bond is 4.85%. The inputs to the Black-Scholes model are: The value from the Black-Scholes model is: The call was trading at $5.80 on March 8, 2001. optst.xls: This spreadsheet allows you to estimate the value of a short-term option when the expected dividends during the option life can be estimated. optlt.xls: This spreadsheet allows you to estimate the value of an option when the underlying asset has a constant dividend yield. Early Exercise
+The Black-Scholes model was designed to value European options that can be exercised only at expiration. In contrast, most options that we encounter in practice are American options and can be exercised at any time until expiration. As mentioned earlier, the possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. In general, though, with traded options, it is almost always better to sell the option to someone else rather than exercise early, since options have a time premium (i.e., they sell for more than their exercise value). There are two exceptions. One occurs when the underlying asset pays large dividends, thus reducing the expected value of the asset. In this case, call options may be exercised just before an ex-dividend date, if the time premium on the options is less than the expected decline in asset value as a consequence of the dividend payment. The other exception arises when an investor holds both the underlying asset and deep in-the-money puts (i.e., puts with strike prices well above the current price of the underlying asset) on that asset at a time when interest rates are high. In this case, the time premium on the put may be less than the potential gain from exercising the put early and earning interest on the exercise price.
+There are two basic ways of dealing with the possibility of early exercise. One is to continue to use the unadjusted Black-Scholes model and to regard the resulting value as a floor or conservative estimate of the true value. The other is to try to adjust the value of the option for the possibility of early exercise. There are two approaches for doing so. One uses the Black-Scholes model to value the option to each potential exercise date. With options on stocks, this basically requires that the investor values options to each ex-dividend day and chooses the maximum of the estimated call values. The second approach is to use a modified version of the binomial model to consider the possibility of early exercise. In this version, the up and the down movements for asset prices in each period can be estimated from the variance. 2
+Approach 1: Pseudo-American Valuation
+Step 1: Define when dividends will be paid and how much the dividends will be.
+Step 2: Value the call option to each ex-dividend date using the dividend-adjusted approach described earlier, where the stock price is reduced by the present value of expected dividends.
+Step 3: Choose the maximum of the call values estimated for each ex-dividend day. ILLUSTRATION 5.5: Using Pseudo-American Option Valuation to Adjust for Early Exercise
+Consider an option with a strike price of $35 on a stock trading at $40. The variance in the ln(stock prices) is 0.05, and the riskless rate is 4%. The option has a remaining life of eight months, and there are three dividends expected during this period: Expected Dividend
+Ex-Dividend Day $0.80
+In 1 month $0.80
+In 4 months $0.80
+In 7 months The call option is first valued to just before the first ex-dividend date: The value from the Black-Scholes model is: The call option is then valued to before the second ex-dividend date: The value of the call based on these parameters is: The call option is then valued to before the third ex-dividend date: The value of the call based on these parameters is: The call option is then valued to expiration: The value of the call based on these parameters is: Approach 2: Using the Binomial Model
+The binomial model is much more capable of handling early exercise because it considers the cash flows at each time period, rather than just at expiration. The biggest limitation of the binomial model is determining what stock prices will be at the end of each period, but this can be overcome by using a variant that allows us to estimate the up and the down movements in stock prices from the estimated variance. There are four steps involved:
+Step 1: If the variance in ln(stock prices) has been estimated for the Black-Scholes valuation, convert these into inputs for the binomial model: where u and d are the up and the down movements per unit time for the binomial, and dt is the number of periods within each year (or unit time).
+Step 2: Specify the period in which the dividends will be paid and make the assumption that the price will drop by the amount of the dividend in that period.
+Step 3: Value the call at each node of the tree, allowing for the possibility of early exercise just before ex-dividend dates. There will be early exercise if the remaining time premium on the option is less than the expected drop in option value as a consequence of the dividend payment.
+Step 4: Value the call at time 0, using the standard binomial approach. bstobin.xls: This spreadsheet allows you to estimate the parameters for a binomial model from the inputs to a Black-Scholes model. Impact of Exercise on Underlying Asset Value
+The Black-Scholes model is based on the assumption that exercising an option does not affect the value of the underlying asset. This may be true for listed options on stocks, but it is not true for some types of options. For instance, the exercise of warrants increases the number of shares outstanding and brings fresh cash into the firm, both of which will affect the stock price.3 The expected negative impact (dilution) of exercise will decrease the value of warrants, compared to otherwise similar call options. The adjustment for dilution to the stock price is fairly simple in the Black-Scholes valuation. The stock price is adjusted for the expected dilution from the exercise of the options. In the case of warrants, for instance: where S = Current value of the stock
+nw = Number of warrants outstanding
+W = Value of warrants outstanding
+ns = Number of shares outstanding
+When the warrants are exercised, the number of shares outstanding will increase, reducing the stock price. The numerator reflects the market value of equity, including both stocks and warrants outstanding. The reduction in S will reduce the value of the call option.
+There is an element of circularity in this analysis, since the value of the warrant is needed to estimate the dilution-adjusted S and the dilution-adjusted S is needed to estimate the value of the warrant. This problem can be resolved by starting the process off with an assumed value for the warrant (e.g., the exercise value or the current market price of the warrant). This will yield a value for the warrant, and this estimated value can then be used as an input to reestimate the warrant’s value until there is convergence. FROM BLACK-SCHOLES TO BINOMIAL
+The process of converting the continuous variance in a Black-Scholes model to a binomial tree is a fairly simple one. Assume, for instance, that you have an asset that is trading at $30 currently and that you estimate the annualized standard deviation in the asset value to be 40 percent; the annualized riskless rate is 5 percent. For simplicity, let us assume that the option that you are valuing has a four-year life and that each period is a year. To estimate the prices at the end of each of the four years, we begin by first estimating the up and down movements in the binomial: Based on these estimates, we can obtain the prices at the end of the first node of the tree (the end of the first year): Progressing through the rest of the tree, we obtain the following numbers: ILLUSTRATION 5.6: Valuing a Warrant on Avatek Corporation
+Avatek Corporation is a real estate firm with 19.637 million shares outstanding, trading at $0.38 a share. In March 2001 the company had 1.8 million options outstanding, with four years to expiration and with an exercise price of $2.25. The stock paid no dividends, and the standard deviation in ln(stock prices) was 93%. The four-year Treasury bond rate was 4.9%. (The options were trading at $0.12 apiece at the time of this analysis.)
+The inputs to the warrant valuation model are as follows: The results of the Black-Scholes valuation of this option are: The options were trading at $0.12 in March 2001. Since the value was equal to the price, there was no need for further iterations. If there had been a difference, we would have reestimated the adjusted stock price and option value. If the options had been non-traded (as is the case with management options), this calculation would have required an iterative process, where the option value is used to get the adjusted value per share and the value per share to get the option value. warrant.xls: This spreadsheet allows you to estimate the value of an option when there is a potential dilution from exercise. The Black-Scholes Model for Valuing Puts
+The value of a put can be derived from the value of a call with the same strike price and the same expiration date: where C is the value of the call and P is the value of the put. This relationship between the call and put values is called put-call parity, and any deviations from parity can be used by investors to make riskless profits. To see why put-call parity holds, consider selling a call and buying a put with exercise price K and expiration date t, and simultaneously buying the underlying asset at the current price S. The payoff from this position is riskless and always yields K at expiration (t). To see this, assume that the stock price at expiration is S*. The payoff on each of the positions in the portfolio can be written as follows: Position
+Payoffs at t if S* > K
+Payoffs at t if S* < K Sell call
+–(S* – K)
+0 Buy put
+0
+K – S* Buy stock
+S*
+S* Total
+K
+K Since this position yields K with certainty, the cost of creating this position must be equal to the present value of K at the riskless rate (K e–rt). Substituting the Black-Scholes equation for the value of an equivalent call into this equation, we get: Thus, the replicating portfolio for a put is created by selling short [1 – N(d1)] shares of stock and investing K e–rt[1 – N(d2)] in the riskless asset. ILLUSTRATION 5.7: Valuing a Put Using Put-Call Parity: Cisco Systems and AT&T
+Consider the call on Cisco Systems that we valued in Illustration 5.2. The call had a strike price of $15 on the stock, had 103 days left to expiration, and was valued at $1.87. The stock was trading at $13.62, and the riskless rate was 4.63%. The put can be valued as follows: The put was trading at $3.38.
+Also, a long-term call on AT&T was valued in Illustration 5.4. The call had a strike price of $20, 1.8333 years left to expiration, and a value of $6.63. The stock was trading at $20.50 and was expected to maintain a dividend yield of 2.51% over the period. The riskless rate was 4.85%. The put value can be estimated as follows: The put was trading at $3.80. Both the call and the put were trading at different prices from our estimates, which may indicate that we have not correctly estimated the stock’s volatility. Jump Process Option Pricing Models
+If price changes remain larger as the time periods in the binomial model are shortened, it can no longer be assumed that prices change continuously. When price changes remain large, a price process that allows for price jumps is much more realistic. Cox and Ross (1976) valued options when prices follow a pure jump process, where the jumps can only be positive. Thus, in the next interval, the stock price will either have a large positive jump with a specified probability or drift downward at a given rate.
+Merton (1976) considered a distribution where there are price jumps superimposed on a continuous price process. He specified the rate at which jumps occur (λ) and the average jump size (k), measured as a percentage of the stock price. The model derived to value options with this process is called a jump diffusion model. In this model, the value of an option is determined by the five variables specified in the Black-Scholes model, and the parameters of the jump process (λ, k). Unfortunately, the estimates of the jump process parameters are so difficult to make for most firms that they overwhelm any advantages that accrue from using a more realistic model. These models, therefore, have seen limited use in practice.
+EXTENSIONS OF OPTION PRICING
+All the option pricing models described so far—the binomial, the Black-Scholes, and the jump process models—are designed to value options with clearly defined exercise prices and maturities on underlying assets that are traded. However, the options we encounter in investment analysis or valuation are often on real assets rather than financial assets. Categorized as real options, they can take much more complicated forms. This section considers some of these variations.
+Capped and Barrier Options
+With a simple call option, there is no specified upper limit on the profits that can be made by the buyer of the call. Asset prices, at least in theory, can keep going up, and the payoffs increase proportionately. In some call options, though, the buyer is entitled to profits up to a specified price but not above it. For instance, consider a call option with a strike price of K1 on an asset. In an unrestricted call option, the payoff on this option will increase as the underlying asset’s price increases above K1. Assume, however, that if the price reaches K2, the payoff is capped at (K2 – K1). The payoff diagram on this option is shown in Figure 5.5. Figure 5.5 Payoff on Capped Call This option is called a capped call. Notice, also, that once the price reaches K2, there is no longer any time premium associated with the option, and the option will therefore be exercised. Capped calls are part of a family of options called barrier options, where the payoff on and the life of the option are functions of whether the underlying asset price reaches a certain level during a specified period.
+The value of a capped call is always lower than the value of the same call without the payoff limit. A simple approximation of this value can be obtained by valuing the call twice, once with the given exercise price and once with the cap, and taking the difference in the two values. In the preceding example, then, the value of the call with an exercise price of K1 and a cap at K2 can be written as: Barrier options can take many forms. In a knockout option, an option ceases to exist if the underlying asset reaches a certain price. In the case of a call option, this knockout price is usually set below the strike price, and this option is called a down-and-out option. In the case of a put option, the knockout price will be set above the exercise price, and this option is called an up-and-out option. Like the capped call, these options are worth less than their unrestricted counterparts. Many real options have limits on potential upside, or knockout provisions, and ignoring these limits can result in the overstatement of the value of these options.
+Compound Options
+Some options derive their value not from an underlying asset, but from other options. These options are called compound options. Compound options can take any of four forms—a call on a call, a put on a put, a call on a put, or a put on a call. Geske (1979) developed the analytical formulation for valuing compound options by replacing the standard normal distribution used in a simple option model with a bivariate normal distribution in the calculation.
+Consider, for instance, the option to expand a project that is discussed in Chapter 30. While we will value this option using a simple option pricing model, in reality there could be multiple stages in expansion, with each stage representing an option for the following stage. In this case, we will undervalue the option by considering it as a simple rather than a compound option.
+Notwithstanding this discussion, the valuation of compound options becomes progressively more difficult as more options are added to the chain. In this case, rather than wreck the valuation on the shoals of estimation error, it may be better to accept the conservative estimate that is provided with a simple valuation model as a floor on the value.
+Rainbow Options
+In a simple option, the uncertainty is about the price of the underlying asset. Some options are exposed to two or more sources of uncertainty, and these options are rainbow options. Using the simple option pricing model to value such options can lead to biased estimates of value. As an example, consider an undeveloped oil reserve as an option, where the firm that owns the reserve has the right to develop the reserve. Here there are two sources of uncertainty. The first is obviously the price of oil, and the second is the quantity of oil that is in the reserve. To value this undeveloped reserve, we can make the simplifying assumption that we know the quantity of oil in the reserve with certainty. In reality, however, uncertainty about the quantity will affect the value of this option and make the decision to exercise more difficult.4
+CONCLUSION
+An option is an asset with payoffs that are contingent on the value of an underlying asset. A call option provides its holder with the right to buy the underlying asset at a fixed price, whereas a put option provides its holder with the right to sell at a fixed price, at any time before the expiration of the option. The value of an option is determined by six variables—the current value of the underlying asset, the variance in this value, the expected dividends on the asset, the strike price and life of the option, and the riskless interest rate. This is illustrated in both the binomial and the Black-Scholes models, which value options by creating replicating portfolios composed of the underlying asset and riskless lending or borrowing. These models can be used to value assets that have option like characteristics.
+QUESTIONS AND SHORT PROBLEMS
+In the problems following, use an equity risk premium of 5.5 percent if none is specified. 1. The following are prices of options traded on Microsoft Corporation, which pays no dividends. The stock is trading at $83, and the annualized riskless rate is 3.8%. The standard deviation in ln(stock prices) (based on historical data) is 30%. a. Estimate the value of a three-month call with a strike price of $85.
+b. Using the inputs from the Black-Scholes model, specify how you would replicate this call.
+c. What is the implied standard deviation in this call?
+d. Assume now that you buy a call with a strike price of $85 and sell a call with a strike price of $90. Draw the payoff diagram on this position.
+e. Using put-call parity, estimate the value of a three-month put with a strike price of $85. 2. You are trying to value three-month call and put options on Merck with a strike price of $30. The stock is trading at $28.75, and the company expects to pay a quarterly dividend per share of $0.28 in two months. The annualized riskless interest rate is 3.6%, and the standard deviation in log stock prices is 20%. a. Estimate the value of the call and put options, using the Black-Scholes model.
+b. What effect does the expected dividend payment have on call values? On put values? Why? 3. There is the possibility that the options on Merck described in the preceding problem could be exercised early. a. Use the pseudo-American call option technique to determine whether this will affect the value of the call.
+b. Why does the possibility of early exercise exist? What types of options are most likely to be exercised early? 4. You have been provided the following information on a three-month call: a. If you wanted to replicate buying this call, how much money would you need to borrow?
+b. If you wanted to replicate buying this call, how many shares of stock would you need to buy? 5. Go Video, a manufacturer of video recorders, was trading at $4 per share in May 1994. There were 11 million shares outstanding. At the same time, it had 550,000 one-year warrants outstanding, with a strike price of $4.25. The stock has had a standard deviation of 60%. The stock does not pay a dividend. The riskless rate is 5%. a. Estimate the value of the warrants, ignoring dilution.
+b. Estimate the value of the warrants, allowing for dilution.
+c. Why does dilution reduce the value of the warrants? 6. You are trying to value a long-term call option on the NYSE Composite index, expiring in five years, with a strike price of 275. The index is currently at 250, and the annualized standard deviation in stock prices is 15%. The average dividend yield on the index is 3% and is expected to remain unchanged over the next five years. The five-year Treasury bond rate is 5%. a. Estimate the value of the long-term call option.
+b. Estimate the value of a put option with the same parameters.
+c. What are the implicit assumptions you are making when you use the Black-Scholes model to value this option? Which of these assumptions are likely to be violated? What are the consequences for your valuation? 7. A new security on AT&T will entitle the investor to all dividends on AT&T over the next three years, limiting upside potential to 20% but also providing downside protection below 10%. AT&T stock is trading at $50, and three-year call and put options are traded on the exchange at the following prices: How much would you be willing to pay for this security? 1 Note, though, that higher variance can reduce the value of the underlying asset. As a call option becomes more in-the-money, the more it resembles the underlying asset. For very deep in-the-money call options, higher variance can reduce the value of the option.
+2 To illustrate, if σ2 is the variance in ln(stock prices), the up and the down movements in the binomial can be estimated as follows: where u and d are the up and down movements per unit time for the binomial, T is the life of the option, and m is the number of periods within that lifetime.
+3 Warrants are call options issued by firms, either as part of management compensation contracts or to raise equity.
+4 The analogy to a listed option on a stock is the case where you do not know with certainty what the stock price is when you exercise the option. The more uncertain you are about the stock price, the more margin for error you have to give yourself when you exercise the option, to ensure that you are in fact earning a profit.
+CHAPTER 5
+Option Pricing Theory and Models
+In general, the value of any asset is the present value of the expected cash flows on that asset. This chapter considers an exception to that rule when it looks at assets with two specific characteristics: 1. The assets derive their value from the values of other assets.
+2. The cash flows on the assets are contingent on the occurrence of specific events. These assets are called options, and the present value of the expected cash flows on these assets will understate their true value. This chapter describes the cash flow characteristics of options, considers the factors that determine their value, and examines how best to value them.
+BASICS OF OPTION PRICING
+An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called a strike price or an exercise price) at or before the expiration date of the option. Since it is a right and not an obligation, the holder can choose not to exercise the right and can allow the option to expire. There are two types of options—call options and put options.
+Call and Put Options: Description and Payoff Diagrams
+A call option gives the buyer of the option the right to buy the underlying asset at the strike price or the exercise price at any time prior to the expiration date of the option. The buyer pays a price for this right. If at expiration the value of the asset is less than the strike price, the option is not exercised and expires worthless. If, however, the value of the asset is greater than the strike price, the option is exercised—the buyer of the option buys the stock at the exercise price, and the difference between the asset value and the exercise price comprises the gross profit on the investment. The net profit on the investment is the difference between the gross profit and the price paid for the call initially.
+A payoff diagram illustrates the cash payoff on an option at expiration. For a call, the net payoff is negative (and equal to the price paid for the call) if the value of the underlying asset is less than the strike price. If the price of the underlying asset exceeds the strike price, the gross payoff is the difference between the value of the underlying asset and the strike price, and the net payoff is the difference between the gross payoff and the price of the call. This is illustrated in Figure 5.1. Figure 5.1 Payoff on Call Option A put option gives the buyer of the option the right to sell the underlying asset at a fixed price, again called the strike or exercise price, at any time prior to the expiration date of the option. The buyer pays a price for this right. If the price of the underlying asset is greater than the strike price, the option will not be exercised and will expire worthless. But if the price of the underlying asset is less than the strike price, the owner of the put option will exercise the option and sell the stock at the strike price, claiming the difference between the strike price and the market value of the asset as the gross profit. Again, netting out the initial cost paid for the put yields the net profit from the transaction.
+A put has a negative net payoff if the value of the underlying asset exceeds the strike price, and has a gross payoff equal to the difference between the strike price and the value of the underlying asset if the asset value is less than the strike price. This is summarized in Figure 5.2. Figure 5.2 Payoff on Put Option DETERMINANTS OF OPTION VALUE
+The value of an option is determined by six variables relating to the underlying asset and financial markets. 1. Current value of the underlying asset. Options are assets that derive value from an underlying asset. Consequently, changes in the value of the underlying asset affect the value of the options on that asset. Since calls provide the right to buy the underlying asset at a fixed price, an increase in the value of the asset will increase the value of the calls. Puts, on the other hand, become less valuable as the value of the asset increases.
+2. Variance in value of the underlying asset. The buyer of an option acquires the right to buy or sell the underlying asset at a fixed price. The higher the variance in the value of the underlying asset, the greater the value of the option.1 This is true for both calls and puts. While it may seem counterintuitive that an increase in a risk measure (variance) should increase value, options are different from other securities since buyers of options can never lose more than the price they pay for them; in fact, they have the potential to earn significant returns from large price movements.
+3. Dividends paid on the underlying asset. The value of the underlying asset can be expected to decrease if dividend payments are made on the asset during the life of the option. Consequently, the value of a call on the asset is a decreasing function of the size of expected dividend payments, and the value of a put is an increasing function of expected dividend payments. A more intuitive way of thinking about dividend payments, for call options, is as a cost of delaying exercise on in-the-money options. To see why, consider an option on a traded stock. Once a call option is in-the-money (i.e., the holder of the option will make a gross payoff by exercising the option), exercising the call option will provide the holder with the stock and entitle him or her to the dividends on the stock in subsequent periods. Failing to exercise the option will mean that these dividends are forgone.
+4. Strike price of the option. A key characteristic used to describe an option is the strike price. In the case of calls, where the holder acquires the right to buy at a fixed price, the value of the call will decline as the strike price increases. In the case of puts, where the holder has the right to sell at a fixed price, the value will increase as the strike price increases.
+5. Time to expiration on the option. Both calls and puts are more valuable the greater the time to expiration. This is because the longer time to expiration provides more time for the value of the underlying asset to move, increasing the value of both types of options. Additionally, in the case of a call, where the buyer has the right to pay a fixed price at expiration, the present value of this fixed price decreases as the life of the option increases, increasing the value of the call.
+6. Riskless interest rate corresponding to life of the option. Since the buyer of an option pays the price of the option up front, an opportunity cost is involved. This cost will depend on the level of interest rates and the time to expiration of the option. The riskless interest rate also enters into the valuation of options when the present value of the exercise price is calculated, since the exercise price does not have to be paid (received) until expiration on calls (puts). Increases in the interest rate will increase the value of calls and reduce the value of puts. Table 5.1 summarizes the variables and their predicted effects on call and put prices.
+Table 5.1 Summary of Variables Affecting Call and Put Prices Effect On Factor
+Call Value
+Put Value Increase in underlying asset’s value
+Increases
+Decreases Increase in variance of underlying asset
+Increases
+Increases Increase in strike price
+Decreases
+Increases Increase in dividends paid
+Decreases
+Increases Increase in time to expiration
+Increases
+Increases Increase in interest rates
+Increases
+Decreases American versus European Options: Variables Relating to Early Exercise
+A primary distinction between American and European options is that an American option can be exercised at any time prior to its expiration, while European options can be exercised only at expiration. The possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. There is one compensating factor that enables the former to be valued using models designed for the latter. In most cases, the time premium associated with the remaining life of an option and transaction costs make early exercise suboptimal. In other words, the holders of in-the-money options generally get much more by selling the options to someone else than by exercising the options.
+OPTION PRICING MODELS
+Option pricing theory has made vast strides since 1972, when Fischer Black and Myron Scholes published their pathbreaking paper that provided a model for valuing dividend-protected European options. Black and Scholes used a “replicating portfolio”—a portfolio composed of the underlying asset and the risk-free asset that had the same cash flows as the option being valued—and the notion of arbitrage to come up with their final formulation. Although their derivation is mathematically complicated, there is a simpler binomial model for valuing options that draws on the same logic.
+Binomial Model
+The binomial option pricing model is based on a simple formulation for the asset price process in which the asset, in any time period, can move to one of two possible prices. The general formulation of a stock price process that follows the binomial path is shown in Figure 5.3. In this figure, S is the current stock price; the price moves up to Su with probability p and down to Sd with probability 1 – p in any time period. Figure 5.3 General Formulation for Binomial Price Path Creating a Replicating Portfolio
+The objective in creating a replicating portfolio is to use a combination of risk-free borrowing/lending and the underlying asset to create the same cash flows as the option being valued. The principles of arbitrage apply then, and the value of the option must be equal to the value of the replicating portfolio. In the case of the general formulation shown in Figure 5.3, where stock prices can move either up to Su or down to Sd in any time period, the replicating portfolio for a call with strike price K will involve borrowing $B and acquiring Δ of the underlying asset, where: where Cu = Value of the call if the stock price is Su
+Cd = Value of the call if the stock price is Sd
+In a multiperiod binomial process, the valuation has to proceed iteratively (i.e., starting with the final time period and moving backward in time until the current point in time). The portfolios replicating the option are created at each step and valued, providing the values for the option in that time period. The final output from the binomial option pricing model is a statement of the value of the option in terms of the replicating portfolio, composed of Δ shares (option delta) of the underlying asset and risk-free borrowing/lending. ILLUSTRATION 5.1: Binomial Option Valuation
+Assume that the objective is to value a call with a strike price of $50, which is expected to expire in two time periods, on an underlying asset whose price currently is $50 and is expected to follow a binomial process: Now assume that the interest rate is 11%. In addition, define: The objective is to combined Δ shares of stock and B dollars of borrowing to replicate the cash flows from the call with a strike price of $50. This can be done iteratively, starting with the last period and working back through the binomial tree.
+Step 1: Start with the end nodes and work backward: Thus, if the stock price is $70 at t = 1, borrowing $45 and buying one share of the stock will give the same cash flows as buying the call. The value of the call at t = 1, if the stock price is $70, is therefore: Considering the other leg of the binomial tree at t = 1, If the stock price is $35 at t = 1, then the call is worth nothing.
+Step 2: Move backward to the earlier time period and create a replicating portfolio that will provide the cash flows the option will provide. In other words, borrowing $22.50 and buying five-sevenths of a share will provide the same cash flows as a call with a strike price of $50 over the call’s lifetime. The value of the call therefore has to be the same as the cost of creating this position. The Determinants of Value
+The binomial model provides insight into the determinants of option value. The value of an option is not determined by the expected price of the asset but by its current price, which, of course, reflects expectations about the future. This is a direct consequence of arbitrage. If the option value deviates from the value of the replicating portfolio, investors can create an arbitrage position (i.e., one that requires no investment, involves no risk, and delivers positive returns). To illustrate, if the portfolio that replicates the call costs more than the call does in the market, an investor could buy the call, sell the replicating portfolio, and be guaranteed the difference as a profit. The cash flows on the two positions will offset each other, leading to no cash flows in subsequent periods. The call option value also increases as the time to expiration is extended, as the price movements (u and d) increase, and with increases in the interest rate.
+While the binomial model provides an intuitive feel for the determinants of option value, it requires a large number of inputs, in terms of expected future prices at each node. As time periods are made shorter in the binomial model, you can make one of two assumptions about asset prices.You can assume that price changes become smaller as periods get shorter; this leads to price changes becoming infinitesimally small as time periods approach zero, leading to a continuous price process. Alternatively, you can assume that price changes stay large even as the period gets shorter; this leads to a jump price process, where prices can jump in any period. This section considers the option pricing models that emerge with each of these assumptions.
+Black-Scholes Model
+When the price process is continuous (i.e., price changes become smaller as time periods get shorter), the binomial model for pricing options converges on the Black-Scholes model. The model, named after its cocreators, Fischer Black and Myron Scholes, allows us to estimate the value of any option using a small number of inputs, and has been shown to be robust in valuing many listed options.
+The Model
+While the derivation of the Black-Scholes model is far too complicated to present here, it is based on the idea of creating a portfolio of the underlying asset and the riskless asset with the same cash flows, and hence the same cost, as the option being valued. The value of a call option in the Black-Scholes model can be written as a function of the five variables: S = Current value of the underlying asset
+K = Strike price of the option
+t = Life to expiration of the option
+r = Riskless interest rate corresponding to the life of the option
+σ2 = Variance in the ln(value) of the underlying asset The value of a call is then: Note that e–rt is the present value factor, and reflects the fact that the exercise price on the call option does not have to be paid until expiration, since the model values European options. N(d1) and N(d2) are probabilities, estimated by using a cumulative standardized normal distribution, and the values of d1 and d2 obtained for an option. The cumulative distribution is shown in Figure 5.4. Figure 5.4 Cumulative Normal Distribution In approximate terms, N(d2) yields the likelihood that an option will generate positive cash flows for its owner at exercise (i.e., that S > K in the case of a call option and that K > S in the case of a put option). The portfolio that replicates the call option is created by buying N(d1) units of the underlying asset, and borrowing Ke–rt N(d2). The portfolio will have the same cash flows as the call option, and thus the same value as the option. N(d1), which is the number of units of the underlying asset that are needed to create the replicating portfolio, is called the option delta. A NOTE ON ESTIMATING THE INPUTS TO THE BLACK-SCHOLES MODEL
+The Black-Scholes model requires inputs that are consistent on time measurement. There are two places where this affects estimates. The first relates to the fact that the model works in continuous time, rather than discrete time. That is why we use the continuous time version of present value (exp–rt) rather than the discrete version, (1 + r)–t. It also means that the inputs such as the riskless rate have to be modified to make them continuous time inputs. For instance, if the one-year Treasury bond rate is 6.2 percent, the risk-free rate that is used in the Black-Scholes model should be: The second relates to the period over which the inputs are estimated. For instance, the preceding rate is an annual rate. The variance that is entered into the model also has to be an annualized variance. The variance, estimated from ln(asset prices), can be annualized easily because variances are linear in time if the serial correlation is zero. Thus, if monthly or weekly prices are used to estimate variance, the variance is annualized by multiplying by 12 or 52, respectively. ILLUSTRATION 5.2: Valuing an Option Using the Black-Scholes Model
+On March 6, 2001, Cisco Systems was trading at $13.62. We will attempt to value a July 2001 call option with a strike price of $15, trading on the CBOE on the same day for $2. The following are the other parameters of the options: The annualized standard deviation in Cisco Systems stock price over the previous year was 81%. This standard deviation is estimated using weekly stock prices over the year, and the resulting number was annualized as follows: The option expiration date is Friday, July 20, 2001. There are 103 days to expiration, and the annualized Treasury bill rate corresponding to this option life is 4.63%. The inputs for the Black-Scholes model are as follows: Current stock price (S) = $13.62
+Strike price on the option = $15
+Option life = 103/365 = 0.2822
+Standard deviation in ln(stock prices) = 81%
+Riskless rate = 4.63% Inputting these numbers into the model, we get: Using the normal distribution, we can estimate the N(d1) and N(d2): The value of the call can now be estimated: Since the call is trading at $2, it is slightly overvalued, assuming that the estimate of standard deviation used is correct. IMPLIED VOLATILITY
+The only input in the Black Scholes on which there can be significant disagreement among investors is the variance. While the variance is often estimated by looking at historical data, the values for options that emerge from using the historical variance can be different from the market prices. For any option, there is some variance at which the estimated value will be equal to the market price. This variance is called an implied variance.
+Consider the Cisco option valued in Illustration 5.2. With a standard deviation of 81 percent, the value of the call option with a strike price of $15 was estimated to be $1.87. Since the market price is higher than the calculated value, we tried higher standard deviations, and at a standard deviation 85.40 percent the value of the option is $2 (which is the market price). This is the implied standard deviation or implied volatility. Model Limitations and Fixes
+The Black-Scholes model was designed to value European options that can be exercised only at maturity and whose underlying assets do not pay dividends. In addition, options are valued based on the assumption that option exercise does not affect the value of the underlying asset. In practice, assets do pay dividends, options sometimes get exercised early, and exercising an option can affect the value of the underlying asset. Adjustments exist that, while not perfect, provide partial corrections to the Black-Scholes model.
+Dividends
+The payment of a dividend reduces the stock price; note that on the ex-dividend day, the stock price generally declines. Consequently, call options become less valuable and put options more valuable as expected dividend payments increase. There are two ways of dealing with dividends in the Black-Scholes model: 1. Short-term options. One approach to dealing with dividends is to estimate the present value of expected dividends that will be paid by the underlying asset during the option life and subtract it from the current value of the asset to use as S in the model. 2. Long-term options. Since it becomes less practical to estimate the present value of dividends the longer the option life, an alternate approach can be used. If the dividend yield (y = Dividends/Current value of the asset) on the underlying asset is expected to remain unchanged during the life of the option, the Black-Scholes model can be modified to take dividends into account. From an intuitive standpoint, the adjustments have two effects. First, the value of the asset is discounted back to the present at the dividend yield to take into account the expected drop in asset value resulting from dividend payments. Second, the interest rate is offset by the dividend yield to reflect the lower carrying cost from holding the asset (in the replicating portfolio). The net effect will be a reduction in the value of calls estimated using this model. ILLUSTRATION 5.3: Valuing a Short-Term Option with Dividend Adjustments—The Black-Scholes Correction
+Assume that it is March 6, 2001, and that AT&T is trading at $20.50 a share. Consider a call option on the stock with a strike price of $20, expiring on July 20, 2001. Using past stock prices, the annualized standard deviation in the log of stock prices for AT&T is estimated at 60%. There is one dividend, amounting to $0.15, and it will be paid in 23 days. The riskless rate is 4.63%. Present value of expected dividend = $0.15/1.046323/365 = $0.15
+Dividend-adjusted stock price = $20.50 – $0.15 = $20.35
+Time to expiration = 103/365 = 0.2822
+Variance in ln(stock prices) = 0.62 = 0.36
+Riskless rate = 4.63% The value from the Black-Scholes model is: The call option was trading at $2.60 on that day. ILLUSTRATION 5.4: Valuing a Long-Term Option with Dividend Adjustments—Primes and Scores
+The CBOE offers longer-term call and put options on some stocks. On March 6, 2001, for instance, you could have purchased an AT&T call expiring on January 17, 2003. The stock price for AT&T is $20.50 (as in the previous example). The following is the valuation of a call option with a strike price of $20. Instead of estimating the present value of dividends over the next two years, assume that AT&T’s dividend yield will remain 2.51% over this period and that the risk-free rate for a two-year Treasury bond is 4.85%. The inputs to the Black-Scholes model are: The value from the Black-Scholes model is: The call was trading at $5.80 on March 8, 2001. optst.xls: This spreadsheet allows you to estimate the value of a short-term option when the expected dividends during the option life can be estimated. optlt.xls: This spreadsheet allows you to estimate the value of an option when the underlying asset has a constant dividend yield. Early Exercise
+The Black-Scholes model was designed to value European options that can be exercised only at expiration. In contrast, most options that we encounter in practice are American options and can be exercised at any time until expiration. As mentioned earlier, the possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. In general, though, with traded options, it is almost always better to sell the option to someone else rather than exercise early, since options have a time premium (i.e., they sell for more than their exercise value). There are two exceptions. One occurs when the underlying asset pays large dividends, thus reducing the expected value of the asset. In this case, call options may be exercised just before an ex-dividend date, if the time premium on the options is less than the expected decline in asset value as a consequence of the dividend payment. The other exception arises when an investor holds both the underlying asset and deep in-the-money puts (i.e., puts with strike prices well above the current price of the underlying asset) on that asset at a time when interest rates are high. In this case, the time premium on the put may be less than the potential gain from exercising the put early and earning interest on the exercise price.
+There are two basic ways of dealing with the possibility of early exercise. One is to continue to use the unadjusted Black-Scholes model and to regard the resulting value as a floor or conservative estimate of the true value. The other is to try to adjust the value of the option for the possibility of early exercise. There are two approaches for doing so. One uses the Black-Scholes model to value the option to each potential exercise date. With options on stocks, this basically requires that the investor values options to each ex-dividend day and chooses the maximum of the estimated call values. The second approach is to use a modified version of the binomial model to consider the possibility of early exercise. In this version, the up and the down movements for asset prices in each period can be estimated from the variance. 2
+Approach 1: Pseudo-American Valuation
+Step 1: Define when dividends will be paid and how much the dividends will be.
+Step 2: Value the call option to each ex-dividend date using the dividend-adjusted approach described earlier, where the stock price is reduced by the present value of expected dividends.
+Step 3: Choose the maximum of the call values estimated for each ex-dividend day. ILLUSTRATION 5.5: Using Pseudo-American Option Valuation to Adjust for Early Exercise
+Consider an option with a strike price of $35 on a stock trading at $40. The variance in the ln(stock prices) is 0.05, and the riskless rate is 4%. The option has a remaining life of eight months, and there are three dividends expected during this period: Expected Dividend
+Ex-Dividend Day $0.80
+In 1 month $0.80
+In 4 months $0.80
+In 7 months The call option is first valued to just before the first ex-dividend date: The value from the Black-Scholes model is: The call option is then valued to before the second ex-dividend date: The value of the call based on these parameters is: The call option is then valued to before the third ex-dividend date: The value of the call based on these parameters is: The call option is then valued to expiration: The value of the call based on these parameters is: Approach 2: Using the Binomial Model
+The binomial model is much more capable of handling early exercise because it considers the cash flows at each time period, rather than just at expiration. The biggest limitation of the binomial model is determining what stock prices will be at the end of each period, but this can be overcome by using a variant that allows us to estimate the up and the down movements in stock prices from the estimated variance. There are four steps involved:
+Step 1: If the variance in ln(stock prices) has been estimated for the Black-Scholes valuation, convert these into inputs for the binomial model: where u and d are the up and the down movements per unit time for the binomial, and dt is the number of periods within each year (or unit time).
+Step 2: Specify the period in which the dividends will be paid and make the assumption that the price will drop by the amount of the dividend in that period.
+Step 3: Value the call at each node of the tree, allowing for the possibility of early exercise just before ex-dividend dates. There will be early exercise if the remaining time premium on the option is less than the expected drop in option value as a consequence of the dividend payment.
+Step 4: Value the call at time 0, using the standard binomial approach. bstobin.xls: This spreadsheet allows you to estimate the parameters for a binomial model from the inputs to a Black-Scholes model. Impact of Exercise on Underlying Asset Value
+The Black-Scholes model is based on the assumption that exercising an option does not affect the value of the underlying asset. This may be true for listed options on stocks, but it is not true for some types of options. For instance, the exercise of warrants increases the number of shares outstanding and brings fresh cash into the firm, both of which will affect the stock price.3 The expected negative impact (dilution) of exercise will decrease the value of warrants, compared to otherwise similar call options. The adjustment for dilution to the stock price is fairly simple in the Black-Scholes valuation. The stock price is adjusted for the expected dilution from the exercise of the options. In the case of warrants, for instance: where S = Current value of the stock
+nw = Number of warrants outstanding
+W = Value of warrants outstanding
+ns = Number of shares outstanding
+When the warrants are exercised, the number of shares outstanding will increase, reducing the stock price. The numerator reflects the market value of equity, including both stocks and warrants outstanding. The reduction in S will reduce the value of the call option.
+There is an element of circularity in this analysis, since the value of the warrant is needed to estimate the dilution-adjusted S and the dilution-adjusted S is needed to estimate the value of the warrant. This problem can be resolved by starting the process off with an assumed value for the warrant (e.g., the exercise value or the current market price of the warrant). This will yield a value for the warrant, and this estimated value can then be used as an input to reestimate the warrant’s value until there is convergence. FROM BLACK-SCHOLES TO BINOMIAL
+The process of converting the continuous variance in a Black-Scholes model to a binomial tree is a fairly simple one. Assume, for instance, that you have an asset that is trading at $30 currently and that you estimate the annualized standard deviation in the asset value to be 40 percent; the annualized riskless rate is 5 percent. For simplicity, let us assume that the option that you are valuing has a four-year life and that each period is a year. To estimate the prices at the end of each of the four years, we begin by first estimating the up and down movements in the binomial: Based on these estimates, we can obtain the prices at the end of the first node of the tree (the end of the first year): Progressing through the rest of the tree, we obtain the following numbers: ILLUSTRATION 5.6: Valuing a Warrant on Avatek Corporation
+Avatek Corporation is a real estate firm with 19.637 million shares outstanding, trading at $0.38 a share. In March 2001 the company had 1.8 million options outstanding, with four years to expiration and with an exercise price of $2.25. The stock paid no dividends, and the standard deviation in ln(stock prices) was 93%. The four-year Treasury bond rate was 4.9%. (The options were trading at $0.12 apiece at the time of this analysis.)
+The inputs to the warrant valuation model are as follows: The results of the Black-Scholes valuation of this option are: The options were trading at $0.12 in March 2001. Since the value was equal to the price, there was no need for further iterations. If there had been a difference, we would have reestimated the adjusted stock price and option value. If the options had been non-traded (as is the case with management options), this calculation would have required an iterative process, where the option value is used to get the adjusted value per share and the value per share to get the option value. warrant.xls: This spreadsheet allows you to estimate the value of an option when there is a potential dilution from exercise. The Black-Scholes Model for Valuing Puts
+The value of a put can be derived from the value of a call with the same strike price and the same expiration date: where C is the value of the call and P is the value of the put. This relationship between the call and put values is called put-call parity, and any deviations from parity can be used by investors to make riskless profits. To see why put-call parity holds, consider selling a call and buying a put with exercise price K and expiration date t, and simultaneously buying the underlying asset at the current price S. The payoff from this position is riskless and always yields K at expiration (t). To see this, assume that the stock price at expiration is S*. The payoff on each of the positions in the portfolio can be written as follows: Position
+Payoffs at t if S* > K
+Payoffs at t if S* < K Sell call
+–(S* – K)
+0 Buy put
+0
+K – S* Buy stock
+S*
+S* Total
+K
+K Since this position yields K with certainty, the cost of creating this position must be equal to the present value of K at the riskless rate (K e–rt). Substituting the Black-Scholes equation for the value of an equivalent call into this equation, we get: Thus, the replicating portfolio for a put is created by selling short [1 – N(d1)] shares of stock and investing K e–rt[1 – N(d2)] in the riskless asset. ILLUSTRATION 5.7: Valuing a Put Using Put-Call Parity: Cisco Systems and AT&T
+Consider the call on Cisco Systems that we valued in Illustration 5.2. The call had a strike price of $15 on the stock, had 103 days left to expiration, and was valued at $1.87. The stock was trading at $13.62, and the riskless rate was 4.63%. The put can be valued as follows: The put was trading at $3.38.
+Also, a long-term call on AT&T was valued in Illustration 5.4. The call had a strike price of $20, 1.8333 years left to expiration, and a value of $6.63. The stock was trading at $20.50 and was expected to maintain a dividend yield of 2.51% over the period. The riskless rate was 4.85%. The put value can be estimated as follows: The put was trading at $3.80. Both the call and the put were trading at different prices from our estimates, which may indicate that we have not correctly estimated the stock’s volatility. Jump Process Option Pricing Models
+If price changes remain larger as the time periods in the binomial model are shortened, it can no longer be assumed that prices change continuously. When price changes remain large, a price process that allows for price jumps is much more realistic. Cox and Ross (1976) valued options when prices follow a pure jump process, where the jumps can only be positive. Thus, in the next interval, the stock price will either have a large positive jump with a specified probability or drift downward at a given rate.
+Merton (1976) considered a distribution where there are price jumps superimposed on a continuous price process. He specified the rate at which jumps occur (λ) and the average jump size (k), measured as a percentage of the stock price. The model derived to value options with this process is called a jump diffusion model. In this model, the value of an option is determined by the five variables specified in the Black-Scholes model, and the parameters of the jump process (λ, k). Unfortunately, the estimates of the jump process parameters are so difficult to make for most firms that they overwhelm any advantages that accrue from using a more realistic model. These models, therefore, have seen limited use in practice.
+EXTENSIONS OF OPTION PRICING
+All the option pricing models described so far—the binomial, the Black-Scholes, and the jump process models—are designed to value options with clearly defined exercise prices and maturities on underlying assets that are traded. However, the options we encounter in investment analysis or valuation are often on real assets rather than financial assets. Categorized as real options, they can take much more complicated forms. This section considers some of these variations.
+Capped and Barrier Options
+With a simple call option, there is no specified upper limit on the profits that can be made by the buyer of the call. Asset prices, at least in theory, can keep going up, and the payoffs increase proportionately. In some call options, though, the buyer is entitled to profits up to a specified price but not above it. For instance, consider a call option with a strike price of K1 on an asset. In an unrestricted call option, the payoff on this option will increase as the underlying asset’s price increases above K1. Assume, however, that if the price reaches K2, the payoff is capped at (K2 – K1). The payoff diagram on this option is shown in Figure 5.5. Figure 5.5 Payoff on Capped Call This option is called a capped call. Notice, also, that once the price reaches K2, there is no longer any time premium associated with the option, and the option will therefore be exercised. Capped calls are part of a family of options called barrier options, where the payoff on and the life of the option are functions of whether the underlying asset price reaches a certain level during a specified period.
+The value of a capped call is always lower than the value of the same call without the payoff limit. A simple approximation of this value can be obtained by valuing the call twice, once with the given exercise price and once with the cap, and taking the difference in the two values. In the preceding example, then, the value of the call with an exercise price of K1 and a cap at K2 can be written as: Barrier options can take many forms. In a knockout option, an option ceases to exist if the underlying asset reaches a certain price. In the case of a call option, this knockout price is usually set below the strike price, and this option is called a down-and-out option. In the case of a put option, the knockout price will be set above the exercise price, and this option is called an up-and-out option. Like the capped call, these options are worth less than their unrestricted counterparts. Many real options have limits on potential upside, or knockout provisions, and ignoring these limits can result in the overstatement of the value of these options.
+Compound Options
+Some options derive their value not from an underlying asset, but from other options. These options are called compound options. Compound options can take any of four forms—a call on a call, a put on a put, a call on a put, or a put on a call. Geske (1979) developed the analytical formulation for valuing compound options by replacing the standard normal distribution used in a simple option model with a bivariate normal distribution in the calculation.
+Consider, for instance, the option to expand a project that is discussed in Chapter 30. While we will value this option using a simple option pricing model, in reality there could be multiple stages in expansion, with each stage representing an option for the following stage. In this case, we will undervalue the option by considering it as a simple rather than a compound option.
+Notwithstanding this discussion, the valuation of compound options becomes progressively more difficult as more options are added to the chain. In this case, rather than wreck the valuation on the shoals of estimation error, it may be better to accept the conservative estimate that is provided with a simple valuation model as a floor on the value.
+Rainbow Options
+In a simple option, the uncertainty is about the price of the underlying asset. Some options are exposed to two or more sources of uncertainty, and these options are rainbow options. Using the simple option pricing model to value such options can lead to biased estimates of value. As an example, consider an undeveloped oil reserve as an option, where the firm that owns the reserve has the right to develop the reserve. Here there are two sources of uncertainty. The first is obviously the price of oil, and the second is the quantity of oil that is in the reserve. To value this undeveloped reserve, we can make the simplifying assumption that we know the quantity of oil in the reserve with certainty. In reality, however, uncertainty about the quantity will affect the value of this option and make the decision to exercise more difficult.4
+CONCLUSION
+An option is an asset with payoffs that are contingent on the value of an underlying asset. A call option provides its holder with the right to buy the underlying asset at a fixed price, whereas a put option provides its holder with the right to sell at a fixed price, at any time before the expiration of the option. The value of an option is determined by six variables—the current value of the underlying asset, the variance in this value, the expected dividends on the asset, the strike price and life of the option, and the riskless interest rate. This is illustrated in both the binomial and the Black-Scholes models, which value options by creating replicating portfolios composed of the underlying asset and riskless lending or borrowing. These models can be used to value assets that have option like characteristics.
+QUESTIONS AND SHORT PROBLEMS
+In the problems following, use an equity risk premium of 5.5 percent if none is specified. 1. The following are prices of options traded on Microsoft Corporation, which pays no dividends. The stock is trading at $83, and the annualized riskless rate is 3.8%. The standard deviation in ln(stock prices) (based on historical data) is 30%. a. Estimate the value of a three-month call with a strike price of $85.
+b. Using the inputs from the Black-Scholes model, specify how you would replicate this call.
+c. What is the implied standard deviation in this call?
+d. Assume now that you buy a call with a strike price of $85 and sell a call with a strike price of $90. Draw the payoff diagram on this position.
+e. Using put-call parity, estimate the value of a three-month put with a strike price of $85. 2. You are trying to value three-month call and put options on Merck with a strike price of $30. The stock is trading at $28.75, and the company expects to pay a quarterly dividend per share of $0.28 in two months. The annualized riskless interest rate is 3.6%, and the standard deviation in log stock prices is 20%. a. Estimate the value of the call and put options, using the Black-Scholes model.
+b. What effect does the expected dividend payment have on call values? On put values? Why? 3. There is the possibility that the options on Merck described in the preceding problem could be exercised early. a. Use the pseudo-American call option technique to determine whether this will affect the value of the call.
+b. Why does the possibility of early exercise exist? What types of options are most likely to be exercised early? 4. You have been provided the following information on a three-month call: a. If you wanted to replicate buying this call, how much money would you need to borrow?
+b. If you wanted to replicate buying this call, how many shares of stock would you need to buy? 5. Go Video, a manufacturer of video recorders, was trading at $4 per share in May 1994. There were 11 million shares outstanding. At the same time, it had 550,000 one-year warrants outstanding, with a strike price of $4.25. The stock has had a standard deviation of 60%. The stock does not pay a dividend. The riskless rate is 5%. a. Estimate the value of the warrants, ignoring dilution.
+b. Estimate the value of the warrants, allowing for dilution.
+c. Why does dilution reduce the value of the warrants? 6. You are trying to value a long-term call option on the NYSE Composite index, expiring in five years, with a strike price of 275. The index is currently at 250, and the annualized standard deviation in stock prices is 15%. The average dividend yield on the index is 3% and is expected to remain unchanged over the next five years. The five-year Treasury bond rate is 5%. a. Estimate the value of the long-term call option.
+b. Estimate the value of a put option with the same parameters.
+c. What are the implicit assumptions you are making when you use the Black-Scholes model to value this option? Which of these assumptions are likely to be violated? What are the consequences for your valuation? 7. A new security on AT&T will entitle the investor to all dividends on AT&T over the next three years, limiting upside potential to 20% but also providing downside protection below 10%. AT&T stock is trading at $50, and three-year call and put options are traded on the exchange at the following prices: How much would you be willing to pay for this security? 1 Note, though, that higher variance can reduce the value of the underlying asset. As a call option becomes more in-the-money, the more it resembles the underlying asset. For very deep in-the-money call options, higher variance can reduce the value of the option.
+2 To illustrate, if σ2 is the variance in ln(stock prices), the up and the down movements in the binomial can be estimated as follows: where u and d are the up and down movements per unit time for the binomial, T is the life of the option, and m is the number of periods within that lifetime.
+3 Warrants are call options issued by firms, either as part of management compensation contracts or to raise equity.
+4 The analogy to a listed option on a stock is the case where you do not know with certainty what the stock price is when you exercise the option. The more uncertain you are about the stock price, the more margin for error you have to give yourself when you exercise the option, to ensure that you are in fact earning a profit.
+CHAPTER 5
+Option Pricing Theory and Models
+In general, the value of any asset is the present value of the expected cash flows on that asset. This chapter considers an exception to that rule when it looks at assets with two specific characteristics: 1. The assets derive their value from the values of other assets.
+2. The cash flows on the assets are contingent on the occurrence of specific events. These assets are called options, and the present value of the expected cash flows on these assets will understate their true value. This chapter describes the cash flow characteristics of options, considers the factors that determine their value, and examines how best to value them.
+BASICS OF OPTION PRICING
+An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called a strike price or an exercise price) at or before the expiration date of the option. Since it is a right and not an obligation, the holder can choose not to exercise the right and can allow the option to expire. There are two types of options—call options and put options.
+Call and Put Options: Description and Payoff Diagrams
+A call option gives the buyer of the option the right to buy the underlying asset at the strike price or the exercise price at any time prior to the expiration date of the option. The buyer pays a price for this right. If at expiration the value of the asset is less than the strike price, the option is not exercised and expires worthless. If, however, the value of the asset is greater than the strike price, the option is exercised—the buyer of the option buys the stock at the exercise price, and the difference between the asset value and the exercise price comprises the gross profit on the investment. The net profit on the investment is the difference between the gross profit and the price paid for the call initially.
+A payoff diagram illustrates the cash payoff on an option at expiration. For a call, the net payoff is negative (and equal to the price paid for the call) if the value of the underlying asset is less than the strike price. If the price of the underlying asset exceeds the strike price, the gross payoff is the difference between the value of the underlying asset and the strike price, and the net payoff is the difference between the gross payoff and the price of the call. This is illustrated in Figure 5.1. Figure 5.1 Payoff on Call Option A put option gives the buyer of the option the right to sell the underlying asset at a fixed price, again called the strike or exercise price, at any time prior to the expiration date of the option. The buyer pays a price for this right. If the price of the underlying asset is greater than the strike price, the option will not be exercised and will expire worthless. But if the price of the underlying asset is less than the strike price, the owner of the put option will exercise the option and sell the stock at the strike price, claiming the difference between the strike price and the market value of the asset as the gross profit. Again, netting out the initial cost paid for the put yields the net profit from the transaction.
+A put has a negative net payoff if the value of the underlying asset exceeds the strike price, and has a gross payoff equal to the difference between the strike price and the value of the underlying asset if the asset value is less than the strike price. This is summarized in Figure 5.2. Figure 5.2 Payoff on Put Option DETERMINANTS OF OPTION VALUE
+The value of an option is determined by six variables relating to the underlying asset and financial markets. 1. Current value of the underlying asset. Options are assets that derive value from an underlying asset. Consequently, changes in the value of the underlying asset affect the value of the options on that asset. Since calls provide the right to buy the underlying asset at a fixed price, an increase in the value of the asset will increase the value of the calls. Puts, on the other hand, become less valuable as the value of the asset increases.
+2. Variance in value of the underlying asset. The buyer of an option acquires the right to buy or sell the underlying asset at a fixed price. The higher the variance in the value of the underlying asset, the greater the value of the option.1 This is true for both calls and puts. While it may seem counterintuitive that an increase in a risk measure (variance) should increase value, options are different from other securities since buyers of options can never lose more than the price they pay for them; in fact, they have the potential to earn significant returns from large price movements.
+3. Dividends paid on the underlying asset. The value of the underlying asset can be expected to decrease if dividend payments are made on the asset during the life of the option. Consequently, the value of a call on the asset is a decreasing function of the size of expected dividend payments, and the value of a put is an increasing function of expected dividend payments. A more intuitive way of thinking about dividend payments, for call options, is as a cost of delaying exercise on in-the-money options. To see why, consider an option on a traded stock. Once a call option is in-the-money (i.e., the holder of the option will make a gross payoff by exercising the option), exercising the call option will provide the holder with the stock and entitle him or her to the dividends on the stock in subsequent periods. Failing to exercise the option will mean that these dividends are forgone.
+4. Strike price of the option. A key characteristic used to describe an option is the strike price. In the case of calls, where the holder acquires the right to buy at a fixed price, the value of the call will decline as the strike price increases. In the case of puts, where the holder has the right to sell at a fixed price, the value will increase as the strike price increases.
+5. Time to expiration on the option. Both calls and puts are more valuable the greater the time to expiration. This is because the longer time to expiration provides more time for the value of the underlying asset to move, increasing the value of both types of options. Additionally, in the case of a call, where the buyer has the right to pay a fixed price at expiration, the present value of this fixed price decreases as the life of the option increases, increasing the value of the call.
+6. Riskless interest rate corresponding to life of the option. Since the buyer of an option pays the price of the option up front, an opportunity cost is involved. This cost will depend on the level of interest rates and the time to expiration of the option. The riskless interest rate also enters into the valuation of options when the present value of the exercise price is calculated, since the exercise price does not have to be paid (received) until expiration on calls (puts). Increases in the interest rate will increase the value of calls and reduce the value of puts. Table 5.1 summarizes the variables and their predicted effects on call and put prices.
+Table 5.1 Summary of Variables Affecting Call and Put Prices Effect On Factor
+Call Value
+Put Value Increase in underlying asset’s value
+Increases
+Decreases Increase in variance of underlying asset
+Increases
+Increases Increase in strike price
+Decreases
+Increases Increase in dividends paid
+Decreases
+Increases Increase in time to expiration
+Increases
+Increases Increase in interest rates
+Increases
+Decreases American versus European Options: Variables Relating to Early Exercise
+A primary distinction between American and European options is that an American option can be exercised at any time prior to its expiration, while European options can be exercised only at expiration. The possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. There is one compensating factor that enables the former to be valued using models designed for the latter. In most cases, the time premium associated with the remaining life of an option and transaction costs make early exercise suboptimal. In other words, the holders of in-the-money options generally get much more by selling the options to someone else than by exercising the options.
+OPTION PRICING MODELS
+Option pricing theory has made vast strides since 1972, when Fischer Black and Myron Scholes published their pathbreaking paper that provided a model for valuing dividend-protected European options. Black and Scholes used a “replicating portfolio”—a portfolio composed of the underlying asset and the risk-free asset that had the same cash flows as the option being valued—and the notion of arbitrage to come up with their final formulation. Although their derivation is mathematically complicated, there is a simpler binomial model for valuing options that draws on the same logic.
+Binomial Model
+The binomial option pricing model is based on a simple formulation for the asset price process in which the asset, in any time period, can move to one of two possible prices. The general formulation of a stock price process that follows the binomial path is shown in Figure 5.3. In this figure, S is the current stock price; the price moves up to Su with probability p and down to Sd with probability 1 – p in any time period. Figure 5.3 General Formulation for Binomial Price Path Creating a Replicating Portfolio
+The objective in creating a replicating portfolio is to use a combination of risk-free borrowing/lending and the underlying asset to create the same cash flows as the option being valued. The principles of arbitrage apply then, and the value of the option must be equal to the value of the replicating portfolio. In the case of the general formulation shown in Figure 5.3, where stock prices can move either up to Su or down to Sd in any time period, the replicating portfolio for a call with strike price K will involve borrowing $B and acquiring Δ of the underlying asset, where: where Cu = Value of the call if the stock price is Su
+Cd = Value of the call if the stock price is Sd
+In a multiperiod binomial process, the valuation has to proceed iteratively (i.e., starting with the final time period and moving backward in time until the current point in time). The portfolios replicating the option are created at each step and valued, providing the values for the option in that time period. The final output from the binomial option pricing model is a statement of the value of the option in terms of the replicating portfolio, composed of Δ shares (option delta) of the underlying asset and risk-free borrowing/lending. ILLUSTRATION 5.1: Binomial Option Valuation
+Assume that the objective is to value a call with a strike price of $50, which is expected to expire in two time periods, on an underlying asset whose price currently is $50 and is expected to follow a binomial process: Now assume that the interest rate is 11%. In addition, define: The objective is to combined Δ shares of stock and B dollars of borrowing to replicate the cash flows from the call with a strike price of $50. This can be done iteratively, starting with the last period and working back through the binomial tree.
+Step 1: Start with the end nodes and work backward: Thus, if the stock price is $70 at t = 1, borrowing $45 and buying one share of the stock will give the same cash flows as buying the call. The value of the call at t = 1, if the stock price is $70, is therefore: Considering the other leg of the binomial tree at t = 1, If the stock price is $35 at t = 1, then the call is worth nothing.
+Step 2: Move backward to the earlier time period and create a replicating portfolio that will provide the cash flows the option will provide. In other words, borrowing $22.50 and buying five-sevenths of a share will provide the same cash flows as a call with a strike price of $50 over the call’s lifetime. The value of the call therefore has to be the same as the cost of creating this position. The Determinants of Value
+The binomial model provides insight into the determinants of option value. The value of an option is not determined by the expected price of the asset but by its current price, which, of course, reflects expectations about the future. This is a direct consequence of arbitrage. If the option value deviates from the value of the replicating portfolio, investors can create an arbitrage position (i.e., one that requires no investment, involves no risk, and delivers positive returns). To illustrate, if the portfolio that replicates the call costs more than the call does in the market, an investor could buy the call, sell the replicating portfolio, and be guaranteed the difference as a profit. The cash flows on the two positions will offset each other, leading to no cash flows in subsequent periods. The call option value also increases as the time to expiration is extended, as the price movements (u and d) increase, and with increases in the interest rate.
+While the binomial model provides an intuitive feel for the determinants of option value, it requires a large number of inputs, in terms of expected future prices at each node. As time periods are made shorter in the binomial model, you can make one of two assumptions about asset prices.You can assume that price changes become smaller as periods get shorter; this leads to price changes becoming infinitesimally small as time periods approach zero, leading to a continuous price process. Alternatively, you can assume that price changes stay large even as the period gets shorter; this leads to a jump price process, where prices can jump in any period. This section considers the option pricing models that emerge with each of these assumptions.
+Black-Scholes Model
+When the price process is continuous (i.e., price changes become smaller as time periods get shorter), the binomial model for pricing options converges on the Black-Scholes model. The model, named after its cocreators, Fischer Black and Myron Scholes, allows us to estimate the value of any option using a small number of inputs, and has been shown to be robust in valuing many listed options.
+The Model
+While the derivation of the Black-Scholes model is far too complicated to present here, it is based on the idea of creating a portfolio of the underlying asset and the riskless asset with the same cash flows, and hence the same cost, as the option being valued. The value of a call option in the Black-Scholes model can be written as a function of the five variables: S = Current value of the underlying asset
+K = Strike price of the option
+t = Life to expiration of the option
+r = Riskless interest rate corresponding to the life of the option
+σ2 = Variance in the ln(value) of the underlying asset The value of a call is then: Note that e–rt is the present value factor, and reflects the fact that the exercise price on the call option does not have to be paid until expiration, since the model values European options. N(d1) and N(d2) are probabilities, estimated by using a cumulative standardized normal distribution, and the values of d1 and d2 obtained for an option. The cumulative distribution is shown in Figure 5.4. Figure 5.4 Cumulative Normal Distribution In approximate terms, N(d2) yields the likelihood that an option will generate positive cash flows for its owner at exercise (i.e., that S > K in the case of a call option and that K > S in the case of a put option). The portfolio that replicates the call option is created by buying N(d1) units of the underlying asset, and borrowing Ke–rt N(d2). The portfolio will have the same cash flows as the call option, and thus the same value as the option. N(d1), which is the number of units of the underlying asset that are needed to create the replicating portfolio, is called the option delta. A NOTE ON ESTIMATING THE INPUTS TO THE BLACK-SCHOLES MODEL
+The Black-Scholes model requires inputs that are consistent on time measurement. There are two places where this affects estimates. The first relates to the fact that the model works in continuous time, rather than discrete time. That is why we use the continuous time version of present value (exp–rt) rather than the discrete version, (1 + r)–t. It also means that the inputs such as the riskless rate have to be modified to make them continuous time inputs. For instance, if the one-year Treasury bond rate is 6.2 percent, the risk-free rate that is used in the Black-Scholes model should be: The second relates to the period over which the inputs are estimated. For instance, the preceding rate is an annual rate. The variance that is entered into the model also has to be an annualized variance. The variance, estimated from ln(asset prices), can be annualized easily because variances are linear in time if the serial correlation is zero. Thus, if monthly or weekly prices are used to estimate variance, the variance is annualized by multiplying by 12 or 52, respectively. ILLUSTRATION 5.2: Valuing an Option Using the Black-Scholes Model
+On March 6, 2001, Cisco Systems was trading at $13.62. We will attempt to value a July 2001 call option with a strike price of $15, trading on the CBOE on the same day for $2. The following are the other parameters of the options: The annualized standard deviation in Cisco Systems stock price over the previous year was 81%. This standard deviation is estimated using weekly stock prices over the year, and the resulting number was annualized as follows: The option expiration date is Friday, July 20, 2001. There are 103 days to expiration, and the annualized Treasury bill rate corresponding to this option life is 4.63%. The inputs for the Black-Scholes model are as follows: Current stock price (S) = $13.62
+Strike price on the option = $15
+Option life = 103/365 = 0.2822
+Standard deviation in ln(stock prices) = 81%
+Riskless rate = 4.63% Inputting these numbers into the model, we get: Using the normal distribution, we can estimate the N(d1) and N(d2): The value of the call can now be estimated: Since the call is trading at $2, it is slightly overvalued, assuming that the estimate of standard deviation used is correct. IMPLIED VOLATILITY
+The only input in the Black Scholes on which there can be significant disagreement among investors is the variance. While the variance is often estimated by looking at historical data, the values for options that emerge from using the historical variance can be different from the market prices. For any option, there is some variance at which the estimated value will be equal to the market price. This variance is called an implied variance.
+Consider the Cisco option valued in Illustration 5.2. With a standard deviation of 81 percent, the value of the call option with a strike price of $15 was estimated to be $1.87. Since the market price is higher than the calculated value, we tried higher standard deviations, and at a standard deviation 85.40 percent the value of the option is $2 (which is the market price). This is the implied standard deviation or implied volatility. Model Limitations and Fixes
+The Black-Scholes model was designed to value European options that can be exercised only at maturity and whose underlying assets do not pay dividends. In addition, options are valued based on the assumption that option exercise does not affect the value of the underlying asset. In practice, assets do pay dividends, options sometimes get exercised early, and exercising an option can affect the value of the underlying asset. Adjustments exist that, while not perfect, provide partial corrections to the Black-Scholes model.
+Dividends
+The payment of a dividend reduces the stock price; note that on the ex-dividend day, the stock price generally declines. Consequently, call options become less valuable and put options more valuable as expected dividend payments increase. There are two ways of dealing with dividends in the Black-Scholes model: 1. Short-term options. One approach to dealing with dividends is to estimate the present value of expected dividends that will be paid by the underlying asset during the option life and subtract it from the current value of the asset to use as S in the model. 2. Long-term options. Since it becomes less practical to estimate the present value of dividends the longer the option life, an alternate approach can be used. If the dividend yield (y = Dividends/Current value of the asset) on the underlying asset is expected to remain unchanged during the life of the option, the Black-Scholes model can be modified to take dividends into account. From an intuitive standpoint, the adjustments have two effects. First, the value of the asset is discounted back to the present at the dividend yield to take into account the expected drop in asset value resulting from dividend payments. Second, the interest rate is offset by the dividend yield to reflect the lower carrying cost from holding the asset (in the replicating portfolio). The net effect will be a reduction in the value of calls estimated using this model. ILLUSTRATION 5.3: Valuing a Short-Term Option with Dividend Adjustments—The Black-Scholes Correction
+Assume that it is March 6, 2001, and that AT&T is trading at $20.50 a share. Consider a call option on the stock with a strike price of $20, expiring on July 20, 2001. Using past stock prices, the annualized standard deviation in the log of stock prices for AT&T is estimated at 60%. There is one dividend, amounting to $0.15, and it will be paid in 23 days. The riskless rate is 4.63%. Present value of expected dividend = $0.15/1.046323/365 = $0.15
+Dividend-adjusted stock price = $20.50 – $0.15 = $20.35
+Time to expiration = 103/365 = 0.2822
+Variance in ln(stock prices) = 0.62 = 0.36
+Riskless rate = 4.63% The value from the Black-Scholes model is: The call option was trading at $2.60 on that day. ILLUSTRATION 5.4: Valuing a Long-Term Option with Dividend Adjustments—Primes and Scores
+The CBOE offers longer-term call and put options on some stocks. On March 6, 2001, for instance, you could have purchased an AT&T call expiring on January 17, 2003. The stock price for AT&T is $20.50 (as in the previous example). The following is the valuation of a call option with a strike price of $20. Instead of estimating the present value of dividends over the next two years, assume that AT&T’s dividend yield will remain 2.51% over this period and that the risk-free rate for a two-year Treasury bond is 4.85%. The inputs to the Black-Scholes model are: The value from the Black-Scholes model is: The call was trading at $5.80 on March 8, 2001. optst.xls: This spreadsheet allows you to estimate the value of a short-term option when the expected dividends during the option life can be estimated. optlt.xls: This spreadsheet allows you to estimate the value of an option when the underlying asset has a constant dividend yield. Early Exercise
+The Black-Scholes model was designed to value European options that can be exercised only at expiration. In contrast, most options that we encounter in practice are American options and can be exercised at any time until expiration. As mentioned earlier, the possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. In general, though, with traded options, it is almost always better to sell the option to someone else rather than exercise early, since options have a time premium (i.e., they sell for more than their exercise value). There are two exceptions. One occurs when the underlying asset pays large dividends, thus reducing the expected value of the asset. In this case, call options may be exercised just before an ex-dividend date, if the time premium on the options is less than the expected decline in asset value as a consequence of the dividend payment. The other exception arises when an investor holds both the underlying asset and deep in-the-money puts (i.e., puts with strike prices well above the current price of the underlying asset) on that asset at a time when interest rates are high. In this case, the time premium on the put may be less than the potential gain from exercising the put early and earning interest on the exercise price.
+There are two basic ways of dealing with the possibility of early exercise. One is to continue to use the unadjusted Black-Scholes model and to regard the resulting value as a floor or conservative estimate of the true value. The other is to try to adjust the value of the option for the possibility of early exercise. There are two approaches for doing so. One uses the Black-Scholes model to value the option to each potential exercise date. With options on stocks, this basically requires that the investor values options to each ex-dividend day and chooses the maximum of the estimated call values. The second approach is to use a modified version of the binomial model to consider the possibility of early exercise. In this version, the up and the down movements for asset prices in each period can be estimated from the variance. 2
+Approach 1: Pseudo-American Valuation
+Step 1: Define when dividends will be paid and how much the dividends will be.
+Step 2: Value the call option to each ex-dividend date using the dividend-adjusted approach described earlier, where the stock price is reduced by the present value of expected dividends.
+Step 3: Choose the maximum of the call values estimated for each ex-dividend day. ILLUSTRATION 5.5: Using Pseudo-American Option Valuation to Adjust for Early Exercise
+Consider an option with a strike price of $35 on a stock trading at $40. The variance in the ln(stock prices) is 0.05, and the riskless rate is 4%. The option has a remaining life of eight months, and there are three dividends expected during this period: Expected Dividend
+Ex-Dividend Day $0.80
+In 1 month $0.80
+In 4 months $0.80
+In 7 months The call option is first valued to just before the first ex-dividend date: The value from the Black-Scholes model is: The call option is then valued to before the second ex-dividend date: The value of the call based on these parameters is: The call option is then valued to before the third ex-dividend date: The value of the call based on these parameters is: The call option is then valued to expiration: The value of the call based on these parameters is: Approach 2: Using the Binomial Model
+The binomial model is much more capable of handling early exercise because it considers the cash flows at each time period, rather than just at expiration. The biggest limitation of the binomial model is determining what stock prices will be at the end of each period, but this can be overcome by using a variant that allows us to estimate the up and the down movements in stock prices from the estimated variance. There are four steps involved:
+Step 1: If the variance in ln(stock prices) has been estimated for the Black-Scholes valuation, convert these into inputs for the binomial model: where u and d are the up and the down movements per unit time for the binomial, and dt is the number of periods within each year (or unit time).
+Step 2: Specify the period in which the dividends will be paid and make the assumption that the price will drop by the amount of the dividend in that period.
+Step 3: Value the call at each node of the tree, allowing for the possibility of early exercise just before ex-dividend dates. There will be early exercise if the remaining time premium on the option is less than the expected drop in option value as a consequence of the dividend payment.
+Step 4: Value the call at time 0, using the standard binomial approach. bstobin.xls: This spreadsheet allows you to estimate the parameters for a binomial model from the inputs to a Black-Scholes model. Impact of Exercise on Underlying Asset Value
+The Black-Scholes model is based on the assumption that exercising an option does not affect the value of the underlying asset. This may be true for listed options on stocks, but it is not true for some types of options. For instance, the exercise of warrants increases the number of shares outstanding and brings fresh cash into the firm, both of which will affect the stock price.3 The expected negative impact (dilution) of exercise will decrease the value of warrants, compared to otherwise similar call options. The adjustment for dilution to the stock price is fairly simple in the Black-Scholes valuation. The stock price is adjusted for the expected dilution from the exercise of the options. In the case of warrants, for instance: where S = Current value of the stock
+nw = Number of warrants outstanding
+W = Value of warrants outstanding
+ns = Number of shares outstanding
+When the warrants are exercised, the number of shares outstanding will increase, reducing the stock price. The numerator reflects the market value of equity, including both stocks and warrants outstanding. The reduction in S will reduce the value of the call option.
+There is an element of circularity in this analysis, since the value of the warrant is needed to estimate the dilution-adjusted S and the dilution-adjusted S is needed to estimate the value of the warrant. This problem can be resolved by starting the process off with an assumed value for the warrant (e.g., the exercise value or the current market price of the warrant). This will yield a value for the warrant, and this estimated value can then be used as an input to reestimate the warrant’s value until there is convergence. FROM BLACK-SCHOLES TO BINOMIAL
+The process of converting the continuous variance in a Black-Scholes model to a binomial tree is a fairly simple one. Assume, for instance, that you have an asset that is trading at $30 currently and that you estimate the annualized standard deviation in the asset value to be 40 percent; the annualized riskless rate is 5 percent. For simplicity, let us assume that the option that you are valuing has a four-year life and that each period is a year. To estimate the prices at the end of each of the four years, we begin by first estimating the up and down movements in the binomial: Based on these estimates, we can obtain the prices at the end of the first node of the tree (the end of the first year): Progressing through the rest of the tree, we obtain the following numbers: ILLUSTRATION 5.6: Valuing a Warrant on Avatek Corporation
+Avatek Corporation is a real estate firm with 19.637 million shares outstanding, trading at $0.38 a share. In March 2001 the company had 1.8 million options outstanding, with four years to expiration and with an exercise price of $2.25. The stock paid no dividends, and the standard deviation in ln(stock prices) was 93%. The four-year Treasury bond rate was 4.9%. (The options were trading at $0.12 apiece at the time of this analysis.)
+The inputs to the warrant valuation model are as follows: The results of the Black-Scholes valuation of this option are: The options were trading at $0.12 in March 2001. Since the value was equal to the price, there was no need for further iterations. If there had been a difference, we would have reestimated the adjusted stock price and option value. If the options had been non-traded (as is the case with management options), this calculation would have required an iterative process, where the option value is used to get the adjusted value per share and the value per share to get the option value. warrant.xls: This spreadsheet allows you to estimate the value of an option when there is a potential dilution from exercise. The Black-Scholes Model for Valuing Puts
+The value of a put can be derived from the value of a call with the same strike price and the same expiration date: where C is the value of the call and P is the value of the put. This relationship between the call and put values is called put-call parity, and any deviations from parity can be used by investors to make riskless profits. To see why put-call parity holds, consider selling a call and buying a put with exercise price K and expiration date t, and simultaneously buying the underlying asset at the current price S. The payoff from this position is riskless and always yields K at expiration (t). To see this, assume that the stock price at expiration is S*. The payoff on each of the positions in the portfolio can be written as follows: Position
+Payoffs at t if S* > K
+Payoffs at t if S* < K Sell call
+–(S* – K)
+0 Buy put
+0
+K – S* Buy stock
+S*
+S* Total
+K
+K Since this position yields K with certainty, the cost of creating this position must be equal to the present value of K at the riskless rate (K e–rt). Substituting the Black-Scholes equation for the value of an equivalent call into this equation, we get: Thus, the replicating portfolio for a put is created by selling short [1 – N(d1)] shares of stock and investing K e–rt[1 – N(d2)] in the riskless asset. ILLUSTRATION 5.7: Valuing a Put Using Put-Call Parity: Cisco Systems and AT&T
+Consider the call on Cisco Systems that we valued in Illustration 5.2. The call had a strike price of $15 on the stock, had 103 days left to expiration, and was valued at $1.87. The stock was trading at $13.62, and the riskless rate was 4.63%. The put can be valued as follows: The put was trading at $3.38.
+Also, a long-term call on AT&T was valued in Illustration 5.4. The call had a strike price of $20, 1.8333 years left to expiration, and a value of $6.63. The stock was trading at $20.50 and was expected to maintain a dividend yield of 2.51% over the period. The riskless rate was 4.85%. The put value can be estimated as follows: The put was trading at $3.80. Both the call and the put were trading at different prices from our estimates, which may indicate that we have not correctly estimated the stock’s volatility. Jump Process Option Pricing Models
+If price changes remain larger as the time periods in the binomial model are shortened, it can no longer be assumed that prices change continuously. When price changes remain large, a price process that allows for price jumps is much more realistic. Cox and Ross (1976) valued options when prices follow a pure jump process, where the jumps can only be positive. Thus, in the next interval, the stock price will either have a large positive jump with a specified probability or drift downward at a given rate.
+Merton (1976) considered a distribution where there are price jumps superimposed on a continuous price process. He specified the rate at which jumps occur (λ) and the average jump size (k), measured as a percentage of the stock price. The model derived to value options with this process is called a jump diffusion model. In this model, the value of an option is determined by the five variables specified in the Black-Scholes model, and the parameters of the jump process (λ, k). Unfortunately, the estimates of the jump process parameters are so difficult to make for most firms that they overwhelm any advantages that accrue from using a more realistic model. These models, therefore, have seen limited use in practice.
+EXTENSIONS OF OPTION PRICING
+All the option pricing models described so far—the binomial, the Black-Scholes, and the jump process models—are designed to value options with clearly defined exercise prices and maturities on underlying assets that are traded. However, the options we encounter in investment analysis or valuation are often on real assets rather than financial assets. Categorized as real options, they can take much more complicated forms. This section considers some of these variations.
+Capped and Barrier Options
+With a simple call option, there is no specified upper limit on the profits that can be made by the buyer of the call. Asset prices, at least in theory, can keep going up, and the payoffs increase proportionately. In some call options, though, the buyer is entitled to profits up to a specified price but not above it. For instance, consider a call option with a strike price of K1 on an asset. In an unrestricted call option, the payoff on this option will increase as the underlying asset’s price increases above K1. Assume, however, that if the price reaches K2, the payoff is capped at (K2 – K1). The payoff diagram on this option is shown in Figure 5.5. Figure 5.5 Payoff on Capped Call This option is called a capped call. Notice, also, that once the price reaches K2, there is no longer any time premium associated with the option, and the option will therefore be exercised. Capped calls are part of a family of options called barrier options, where the payoff on and the life of the option are functions of whether the underlying asset price reaches a certain level during a specified period.
+The value of a capped call is always lower than the value of the same call without the payoff limit. A simple approximation of this value can be obtained by valuing the call twice, once with the given exercise price and once with the cap, and taking the difference in the two values. In the preceding example, then, the value of the call with an exercise price of K1 and a cap at K2 can be written as: Barrier options can take many forms. In a knockout option, an option ceases to exist if the underlying asset reaches a certain price. In the case of a call option, this knockout price is usually set below the strike price, and this option is called a down-and-out option. In the case of a put option, the knockout price will be set above the exercise price, and this option is called an up-and-out option. Like the capped call, these options are worth less than their unrestricted counterparts. Many real options have limits on potential upside, or knockout provisions, and ignoring these limits can result in the overstatement of the value of these options.
+Compound Options
+Some options derive their value not from an underlying asset, but from other options. These options are called compound options. Compound options can take any of four forms—a call on a call, a put on a put, a call on a put, or a put on a call. Geske (1979) developed the analytical formulation for valuing compound options by replacing the standard normal distribution used in a simple option model with a bivariate normal distribution in the calculation.
+Consider, for instance, the option to expand a project that is discussed in Chapter 30. While we will value this option using a simple option pricing model, in reality there could be multiple stages in expansion, with each stage representing an option for the following stage. In this case, we will undervalue the option by considering it as a simple rather than a compound option.
+Notwithstanding this discussion, the valuation of compound options becomes progressively more difficult as more options are added to the chain. In this case, rather than wreck the valuation on the shoals of estimation error, it may be better to accept the conservative estimate that is provided with a simple valuation model as a floor on the value.
+Rainbow Options
+In a simple option, the uncertainty is about the price of the underlying asset. Some options are exposed to two or more sources of uncertainty, and these options are rainbow options. Using the simple option pricing model to value such options can lead to biased estimates of value. As an example, consider an undeveloped oil reserve as an option, where the firm that owns the reserve has the right to develop the reserve. Here there are two sources of uncertainty. The first is obviously the price of oil, and the second is the quantity of oil that is in the reserve. To value this undeveloped reserve, we can make the simplifying assumption that we know the quantity of oil in the reserve with certainty. In reality, however, uncertainty about the quantity will affect the value of this option and make the decision to exercise more difficult.4
+CONCLUSION
+An option is an asset with payoffs that are contingent on the value of an underlying asset. A call option provides its holder with the right to buy the underlying asset at a fixed price, whereas a put option provides its holder with the right to sell at a fixed price, at any time before the expiration of the option. The value of an option is determined by six variables—the current value of the underlying asset, the variance in this value, the expected dividends on the asset, the strike price and life of the option, and the riskless interest rate. This is illustrated in both the binomial and the Black-Scholes models, which value options by creating replicating portfolios composed of the underlying asset and riskless lending or borrowing. These models can be used to value assets that have option like characteristics.
+QUESTIONS AND SHORT PROBLEMS
+In the problems following, use an equity risk premium of 5.5 percent if none is specified. 1. The following are prices of options traded on Microsoft Corporation, which pays no dividends. The stock is trading at $83, and the annualized riskless rate is 3.8%. The standard deviation in ln(stock prices) (based on historical data) is 30%. a. Estimate the value of a three-month call with a strike price of $85.
+b. Using the inputs from the Black-Scholes model, specify how you would replicate this call.
+c. What is the implied standard deviation in this call?
+d. Assume now that you buy a call with a strike price of $85 and sell a call with a strike price of $90. Draw the payoff diagram on this position.
+e. Using put-call parity, estimate the value of a three-month put with a strike price of $85. 2. You are trying to value three-month call and put options on Merck with a strike price of $30. The stock is trading at $28.75, and the company expects to pay a quarterly dividend per share of $0.28 in two months. The annualized riskless interest rate is 3.6%, and the standard deviation in log stock prices is 20%. a. Estimate the value of the call and put options, using the Black-Scholes model.
+b. What effect does the expected dividend payment have on call values? On put values? Why? 3. There is the possibility that the options on Merck described in the preceding problem could be exercised early. a. Use the pseudo-American call option technique to determine whether this will affect the value of the call.
+b. Why does the possibility of early exercise exist? What types of options are most likely to be exercised early? 4. You have been provided the following information on a three-month call: a. If you wanted to replicate buying this call, how much money would you need to borrow?
+b. If you wanted to replicate buying this call, how many shares of stock would you need to buy? 5. Go Video, a manufacturer of video recorders, was trading at $4 per share in May 1994. There were 11 million shares outstanding. At the same time, it had 550,000 one-year warrants outstanding, with a strike price of $4.25. The stock has had a standard deviation of 60%. The stock does not pay a dividend. The riskless rate is 5%. a. Estimate the value of the warrants, ignoring dilution.
+b. Estimate the value of the warrants, allowing for dilution.
+c. Why does dilution reduce the value of the warrants? 6. You are trying to value a long-term call option on the NYSE Composite index, expiring in five years, with a strike price of 275. The index is currently at 250, and the annualized standard deviation in stock prices is 15%. The average dividend yield on the index is 3% and is expected to remain unchanged over the next five years. The five-year Treasury bond rate is 5%. a. Estimate the value of the long-term call option.
+b. Estimate the value of a put option with the same parameters.
+c. What are the implicit assumptions you are making when you use the Black-Scholes model to value this option? Which of these assumptions are likely to be violated? What are the consequences for your valuation? 7. A new security on AT&T will entitle the investor to all dividends on AT&T over the next three years, limiting upside potential to 20% but also providing downside protection below 10%. AT&T stock is trading at $50, and three-year call and put options are traded on the exchange at the following prices: How much would you be willing to pay for this security? 1 Note, though, that higher variance can reduce the value of the underlying asset. As a call option becomes more in-the-money, the more it resembles the underlying asset. For very deep in-the-money call options, higher variance can reduce the value of the option.
+2 To illustrate, if σ2 is the variance in ln(stock prices), the up and the down movements in the binomial can be estimated as follows: where u and d are the up and down movements per unit time for the binomial, T is the life of the option, and m is the number of periods within that lifetime.
+3 Warrants are call options issued by firms, either as part of management compensation contracts or to raise equity.
+4 The analogy to a listed option on a stock is the case where you do not know with certainty what the stock price is when you exercise the option. The more uncertain you are about the stock price, the more margin for error you have to give yourself when you exercise the option, to ensure that you are in fact earning a profit.
+CHAPTER 5
+Option Pricing Theory and Models
+In general, the value of any asset is the present value of the expected cash flows on that asset. This chapter considers an exception to that rule when it looks at assets with two specific characteristics: 1. The assets derive their value from the values of other assets.
+2. The cash flows on the assets are contingent on the occurrence of specific events. These assets are called options, and the present value of the expected cash flows on these assets will understate their true value. This chapter describes the cash flow characteristics of options, considers the factors that determine their value, and examines how best to value them.
+BASICS OF OPTION PRICING
+An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called a strike price or an exercise price) at or before the expiration date of the option. Since it is a right and not an obligation, the holder can choose not to exercise the right and can allow the option to expire. There are two types of options—call options and put options.
+Call and Put Options: Description and Payoff Diagrams
+A call option gives the buyer of the option the right to buy the underlying asset at the strike price or the exercise price at any time prior to the expiration date of the option. The buyer pays a price for this right. If at expiration the value of the asset is less than the strike price, the option is not exercised and expires worthless. If, however, the value of the asset is greater than the strike price, the option is exercised—the buyer of the option buys the stock at the exercise price, and the difference between the asset value and the exercise price comprises the gross profit on the investment. The net profit on the investment is the difference between the gross profit and the price paid for the call initially.
+A payoff diagram illustrates the cash payoff on an option at expiration. For a call, the net payoff is negative (and equal to the price paid for the call) if the value of the underlying asset is less than the strike price. If the price of the underlying asset exceeds the strike price, the gross payoff is the difference between the value of the underlying asset and the strike price, and the net payoff is the difference between the gross payoff and the price of the call. This is illustrated in Figure 5.1. Figure 5.1 Payoff on Call Option A put option gives the buyer of the option the right to sell the underlying asset at a fixed price, again called the strike or exercise price, at any time prior to the expiration date of the option. The buyer pays a price for this right. If the price of the underlying asset is greater than the strike price, the option will not be exercised and will expire worthless. But if the price of the underlying asset is less than the strike price, the owner of the put option will exercise the option and sell the stock at the strike price, claiming the difference between the strike price and the market value of the asset as the gross profit. Again, netting out the initial cost paid for the put yields the net profit from the transaction.
+A put has a negative net payoff if the value of the underlying asset exceeds the strike price, and has a gross payoff equal to the difference between the strike price and the value of the underlying asset if the asset value is less than the strike price. This is summarized in Figure 5.2. Figure 5.2 Payoff on Put Option DETERMINANTS OF OPTION VALUE
+The value of an option is determined by six variables relating to the underlying asset and financial markets. 1. Current value of the underlying asset. Options are assets that derive value from an underlying asset. Consequently, changes in the value of the underlying asset affect the value of the options on that asset. Since calls provide the right to buy the underlying asset at a fixed price, an increase in the value of the asset will increase the value of the calls. Puts, on the other hand, become less valuable as the value of the asset increases.
+2. Variance in value of the underlying asset. The buyer of an option acquires the right to buy or sell the underlying asset at a fixed price. The higher the variance in the value of the underlying asset, the greater the value of the option.1 This is true for both calls and puts. While it may seem counterintuitive that an increase in a risk measure (variance) should increase value, options are different from other securities since buyers of options can never lose more than the price they pay for them; in fact, they have the potential to earn significant returns from large price movements.
+3. Dividends paid on the underlying asset. The value of the underlying asset can be expected to decrease if dividend payments are made on the asset during the life of the option. Consequently, the value of a call on the asset is a decreasing function of the size of expected dividend payments, and the value of a put is an increasing function of expected dividend payments. A more intuitive way of thinking about dividend payments, for call options, is as a cost of delaying exercise on in-the-money options. To see why, consider an option on a traded stock. Once a call option is in-the-money (i.e., the holder of the option will make a gross payoff by exercising the option), exercising the call option will provide the holder with the stock and entitle him or her to the dividends on the stock in subsequent periods. Failing to exercise the option will mean that these dividends are forgone.
+4. Strike price of the option. A key characteristic used to describe an option is the strike price. In the case of calls, where the holder acquires the right to buy at a fixed price, the value of the call will decline as the strike price increases. In the case of puts, where the holder has the right to sell at a fixed price, the value will increase as the strike price increases.
+5. Time to expiration on the option. Both calls and puts are more valuable the greater the time to expiration. This is because the longer time to expiration provides more time for the value of the underlying asset to move, increasing the value of both types of options. Additionally, in the case of a call, where the buyer has the right to pay a fixed price at expiration, the present value of this fixed price decreases as the life of the option increases, increasing the value of the call.
+6. Riskless interest rate corresponding to life of the option. Since the buyer of an option pays the price of the option up front, an opportunity cost is involved. This cost will depend on the level of interest rates and the time to expiration of the option. The riskless interest rate also enters into the valuation of options when the present value of the exercise price is calculated, since the exercise price does not have to be paid (received) until expiration on calls (puts). Increases in the interest rate will increase the value of calls and reduce the value of puts. Table 5.1 summarizes the variables and their predicted effects on call and put prices.
+Table 5.1 Summary of Variables Affecting Call and Put Prices Effect On Factor
+Call Value
+Put Value Increase in underlying asset’s value
+Increases
+Decreases Increase in variance of underlying asset
+Increases
+Increases Increase in strike price
+Decreases
+Increases Increase in dividends paid
+Decreases
+Increases Increase in time to expiration
+Increases
+Increases Increase in interest rates
+Increases
+Decreases American versus European Options: Variables Relating to Early Exercise
+A primary distinction between American and European options is that an American option can be exercised at any time prior to its expiration, while European options can be exercised only at expiration. The possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. There is one compensating factor that enables the former to be valued using models designed for the latter. In most cases, the time premium associated with the remaining life of an option and transaction costs make early exercise suboptimal. In other words, the holders of in-the-money options generally get much more by selling the options to someone else than by exercising the options.
+OPTION PRICING MODELS
+Option pricing theory has made vast strides since 1972, when Fischer Black and Myron Scholes published their pathbreaking paper that provided a model for valuing dividend-protected European options. Black and Scholes used a “replicating portfolio”—a portfolio composed of the underlying asset and the risk-free asset that had the same cash flows as the option being valued—and the notion of arbitrage to come up with their final formulation. Although their derivation is mathematically complicated, there is a simpler binomial model for valuing options that draws on the same logic.
+Binomial Model
+The binomial option pricing model is based on a simple formulation for the asset price process in which the asset, in any time period, can move to one of two possible prices. The general formulation of a stock price process that follows the binomial path is shown in Figure 5.3. In this figure, S is the current stock price; the price moves up to Su with probability p and down to Sd with probability 1 – p in any time period. Figure 5.3 General Formulation for Binomial Price Path Creating a Replicating Portfolio
+The objective in creating a replicating portfolio is to use a combination of risk-free borrowing/lending and the underlying asset to create the same cash flows as the option being valued. The principles of arbitrage apply then, and the value of the option must be equal to the value of the replicating portfolio. In the case of the general formulation shown in Figure 5.3, where stock prices can move either up to Su or down to Sd in any time period, the replicating portfolio for a call with strike price K will involve borrowing $B and acquiring Δ of the underlying asset, where: where Cu = Value of the call if the stock price is Su
+Cd = Value of the call if the stock price is Sd
+In a multiperiod binomial process, the valuation has to proceed iteratively (i.e., starting with the final time period and moving backward in time until the current point in time). The portfolios replicating the option are created at each step and valued, providing the values for the option in that time period. The final output from the binomial option pricing model is a statement of the value of the option in terms of the replicating portfolio, composed of Δ shares (option delta) of the underlying asset and risk-free borrowing/lending. ILLUSTRATION 5.1: Binomial Option Valuation
+Assume that the objective is to value a call with a strike price of $50, which is expected to expire in two time periods, on an underlying asset whose price currently is $50 and is expected to follow a binomial process: Now assume that the interest rate is 11%. In addition, define: The objective is to combined Δ shares of stock and B dollars of borrowing to replicate the cash flows from the call with a strike price of $50. This can be done iteratively, starting with the last period and working back through the binomial tree.
+Step 1: Start with the end nodes and work backward: Thus, if the stock price is $70 at t = 1, borrowing $45 and buying one share of the stock will give the same cash flows as buying the call. The value of the call at t = 1, if the stock price is $70, is therefore: Considering the other leg of the binomial tree at t = 1, If the stock price is $35 at t = 1, then the call is worth nothing.
+Step 2: Move backward to the earlier time period and create a replicating portfolio that will provide the cash flows the option will provide. In other words, borrowing $22.50 and buying five-sevenths of a share will provide the same cash flows as a call with a strike price of $50 over the call’s lifetime. The value of the call therefore has to be the same as the cost of creating this position. The Determinants of Value
+The binomial model provides insight into the determinants of option value. The value of an option is not determined by the expected price of the asset but by its current price, which, of course, reflects expectations about the future. This is a direct consequence of arbitrage. If the option value deviates from the value of the replicating portfolio, investors can create an arbitrage position (i.e., one that requires no investment, involves no risk, and delivers positive returns). To illustrate, if the portfolio that replicates the call costs more than the call does in the market, an investor could buy the call, sell the replicating portfolio, and be guaranteed the difference as a profit. The cash flows on the two positions will offset each other, leading to no cash flows in subsequent periods. The call option value also increases as the time to expiration is extended, as the price movements (u and d) increase, and with increases in the interest rate.
+While the binomial model provides an intuitive feel for the determinants of option value, it requires a large number of inputs, in terms of expected future prices at each node. As time periods are made shorter in the binomial model, you can make one of two assumptions about asset prices.You can assume that price changes become smaller as periods get shorter; this leads to price changes becoming infinitesimally small as time periods approach zero, leading to a continuous price process. Alternatively, you can assume that price changes stay large even as the period gets shorter; this leads to a jump price process, where prices can jump in any period. This section considers the option pricing models that emerge with each of these assumptions.
+Black-Scholes Model
+When the price process is continuous (i.e., price changes become smaller as time periods get shorter), the binomial model for pricing options converges on the Black-Scholes model. The model, named after its cocreators, Fischer Black and Myron Scholes, allows us to estimate the value of any option using a small number of inputs, and has been shown to be robust in valuing many listed options.
+The Model
+While the derivation of the Black-Scholes model is far too complicated to present here, it is based on the idea of creating a portfolio of the underlying asset and the riskless asset with the same cash flows, and hence the same cost, as the option being valued. The value of a call option in the Black-Scholes model can be written as a function of the five variables: S = Current value of the underlying asset
+K = Strike price of the option
+t = Life to expiration of the option
+r = Riskless interest rate corresponding to the life of the option
+σ2 = Variance in the ln(value) of the underlying asset The value of a call is then: Note that e–rt is the present value factor, and reflects the fact that the exercise price on the call option does not have to be paid until expiration, since the model values European options. N(d1) and N(d2) are probabilities, estimated by using a cumulative standardized normal distribution, and the values of d1 and d2 obtained for an option. The cumulative distribution is shown in Figure 5.4. Figure 5.4 Cumulative Normal Distribution In approximate terms, N(d2) yields the likelihood that an option will generate positive cash flows for its owner at exercise (i.e., that S > K in the case of a call option and that K > S in the case of a put option). The portfolio that replicates the call option is created by buying N(d1) units of the underlying asset, and borrowing Ke–rt N(d2). The portfolio will have the same cash flows as the call option, and thus the same value as the option. N(d1), which is the number of units of the underlying asset that are needed to create the replicating portfolio, is called the option delta. A NOTE ON ESTIMATING THE INPUTS TO THE BLACK-SCHOLES MODEL
+The Black-Scholes model requires inputs that are consistent on time measurement. There are two places where this affects estimates. The first relates to the fact that the model works in continuous time, rather than discrete time. That is why we use the continuous time version of present value (exp–rt) rather than the discrete version, (1 + r)–t. It also means that the inputs such as the riskless rate have to be modified to make them continuous time inputs. For instance, if the one-year Treasury bond rate is 6.2 percent, the risk-free rate that is used in the Black-Scholes model should be: The second relates to the period over which the inputs are estimated. For instance, the preceding rate is an annual rate. The variance that is entered into the model also has to be an annualized variance. The variance, estimated from ln(asset prices), can be annualized easily because variances are linear in time if the serial correlation is zero. Thus, if monthly or weekly prices are used to estimate variance, the variance is annualized by multiplying by 12 or 52, respectively. ILLUSTRATION 5.2: Valuing an Option Using the Black-Scholes Model
+On March 6, 2001, Cisco Systems was trading at $13.62. We will attempt to value a July 2001 call option with a strike price of $15, trading on the CBOE on the same day for $2. The following are the other parameters of the options: The annualized standard deviation in Cisco Systems stock price over the previous year was 81%. This standard deviation is estimated using weekly stock prices over the year, and the resulting number was annualized as follows: The option expiration date is Friday, July 20, 2001. There are 103 days to expiration, and the annualized Treasury bill rate corresponding to this option life is 4.63%. The inputs for the Black-Scholes model are as follows: Current stock price (S) = $13.62
+Strike price on the option = $15
+Option life = 103/365 = 0.2822
+Standard deviation in ln(stock prices) = 81%
+Riskless rate = 4.63% Inputting these numbers into the model, we get: Using the normal distribution, we can estimate the N(d1) and N(d2): The value of the call can now be estimated: Since the call is trading at $2, it is slightly overvalued, assuming that the estimate of standard deviation used is correct. IMPLIED VOLATILITY
+The only input in the Black Scholes on which there can be significant disagreement among investors is the variance. While the variance is often estimated by looking at historical data, the values for options that emerge from using the historical variance can be different from the market prices. For any option, there is some variance at which the estimated value will be equal to the market price. This variance is called an implied variance.
+Consider the Cisco option valued in Illustration 5.2. With a standard deviation of 81 percent, the value of the call option with a strike price of $15 was estimated to be $1.87. Since the market price is higher than the calculated value, we tried higher standard deviations, and at a standard deviation 85.40 percent the value of the option is $2 (which is the market price). This is the implied standard deviation or implied volatility. Model Limitations and Fixes
+The Black-Scholes model was designed to value European options that can be exercised only at maturity and whose underlying assets do not pay dividends. In addition, options are valued based on the assumption that option exercise does not affect the value of the underlying asset. In practice, assets do pay dividends, options sometimes get exercised early, and exercising an option can affect the value of the underlying asset. Adjustments exist that, while not perfect, provide partial corrections to the Black-Scholes model.
+Dividends
+The payment of a dividend reduces the stock price; note that on the ex-dividend day, the stock price generally declines. Consequently, call options become less valuable and put options more valuable as expected dividend payments increase. There are two ways of dealing with dividends in the Black-Scholes model: 1. Short-term options. One approach to dealing with dividends is to estimate the present value of expected dividends that will be paid by the underlying asset during the option life and subtract it from the current value of the asset to use as S in the model. 2. Long-term options. Since it becomes less practical to estimate the present value of dividends the longer the option life, an alternate approach can be used. If the dividend yield (y = Dividends/Current value of the asset) on the underlying asset is expected to remain unchanged during the life of the option, the Black-Scholes model can be modified to take dividends into account. From an intuitive standpoint, the adjustments have two effects. First, the value of the asset is discounted back to the present at the dividend yield to take into account the expected drop in asset value resulting from dividend payments. Second, the interest rate is offset by the dividend yield to reflect the lower carrying cost from holding the asset (in the replicating portfolio). The net effect will be a reduction in the value of calls estimated using this model. ILLUSTRATION 5.3: Valuing a Short-Term Option with Dividend Adjustments—The Black-Scholes Correction
+Assume that it is March 6, 2001, and that AT&T is trading at $20.50 a share. Consider a call option on the stock with a strike price of $20, expiring on July 20, 2001. Using past stock prices, the annualized standard deviation in the log of stock prices for AT&T is estimated at 60%. There is one dividend, amounting to $0.15, and it will be paid in 23 days. The riskless rate is 4.63%. Present value of expected dividend = $0.15/1.046323/365 = $0.15
+Dividend-adjusted stock price = $20.50 – $0.15 = $20.35
+Time to expiration = 103/365 = 0.2822
+Variance in ln(stock prices) = 0.62 = 0.36
+Riskless rate = 4.63% The value from the Black-Scholes model is: The call option was trading at $2.60 on that day. ILLUSTRATION 5.4: Valuing a Long-Term Option with Dividend Adjustments—Primes and Scores
+The CBOE offers longer-term call and put options on some stocks. On March 6, 2001, for instance, you could have purchased an AT&T call expiring on January 17, 2003. The stock price for AT&T is $20.50 (as in the previous example). The following is the valuation of a call option with a strike price of $20. Instead of estimating the present value of dividends over the next two years, assume that AT&T’s dividend yield will remain 2.51% over this period and that the risk-free rate for a two-year Treasury bond is 4.85%. The inputs to the Black-Scholes model are: The value from the Black-Scholes model is: The call was trading at $5.80 on March 8, 2001. optst.xls: This spreadsheet allows you to estimate the value of a short-term option when the expected dividends during the option life can be estimated. optlt.xls: This spreadsheet allows you to estimate the value of an option when the underlying asset has a constant dividend yield. Early Exercise
+The Black-Scholes model was designed to value European options that can be exercised only at expiration. In contrast, most options that we encounter in practice are American options and can be exercised at any time until expiration. As mentioned earlier, the possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. In general, though, with traded options, it is almost always better to sell the option to someone else rather than exercise early, since options have a time premium (i.e., they sell for more than their exercise value). There are two exceptions. One occurs when the underlying asset pays large dividends, thus reducing the expected value of the asset. In this case, call options may be exercised just before an ex-dividend date, if the time premium on the options is less than the expected decline in asset value as a consequence of the dividend payment. The other exception arises when an investor holds both the underlying asset and deep in-the-money puts (i.e., puts with strike prices well above the current price of the underlying asset) on that asset at a time when interest rates are high. In this case, the time premium on the put may be less than the potential gain from exercising the put early and earning interest on the exercise price.
+There are two basic ways of dealing with the possibility of early exercise. One is to continue to use the unadjusted Black-Scholes model and to regard the resulting value as a floor or conservative estimate of the true value. The other is to try to adjust the value of the option for the possibility of early exercise. There are two approaches for doing so. One uses the Black-Scholes model to value the option to each potential exercise date. With options on stocks, this basically requires that the investor values options to each ex-dividend day and chooses the maximum of the estimated call values. The second approach is to use a modified version of the binomial model to consider the possibility of early exercise. In this version, the up and the down movements for asset prices in each period can be estimated from the variance. 2
+Approach 1: Pseudo-American Valuation
+Step 1: Define when dividends will be paid and how much the dividends will be.
+Step 2: Value the call option to each ex-dividend date using the dividend-adjusted approach described earlier, where the stock price is reduced by the present value of expected dividends.
+Step 3: Choose the maximum of the call values estimated for each ex-dividend day. ILLUSTRATION 5.5: Using Pseudo-American Option Valuation to Adjust for Early Exercise
+Consider an option with a strike price of $35 on a stock trading at $40. The variance in the ln(stock prices) is 0.05, and the riskless rate is 4%. The option has a remaining life of eight months, and there are three dividends expected during this period: Expected Dividend
+Ex-Dividend Day $0.80
+In 1 month $0.80
+In 4 months $0.80
+In 7 months The call option is first valued to just before the first ex-dividend date: The value from the Black-Scholes model is: The call option is then valued to before the second ex-dividend date: The value of the call based on these parameters is: The call option is then valued to before the third ex-dividend date: The value of the call based on these parameters is: The call option is then valued to expiration: The value of the call based on these parameters is: Approach 2: Using the Binomial Model
+The binomial model is much more capable of handling early exercise because it considers the cash flows at each time period, rather than just at expiration. The biggest limitation of the binomial model is determining what stock prices will be at the end of each period, but this can be overcome by using a variant that allows us to estimate the up and the down movements in stock prices from the estimated variance. There are four steps involved:
+Step 1: If the variance in ln(stock prices) has been estimated for the Black-Scholes valuation, convert these into inputs for the binomial model: where u and d are the up and the down movements per unit time for the binomial, and dt is the number of periods within each year (or unit time).
+Step 2: Specify the period in which the dividends will be paid and make the assumption that the price will drop by the amount of the dividend in that period.
+Step 3: Value the call at each node of the tree, allowing for the possibility of early exercise just before ex-dividend dates. There will be early exercise if the remaining time premium on the option is less than the expected drop in option value as a consequence of the dividend payment.
+Step 4: Value the call at time 0, using the standard binomial approach. bstobin.xls: This spreadsheet allows you to estimate the parameters for a binomial model from the inputs to a Black-Scholes model. Impact of Exercise on Underlying Asset Value
+The Black-Scholes model is based on the assumption that exercising an option does not affect the value of the underlying asset. This may be true for listed options on stocks, but it is not true for some types of options. For instance, the exercise of warrants increases the number of shares outstanding and brings fresh cash into the firm, both of which will affect the stock price.3 The expected negative impact (dilution) of exercise will decrease the value of warrants, compared to otherwise similar call options. The adjustment for dilution to the stock price is fairly simple in the Black-Scholes valuation. The stock price is adjusted for the expected dilution from the exercise of the options. In the case of warrants, for instance: where S = Current value of the stock
+nw = Number of warrants outstanding
+W = Value of warrants outstanding
+ns = Number of shares outstanding
+When the warrants are exercised, the number of shares outstanding will increase, reducing the stock price. The numerator reflects the market value of equity, including both stocks and warrants outstanding. The reduction in S will reduce the value of the call option.
+There is an element of circularity in this analysis, since the value of the warrant is needed to estimate the dilution-adjusted S and the dilution-adjusted S is needed to estimate the value of the warrant. This problem can be resolved by starting the process off with an assumed value for the warrant (e.g., the exercise value or the current market price of the warrant). This will yield a value for the warrant, and this estimated value can then be used as an input to reestimate the warrant’s value until there is convergence. FROM BLACK-SCHOLES TO BINOMIAL
+The process of converting the continuous variance in a Black-Scholes model to a binomial tree is a fairly simple one. Assume, for instance, that you have an asset that is trading at $30 currently and that you estimate the annualized standard deviation in the asset value to be 40 percent; the annualized riskless rate is 5 percent. For simplicity, let us assume that the option that you are valuing has a four-year life and that each period is a year. To estimate the prices at the end of each of the four years, we begin by first estimating the up and down movements in the binomial: Based on these estimates, we can obtain the prices at the end of the first node of the tree (the end of the first year): Progressing through the rest of the tree, we obtain the following numbers: ILLUSTRATION 5.6: Valuing a Warrant on Avatek Corporation
+Avatek Corporation is a real estate firm with 19.637 million shares outstanding, trading at $0.38 a share. In March 2001 the company had 1.8 million options outstanding, with four years to expiration and with an exercise price of $2.25. The stock paid no dividends, and the standard deviation in ln(stock prices) was 93%. The four-year Treasury bond rate was 4.9%. (The options were trading at $0.12 apiece at the time of this analysis.)
+The inputs to the warrant valuation model are as follows: The results of the Black-Scholes valuation of this option are: The options were trading at $0.12 in March 2001. Since the value was equal to the price, there was no need for further iterations. If there had been a difference, we would have reestimated the adjusted stock price and option value. If the options had been non-traded (as is the case with management options), this calculation would have required an iterative process, where the option value is used to get the adjusted value per share and the value per share to get the option value. warrant.xls: This spreadsheet allows you to estimate the value of an option when there is a potential dilution from exercise. The Black-Scholes Model for Valuing Puts
+The value of a put can be derived from the value of a call with the same strike price and the same expiration date: where C is the value of the call and P is the value of the put. This relationship between the call and put values is called put-call parity, and any deviations from parity can be used by investors to make riskless profits. To see why put-call parity holds, consider selling a call and buying a put with exercise price K and expiration date t, and simultaneously buying the underlying asset at the current price S. The payoff from this position is riskless and always yields K at expiration (t). To see this, assume that the stock price at expiration is S*. The payoff on each of the positions in the portfolio can be written as follows: Position
+Payoffs at t if S* > K
+Payoffs at t if S* < K Sell call
+–(S* – K)
+0 Buy put
+0
+K – S* Buy stock
+S*
+S* Total
+K
+K Since this position yields K with certainty, the cost of creating this position must be equal to the present value of K at the riskless rate (K e–rt). Substituting the Black-Scholes equation for the value of an equivalent call into this equation, we get: Thus, the replicating portfolio for a put is created by selling short [1 – N(d1)] shares of stock and investing K e–rt[1 – N(d2)] in the riskless asset. ILLUSTRATION 5.7: Valuing a Put Using Put-Call Parity: Cisco Systems and AT&T
+Consider the call on Cisco Systems that we valued in Illustration 5.2. The call had a strike price of $15 on the stock, had 103 days left to expiration, and was valued at $1.87. The stock was trading at $13.62, and the riskless rate was 4.63%. The put can be valued as follows: The put was trading at $3.38.
+Also, a long-term call on AT&T was valued in Illustration 5.4. The call had a strike price of $20, 1.8333 years left to expiration, and a value of $6.63. The stock was trading at $20.50 and was expected to maintain a dividend yield of 2.51% over the period. The riskless rate was 4.85%. The put value can be estimated as follows: The put was trading at $3.80. Both the call and the put were trading at different prices from our estimates, which may indicate that we have not correctly estimated the stock’s volatility. Jump Process Option Pricing Models
+If price changes remain larger as the time periods in the binomial model are shortened, it can no longer be assumed that prices change continuously. When price changes remain large, a price process that allows for price jumps is much more realistic. Cox and Ross (1976) valued options when prices follow a pure jump process, where the jumps can only be positive. Thus, in the next interval, the stock price will either have a large positive jump with a specified probability or drift downward at a given rate.
+Merton (1976) considered a distribution where there are price jumps superimposed on a continuous price process. He specified the rate at which jumps occur (λ) and the average jump size (k), measured as a percentage of the stock price. The model derived to value options with this process is called a jump diffusion model. In this model, the value of an option is determined by the five variables specified in the Black-Scholes model, and the parameters of the jump process (λ, k). Unfortunately, the estimates of the jump process parameters are so difficult to make for most firms that they overwhelm any advantages that accrue from using a more realistic model. These models, therefore, have seen limited use in practice.
+EXTENSIONS OF OPTION PRICING
+All the option pricing models described so far—the binomial, the Black-Scholes, and the jump process models—are designed to value options with clearly defined exercise prices and maturities on underlying assets that are traded. However, the options we encounter in investment analysis or valuation are often on real assets rather than financial assets. Categorized as real options, they can take much more complicated forms. This section considers some of these variations.
+Capped and Barrier Options
+With a simple call option, there is no specified upper limit on the profits that can be made by the buyer of the call. Asset prices, at least in theory, can keep going up, and the payoffs increase proportionately. In some call options, though, the buyer is entitled to profits up to a specified price but not above it. For instance, consider a call option with a strike price of K1 on an asset. In an unrestricted call option, the payoff on this option will increase as the underlying asset’s price increases above K1. Assume, however, that if the price reaches K2, the payoff is capped at (K2 – K1). The payoff diagram on this option is shown in Figure 5.5. Figure 5.5 Payoff on Capped Call This option is called a capped call. Notice, also, that once the price reaches K2, there is no longer any time premium associated with the option, and the option will therefore be exercised. Capped calls are part of a family of options called barrier options, where the payoff on and the life of the option are functions of whether the underlying asset price reaches a certain level during a specified period.
+The value of a capped call is always lower than the value of the same call without the payoff limit. A simple approximation of this value can be obtained by valuing the call twice, once with the given exercise price and once with the cap, and taking the difference in the two values. In the preceding example, then, the value of the call with an exercise price of K1 and a cap at K2 can be written as: Barrier options can take many forms. In a knockout option, an option ceases to exist if the underlying asset reaches a certain price. In the case of a call option, this knockout price is usually set below the strike price, and this option is called a down-and-out option. In the case of a put option, the knockout price will be set above the exercise price, and this option is called an up-and-out option. Like the capped call, these options are worth less than their unrestricted counterparts. Many real options have limits on potential upside, or knockout provisions, and ignoring these limits can result in the overstatement of the value of these options.
+Compound Options
+Some options derive their value not from an underlying asset, but from other options. These options are called compound options. Compound options can take any of four forms—a call on a call, a put on a put, a call on a put, or a put on a call. Geske (1979) developed the analytical formulation for valuing compound options by replacing the standard normal distribution used in a simple option model with a bivariate normal distribution in the calculation.
+Consider, for instance, the option to expand a project that is discussed in Chapter 30. While we will value this option using a simple option pricing model, in reality there could be multiple stages in expansion, with each stage representing an option for the following stage. In this case, we will undervalue the option by considering it as a simple rather than a compound option.
+Notwithstanding this discussion, the valuation of compound options becomes progressively more difficult as more options are added to the chain. In this case, rather than wreck the valuation on the shoals of estimation error, it may be better to accept the conservative estimate that is provided with a simple valuation model as a floor on the value.
+Rainbow Options
+In a simple option, the uncertainty is about the price of the underlying asset. Some options are exposed to two or more sources of uncertainty, and these options are rainbow options. Using the simple option pricing model to value such options can lead to biased estimates of value. As an example, consider an undeveloped oil reserve as an option, where the firm that owns the reserve has the right to develop the reserve. Here there are two sources of uncertainty. The first is obviously the price of oil, and the second is the quantity of oil that is in the reserve. To value this undeveloped reserve, we can make the simplifying assumption that we know the quantity of oil in the reserve with certainty. In reality, however, uncertainty about the quantity will affect the value of this option and make the decision to exercise more difficult.4
+CONCLUSION
+An option is an asset with payoffs that are contingent on the value of an underlying asset. A call option provides its holder with the right to buy the underlying asset at a fixed price, whereas a put option provides its holder with the right to sell at a fixed price, at any time before the expiration of the option. The value of an option is determined by six variables—the current value of the underlying asset, the variance in this value, the expected dividends on the asset, the strike price and life of the option, and the riskless interest rate. This is illustrated in both the binomial and the Black-Scholes models, which value options by creating replicating portfolios composed of the underlying asset and riskless lending or borrowing. These models can be used to value assets that have option like characteristics.
+QUESTIONS AND SHORT PROBLEMS
+In the problems following, use an equity risk premium of 5.5 percent if none is specified. 1. The following are prices of options traded on Microsoft Corporation, which pays no dividends. The stock is trading at $83, and the annualized riskless rate is 3.8%. The standard deviation in ln(stock prices) (based on historical data) is 30%. a. Estimate the value of a three-month call with a strike price of $85.
+b. Using the inputs from the Black-Scholes model, specify how you would replicate this call.
+c. What is the implied standard deviation in this call?
+d. Assume now that you buy a call with a strike price of $85 and sell a call with a strike price of $90. Draw the payoff diagram on this position.
+e. Using put-call parity, estimate the value of a three-month put with a strike price of $85. 2. You are trying to value three-month call and put options on Merck with a strike price of $30. The stock is trading at $28.75, and the company expects to pay a quarterly dividend per share of $0.28 in two months. The annualized riskless interest rate is 3.6%, and the standard deviation in log stock prices is 20%. a. Estimate the value of the call and put options, using the Black-Scholes model.
+b. What effect does the expected dividend payment have on call values? On put values? Why? 3. There is the possibility that the options on Merck described in the preceding problem could be exercised early. a. Use the pseudo-American call option technique to determine whether this will affect the value of the call.
+b. Why does the possibility of early exercise exist? What types of options are most likely to be exercised early? 4. You have been provided the following information on a three-month call: a. If you wanted to replicate buying this call, how much money would you need to borrow?
+b. If you wanted to replicate buying this call, how many shares of stock would you need to buy? 5. Go Video, a manufacturer of video recorders, was trading at $4 per share in May 1994. There were 11 million shares outstanding. At the same time, it had 550,000 one-year warrants outstanding, with a strike price of $4.25. The stock has had a standard deviation of 60%. The stock does not pay a dividend. The riskless rate is 5%. a. Estimate the value of the warrants, ignoring dilution.
+b. Estimate the value of the warrants, allowing for dilution.
+c. Why does dilution reduce the value of the warrants? 6. You are trying to value a long-term call option on the NYSE Composite index, expiring in five years, with a strike price of 275. The index is currently at 250, and the annualized standard deviation in stock prices is 15%. The average dividend yield on the index is 3% and is expected to remain unchanged over the next five years. The five-year Treasury bond rate is 5%. a. Estimate the value of the long-term call option.
+b. Estimate the value of a put option with the same parameters.
+c. What are the implicit assumptions you are making when you use the Black-Scholes model to value this option? Which of these assumptions are likely to be violated? What are the consequences for your valuation? 7. A new security on AT&T will entitle the investor to all dividends on AT&T over the next three years, limiting upside potential to 20% but also providing downside protection below 10%. AT&T stock is trading at $50, and three-year call and put options are traded on the exchange at the following prices: How much would you be willing to pay for this security? 1 Note, though, that higher variance can reduce the value of the underlying asset. As a call option becomes more in-the-money, the more it resembles the underlying asset. For very deep in-the-money call options, higher variance can reduce the value of the option.
+2 To illustrate, if σ2 is the variance in ln(stock prices), the up and the down movements in the binomial can be estimated as follows: where u and d are the up and down movements per unit time for the binomial, T is the life of the option, and m is the number of periods within that lifetime.
+3 Warrants are call options issued by firms, either as part of management compensation contracts or to raise equity.
+4 The analogy to a listed option on a stock is the case where you do not know with certainty what the stock price is when you exercise the option. The more uncertain you are about the stock price, the more margin for error you have to give yourself when you exercise the option, to ensure that you are in fact earning a profit.
+CHAPTER 5
+Option Pricing Theory and Models
+In general, the value of any asset is the present value of the expected cash flows on that asset. This chapter considers an exception to that rule when it looks at assets with two specific characteristics: 1. The assets derive their value from the values of other assets.
+2. The cash flows on the assets are contingent on the occurrence of specific events. These assets are called options, and the present value of the expected cash flows on these assets will understate their true value. This chapter describes the cash flow characteristics of options, considers the factors that determine their value, and examines how best to value them.
+BASICS OF OPTION PRICING
+An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called a strike price or an exercise price) at or before the expiration date of the option. Since it is a right and not an obligation, the holder can choose not to exercise the right and can allow the option to expire. There are two types of options—call options and put options.
+Call and Put Options: Description and Payoff Diagrams
+A call option gives the buyer of the option the right to buy the underlying asset at the strike price or the exercise price at any time prior to the expiration date of the option. The buyer pays a price for this right. If at expiration the value of the asset is less than the strike price, the option is not exercised and expires worthless. If, however, the value of the asset is greater than the strike price, the option is exercised—the buyer of the option buys the stock at the exercise price, and the difference between the asset value and the exercise price comprises the gross profit on the investment. The net profit on the investment is the difference between the gross profit and the price paid for the call initially.
+A payoff diagram illustrates the cash payoff on an option at expiration. For a call, the net payoff is negative (and equal to the price paid for the call) if the value of the underlying asset is less than the strike price. If the price of the underlying asset exceeds the strike price, the gross payoff is the difference between the value of the underlying asset and the strike price, and the net payoff is the difference between the gross payoff and the price of the call. This is illustrated in Figure 5.1. Figure 5.1 Payoff on Call Option A put option gives the buyer of the option the right to sell the underlying asset at a fixed price, again called the strike or exercise price, at any time prior to the expiration date of the option. The buyer pays a price for this right. If the price of the underlying asset is greater than the strike price, the option will not be exercised and will expire worthless. But if the price of the underlying asset is less than the strike price, the owner of the put option will exercise the option and sell the stock at the strike price, claiming the difference between the strike price and the market value of the asset as the gross profit. Again, netting out the initial cost paid for the put yields the net profit from the transaction.
+A put has a negative net payoff if the value of the underlying asset exceeds the strike price, and has a gross payoff equal to the difference between the strike price and the value of the underlying asset if the asset value is less than the strike price. This is summarized in Figure 5.2. Figure 5.2 Payoff on Put Option DETERMINANTS OF OPTION VALUE
+The value of an option is determined by six variables relating to the underlying asset and financial markets. 1. Current value of the underlying asset. Options are assets that derive value from an underlying asset. Consequently, changes in the value of the underlying asset affect the value of the options on that asset. Since calls provide the right to buy the underlying asset at a fixed price, an increase in the value of the asset will increase the value of the calls. Puts, on the other hand, become less valuable as the value of the asset increases.
+2. Variance in value of the underlying asset. The buyer of an option acquires the right to buy or sell the underlying asset at a fixed price. The higher the variance in the value of the underlying asset, the greater the value of the option.1 This is true for both calls and puts. While it may seem counterintuitive that an increase in a risk measure (variance) should increase value, options are different from other securities since buyers of options can never lose more than the price they pay for them; in fact, they have the potential to earn significant returns from large price movements.
+3. Dividends paid on the underlying asset. The value of the underlying asset can be expected to decrease if dividend payments are made on the asset during the life of the option. Consequently, the value of a call on the asset is a decreasing function of the size of expected dividend payments, and the value of a put is an increasing function of expected dividend payments. A more intuitive way of thinking about dividend payments, for call options, is as a cost of delaying exercise on in-the-money options. To see why, consider an option on a traded stock. Once a call option is in-the-money (i.e., the holder of the option will make a gross payoff by exercising the option), exercising the call option will provide the holder with the stock and entitle him or her to the dividends on the stock in subsequent periods. Failing to exercise the option will mean that these dividends are forgone.
+4. Strike price of the option. A key characteristic used to describe an option is the strike price. In the case of calls, where the holder acquires the right to buy at a fixed price, the value of the call will decline as the strike price increases. In the case of puts, where the holder has the right to sell at a fixed price, the value will increase as the strike price increases.
+5. Time to expiration on the option. Both calls and puts are more valuable the greater the time to expiration. This is because the longer time to expiration provides more time for the value of the underlying asset to move, increasing the value of both types of options. Additionally, in the case of a call, where the buyer has the right to pay a fixed price at expiration, the present value of this fixed price decreases as the life of the option increases, increasing the value of the call.
+6. Riskless interest rate corresponding to life of the option. Since the buyer of an option pays the price of the option up front, an opportunity cost is involved. This cost will depend on the level of interest rates and the time to expiration of the option. The riskless interest rate also enters into the valuation of options when the present value of the exercise price is calculated, since the exercise price does not have to be paid (received) until expiration on calls (puts). Increases in the interest rate will increase the value of calls and reduce the value of puts. Table 5.1 summarizes the variables and their predicted effects on call and put prices.
+Table 5.1 Summary of Variables Affecting Call and Put Prices Effect On Factor
+Call Value
+Put Value Increase in underlying asset’s value
+Increases
+Decreases Increase in variance of underlying asset
+Increases
+Increases Increase in strike price
+Decreases
+Increases Increase in dividends paid
+Decreases
+Increases Increase in time to expiration
+Increases
+Increases Increase in interest rates
+Increases
+Decreases American versus European Options: Variables Relating to Early Exercise
+A primary distinction between American and European options is that an American option can be exercised at any time prior to its expiration, while European options can be exercised only at expiration. The possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. There is one compensating factor that enables the former to be valued using models designed for the latter. In most cases, the time premium associated with the remaining life of an option and transaction costs make early exercise suboptimal. In other words, the holders of in-the-money options generally get much more by selling the options to someone else than by exercising the options.
+OPTION PRICING MODELS
+Option pricing theory has made vast strides since 1972, when Fischer Black and Myron Scholes published their pathbreaking paper that provided a model for valuing dividend-protected European options. Black and Scholes used a “replicating portfolio”—a portfolio composed of the underlying asset and the risk-free asset that had the same cash flows as the option being valued—and the notion of arbitrage to come up with their final formulation. Although their derivation is mathematically complicated, there is a simpler binomial model for valuing options that draws on the same logic.
+Binomial Model
+The binomial option pricing model is based on a simple formulation for the asset price process in which the asset, in any time period, can move to one of two possible prices. The general formulation of a stock price process that follows the binomial path is shown in Figure 5.3. In this figure, S is the current stock price; the price moves up to Su with probability p and down to Sd with probability 1 – p in any time period. Figure 5.3 General Formulation for Binomial Price Path Creating a Replicating Portfolio
+The objective in creating a replicating portfolio is to use a combination of risk-free borrowing/lending and the underlying asset to create the same cash flows as the option being valued. The principles of arbitrage apply then, and the value of the option must be equal to the value of the replicating portfolio. In the case of the general formulation shown in Figure 5.3, where stock prices can move either up to Su or down to Sd in any time period, the replicating portfolio for a call with strike price K will involve borrowing $B and acquiring Δ of the underlying asset, where: where Cu = Value of the call if the stock price is Su
+Cd = Value of the call if the stock price is Sd
+In a multiperiod binomial process, the valuation has to proceed iteratively (i.e., starting with the final time period and moving backward in time until the current point in time). The portfolios replicating the option are created at each step and valued, providing the values for the option in that time period. The final output from the binomial option pricing model is a statement of the value of the option in terms of the replicating portfolio, composed of Δ shares (option delta) of the underlying asset and risk-free borrowing/lending. ILLUSTRATION 5.1: Binomial Option Valuation
+Assume that the objective is to value a call with a strike price of $50, which is expected to expire in two time periods, on an underlying asset whose price currently is $50 and is expected to follow a binomial process: Now assume that the interest rate is 11%. In addition, define: The objective is to combined Δ shares of stock and B dollars of borrowing to replicate the cash flows from the call with a strike price of $50. This can be done iteratively, starting with the last period and working back through the binomial tree.
+Step 1: Start with the end nodes and work backward: Thus, if the stock price is $70 at t = 1, borrowing $45 and buying one share of the stock will give the same cash flows as buying the call. The value of the call at t = 1, if the stock price is $70, is therefore: Considering the other leg of the binomial tree at t = 1, If the stock price is $35 at t = 1, then the call is worth nothing.
+Step 2: Move backward to the earlier time period and create a replicating portfolio that will provide the cash flows the option will provide. In other words, borrowing $22.50 and buying five-sevenths of a share will provide the same cash flows as a call with a strike price of $50 over the call’s lifetime. The value of the call therefore has to be the same as the cost of creating this position. The Determinants of Value
+The binomial model provides insight into the determinants of option value. The value of an option is not determined by the expected price of the asset but by its current price, which, of course, reflects expectations about the future. This is a direct consequence of arbitrage. If the option value deviates from the value of the replicating portfolio, investors can create an arbitrage position (i.e., one that requires no investment, involves no risk, and delivers positive returns). To illustrate, if the portfolio that replicates the call costs more than the call does in the market, an investor could buy the call, sell the replicating portfolio, and be guaranteed the difference as a profit. The cash flows on the two positions will offset each other, leading to no cash flows in subsequent periods. The call option value also increases as the time to expiration is extended, as the price movements (u and d) increase, and with increases in the interest rate.
+While the binomial model provides an intuitive feel for the determinants of option value, it requires a large number of inputs, in terms of expected future prices at each node. As time periods are made shorter in the binomial model, you can make one of two assumptions about asset prices.You can assume that price changes become smaller as periods get shorter; this leads to price changes becoming infinitesimally small as time periods approach zero, leading to a continuous price process. Alternatively, you can assume that price changes stay large even as the period gets shorter; this leads to a jump price process, where prices can jump in any period. This section considers the option pricing models that emerge with each of these assumptions.
+Black-Scholes Model
+When the price process is continuous (i.e., price changes become smaller as time periods get shorter), the binomial model for pricing options converges on the Black-Scholes model. The model, named after its cocreators, Fischer Black and Myron Scholes, allows us to estimate the value of any option using a small number of inputs, and has been shown to be robust in valuing many listed options.
+The Model
+While the derivation of the Black-Scholes model is far too complicated to present here, it is based on the idea of creating a portfolio of the underlying asset and the riskless asset with the same cash flows, and hence the same cost, as the option being valued. The value of a call option in the Black-Scholes model can be written as a function of the five variables: S = Current value of the underlying asset
+K = Strike price of the option
+t = Life to expiration of the option
+r = Riskless interest rate corresponding to the life of the option
+σ2 = Variance in the ln(value) of the underlying asset The value of a call is then: Note that e–rt is the present value factor, and reflects the fact that the exercise price on the call option does not have to be paid until expiration, since the model values European options. N(d1) and N(d2) are probabilities, estimated by using a cumulative standardized normal distribution, and the values of d1 and d2 obtained for an option. The cumulative distribution is shown in Figure 5.4. Figure 5.4 Cumulative Normal Distribution In approximate terms, N(d2) yields the likelihood that an option will generate positive cash flows for its owner at exercise (i.e., that S > K in the case of a call option and that K > S in the case of a put option). The portfolio that replicates the call option is created by buying N(d1) units of the underlying asset, and borrowing Ke–rt N(d2). The portfolio will have the same cash flows as the call option, and thus the same value as the option. N(d1), which is the number of units of the underlying asset that are needed to create the replicating portfolio, is called the option delta. A NOTE ON ESTIMATING THE INPUTS TO THE BLACK-SCHOLES MODEL
+The Black-Scholes model requires inputs that are consistent on time measurement. There are two places where this affects estimates. The first relates to the fact that the model works in continuous time, rather than discrete time. That is why we use the continuous time version of present value (exp–rt) rather than the discrete version, (1 + r)–t. It also means that the inputs such as the riskless rate have to be modified to make them continuous time inputs. For instance, if the one-year Treasury bond rate is 6.2 percent, the risk-free rate that is used in the Black-Scholes model should be: The second relates to the period over which the inputs are estimated. For instance, the preceding rate is an annual rate. The variance that is entered into the model also has to be an annualized variance. The variance, estimated from ln(asset prices), can be annualized easily because variances are linear in time if the serial correlation is zero. Thus, if monthly or weekly prices are used to estimate variance, the variance is annualized by multiplying by 12 or 52, respectively. ILLUSTRATION 5.2: Valuing an Option Using the Black-Scholes Model
+On March 6, 2001, Cisco Systems was trading at $13.62. We will attempt to value a July 2001 call option with a strike price of $15, trading on the CBOE on the same day for $2. The following are the other parameters of the options: The annualized standard deviation in Cisco Systems stock price over the previous year was 81%. This standard deviation is estimated using weekly stock prices over the year, and the resulting number was annualized as follows: The option expiration date is Friday, July 20, 2001. There are 103 days to expiration, and the annualized Treasury bill rate corresponding to this option life is 4.63%. The inputs for the Black-Scholes model are as follows: Current stock price (S) = $13.62
+Strike price on the option = $15
+Option life = 103/365 = 0.2822
+Standard deviation in ln(stock prices) = 81%
+Riskless rate = 4.63% Inputting these numbers into the model, we get: Using the normal distribution, we can estimate the N(d1) and N(d2): The value of the call can now be estimated: Since the call is trading at $2, it is slightly overvalued, assuming that the estimate of standard deviation used is correct. IMPLIED VOLATILITY
+The only input in the Black Scholes on which there can be significant disagreement among investors is the variance. While the variance is often estimated by looking at historical data, the values for options that emerge from using the historical variance can be different from the market prices. For any option, there is some variance at which the estimated value will be equal to the market price. This variance is called an implied variance.
+Consider the Cisco option valued in Illustration 5.2. With a standard deviation of 81 percent, the value of the call option with a strike price of $15 was estimated to be $1.87. Since the market price is higher than the calculated value, we tried higher standard deviations, and at a standard deviation 85.40 percent the value of the option is $2 (which is the market price). This is the implied standard deviation or implied volatility. Model Limitations and Fixes
+The Black-Scholes model was designed to value European options that can be exercised only at maturity and whose underlying assets do not pay dividends. In addition, options are valued based on the assumption that option exercise does not affect the value of the underlying asset. In practice, assets do pay dividends, options sometimes get exercised early, and exercising an option can affect the value of the underlying asset. Adjustments exist that, while not perfect, provide partial corrections to the Black-Scholes model.
+Dividends
+The payment of a dividend reduces the stock price; note that on the ex-dividend day, the stock price generally declines. Consequently, call options become less valuable and put options more valuable as expected dividend payments increase. There are two ways of dealing with dividends in the Black-Scholes model: 1. Short-term options. One approach to dealing with dividends is to estimate the present value of expected dividends that will be paid by the underlying asset during the option life and subtract it from the current value of the asset to use as S in the model. 2. Long-term options. Since it becomes less practical to estimate the present value of dividends the longer the option life, an alternate approach can be used. If the dividend yield (y = Dividends/Current value of the asset) on the underlying asset is expected to remain unchanged during the life of the option, the Black-Scholes model can be modified to take dividends into account. From an intuitive standpoint, the adjustments have two effects. First, the value of the asset is discounted back to the present at the dividend yield to take into account the expected drop in asset value resulting from dividend payments. Second, the interest rate is offset by the dividend yield to reflect the lower carrying cost from holding the asset (in the replicating portfolio). The net effect will be a reduction in the value of calls estimated using this model. ILLUSTRATION 5.3: Valuing a Short-Term Option with Dividend Adjustments—The Black-Scholes Correction
+Assume that it is March 6, 2001, and that AT&T is trading at $20.50 a share. Consider a call option on the stock with a strike price of $20, expiring on July 20, 2001. Using past stock prices, the annualized standard deviation in the log of stock prices for AT&T is estimated at 60%. There is one dividend, amounting to $0.15, and it will be paid in 23 days. The riskless rate is 4.63%. Present value of expected dividend = $0.15/1.046323/365 = $0.15
+Dividend-adjusted stock price = $20.50 – $0.15 = $20.35
+Time to expiration = 103/365 = 0.2822
+Variance in ln(stock prices) = 0.62 = 0.36
+Riskless rate = 4.63% The value from the Black-Scholes model is: The call option was trading at $2.60 on that day. ILLUSTRATION 5.4: Valuing a Long-Term Option with Dividend Adjustments—Primes and Scores
+The CBOE offers longer-term call and put options on some stocks. On March 6, 2001, for instance, you could have purchased an AT&T call expiring on January 17, 2003. The stock price for AT&T is $20.50 (as in the previous example). The following is the valuation of a call option with a strike price of $20. Instead of estimating the present value of dividends over the next two years, assume that AT&T’s dividend yield will remain 2.51% over this period and that the risk-free rate for a two-year Treasury bond is 4.85%. The inputs to the Black-Scholes model are: The value from the Black-Scholes model is: The call was trading at $5.80 on March 8, 2001. optst.xls: This spreadsheet allows you to estimate the value of a short-term option when the expected dividends during the option life can be estimated. optlt.xls: This spreadsheet allows you to estimate the value of an option when the underlying asset has a constant dividend yield. Early Exercise
+The Black-Scholes model was designed to value European options that can be exercised only at expiration. In contrast, most options that we encounter in practice are American options and can be exercised at any time until expiration. As mentioned earlier, the possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. In general, though, with traded options, it is almost always better to sell the option to someone else rather than exercise early, since options have a time premium (i.e., they sell for more than their exercise value). There are two exceptions. One occurs when the underlying asset pays large dividends, thus reducing the expected value of the asset. In this case, call options may be exercised just before an ex-dividend date, if the time premium on the options is less than the expected decline in asset value as a consequence of the dividend payment. The other exception arises when an investor holds both the underlying asset and deep in-the-money puts (i.e., puts with strike prices well above the current price of the underlying asset) on that asset at a time when interest rates are high. In this case, the time premium on the put may be less than the potential gain from exercising the put early and earning interest on the exercise price.
+There are two basic ways of dealing with the possibility of early exercise. One is to continue to use the unadjusted Black-Scholes model and to regard the resulting value as a floor or conservative estimate of the true value. The other is to try to adjust the value of the option for the possibility of early exercise. There are two approaches for doing so. One uses the Black-Scholes model to value the option to each potential exercise date. With options on stocks, this basically requires that the investor values options to each ex-dividend day and chooses the maximum of the estimated call values. The second approach is to use a modified version of the binomial model to consider the possibility of early exercise. In this version, the up and the down movements for asset prices in each period can be estimated from the variance. 2
+Approach 1: Pseudo-American Valuation
+Step 1: Define when dividends will be paid and how much the dividends will be.
+Step 2: Value the call option to each ex-dividend date using the dividend-adjusted approach described earlier, where the stock price is reduced by the present value of expected dividends.
+Step 3: Choose the maximum of the call values estimated for each ex-dividend day. ILLUSTRATION 5.5: Using Pseudo-American Option Valuation to Adjust for Early Exercise
+Consider an option with a strike price of $35 on a stock trading at $40. The variance in the ln(stock prices) is 0.05, and the riskless rate is 4%. The option has a remaining life of eight months, and there are three dividends expected during this period: Expected Dividend
+Ex-Dividend Day $0.80
+In 1 month $0.80
+In 4 months $0.80
+In 7 months The call option is first valued to just before the first ex-dividend date: The value from the Black-Scholes model is: The call option is then valued to before the second ex-dividend date: The value of the call based on these parameters is: The call option is then valued to before the third ex-dividend date: The value of the call based on these parameters is: The call option is then valued to expiration: The value of the call based on these parameters is: Approach 2: Using the Binomial Model
+The binomial model is much more capable of handling early exercise because it considers the cash flows at each time period, rather than just at expiration. The biggest limitation of the binomial model is determining what stock prices will be at the end of each period, but this can be overcome by using a variant that allows us to estimate the up and the down movements in stock prices from the estimated variance. There are four steps involved:
+Step 1: If the variance in ln(stock prices) has been estimated for the Black-Scholes valuation, convert these into inputs for the binomial model: where u and d are the up and the down movements per unit time for the binomial, and dt is the number of periods within each year (or unit time).
+Step 2: Specify the period in which the dividends will be paid and make the assumption that the price will drop by the amount of the dividend in that period.
+Step 3: Value the call at each node of the tree, allowing for the possibility of early exercise just before ex-dividend dates. There will be early exercise if the remaining time premium on the option is less than the expected drop in option value as a consequence of the dividend payment.
+Step 4: Value the call at time 0, using the standard binomial approach. bstobin.xls: This spreadsheet allows you to estimate the parameters for a binomial model from the inputs to a Black-Scholes model. Impact of Exercise on Underlying Asset Value
+The Black-Scholes model is based on the assumption that exercising an option does not affect the value of the underlying asset. This may be true for listed options on stocks, but it is not true for some types of options. For instance, the exercise of warrants increases the number of shares outstanding and brings fresh cash into the firm, both of which will affect the stock price.3 The expected negative impact (dilution) of exercise will decrease the value of warrants, compared to otherwise similar call options. The adjustment for dilution to the stock price is fairly simple in the Black-Scholes valuation. The stock price is adjusted for the expected dilution from the exercise of the options. In the case of warrants, for instance: where S = Current value of the stock
+nw = Number of warrants outstanding
+W = Value of warrants outstanding
+ns = Number of shares outstanding
+When the warrants are exercised, the number of shares outstanding will increase, reducing the stock price. The numerator reflects the market value of equity, including both stocks and warrants outstanding. The reduction in S will reduce the value of the call option.
+There is an element of circularity in this analysis, since the value of the warrant is needed to estimate the dilution-adjusted S and the dilution-adjusted S is needed to estimate the value of the warrant. This problem can be resolved by starting the process off with an assumed value for the warrant (e.g., the exercise value or the current market price of the warrant). This will yield a value for the warrant, and this estimated value can then be used as an input to reestimate the warrant’s value until there is convergence. FROM BLACK-SCHOLES TO BINOMIAL
+The process of converting the continuous variance in a Black-Scholes model to a binomial tree is a fairly simple one. Assume, for instance, that you have an asset that is trading at $30 currently and that you estimate the annualized standard deviation in the asset value to be 40 percent; the annualized riskless rate is 5 percent. For simplicity, let us assume that the option that you are valuing has a four-year life and that each period is a year. To estimate the prices at the end of each of the four years, we begin by first estimating the up and down movements in the binomial: Based on these estimates, we can obtain the prices at the end of the first node of the tree (the end of the first year): Progressing through the rest of the tree, we obtain the following numbers: ILLUSTRATION 5.6: Valuing a Warrant on Avatek Corporation
+Avatek Corporation is a real estate firm with 19.637 million shares outstanding, trading at $0.38 a share. In March 2001 the company had 1.8 million options outstanding, with four years to expiration and with an exercise price of $2.25. The stock paid no dividends, and the standard deviation in ln(stock prices) was 93%. The four-year Treasury bond rate was 4.9%. (The options were trading at $0.12 apiece at the time of this analysis.)
+The inputs to the warrant valuation model are as follows: The results of the Black-Scholes valuation of this option are: The options were trading at $0.12 in March 2001. Since the value was equal to the price, there was no need for further iterations. If there had been a difference, we would have reestimated the adjusted stock price and option value. If the options had been non-traded (as is the case with management options), this calculation would have required an iterative process, where the option value is used to get the adjusted value per share and the value per share to get the option value. warrant.xls: This spreadsheet allows you to estimate the value of an option when there is a potential dilution from exercise. The Black-Scholes Model for Valuing Puts
+The value of a put can be derived from the value of a call with the same strike price and the same expiration date: where C is the value of the call and P is the value of the put. This relationship between the call and put values is called put-call parity, and any deviations from parity can be used by investors to make riskless profits. To see why put-call parity holds, consider selling a call and buying a put with exercise price K and expiration date t, and simultaneously buying the underlying asset at the current price S. The payoff from this position is riskless and always yields K at expiration (t). To see this, assume that the stock price at expiration is S*. The payoff on each of the positions in the portfolio can be written as follows: Position
+Payoffs at t if S* > K
+Payoffs at t if S* < K Sell call
+–(S* – K)
+0 Buy put
+0
+K – S* Buy stock
+S*
+S* Total
+K
+K Since this position yields K with certainty, the cost of creating this position must be equal to the present value of K at the riskless rate (K e–rt). Substituting the Black-Scholes equation for the value of an equivalent call into this equation, we get: Thus, the replicating portfolio for a put is created by selling short [1 – N(d1)] shares of stock and investing K e–rt[1 – N(d2)] in the riskless asset. ILLUSTRATION 5.7: Valuing a Put Using Put-Call Parity: Cisco Systems and AT&T
+Consider the call on Cisco Systems that we valued in Illustration 5.2. The call had a strike price of $15 on the stock, had 103 days left to expiration, and was valued at $1.87. The stock was trading at $13.62, and the riskless rate was 4.63%. The put can be valued as follows: The put was trading at $3.38.
+Also, a long-term call on AT&T was valued in Illustration 5.4. The call had a strike price of $20, 1.8333 years left to expiration, and a value of $6.63. The stock was trading at $20.50 and was expected to maintain a dividend yield of 2.51% over the period. The riskless rate was 4.85%. The put value can be estimated as follows: The put was trading at $3.80. Both the call and the put were trading at different prices from our estimates, which may indicate that we have not correctly estimated the stock’s volatility. Jump Process Option Pricing Models
+If price changes remain larger as the time periods in the binomial model are shortened, it can no longer be assumed that prices change continuously. When price changes remain large, a price process that allows for price jumps is much more realistic. Cox and Ross (1976) valued options when prices follow a pure jump process, where the jumps can only be positive. Thus, in the next interval, the stock price will either have a large positive jump with a specified probability or drift downward at a given rate.
+Merton (1976) considered a distribution where there are price jumps superimposed on a continuous price process. He specified the rate at which jumps occur (λ) and the average jump size (k), measured as a percentage of the stock price. The model derived to value options with this process is called a jump diffusion model. In this model, the value of an option is determined by the five variables specified in the Black-Scholes model, and the parameters of the jump process (λ, k). Unfortunately, the estimates of the jump process parameters are so difficult to make for most firms that they overwhelm any advantages that accrue from using a more realistic model. These models, therefore, have seen limited use in practice.
+EXTENSIONS OF OPTION PRICING
+All the option pricing models described so far—the binomial, the Black-Scholes, and the jump process models—are designed to value options with clearly defined exercise prices and maturities on underlying assets that are traded. However, the options we encounter in investment analysis or valuation are often on real assets rather than financial assets. Categorized as real options, they can take much more complicated forms. This section considers some of these variations.
+Capped and Barrier Options
+With a simple call option, there is no specified upper limit on the profits that can be made by the buyer of the call. Asset prices, at least in theory, can keep going up, and the payoffs increase proportionately. In some call options, though, the buyer is entitled to profits up to a specified price but not above it. For instance, consider a call option with a strike price of K1 on an asset. In an unrestricted call option, the payoff on this option will increase as the underlying asset’s price increases above K1. Assume, however, that if the price reaches K2, the payoff is capped at (K2 – K1). The payoff diagram on this option is shown in Figure 5.5. Figure 5.5 Payoff on Capped Call This option is called a capped call. Notice, also, that once the price reaches K2, there is no longer any time premium associated with the option, and the option will therefore be exercised. Capped calls are part of a family of options called barrier options, where the payoff on and the life of the option are functions of whether the underlying asset price reaches a certain level during a specified period.
+The value of a capped call is always lower than the value of the same call without the payoff limit. A simple approximation of this value can be obtained by valuing the call twice, once with the given exercise price and once with the cap, and taking the difference in the two values. In the preceding example, then, the value of the call with an exercise price of K1 and a cap at K2 can be written as: Barrier options can take many forms. In a knockout option, an option ceases to exist if the underlying asset reaches a certain price. In the case of a call option, this knockout price is usually set below the strike price, and this option is called a down-and-out option. In the case of a put option, the knockout price will be set above the exercise price, and this option is called an up-and-out option. Like the capped call, these options are worth less than their unrestricted counterparts. Many real options have limits on potential upside, or knockout provisions, and ignoring these limits can result in the overstatement of the value of these options.
+Compound Options
+Some options derive their value not from an underlying asset, but from other options. These options are called compound options. Compound options can take any of four forms—a call on a call, a put on a put, a call on a put, or a put on a call. Geske (1979) developed the analytical formulation for valuing compound options by replacing the standard normal distribution used in a simple option model with a bivariate normal distribution in the calculation.
+Consider, for instance, the option to expand a project that is discussed in Chapter 30. While we will value this option using a simple option pricing model, in reality there could be multiple stages in expansion, with each stage representing an option for the following stage. In this case, we will undervalue the option by considering it as a simple rather than a compound option.
+Notwithstanding this discussion, the valuation of compound options becomes progressively more difficult as more options are added to the chain. In this case, rather than wreck the valuation on the shoals of estimation error, it may be better to accept the conservative estimate that is provided with a simple valuation model as a floor on the value.
+Rainbow Options
+In a simple option, the uncertainty is about the price of the underlying asset. Some options are exposed to two or more sources of uncertainty, and these options are rainbow options. Using the simple option pricing model to value such options can lead to biased estimates of value. As an example, consider an undeveloped oil reserve as an option, where the firm that owns the reserve has the right to develop the reserve. Here there are two sources of uncertainty. The first is obviously the price of oil, and the second is the quantity of oil that is in the reserve. To value this undeveloped reserve, we can make the simplifying assumption that we know the quantity of oil in the reserve with certainty. In reality, however, uncertainty about the quantity will affect the value of this option and make the decision to exercise more difficult.4
+CONCLUSION
+An option is an asset with payoffs that are contingent on the value of an underlying asset. A call option provides its holder with the right to buy the underlying asset at a fixed price, whereas a put option provides its holder with the right to sell at a fixed price, at any time before the expiration of the option. The value of an option is determined by six variables—the current value of the underlying asset, the variance in this value, the expected dividends on the asset, the strike price and life of the option, and the riskless interest rate. This is illustrated in both the binomial and the Black-Scholes models, which value options by creating replicating portfolios composed of the underlying asset and riskless lending or borrowing. These models can be used to value assets that have option like characteristics.
+QUESTIONS AND SHORT PROBLEMS
+In the problems following, use an equity risk premium of 5.5 percent if none is specified. 1. The following are prices of options traded on Microsoft Corporation, which pays no dividends. The stock is trading at $83, and the annualized riskless rate is 3.8%. The standard deviation in ln(stock prices) (based on historical data) is 30%. a. Estimate the value of a three-month call with a strike price of $85.
+b. Using the inputs from the Black-Scholes model, specify how you would replicate this call.
+c. What is the implied standard deviation in this call?
+d. Assume now that you buy a call with a strike price of $85 and sell a call with a strike price of $90. Draw the payoff diagram on this position.
+e. Using put-call parity, estimate the value of a three-month put with a strike price of $85. 2. You are trying to value three-month call and put options on Merck with a strike price of $30. The stock is trading at $28.75, and the company expects to pay a quarterly dividend per share of $0.28 in two months. The annualized riskless interest rate is 3.6%, and the standard deviation in log stock prices is 20%. a. Estimate the value of the call and put options, using the Black-Scholes model.
+b. What effect does the expected dividend payment have on call values? On put values? Why? 3. There is the possibility that the options on Merck described in the preceding problem could be exercised early. a. Use the pseudo-American call option technique to determine whether this will affect the value of the call.
+b. Why does the possibility of early exercise exist? What types of options are most likely to be exercised early? 4. You have been provided the following information on a three-month call: a. If you wanted to replicate buying this call, how much money would you need to borrow?
+b. If you wanted to replicate buying this call, how many shares of stock would you need to buy? 5. Go Video, a manufacturer of video recorders, was trading at $4 per share in May 1994. There were 11 million shares outstanding. At the same time, it had 550,000 one-year warrants outstanding, with a strike price of $4.25. The stock has had a standard deviation of 60%. The stock does not pay a dividend. The riskless rate is 5%. a. Estimate the value of the warrants, ignoring dilution.
+b. Estimate the value of the warrants, allowing for dilution.
+c. Why does dilution reduce the value of the warrants? 6. You are trying to value a long-term call option on the NYSE Composite index, expiring in five years, with a strike price of 275. The index is currently at 250, and the annualized standard deviation in stock prices is 15%. The average dividend yield on the index is 3% and is expected to remain unchanged over the next five years. The five-year Treasury bond rate is 5%. a. Estimate the value of the long-term call option.
+b. Estimate the value of a put option with the same parameters.
+c. What are the implicit assumptions you are making when you use the Black-Scholes model to value this option? Which of these assumptions are likely to be violated? What are the consequences for your valuation? 7. A new security on AT&T will entitle the investor to all dividends on AT&T over the next three years, limiting upside potential to 20% but also providing downside protection below 10%. AT&T stock is trading at $50, and three-year call and put options are traded on the exchange at the following prices: How much would you be willing to pay for this security? 1 Note, though, that higher variance can reduce the value of the underlying asset. As a call option becomes more in-the-money, the more it resembles the underlying asset. For very deep in-the-money call options, higher variance can reduce the value of the option.
+2 To illustrate, if σ2 is the variance in ln(stock prices), the up and the down movements in the binomial can be estimated as follows: where u and d are the up and down movements per unit time for the binomial, T is the life of the option, and m is the number of periods within that lifetime.
+3 Warrants are call options issued by firms, either as part of management compensation contracts or to raise equity.
+4 The analogy to a listed option on a stock is the case where you do not know with certainty what the stock price is when you exercise the option. The more uncertain you are about the stock price, the more margin for error you have to give yourself when you exercise the option, to ensure that you are in fact earning a profit.
+CHAPTER 5
+Option Pricing Theory and Models
+In general, the value of any asset is the present value of the expected cash flows on that asset. This chapter considers an exception to that rule when it looks at assets with two specific characteristics: 1. The assets derive their value from the values of other assets.
+2. The cash flows on the assets are contingent on the occurrence of specific events. These assets are called options, and the present value of the expected cash flows on these assets will understate their true value. This chapter describes the cash flow characteristics of options, considers the factors that determine their value, and examines how best to value them.
+BASICS OF OPTION PRICING
+An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called a strike price or an exercise price) at or before the expiration date of the option. Since it is a right and not an obligation, the holder can choose not to exercise the right and can allow the option to expire. There are two types of options—call options and put options.
+Call and Put Options: Description and Payoff Diagrams
+A call option gives the buyer of the option the right to buy the underlying asset at the strike price or the exercise price at any time prior to the expiration date of the option. The buyer pays a price for this right. If at expiration the value of the asset is less than the strike price, the option is not exercised and expires worthless. If, however, the value of the asset is greater than the strike price, the option is exercised—the buyer of the option buys the stock at the exercise price, and the difference between the asset value and the exercise price comprises the gross profit on the investment. The net profit on the investment is the difference between the gross profit and the price paid for the call initially.
+A payoff diagram illustrates the cash payoff on an option at expiration. For a call, the net payoff is negative (and equal to the price paid for the call) if the value of the underlying asset is less than the strike price. If the price of the underlying asset exceeds the strike price, the gross payoff is the difference between the value of the underlying asset and the strike price, and the net payoff is the difference between the gross payoff and the price of the call. This is illustrated in Figure 5.1. Figure 5.1 Payoff on Call Option A put option gives the buyer of the option the right to sell the underlying asset at a fixed price, again called the strike or exercise price, at any time prior to the expiration date of the option. The buyer pays a price for this right. If the price of the underlying asset is greater than the strike price, the option will not be exercised and will expire worthless. But if the price of the underlying asset is less than the strike price, the owner of the put option will exercise the option and sell the stock at the strike price, claiming the difference between the strike price and the market value of the asset as the gross profit. Again, netting out the initial cost paid for the put yields the net profit from the transaction.
+A put has a negative net payoff if the value of the underlying asset exceeds the strike price, and has a gross payoff equal to the difference between the strike price and the value of the underlying asset if the asset value is less than the strike price. This is summarized in Figure 5.2. Figure 5.2 Payoff on Put Option DETERMINANTS OF OPTION VALUE
+The value of an option is determined by six variables relating to the underlying asset and financial markets. 1. Current value of the underlying asset. Options are assets that derive value from an underlying asset. Consequently, changes in the value of the underlying asset affect the value of the options on that asset. Since calls provide the right to buy the underlying asset at a fixed price, an increase in the value of the asset will increase the value of the calls. Puts, on the other hand, become less valuable as the value of the asset increases.
+2. Variance in value of the underlying asset. The buyer of an option acquires the right to buy or sell the underlying asset at a fixed price. The higher the variance in the value of the underlying asset, the greater the value of the option.1 This is true for both calls and puts. While it may seem counterintuitive that an increase in a risk measure (variance) should increase value, options are different from other securities since buyers of options can never lose more than the price they pay for them; in fact, they have the potential to earn significant returns from large price movements.
+3. Dividends paid on the underlying asset. The value of the underlying asset can be expected to decrease if dividend payments are made on the asset during the life of the option. Consequently, the value of a call on the asset is a decreasing function of the size of expected dividend payments, and the value of a put is an increasing function of expected dividend payments. A more intuitive way of thinking about dividend payments, for call options, is as a cost of delaying exercise on in-the-money options. To see why, consider an option on a traded stock. Once a call option is in-the-money (i.e., the holder of the option will make a gross payoff by exercising the option), exercising the call option will provide the holder with the stock and entitle him or her to the dividends on the stock in subsequent periods. Failing to exercise the option will mean that these dividends are forgone.
+4. Strike price of the option. A key characteristic used to describe an option is the strike price. In the case of calls, where the holder acquires the right to buy at a fixed price, the value of the call will decline as the strike price increases. In the case of puts, where the holder has the right to sell at a fixed price, the value will increase as the strike price increases.
+5. Time to expiration on the option. Both calls and puts are more valuable the greater the time to expiration. This is because the longer time to expiration provides more time for the value of the underlying asset to move, increasing the value of both types of options. Additionally, in the case of a call, where the buyer has the right to pay a fixed price at expiration, the present value of this fixed price decreases as the life of the option increases, increasing the value of the call.
+6. Riskless interest rate corresponding to life of the option. Since the buyer of an option pays the price of the option up front, an opportunity cost is involved. This cost will depend on the level of interest rates and the time to expiration of the option. The riskless interest rate also enters into the valuation of options when the present value of the exercise price is calculated, since the exercise price does not have to be paid (received) until expiration on calls (puts). Increases in the interest rate will increase the value of calls and reduce the value of puts. Table 5.1 summarizes the variables and their predicted effects on call and put prices.
+Table 5.1 Summary of Variables Affecting Call and Put Prices Effect On Factor
+Call Value
+Put Value Increase in underlying asset’s value
+Increases
+Decreases Increase in variance of underlying asset
+Increases
+Increases Increase in strike price
+Decreases
+Increases Increase in dividends paid
+Decreases
+Increases Increase in time to expiration
+Increases
+Increases Increase in interest rates
+Increases
+Decreases American versus European Options: Variables Relating to Early Exercise
+A primary distinction between American and European options is that an American option can be exercised at any time prior to its expiration, while European options can be exercised only at expiration. The possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. There is one compensating factor that enables the former to be valued using models designed for the latter. In most cases, the time premium associated with the remaining life of an option and transaction costs make early exercise suboptimal. In other words, the holders of in-the-money options generally get much more by selling the options to someone else than by exercising the options.
+OPTION PRICING MODELS
+Option pricing theory has made vast strides since 1972, when Fischer Black and Myron Scholes published their pathbreaking paper that provided a model for valuing dividend-protected European options. Black and Scholes used a “replicating portfolio”—a portfolio composed of the underlying asset and the risk-free asset that had the same cash flows as the option being valued—and the notion of arbitrage to come up with their final formulation. Although their derivation is mathematically complicated, there is a simpler binomial model for valuing options that draws on the same logic.
+Binomial Model
+The binomial option pricing model is based on a simple formulation for the asset price process in which the asset, in any time period, can move to one of two possible prices. The general formulation of a stock price process that follows the binomial path is shown in Figure 5.3. In this figure, S is the current stock price; the price moves up to Su with probability p and down to Sd with probability 1 – p in any time period. Figure 5.3 General Formulation for Binomial Price Path Creating a Replicating Portfolio
+The objective in creating a replicating portfolio is to use a combination of risk-free borrowing/lending and the underlying asset to create the same cash flows as the option being valued. The principles of arbitrage apply then, and the value of the option must be equal to the value of the replicating portfolio. In the case of the general formulation shown in Figure 5.3, where stock prices can move either up to Su or down to Sd in any time period, the replicating portfolio for a call with strike price K will involve borrowing $B and acquiring Δ of the underlying asset, where: where Cu = Value of the call if the stock price is Su
+Cd = Value of the call if the stock price is Sd
+In a multiperiod binomial process, the valuation has to proceed iteratively (i.e., starting with the final time period and moving backward in time until the current point in time). The portfolios replicating the option are created at each step and valued, providing the values for the option in that time period. The final output from the binomial option pricing model is a statement of the value of the option in terms of the replicating portfolio, composed of Δ shares (option delta) of the underlying asset and risk-free borrowing/lending. ILLUSTRATION 5.1: Binomial Option Valuation
+Assume that the objective is to value a call with a strike price of $50, which is expected to expire in two time periods, on an underlying asset whose price currently is $50 and is expected to follow a binomial process: Now assume that the interest rate is 11%. In addition, define: The objective is to combined Δ shares of stock and B dollars of borrowing to replicate the cash flows from the call with a strike price of $50. This can be done iteratively, starting with the last period and working back through the binomial tree.
+Step 1: Start with the end nodes and work backward: Thus, if the stock price is $70 at t = 1, borrowing $45 and buying one share of the stock will give the same cash flows as buying the call. The value of the call at t = 1, if the stock price is $70, is therefore: Considering the other leg of the binomial tree at t = 1, If the stock price is $35 at t = 1, then the call is worth nothing.
+Step 2: Move backward to the earlier time period and create a replicating portfolio that will provide the cash flows the option will provide. In other words, borrowing $22.50 and buying five-sevenths of a share will provide the same cash flows as a call with a strike price of $50 over the call’s lifetime. The value of the call therefore has to be the same as the cost of creating this position. The Determinants of Value
+The binomial model provides insight into the determinants of option value. The value of an option is not determined by the expected price of the asset but by its current price, which, of course, reflects expectations about the future. This is a direct consequence of arbitrage. If the option value deviates from the value of the replicating portfolio, investors can create an arbitrage position (i.e., one that requires no investment, involves no risk, and delivers positive returns). To illustrate, if the portfolio that replicates the call costs more than the call does in the market, an investor could buy the call, sell the replicating portfolio, and be guaranteed the difference as a profit. The cash flows on the two positions will offset each other, leading to no cash flows in subsequent periods. The call option value also increases as the time to expiration is extended, as the price movements (u and d) increase, and with increases in the interest rate.
+While the binomial model provides an intuitive feel for the determinants of option value, it requires a large number of inputs, in terms of expected future prices at each node. As time periods are made shorter in the binomial model, you can make one of two assumptions about asset prices.You can assume that price changes become smaller as periods get shorter; this leads to price changes becoming infinitesimally small as time periods approach zero, leading to a continuous price process. Alternatively, you can assume that price changes stay large even as the period gets shorter; this leads to a jump price process, where prices can jump in any period. This section considers the option pricing models that emerge with each of these assumptions.
+Black-Scholes Model
+When the price process is continuous (i.e., price changes become smaller as time periods get shorter), the binomial model for pricing options converges on the Black-Scholes model. The model, named after its cocreators, Fischer Black and Myron Scholes, allows us to estimate the value of any option using a small number of inputs, and has been shown to be robust in valuing many listed options.
+The Model
+While the derivation of the Black-Scholes model is far too complicated to present here, it is based on the idea of creating a portfolio of the underlying asset and the riskless asset with the same cash flows, and hence the same cost, as the option being valued. The value of a call option in the Black-Scholes model can be written as a function of the five variables: S = Current value of the underlying asset
+K = Strike price of the option
+t = Life to expiration of the option
+r = Riskless interest rate corresponding to the life of the option
+σ2 = Variance in the ln(value) of the underlying asset The value of a call is then: Note that e–rt is the present value factor, and reflects the fact that the exercise price on the call option does not have to be paid until expiration, since the model values European options. N(d1) and N(d2) are probabilities, estimated by using a cumulative standardized normal distribution, and the values of d1 and d2 obtained for an option. The cumulative distribution is shown in Figure 5.4. Figure 5.4 Cumulative Normal Distribution In approximate terms, N(d2) yields the likelihood that an option will generate positive cash flows for its owner at exercise (i.e., that S > K in the case of a call option and that K > S in the case of a put option). The portfolio that replicates the call option is created by buying N(d1) units of the underlying asset, and borrowing Ke–rt N(d2). The portfolio will have the same cash flows as the call option, and thus the same value as the option. N(d1), which is the number of units of the underlying asset that are needed to create the replicating portfolio, is called the option delta. A NOTE ON ESTIMATING THE INPUTS TO THE BLACK-SCHOLES MODEL
+The Black-Scholes model requires inputs that are consistent on time measurement. There are two places where this affects estimates. The first relates to the fact that the model works in continuous time, rather than discrete time. That is why we use the continuous time version of present value (exp–rt) rather than the discrete version, (1 + r)–t. It also means that the inputs such as the riskless rate have to be modified to make them continuous time inputs. For instance, if the one-year Treasury bond rate is 6.2 percent, the risk-free rate that is used in the Black-Scholes model should be: The second relates to the period over which the inputs are estimated. For instance, the preceding rate is an annual rate. The variance that is entered into the model also has to be an annualized variance. The variance, estimated from ln(asset prices), can be annualized easily because variances are linear in time if the serial correlation is zero. Thus, if monthly or weekly prices are used to estimate variance, the variance is annualized by multiplying by 12 or 52, respectively. ILLUSTRATION 5.2: Valuing an Option Using the Black-Scholes Model
+On March 6, 2001, Cisco Systems was trading at $13.62. We will attempt to value a July 2001 call option with a strike price of $15, trading on the CBOE on the same day for $2. The following are the other parameters of the options: The annualized standard deviation in Cisco Systems stock price over the previous year was 81%. This standard deviation is estimated using weekly stock prices over the year, and the resulting number was annualized as follows: The option expiration date is Friday, July 20, 2001. There are 103 days to expiration, and the annualized Treasury bill rate corresponding to this option life is 4.63%. The inputs for the Black-Scholes model are as follows: Current stock price (S) = $13.62
+Strike price on the option = $15
+Option life = 103/365 = 0.2822
+Standard deviation in ln(stock prices) = 81%
+Riskless rate = 4.63% Inputting these numbers into the model, we get: Using the normal distribution, we can estimate the N(d1) and N(d2): The value of the call can now be estimated: Since the call is trading at $2, it is slightly overvalued, assuming that the estimate of standard deviation used is correct. IMPLIED VOLATILITY
+The only input in the Black Scholes on which there can be significant disagreement among investors is the variance. While the variance is often estimated by looking at historical data, the values for options that emerge from using the historical variance can be different from the market prices. For any option, there is some variance at which the estimated value will be equal to the market price. This variance is called an implied variance.
+Consider the Cisco option valued in Illustration 5.2. With a standard deviation of 81 percent, the value of the call option with a strike price of $15 was estimated to be $1.87. Since the market price is higher than the calculated value, we tried higher standard deviations, and at a standard deviation 85.40 percent the value of the option is $2 (which is the market price). This is the implied standard deviation or implied volatility. Model Limitations and Fixes
+The Black-Scholes model was designed to value European options that can be exercised only at maturity and whose underlying assets do not pay dividends. In addition, options are valued based on the assumption that option exercise does not affect the value of the underlying asset. In practice, assets do pay dividends, options sometimes get exercised early, and exercising an option can affect the value of the underlying asset. Adjustments exist that, while not perfect, provide partial corrections to the Black-Scholes model.
+Dividends
+The payment of a dividend reduces the stock price; note that on the ex-dividend day, the stock price generally declines. Consequently, call options become less valuable and put options more valuable as expected dividend payments increase. There are two ways of dealing with dividends in the Black-Scholes model: 1. Short-term options. One approach to dealing with dividends is to estimate the present value of expected dividends that will be paid by the underlying asset during the option life and subtract it from the current value of the asset to use as S in the model. 2. Long-term options. Since it becomes less practical to estimate the present value of dividends the longer the option life, an alternate approach can be used. If the dividend yield (y = Dividends/Current value of the asset) on the underlying asset is expected to remain unchanged during the life of the option, the Black-Scholes model can be modified to take dividends into account. From an intuitive standpoint, the adjustments have two effects. First, the value of the asset is discounted back to the present at the dividend yield to take into account the expected drop in asset value resulting from dividend payments. Second, the interest rate is offset by the dividend yield to reflect the lower carrying cost from holding the asset (in the replicating portfolio). The net effect will be a reduction in the value of calls estimated using this model. ILLUSTRATION 5.3: Valuing a Short-Term Option with Dividend Adjustments—The Black-Scholes Correction
+Assume that it is March 6, 2001, and that AT&T is trading at $20.50 a share. Consider a call option on the stock with a strike price of $20, expiring on July 20, 2001. Using past stock prices, the annualized standard deviation in the log of stock prices for AT&T is estimated at 60%. There is one dividend, amounting to $0.15, and it will be paid in 23 days. The riskless rate is 4.63%. Present value of expected dividend = $0.15/1.046323/365 = $0.15
+Dividend-adjusted stock price = $20.50 – $0.15 = $20.35
+Time to expiration = 103/365 = 0.2822
+Variance in ln(stock prices) = 0.62 = 0.36
+Riskless rate = 4.63% The value from the Black-Scholes model is: The call option was trading at $2.60 on that day. ILLUSTRATION 5.4: Valuing a Long-Term Option with Dividend Adjustments—Primes and Scores
+The CBOE offers longer-term call and put options on some stocks. On March 6, 2001, for instance, you could have purchased an AT&T call expiring on January 17, 2003. The stock price for AT&T is $20.50 (as in the previous example). The following is the valuation of a call option with a strike price of $20. Instead of estimating the present value of dividends over the next two years, assume that AT&T’s dividend yield will remain 2.51% over this period and that the risk-free rate for a two-year Treasury bond is 4.85%. The inputs to the Black-Scholes model are: The value from the Black-Scholes model is: The call was trading at $5.80 on March 8, 2001. optst.xls: This spreadsheet allows you to estimate the value of a short-term option when the expected dividends during the option life can be estimated. optlt.xls: This spreadsheet allows you to estimate the value of an option when the underlying asset has a constant dividend yield. Early Exercise
+The Black-Scholes model was designed to value European options that can be exercised only at expiration. In contrast, most options that we encounter in practice are American options and can be exercised at any time until expiration. As mentioned earlier, the possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. In general, though, with traded options, it is almost always better to sell the option to someone else rather than exercise early, since options have a time premium (i.e., they sell for more than their exercise value). There are two exceptions. One occurs when the underlying asset pays large dividends, thus reducing the expected value of the asset. In this case, call options may be exercised just before an ex-dividend date, if the time premium on the options is less than the expected decline in asset value as a consequence of the dividend payment. The other exception arises when an investor holds both the underlying asset and deep in-the-money puts (i.e., puts with strike prices well above the current price of the underlying asset) on that asset at a time when interest rates are high. In this case, the time premium on the put may be less than the potential gain from exercising the put early and earning interest on the exercise price.
+There are two basic ways of dealing with the possibility of early exercise. One is to continue to use the unadjusted Black-Scholes model and to regard the resulting value as a floor or conservative estimate of the true value. The other is to try to adjust the value of the option for the possibility of early exercise. There are two approaches for doing so. One uses the Black-Scholes model to value the option to each potential exercise date. With options on stocks, this basically requires that the investor values options to each ex-dividend day and chooses the maximum of the estimated call values. The second approach is to use a modified version of the binomial model to consider the possibility of early exercise. In this version, the up and the down movements for asset prices in each period can be estimated from the variance. 2
+Approach 1: Pseudo-American Valuation
+Step 1: Define when dividends will be paid and how much the dividends will be.
+Step 2: Value the call option to each ex-dividend date using the dividend-adjusted approach described earlier, where the stock price is reduced by the present value of expected dividends.
+Step 3: Choose the maximum of the call values estimated for each ex-dividend day. ILLUSTRATION 5.5: Using Pseudo-American Option Valuation to Adjust for Early Exercise
+Consider an option with a strike price of $35 on a stock trading at $40. The variance in the ln(stock prices) is 0.05, and the riskless rate is 4%. The option has a remaining life of eight months, and there are three dividends expected during this period: Expected Dividend
+Ex-Dividend Day $0.80
+In 1 month $0.80
+In 4 months $0.80
+In 7 months The call option is first valued to just before the first ex-dividend date: The value from the Black-Scholes model is: The call option is then valued to before the second ex-dividend date: The value of the call based on these parameters is: The call option is then valued to before the third ex-dividend date: The value of the call based on these parameters is: The call option is then valued to expiration: The value of the call based on these parameters is: Approach 2: Using the Binomial Model
+The binomial model is much more capable of handling early exercise because it considers the cash flows at each time period, rather than just at expiration. The biggest limitation of the binomial model is determining what stock prices will be at the end of each period, but this can be overcome by using a variant that allows us to estimate the up and the down movements in stock prices from the estimated variance. There are four steps involved:
+Step 1: If the variance in ln(stock prices) has been estimated for the Black-Scholes valuation, convert these into inputs for the binomial model: where u and d are the up and the down movements per unit time for the binomial, and dt is the number of periods within each year (or unit time).
+Step 2: Specify the period in which the dividends will be paid and make the assumption that the price will drop by the amount of the dividend in that period.
+Step 3: Value the call at each node of the tree, allowing for the possibility of early exercise just before ex-dividend dates. There will be early exercise if the remaining time premium on the option is less than the expected drop in option value as a consequence of the dividend payment.
+Step 4: Value the call at time 0, using the standard binomial approach. bstobin.xls: This spreadsheet allows you to estimate the parameters for a binomial model from the inputs to a Black-Scholes model. Impact of Exercise on Underlying Asset Value
+The Black-Scholes model is based on the assumption that exercising an option does not affect the value of the underlying asset. This may be true for listed options on stocks, but it is not true for some types of options. For instance, the exercise of warrants increases the number of shares outstanding and brings fresh cash into the firm, both of which will affect the stock price.3 The expected negative impact (dilution) of exercise will decrease the value of warrants, compared to otherwise similar call options. The adjustment for dilution to the stock price is fairly simple in the Black-Scholes valuation. The stock price is adjusted for the expected dilution from the exercise of the options. In the case of warrants, for instance: where S = Current value of the stock
+nw = Number of warrants outstanding
+W = Value of warrants outstanding
+ns = Number of shares outstanding
+When the warrants are exercised, the number of shares outstanding will increase, reducing the stock price. The numerator reflects the market value of equity, including both stocks and warrants outstanding. The reduction in S will reduce the value of the call option.
+There is an element of circularity in this analysis, since the value of the warrant is needed to estimate the dilution-adjusted S and the dilution-adjusted S is needed to estimate the value of the warrant. This problem can be resolved by starting the process off with an assumed value for the warrant (e.g., the exercise value or the current market price of the warrant). This will yield a value for the warrant, and this estimated value can then be used as an input to reestimate the warrant’s value until there is convergence. FROM BLACK-SCHOLES TO BINOMIAL
+The process of converting the continuous variance in a Black-Scholes model to a binomial tree is a fairly simple one. Assume, for instance, that you have an asset that is trading at $30 currently and that you estimate the annualized standard deviation in the asset value to be 40 percent; the annualized riskless rate is 5 percent. For simplicity, let us assume that the option that you are valuing has a four-year life and that each period is a year. To estimate the prices at the end of each of the four years, we begin by first estimating the up and down movements in the binomial: Based on these estimates, we can obtain the prices at the end of the first node of the tree (the end of the first year): Progressing through the rest of the tree, we obtain the following numbers: ILLUSTRATION 5.6: Valuing a Warrant on Avatek Corporation
+Avatek Corporation is a real estate firm with 19.637 million shares outstanding, trading at $0.38 a share. In March 2001 the company had 1.8 million options outstanding, with four years to expiration and with an exercise price of $2.25. The stock paid no dividends, and the standard deviation in ln(stock prices) was 93%. The four-year Treasury bond rate was 4.9%. (The options were trading at $0.12 apiece at the time of this analysis.)
+The inputs to the warrant valuation model are as follows: The results of the Black-Scholes valuation of this option are: The options were trading at $0.12 in March 2001. Since the value was equal to the price, there was no need for further iterations. If there had been a difference, we would have reestimated the adjusted stock price and option value. If the options had been non-traded (as is the case with management options), this calculation would have required an iterative process, where the option value is used to get the adjusted value per share and the value per share to get the option value. warrant.xls: This spreadsheet allows you to estimate the value of an option when there is a potential dilution from exercise. The Black-Scholes Model for Valuing Puts
+The value of a put can be derived from the value of a call with the same strike price and the same expiration date: where C is the value of the call and P is the value of the put. This relationship between the call and put values is called put-call parity, and any deviations from parity can be used by investors to make riskless profits. To see why put-call parity holds, consider selling a call and buying a put with exercise price K and expiration date t, and simultaneously buying the underlying asset at the current price S. The payoff from this position is riskless and always yields K at expiration (t). To see this, assume that the stock price at expiration is S*. The payoff on each of the positions in the portfolio can be written as follows: Position
+Payoffs at t if S* > K
+Payoffs at t if S* < K Sell call
+–(S* – K)
+0 Buy put
+0
+K – S* Buy stock
+S*
+S* Total
+K
+K Since this position yields K with certainty, the cost of creating this position must be equal to the present value of K at the riskless rate (K e–rt). Substituting the Black-Scholes equation for the value of an equivalent call into this equation, we get: Thus, the replicating portfolio for a put is created by selling short [1 – N(d1)] shares of stock and investing K e–rt[1 – N(d2)] in the riskless asset. ILLUSTRATION 5.7: Valuing a Put Using Put-Call Parity: Cisco Systems and AT&T
+Consider the call on Cisco Systems that we valued in Illustration 5.2. The call had a strike price of $15 on the stock, had 103 days left to expiration, and was valued at $1.87. The stock was trading at $13.62, and the riskless rate was 4.63%. The put can be valued as follows: The put was trading at $3.38.
+Also, a long-term call on AT&T was valued in Illustration 5.4. The call had a strike price of $20, 1.8333 years left to expiration, and a value of $6.63. The stock was trading at $20.50 and was expected to maintain a dividend yield of 2.51% over the period. The riskless rate was 4.85%. The put value can be estimated as follows: The put was trading at $3.80. Both the call and the put were trading at different prices from our estimates, which may indicate that we have not correctly estimated the stock’s volatility. Jump Process Option Pricing Models
+If price changes remain larger as the time periods in the binomial model are shortened, it can no longer be assumed that prices change continuously. When price changes remain large, a price process that allows for price jumps is much more realistic. Cox and Ross (1976) valued options when prices follow a pure jump process, where the jumps can only be positive. Thus, in the next interval, the stock price will either have a large positive jump with a specified probability or drift downward at a given rate.
+Merton (1976) considered a distribution where there are price jumps superimposed on a continuous price process. He specified the rate at which jumps occur (λ) and the average jump size (k), measured as a percentage of the stock price. The model derived to value options with this process is called a jump diffusion model. In this model, the value of an option is determined by the five variables specified in the Black-Scholes model, and the parameters of the jump process (λ, k). Unfortunately, the estimates of the jump process parameters are so difficult to make for most firms that they overwhelm any advantages that accrue from using a more realistic model. These models, therefore, have seen limited use in practice.
+EXTENSIONS OF OPTION PRICING
+All the option pricing models described so far—the binomial, the Black-Scholes, and the jump process models—are designed to value options with clearly defined exercise prices and maturities on underlying assets that are traded. However, the options we encounter in investment analysis or valuation are often on real assets rather than financial assets. Categorized as real options, they can take much more complicated forms. This section considers some of these variations.
+Capped and Barrier Options
+With a simple call option, there is no specified upper limit on the profits that can be made by the buyer of the call. Asset prices, at least in theory, can keep going up, and the payoffs increase proportionately. In some call options, though, the buyer is entitled to profits up to a specified price but not above it. For instance, consider a call option with a strike price of K1 on an asset. In an unrestricted call option, the payoff on this option will increase as the underlying asset’s price increases above K1. Assume, however, that if the price reaches K2, the payoff is capped at (K2 – K1). The payoff diagram on this option is shown in Figure 5.5. Figure 5.5 Payoff on Capped Call This option is called a capped call. Notice, also, that once the price reaches K2, there is no longer any time premium associated with the option, and the option will therefore be exercised. Capped calls are part of a family of options called barrier options, where the payoff on and the life of the option are functions of whether the underlying asset price reaches a certain level during a specified period.
+The value of a capped call is always lower than the value of the same call without the payoff limit. A simple approximation of this value can be obtained by valuing the call twice, once with the given exercise price and once with the cap, and taking the difference in the two values. In the preceding example, then, the value of the call with an exercise price of K1 and a cap at K2 can be written as: Barrier options can take many forms. In a knockout option, an option ceases to exist if the underlying asset reaches a certain price. In the case of a call option, this knockout price is usually set below the strike price, and this option is called a down-and-out option. In the case of a put option, the knockout price will be set above the exercise price, and this option is called an up-and-out option. Like the capped call, these options are worth less than their unrestricted counterparts. Many real options have limits on potential upside, or knockout provisions, and ignoring these limits can result in the overstatement of the value of these options.
+Compound Options
+Some options derive their value not from an underlying asset, but from other options. These options are called compound options. Compound options can take any of four forms—a call on a call, a put on a put, a call on a put, or a put on a call. Geske (1979) developed the analytical formulation for valuing compound options by replacing the standard normal distribution used in a simple option model with a bivariate normal distribution in the calculation.
+Consider, for instance, the option to expand a project that is discussed in Chapter 30. While we will value this option using a simple option pricing model, in reality there could be multiple stages in expansion, with each stage representing an option for the following stage. In this case, we will undervalue the option by considering it as a simple rather than a compound option.
+Notwithstanding this discussion, the valuation of compound options becomes progressively more difficult as more options are added to the chain. In this case, rather than wreck the valuation on the shoals of estimation error, it may be better to accept the conservative estimate that is provided with a simple valuation model as a floor on the value.
+Rainbow Options
+In a simple option, the uncertainty is about the price of the underlying asset. Some options are exposed to two or more sources of uncertainty, and these options are rainbow options. Using the simple option pricing model to value such options can lead to biased estimates of value. As an example, consider an undeveloped oil reserve as an option, where the firm that owns the reserve has the right to develop the reserve. Here there are two sources of uncertainty. The first is obviously the price of oil, and the second is the quantity of oil that is in the reserve. To value this undeveloped reserve, we can make the simplifying assumption that we know the quantity of oil in the reserve with certainty. In reality, however, uncertainty about the quantity will affect the value of this option and make the decision to exercise more difficult.4
+CONCLUSION
+An option is an asset with payoffs that are contingent on the value of an underlying asset. A call option provides its holder with the right to buy the underlying asset at a fixed price, whereas a put option provides its holder with the right to sell at a fixed price, at any time before the expiration of the option. The value of an option is determined by six variables—the current value of the underlying asset, the variance in this value, the expected dividends on the asset, the strike price and life of the option, and the riskless interest rate. This is illustrated in both the binomial and the Black-Scholes models, which value options by creating replicating portfolios composed of the underlying asset and riskless lending or borrowing. These models can be used to value assets that have option like characteristics.
+QUESTIONS AND SHORT PROBLEMS
+In the problems following, use an equity risk premium of 5.5 percent if none is specified. 1. The following are prices of options traded on Microsoft Corporation, which pays no dividends. The stock is trading at $83, and the annualized riskless rate is 3.8%. The standard deviation in ln(stock prices) (based on historical data) is 30%. a. Estimate the value of a three-month call with a strike price of $85.
+b. Using the inputs from the Black-Scholes model, specify how you would replicate this call.
+c. What is the implied standard deviation in this call?
+d. Assume now that you buy a call with a strike price of $85 and sell a call with a strike price of $90. Draw the payoff diagram on this position.
+e. Using put-call parity, estimate the value of a three-month put with a strike price of $85. 2. You are trying to value three-month call and put options on Merck with a strike price of $30. The stock is trading at $28.75, and the company expects to pay a quarterly dividend per share of $0.28 in two months. The annualized riskless interest rate is 3.6%, and the standard deviation in log stock prices is 20%. a. Estimate the value of the call and put options, using the Black-Scholes model.
+b. What effect does the expected dividend payment have on call values? On put values? Why? 3. There is the possibility that the options on Merck described in the preceding problem could be exercised early. a. Use the pseudo-American call option technique to determine whether this will affect the value of the call.
+b. Why does the possibility of early exercise exist? What types of options are most likely to be exercised early? 4. You have been provided the following information on a three-month call: a. If you wanted to replicate buying this call, how much money would you need to borrow?
+b. If you wanted to replicate buying this call, how many shares of stock would you need to buy? 5. Go Video, a manufacturer of video recorders, was trading at $4 per share in May 1994. There were 11 million shares outstanding. At the same time, it had 550,000 one-year warrants outstanding, with a strike price of $4.25. The stock has had a standard deviation of 60%. The stock does not pay a dividend. The riskless rate is 5%. a. Estimate the value of the warrants, ignoring dilution.
+b. Estimate the value of the warrants, allowing for dilution.
+c. Why does dilution reduce the value of the warrants? 6. You are trying to value a long-term call option on the NYSE Composite index, expiring in five years, with a strike price of 275. The index is currently at 250, and the annualized standard deviation in stock prices is 15%. The average dividend yield on the index is 3% and is expected to remain unchanged over the next five years. The five-year Treasury bond rate is 5%. a. Estimate the value of the long-term call option.
+b. Estimate the value of a put option with the same parameters.
+c. What are the implicit assumptions you are making when you use the Black-Scholes model to value this option? Which of these assumptions are likely to be violated? What are the consequences for your valuation? 7. A new security on AT&T will entitle the investor to all dividends on AT&T over the next three years, limiting upside potential to 20% but also providing downside protection below 10%. AT&T stock is trading at $50, and three-year call and put options are traded on the exchange at the following prices: How much would you be willing to pay for this security? 1 Note, though, that higher variance can reduce the value of the underlying asset. As a call option becomes more in-the-money, the more it resembles the underlying asset. For very deep in-the-money call options, higher variance can reduce the value of the option.
+2 To illustrate, if σ2 is the variance in ln(stock prices), the up and the down movements in the binomial can be estimated as follows: where u and d are the up and down movements per unit time for the binomial, T is the life of the option, and m is the number of periods within that lifetime.
+3 Warrants are call options issued by firms, either as part of management compensation contracts or to raise equity.
+4 The analogy to a listed option on a stock is the case where you do not know with certainty what the stock price is when you exercise the option. The more uncertain you are about the stock price, the more margin for error you have to give yourself when you exercise the option, to ensure that you are in fact earning a profit.
+CHAPTER 5
+Option Pricing Theory and Models
+In general, the value of any asset is the present value of the expected cash flows on that asset. This chapter considers an exception to that rule when it looks at assets with two specific characteristics: 1. The assets derive their value from the values of other assets.
+2. The cash flows on the assets are contingent on the occurrence of specific events. These assets are called options, and the present value of the expected cash flows on these assets will understate their true value. This chapter describes the cash flow characteristics of options, considers the factors that determine their value, and examines how best to value them.
+BASICS OF OPTION PRICING
+An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called a strike price or an exercise price) at or before the expiration date of the option. Since it is a right and not an obligation, the holder can choose not to exercise the right and can allow the option to expire. There are two types of options—call options and put options.
+Call and Put Options: Description and Payoff Diagrams
+A call option gives the buyer of the option the right to buy the underlying asset at the strike price or the exercise price at any time prior to the expiration date of the option. The buyer pays a price for this right. If at expiration the value of the asset is less than the strike price, the option is not exercised and expires worthless. If, however, the value of the asset is greater than the strike price, the option is exercised—the buyer of the option buys the stock at the exercise price, and the difference between the asset value and the exercise price comprises the gross profit on the investment. The net profit on the investment is the difference between the gross profit and the price paid for the call initially.
+A payoff diagram illustrates the cash payoff on an option at expiration. For a call, the net payoff is negative (and equal to the price paid for the call) if the value of the underlying asset is less than the strike price. If the price of the underlying asset exceeds the strike price, the gross payoff is the difference between the value of the underlying asset and the strike price, and the net payoff is the difference between the gross payoff and the price of the call. This is illustrated in Figure 5.1. Figure 5.1 Payoff on Call Option A put option gives the buyer of the option the right to sell the underlying asset at a fixed price, again called the strike or exercise price, at any time prior to the expiration date of the option. The buyer pays a price for this right. If the price of the underlying asset is greater than the strike price, the option will not be exercised and will expire worthless. But if the price of the underlying asset is less than the strike price, the owner of the put option will exercise the option and sell the stock at the strike price, claiming the difference between the strike price and the market value of the asset as the gross profit. Again, netting out the initial cost paid for the put yields the net profit from the transaction.
+A put has a negative net payoff if the value of the underlying asset exceeds the strike price, and has a gross payoff equal to the difference between the strike price and the value of the underlying asset if the asset value is less than the strike price. This is summarized in Figure 5.2. Figure 5.2 Payoff on Put Option DETERMINANTS OF OPTION VALUE
+The value of an option is determined by six variables relating to the underlying asset and financial markets. 1. Current value of the underlying asset. Options are assets that derive value from an underlying asset. Consequently, changes in the value of the underlying asset affect the value of the options on that asset. Since calls provide the right to buy the underlying asset at a fixed price, an increase in the value of the asset will increase the value of the calls. Puts, on the other hand, become less valuable as the value of the asset increases.
+2. Variance in value of the underlying asset. The buyer of an option acquires the right to buy or sell the underlying asset at a fixed price. The higher the variance in the value of the underlying asset, the greater the value of the option.1 This is true for both calls and puts. While it may seem counterintuitive that an increase in a risk measure (variance) should increase value, options are different from other securities since buyers of options can never lose more than the price they pay for them; in fact, they have the potential to earn significant returns from large price movements.
+3. Dividends paid on the underlying asset. The value of the underlying asset can be expected to decrease if dividend payments are made on the asset during the life of the option. Consequently, the value of a call on the asset is a decreasing function of the size of expected dividend payments, and the value of a put is an increasing function of expected dividend payments. A more intuitive way of thinking about dividend payments, for call options, is as a cost of delaying exercise on in-the-money options. To see why, consider an option on a traded stock. Once a call option is in-the-money (i.e., the holder of the option will make a gross payoff by exercising the option), exercising the call option will provide the holder with the stock and entitle him or her to the dividends on the stock in subsequent periods. Failing to exercise the option will mean that these dividends are forgone.
+4. Strike price of the option. A key characteristic used to describe an option is the strike price. In the case of calls, where the holder acquires the right to buy at a fixed price, the value of the call will decline as the strike price increases. In the case of puts, where the holder has the right to sell at a fixed price, the value will increase as the strike price increases.
+5. Time to expiration on the option. Both calls and puts are more valuable the greater the time to expiration. This is because the longer time to expiration provides more time for the value of the underlying asset to move, increasing the value of both types of options. Additionally, in the case of a call, where the buyer has the right to pay a fixed price at expiration, the present value of this fixed price decreases as the life of the option increases, increasing the value of the call.
+6. Riskless interest rate corresponding to life of the option. Since the buyer of an option pays the price of the option up front, an opportunity cost is involved. This cost will depend on the level of interest rates and the time to expiration of the option. The riskless interest rate also enters into the valuation of options when the present value of the exercise price is calculated, since the exercise price does not have to be paid (received) until expiration on calls (puts). Increases in the interest rate will increase the value of calls and reduce the value of puts. Table 5.1 summarizes the variables and their predicted effects on call and put prices.
+Table 5.1 Summary of Variables Affecting Call and Put Prices Effect On Factor
+Call Value
+Put Value Increase in underlying asset’s value
+Increases
+Decreases Increase in variance of underlying asset
+Increases
+Increases Increase in strike price
+Decreases
+Increases Increase in dividends paid
+Decreases
+Increases Increase in time to expiration
+Increases
+Increases Increase in interest rates
+Increases
+Decreases American versus European Options: Variables Relating to Early Exercise
+A primary distinction between American and European options is that an American option can be exercised at any time prior to its expiration, while European options can be exercised only at expiration. The possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. There is one compensating factor that enables the former to be valued using models designed for the latter. In most cases, the time premium associated with the remaining life of an option and transaction costs make early exercise suboptimal. In other words, the holders of in-the-money options generally get much more by selling the options to someone else than by exercising the options.
+OPTION PRICING MODELS
+Option pricing theory has made vast strides since 1972, when Fischer Black and Myron Scholes published their pathbreaking paper that provided a model for valuing dividend-protected European options. Black and Scholes used a “replicating portfolio”—a portfolio composed of the underlying asset and the risk-free asset that had the same cash flows as the option being valued—and the notion of arbitrage to come up with their final formulation. Although their derivation is mathematically complicated, there is a simpler binomial model for valuing options that draws on the same logic.
+Binomial Model
+The binomial option pricing model is based on a simple formulation for the asset price process in which the asset, in any time period, can move to one of two possible prices. The general formulation of a stock price process that follows the binomial path is shown in Figure 5.3. In this figure, S is the current stock price; the price moves up to Su with probability p and down to Sd with probability 1 – p in any time period. Figure 5.3 General Formulation for Binomial Price Path Creating a Replicating Portfolio
+The objective in creating a replicating portfolio is to use a combination of risk-free borrowing/lending and the underlying asset to create the same cash flows as the option being valued. The principles of arbitrage apply then, and the value of the option must be equal to the value of the replicating portfolio. In the case of the general formulation shown in Figure 5.3, where stock prices can move either up to Su or down to Sd in any time period, the replicating portfolio for a call with strike price K will involve borrowing $B and acquiring Δ of the underlying asset, where: where Cu = Value of the call if the stock price is Su
+Cd = Value of the call if the stock price is Sd
+In a multiperiod binomial process, the valuation has to proceed iteratively (i.e., starting with the final time period and moving backward in time until the current point in time). The portfolios replicating the option are created at each step and valued, providing the values for the option in that time period. The final output from the binomial option pricing model is a statement of the value of the option in terms of the replicating portfolio, composed of Δ shares (option delta) of the underlying asset and risk-free borrowing/lending. ILLUSTRATION 5.1: Binomial Option Valuation
+Assume that the objective is to value a call with a strike price of $50, which is expected to expire in two time periods, on an underlying asset whose price currently is $50 and is expected to follow a binomial process: Now assume that the interest rate is 11%. In addition, define: The objective is to combined Δ shares of stock and B dollars of borrowing to replicate the cash flows from the call with a strike price of $50. This can be done iteratively, starting with the last period and working back through the binomial tree.
+Step 1: Start with the end nodes and work backward: Thus, if the stock price is $70 at t = 1, borrowing $45 and buying one share of the stock will give the same cash flows as buying the call. The value of the call at t = 1, if the stock price is $70, is therefore: Considering the other leg of the binomial tree at t = 1, If the stock price is $35 at t = 1, then the call is worth nothing.
+Step 2: Move backward to the earlier time period and create a replicating portfolio that will provide the cash flows the option will provide. In other words, borrowing $22.50 and buying five-sevenths of a share will provide the same cash flows as a call with a strike price of $50 over the call’s lifetime. The value of the call therefore has to be the same as the cost of creating this position. The Determinants of Value
+The binomial model provides insight into the determinants of option value. The value of an option is not determined by the expected price of the asset but by its current price, which, of course, reflects expectations about the future. This is a direct consequence of arbitrage. If the option value deviates from the value of the replicating portfolio, investors can create an arbitrage position (i.e., one that requires no investment, involves no risk, and delivers positive returns). To illustrate, if the portfolio that replicates the call costs more than the call does in the market, an investor could buy the call, sell the replicating portfolio, and be guaranteed the difference as a profit. The cash flows on the two positions will offset each other, leading to no cash flows in subsequent periods. The call option value also increases as the time to expiration is extended, as the price movements (u and d) increase, and with increases in the interest rate.
+While the binomial model provides an intuitive feel for the determinants of option value, it requires a large number of inputs, in terms of expected future prices at each node. As time periods are made shorter in the binomial model, you can make one of two assumptions about asset prices.You can assume that price changes become smaller as periods get shorter; this leads to price changes becoming infinitesimally small as time periods approach zero, leading to a continuous price process. Alternatively, you can assume that price changes stay large even as the period gets shorter; this leads to a jump price process, where prices can jump in any period. This section considers the option pricing models that emerge with each of these assumptions.
+Black-Scholes Model
+When the price process is continuous (i.e., price changes become smaller as time periods get shorter), the binomial model for pricing options converges on the Black-Scholes model. The model, named after its cocreators, Fischer Black and Myron Scholes, allows us to estimate the value of any option using a small number of inputs, and has been shown to be robust in valuing many listed options.
+The Model
+While the derivation of the Black-Scholes model is far too complicated to present here, it is based on the idea of creating a portfolio of the underlying asset and the riskless asset with the same cash flows, and hence the same cost, as the option being valued. The value of a call option in the Black-Scholes model can be written as a function of the five variables: S = Current value of the underlying asset
+K = Strike price of the option
+t = Life to expiration of the option
+r = Riskless interest rate corresponding to the life of the option
+σ2 = Variance in the ln(value) of the underlying asset The value of a call is then: Note that e–rt is the present value factor, and reflects the fact that the exercise price on the call option does not have to be paid until expiration, since the model values European options. N(d1) and N(d2) are probabilities, estimated by using a cumulative standardized normal distribution, and the values of d1 and d2 obtained for an option. The cumulative distribution is shown in Figure 5.4. Figure 5.4 Cumulative Normal Distribution In approximate terms, N(d2) yields the likelihood that an option will generate positive cash flows for its owner at exercise (i.e., that S > K in the case of a call option and that K > S in the case of a put option). The portfolio that replicates the call option is created by buying N(d1) units of the underlying asset, and borrowing Ke–rt N(d2). The portfolio will have the same cash flows as the call option, and thus the same value as the option. N(d1), which is the number of units of the underlying asset that are needed to create the replicating portfolio, is called the option delta. A NOTE ON ESTIMATING THE INPUTS TO THE BLACK-SCHOLES MODEL
+The Black-Scholes model requires inputs that are consistent on time measurement. There are two places where this affects estimates. The first relates to the fact that the model works in continuous time, rather than discrete time. That is why we use the continuous time version of present value (exp–rt) rather than the discrete version, (1 + r)–t. It also means that the inputs such as the riskless rate have to be modified to make them continuous time inputs. For instance, if the one-year Treasury bond rate is 6.2 percent, the risk-free rate that is used in the Black-Scholes model should be: The second relates to the period over which the inputs are estimated. For instance, the preceding rate is an annual rate. The variance that is entered into the model also has to be an annualized variance. The variance, estimated from ln(asset prices), can be annualized easily because variances are linear in time if the serial correlation is zero. Thus, if monthly or weekly prices are used to estimate variance, the variance is annualized by multiplying by 12 or 52, respectively. ILLUSTRATION 5.2: Valuing an Option Using the Black-Scholes Model
+On March 6, 2001, Cisco Systems was trading at $13.62. We will attempt to value a July 2001 call option with a strike price of $15, trading on the CBOE on the same day for $2. The following are the other parameters of the options: The annualized standard deviation in Cisco Systems stock price over the previous year was 81%. This standard deviation is estimated using weekly stock prices over the year, and the resulting number was annualized as follows: The option expiration date is Friday, July 20, 2001. There are 103 days to expiration, and the annualized Treasury bill rate corresponding to this option life is 4.63%. The inputs for the Black-Scholes model are as follows: Current stock price (S) = $13.62
+Strike price on the option = $15
+Option life = 103/365 = 0.2822
+Standard deviation in ln(stock prices) = 81%
+Riskless rate = 4.63% Inputting these numbers into the model, we get: Using the normal distribution, we can estimate the N(d1) and N(d2): The value of the call can now be estimated: Since the call is trading at $2, it is slightly overvalued, assuming that the estimate of standard deviation used is correct. IMPLIED VOLATILITY
+The only input in the Black Scholes on which there can be significant disagreement among investors is the variance. While the variance is often estimated by looking at historical data, the values for options that emerge from using the historical variance can be different from the market prices. For any option, there is some variance at which the estimated value will be equal to the market price. This variance is called an implied variance.
+Consider the Cisco option valued in Illustration 5.2. With a standard deviation of 81 percent, the value of the call option with a strike price of $15 was estimated to be $1.87. Since the market price is higher than the calculated value, we tried higher standard deviations, and at a standard deviation 85.40 percent the value of the option is $2 (which is the market price). This is the implied standard deviation or implied volatility. Model Limitations and Fixes
+The Black-Scholes model was designed to value European options that can be exercised only at maturity and whose underlying assets do not pay dividends. In addition, options are valued based on the assumption that option exercise does not affect the value of the underlying asset. In practice, assets do pay dividends, options sometimes get exercised early, and exercising an option can affect the value of the underlying asset. Adjustments exist that, while not perfect, provide partial corrections to the Black-Scholes model.
+Dividends
+The payment of a dividend reduces the stock price; note that on the ex-dividend day, the stock price generally declines. Consequently, call options become less valuable and put options more valuable as expected dividend payments increase. There are two ways of dealing with dividends in the Black-Scholes model: 1. Short-term options. One approach to dealing with dividends is to estimate the present value of expected dividends that will be paid by the underlying asset during the option life and subtract it from the current value of the asset to use as S in the model. 2. Long-term options. Since it becomes less practical to estimate the present value of dividends the longer the option life, an alternate approach can be used. If the dividend yield (y = Dividends/Current value of the asset) on the underlying asset is expected to remain unchanged during the life of the option, the Black-Scholes model can be modified to take dividends into account. From an intuitive standpoint, the adjustments have two effects. First, the value of the asset is discounted back to the present at the dividend yield to take into account the expected drop in asset value resulting from dividend payments. Second, the interest rate is offset by the dividend yield to reflect the lower carrying cost from holding the asset (in the replicating portfolio). The net effect will be a reduction in the value of calls estimated using this model. ILLUSTRATION 5.3: Valuing a Short-Term Option with Dividend Adjustments—The Black-Scholes Correction
+Assume that it is March 6, 2001, and that AT&T is trading at $20.50 a share. Consider a call option on the stock with a strike price of $20, expiring on July 20, 2001. Using past stock prices, the annualized standard deviation in the log of stock prices for AT&T is estimated at 60%. There is one dividend, amounting to $0.15, and it will be paid in 23 days. The riskless rate is 4.63%. Present value of expected dividend = $0.15/1.046323/365 = $0.15
+Dividend-adjusted stock price = $20.50 – $0.15 = $20.35
+Time to expiration = 103/365 = 0.2822
+Variance in ln(stock prices) = 0.62 = 0.36
+Riskless rate = 4.63% The value from the Black-Scholes model is: The call option was trading at $2.60 on that day. ILLUSTRATION 5.4: Valuing a Long-Term Option with Dividend Adjustments—Primes and Scores
+The CBOE offers longer-term call and put options on some stocks. On March 6, 2001, for instance, you could have purchased an AT&T call expiring on January 17, 2003. The stock price for AT&T is $20.50 (as in the previous example). The following is the valuation of a call option with a strike price of $20. Instead of estimating the present value of dividends over the next two years, assume that AT&T’s dividend yield will remain 2.51% over this period and that the risk-free rate for a two-year Treasury bond is 4.85%. The inputs to the Black-Scholes model are: The value from the Black-Scholes model is: The call was trading at $5.80 on March 8, 2001. optst.xls: This spreadsheet allows you to estimate the value of a short-term option when the expected dividends during the option life can be estimated. optlt.xls: This spreadsheet allows you to estimate the value of an option when the underlying asset has a constant dividend yield. Early Exercise
+The Black-Scholes model was designed to value European options that can be exercised only at expiration. In contrast, most options that we encounter in practice are American options and can be exercised at any time until expiration. As mentioned earlier, the possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. In general, though, with traded options, it is almost always better to sell the option to someone else rather than exercise early, since options have a time premium (i.e., they sell for more than their exercise value). There are two exceptions. One occurs when the underlying asset pays large dividends, thus reducing the expected value of the asset. In this case, call options may be exercised just before an ex-dividend date, if the time premium on the options is less than the expected decline in asset value as a consequence of the dividend payment. The other exception arises when an investor holds both the underlying asset and deep in-the-money puts (i.e., puts with strike prices well above the current price of the underlying asset) on that asset at a time when interest rates are high. In this case, the time premium on the put may be less than the potential gain from exercising the put early and earning interest on the exercise price.
+There are two basic ways of dealing with the possibility of early exercise. One is to continue to use the unadjusted Black-Scholes model and to regard the resulting value as a floor or conservative estimate of the true value. The other is to try to adjust the value of the option for the possibility of early exercise. There are two approaches for doing so. One uses the Black-Scholes model to value the option to each potential exercise date. With options on stocks, this basically requires that the investor values options to each ex-dividend day and chooses the maximum of the estimated call values. The second approach is to use a modified version of the binomial model to consider the possibility of early exercise. In this version, the up and the down movements for asset prices in each period can be estimated from the variance. 2
+Approach 1: Pseudo-American Valuation
+Step 1: Define when dividends will be paid and how much the dividends will be.
+Step 2: Value the call option to each ex-dividend date using the dividend-adjusted approach described earlier, where the stock price is reduced by the present value of expected dividends.
+Step 3: Choose the maximum of the call values estimated for each ex-dividend day. ILLUSTRATION 5.5: Using Pseudo-American Option Valuation to Adjust for Early Exercise
+Consider an option with a strike price of $35 on a stock trading at $40. The variance in the ln(stock prices) is 0.05, and the riskless rate is 4%. The option has a remaining life of eight months, and there are three dividends expected during this period: Expected Dividend
+Ex-Dividend Day $0.80
+In 1 month $0.80
+In 4 months $0.80
+In 7 months The call option is first valued to just before the first ex-dividend date: The value from the Black-Scholes model is: The call option is then valued to before the second ex-dividend date: The value of the call based on these parameters is: The call option is then valued to before the third ex-dividend date: The value of the call based on these parameters is: The call option is then valued to expiration: The value of the call based on these parameters is: Approach 2: Using the Binomial Model
+The binomial model is much more capable of handling early exercise because it considers the cash flows at each time period, rather than just at expiration. The biggest limitation of the binomial model is determining what stock prices will be at the end of each period, but this can be overcome by using a variant that allows us to estimate the up and the down movements in stock prices from the estimated variance. There are four steps involved:
+Step 1: If the variance in ln(stock prices) has been estimated for the Black-Scholes valuation, convert these into inputs for the binomial model: where u and d are the up and the down movements per unit time for the binomial, and dt is the number of periods within each year (or unit time).
+Step 2: Specify the period in which the dividends will be paid and make the assumption that the price will drop by the amount of the dividend in that period.
+Step 3: Value the call at each node of the tree, allowing for the possibility of early exercise just before ex-dividend dates. There will be early exercise if the remaining time premium on the option is less than the expected drop in option value as a consequence of the dividend payment.
+Step 4: Value the call at time 0, using the standard binomial approach. bstobin.xls: This spreadsheet allows you to estimate the parameters for a binomial model from the inputs to a Black-Scholes model. Impact of Exercise on Underlying Asset Value
+The Black-Scholes model is based on the assumption that exercising an option does not affect the value of the underlying asset. This may be true for listed options on stocks, but it is not true for some types of options. For instance, the exercise of warrants increases the number of shares outstanding and brings fresh cash into the firm, both of which will affect the stock price.3 The expected negative impact (dilution) of exercise will decrease the value of warrants, compared to otherwise similar call options. The adjustment for dilution to the stock price is fairly simple in the Black-Scholes valuation. The stock price is adjusted for the expected dilution from the exercise of the options. In the case of warrants, for instance: where S = Current value of the stock
+nw = Number of warrants outstanding
+W = Value of warrants outstanding
+ns = Number of shares outstanding
+When the warrants are exercised, the number of shares outstanding will increase, reducing the stock price. The numerator reflects the market value of equity, including both stocks and warrants outstanding. The reduction in S will reduce the value of the call option.
+There is an element of circularity in this analysis, since the value of the warrant is needed to estimate the dilution-adjusted S and the dilution-adjusted S is needed to estimate the value of the warrant. This problem can be resolved by starting the process off with an assumed value for the warrant (e.g., the exercise value or the current market price of the warrant). This will yield a value for the warrant, and this estimated value can then be used as an input to reestimate the warrant’s value until there is convergence. FROM BLACK-SCHOLES TO BINOMIAL
+The process of converting the continuous variance in a Black-Scholes model to a binomial tree is a fairly simple one. Assume, for instance, that you have an asset that is trading at $30 currently and that you estimate the annualized standard deviation in the asset value to be 40 percent; the annualized riskless rate is 5 percent. For simplicity, let us assume that the option that you are valuing has a four-year life and that each period is a year. To estimate the prices at the end of each of the four years, we begin by first estimating the up and down movements in the binomial: Based on these estimates, we can obtain the prices at the end of the first node of the tree (the end of the first year): Progressing through the rest of the tree, we obtain the following numbers: ILLUSTRATION 5.6: Valuing a Warrant on Avatek Corporation
+Avatek Corporation is a real estate firm with 19.637 million shares outstanding, trading at $0.38 a share. In March 2001 the company had 1.8 million options outstanding, with four years to expiration and with an exercise price of $2.25. The stock paid no dividends, and the standard deviation in ln(stock prices) was 93%. The four-year Treasury bond rate was 4.9%. (The options were trading at $0.12 apiece at the time of this analysis.)
+The inputs to the warrant valuation model are as follows: The results of the Black-Scholes valuation of this option are: The options were trading at $0.12 in March 2001. Since the value was equal to the price, there was no need for further iterations. If there had been a difference, we would have reestimated the adjusted stock price and option value. If the options had been non-traded (as is the case with management options), this calculation would have required an iterative process, where the option value is used to get the adjusted value per share and the value per share to get the option value. warrant.xls: This spreadsheet allows you to estimate the value of an option when there is a potential dilution from exercise. The Black-Scholes Model for Valuing Puts
+The value of a put can be derived from the value of a call with the same strike price and the same expiration date: where C is the value of the call and P is the value of the put. This relationship between the call and put values is called put-call parity, and any deviations from parity can be used by investors to make riskless profits. To see why put-call parity holds, consider selling a call and buying a put with exercise price K and expiration date t, and simultaneously buying the underlying asset at the current price S. The payoff from this position is riskless and always yields K at expiration (t). To see this, assume that the stock price at expiration is S*. The payoff on each of the positions in the portfolio can be written as follows: Position
+Payoffs at t if S* > K
+Payoffs at t if S* < K Sell call
+–(S* – K)
+0 Buy put
+0
+K – S* Buy stock
+S*
+S* Total
+K
+K Since this position yields K with certainty, the cost of creating this position must be equal to the present value of K at the riskless rate (K e–rt). Substituting the Black-Scholes equation for the value of an equivalent call into this equation, we get: Thus, the replicating portfolio for a put is created by selling short [1 – N(d1)] shares of stock and investing K e–rt[1 – N(d2)] in the riskless asset. ILLUSTRATION 5.7: Valuing a Put Using Put-Call Parity: Cisco Systems and AT&T
+Consider the call on Cisco Systems that we valued in Illustration 5.2. The call had a strike price of $15 on the stock, had 103 days left to expiration, and was valued at $1.87. The stock was trading at $13.62, and the riskless rate was 4.63%. The put can be valued as follows: The put was trading at $3.38.
+Also, a long-term call on AT&T was valued in Illustration 5.4. The call had a strike price of $20, 1.8333 years left to expiration, and a value of $6.63. The stock was trading at $20.50 and was expected to maintain a dividend yield of 2.51% over the period. The riskless rate was 4.85%. The put value can be estimated as follows: The put was trading at $3.80. Both the call and the put were trading at different prices from our estimates, which may indicate that we have not correctly estimated the stock’s volatility. Jump Process Option Pricing Models
+If price changes remain larger as the time periods in the binomial model are shortened, it can no longer be assumed that prices change continuously. When price changes remain large, a price process that allows for price jumps is much more realistic. Cox and Ross (1976) valued options when prices follow a pure jump process, where the jumps can only be positive. Thus, in the next interval, the stock price will either have a large positive jump with a specified probability or drift downward at a given rate.
+Merton (1976) considered a distribution where there are price jumps superimposed on a continuous price process. He specified the rate at which jumps occur (λ) and the average jump size (k), measured as a percentage of the stock price. The model derived to value options with this process is called a jump diffusion model. In this model, the value of an option is determined by the five variables specified in the Black-Scholes model, and the parameters of the jump process (λ, k). Unfortunately, the estimates of the jump process parameters are so difficult to make for most firms that they overwhelm any advantages that accrue from using a more realistic model. These models, therefore, have seen limited use in practice.
+EXTENSIONS OF OPTION PRICING
+All the option pricing models described so far—the binomial, the Black-Scholes, and the jump process models—are designed to value options with clearly defined exercise prices and maturities on underlying assets that are traded. However, the options we encounter in investment analysis or valuation are often on real assets rather than financial assets. Categorized as real options, they can take much more complicated forms. This section considers some of these variations.
+Capped and Barrier Options
+With a simple call option, there is no specified upper limit on the profits that can be made by the buyer of the call. Asset prices, at least in theory, can keep going up, and the payoffs increase proportionately. In some call options, though, the buyer is entitled to profits up to a specified price but not above it. For instance, consider a call option with a strike price of K1 on an asset. In an unrestricted call option, the payoff on this option will increase as the underlying asset’s price increases above K1. Assume, however, that if the price reaches K2, the payoff is capped at (K2 – K1). The payoff diagram on this option is shown in Figure 5.5. Figure 5.5 Payoff on Capped Call This option is called a capped call. Notice, also, that once the price reaches K2, there is no longer any time premium associated with the option, and the option will therefore be exercised. Capped calls are part of a family of options called barrier options, where the payoff on and the life of the option are functions of whether the underlying asset price reaches a certain level during a specified period.
+The value of a capped call is always lower than the value of the same call without the payoff limit. A simple approximation of this value can be obtained by valuing the call twice, once with the given exercise price and once with the cap, and taking the difference in the two values. In the preceding example, then, the value of the call with an exercise price of K1 and a cap at K2 can be written as: Barrier options can take many forms. In a knockout option, an option ceases to exist if the underlying asset reaches a certain price. In the case of a call option, this knockout price is usually set below the strike price, and this option is called a down-and-out option. In the case of a put option, the knockout price will be set above the exercise price, and this option is called an up-and-out option. Like the capped call, these options are worth less than their unrestricted counterparts. Many real options have limits on potential upside, or knockout provisions, and ignoring these limits can result in the overstatement of the value of these options.
+Compound Options
+Some options derive their value not from an underlying asset, but from other options. These options are called compound options. Compound options can take any of four forms—a call on a call, a put on a put, a call on a put, or a put on a call. Geske (1979) developed the analytical formulation for valuing compound options by replacing the standard normal distribution used in a simple option model with a bivariate normal distribution in the calculation.
+Consider, for instance, the option to expand a project that is discussed in Chapter 30. While we will value this option using a simple option pricing model, in reality there could be multiple stages in expansion, with each stage representing an option for the following stage. In this case, we will undervalue the option by considering it as a simple rather than a compound option.
+Notwithstanding this discussion, the valuation of compound options becomes progressively more difficult as more options are added to the chain. In this case, rather than wreck the valuation on the shoals of estimation error, it may be better to accept the conservative estimate that is provided with a simple valuation model as a floor on the value.
+Rainbow Options
+In a simple option, the uncertainty is about the price of the underlying asset. Some options are exposed to two or more sources of uncertainty, and these options are rainbow options. Using the simple option pricing model to value such options can lead to biased estimates of value. As an example, consider an undeveloped oil reserve as an option, where the firm that owns the reserve has the right to develop the reserve. Here there are two sources of uncertainty. The first is obviously the price of oil, and the second is the quantity of oil that is in the reserve. To value this undeveloped reserve, we can make the simplifying assumption that we know the quantity of oil in the reserve with certainty. In reality, however, uncertainty about the quantity will affect the value of this option and make the decision to exercise more difficult.4
+CONCLUSION
+An option is an asset with payoffs that are contingent on the value of an underlying asset. A call option provides its holder with the right to buy the underlying asset at a fixed price, whereas a put option provides its holder with the right to sell at a fixed price, at any time before the expiration of the option. The value of an option is determined by six variables—the current value of the underlying asset, the variance in this value, the expected dividends on the asset, the strike price and life of the option, and the riskless interest rate. This is illustrated in both the binomial and the Black-Scholes models, which value options by creating replicating portfolios composed of the underlying asset and riskless lending or borrowing. These models can be used to value assets that have option like characteristics.
+QUESTIONS AND SHORT PROBLEMS
+In the problems following, use an equity risk premium of 5.5 percent if none is specified. 1. The following are prices of options traded on Microsoft Corporation, which pays no dividends. The stock is trading at $83, and the annualized riskless rate is 3.8%. The standard deviation in ln(stock prices) (based on historical data) is 30%. a. Estimate the value of a three-month call with a strike price of $85.
+b. Using the inputs from the Black-Scholes model, specify how you would replicate this call.
+c. What is the implied standard deviation in this call?
+d. Assume now that you buy a call with a strike price of $85 and sell a call with a strike price of $90. Draw the payoff diagram on this position.
+e. Using put-call parity, estimate the value of a three-month put with a strike price of $85. 2. You are trying to value three-month call and put options on Merck with a strike price of $30. The stock is trading at $28.75, and the company expects to pay a quarterly dividend per share of $0.28 in two months. The annualized riskless interest rate is 3.6%, and the standard deviation in log stock prices is 20%. a. Estimate the value of the call and put options, using the Black-Scholes model.
+b. What effect does the expected dividend payment have on call values? On put values? Why? 3. There is the possibility that the options on Merck described in the preceding problem could be exercised early. a. Use the pseudo-American call option technique to determine whether this will affect the value of the call.
+b. Why does the possibility of early exercise exist? What types of options are most likely to be exercised early? 4. You have been provided the following information on a three-month call: a. If you wanted to replicate buying this call, how much money would you need to borrow?
+b. If you wanted to replicate buying this call, how many shares of stock would you need to buy? 5. Go Video, a manufacturer of video recorders, was trading at $4 per share in May 1994. There were 11 million shares outstanding. At the same time, it had 550,000 one-year warrants outstanding, with a strike price of $4.25. The stock has had a standard deviation of 60%. The stock does not pay a dividend. The riskless rate is 5%. a. Estimate the value of the warrants, ignoring dilution.
+b. Estimate the value of the warrants, allowing for dilution.
+c. Why does dilution reduce the value of the warrants? 6. You are trying to value a long-term call option on the NYSE Composite index, expiring in five years, with a strike price of 275. The index is currently at 250, and the annualized standard deviation in stock prices is 15%. The average dividend yield on the index is 3% and is expected to remain unchanged over the next five years. The five-year Treasury bond rate is 5%. a. Estimate the value of the long-term call option.
+b. Estimate the value of a put option with the same parameters.
+c. What are the implicit assumptions you are making when you use the Black-Scholes model to value this option? Which of these assumptions are likely to be violated? What are the consequences for your valuation? 7. A new security on AT&T will entitle the investor to all dividends on AT&T over the next three years, limiting upside potential to 20% but also providing downside protection below 10%. AT&T stock is trading at $50, and three-year call and put options are traded on the exchange at the following prices: How much would you be willing to pay for this security? 1 Note, though, that higher variance can reduce the value of the underlying asset. As a call option becomes more in-the-money, the more it resembles the underlying asset. For very deep in-the-money call options, higher variance can reduce the value of the option.
+2 To illustrate, if σ2 is the variance in ln(stock prices), the up and the down movements in the binomial can be estimated as follows: where u and d are the up and down movements per unit time for the binomial, T is the life of the option, and m is the number of periods within that lifetime.
+3 Warrants are call options issued by firms, either as part of management compensation contracts or to raise equity.
+4 The analogy to a listed option on a stock is the case where you do not know with certainty what the stock price is when you exercise the option. The more uncertain you are about the stock price, the more margin for error you have to give yourself when you exercise the option, to ensure that you are in fact earning a profit.
+CHAPTER 5
+Option Pricing Theory and Models
+In general, the value of any asset is the present value of the expected cash flows on that asset. This chapter considers an exception to that rule when it looks at assets with two specific characteristics: 1. The assets derive their value from the values of other assets.
+2. The cash flows on the assets are contingent on the occurrence of specific events. These assets are called options, and the present value of the expected cash flows on these assets will understate their true value. This chapter describes the cash flow characteristics of options, considers the factors that determine their value, and examines how best to value them.
+BASICS OF OPTION PRICING
+An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called a strike price or an exercise price) at or before the expiration date of the option. Since it is a right and not an obligation, the holder can choose not to exercise the right and can allow the option to expire. There are two types of options—call options and put options.
+Call and Put Options: Description and Payoff Diagrams
+A call option gives the buyer of the option the right to buy the underlying asset at the strike price or the exercise price at any time prior to the expiration date of the option. The buyer pays a price for this right. If at expiration the value of the asset is less than the strike price, the option is not exercised and expires worthless. If, however, the value of the asset is greater than the strike price, the option is exercised—the buyer of the option buys the stock at the exercise price, and the difference between the asset value and the exercise price comprises the gross profit on the investment. The net profit on the investment is the difference between the gross profit and the price paid for the call initially.
+A payoff diagram illustrates the cash payoff on an option at expiration. For a call, the net payoff is negative (and equal to the price paid for the call) if the value of the underlying asset is less than the strike price. If the price of the underlying asset exceeds the strike price, the gross payoff is the difference between the value of the underlying asset and the strike price, and the net payoff is the difference between the gross payoff and the price of the call. This is illustrated in Figure 5.1. Figure 5.1 Payoff on Call Option A put option gives the buyer of the option the right to sell the underlying asset at a fixed price, again called the strike or exercise price, at any time prior to the expiration date of the option. The buyer pays a price for this right. If the price of the underlying asset is greater than the strike price, the option will not be exercised and will expire worthless. But if the price of the underlying asset is less than the strike price, the owner of the put option will exercise the option and sell the stock at the strike price, claiming the difference between the strike price and the market value of the asset as the gross profit. Again, netting out the initial cost paid for the put yields the net profit from the transaction.
+A put has a negative net payoff if the value of the underlying asset exceeds the strike price, and has a gross payoff equal to the difference between the strike price and the value of the underlying asset if the asset value is less than the strike price. This is summarized in Figure 5.2. Figure 5.2 Payoff on Put Option DETERMINANTS OF OPTION VALUE
+The value of an option is determined by six variables relating to the underlying asset and financial markets. 1. Current value of the underlying asset. Options are assets that derive value from an underlying asset. Consequently, changes in the value of the underlying asset affect the value of the options on that asset. Since calls provide the right to buy the underlying asset at a fixed price, an increase in the value of the asset will increase the value of the calls. Puts, on the other hand, become less valuable as the value of the asset increases.
+2. Variance in value of the underlying asset. The buyer of an option acquires the right to buy or sell the underlying asset at a fixed price. The higher the variance in the value of the underlying asset, the greater the value of the option.1 This is true for both calls and puts. While it may seem counterintuitive that an increase in a risk measure (variance) should increase value, options are different from other securities since buyers of options can never lose more than the price they pay for them; in fact, they have the potential to earn significant returns from large price movements.
+3. Dividends paid on the underlying asset. The value of the underlying asset can be expected to decrease if dividend payments are made on the asset during the life of the option. Consequently, the value of a call on the asset is a decreasing function of the size of expected dividend payments, and the value of a put is an increasing function of expected dividend payments. A more intuitive way of thinking about dividend payments, for call options, is as a cost of delaying exercise on in-the-money options. To see why, consider an option on a traded stock. Once a call option is in-the-money (i.e., the holder of the option will make a gross payoff by exercising the option), exercising the call option will provide the holder with the stock and entitle him or her to the dividends on the stock in subsequent periods. Failing to exercise the option will mean that these dividends are forgone.
+4. Strike price of the option. A key characteristic used to describe an option is the strike price. In the case of calls, where the holder acquires the right to buy at a fixed price, the value of the call will decline as the strike price increases. In the case of puts, where the holder has the right to sell at a fixed price, the value will increase as the strike price increases.
+5. Time to expiration on the option. Both calls and puts are more valuable the greater the time to expiration. This is because the longer time to expiration provides more time for the value of the underlying asset to move, increasing the value of both types of options. Additionally, in the case of a call, where the buyer has the right to pay a fixed price at expiration, the present value of this fixed price decreases as the life of the option increases, increasing the value of the call.
+6. Riskless interest rate corresponding to life of the option. Since the buyer of an option pays the price of the option up front, an opportunity cost is involved. This cost will depend on the level of interest rates and the time to expiration of the option. The riskless interest rate also enters into the valuation of options when the present value of the exercise price is calculated, since the exercise price does not have to be paid (received) until expiration on calls (puts). Increases in the interest rate will increase the value of calls and reduce the value of puts. Table 5.1 summarizes the variables and their predicted effects on call and put prices.
+Table 5.1 Summary of Variables Affecting Call and Put Prices Effect On Factor
+Call Value
+Put Value Increase in underlying asset’s value
+Increases
+Decreases Increase in variance of underlying asset
+Increases
+Increases Increase in strike price
+Decreases
+Increases Increase in dividends paid
+Decreases
+Increases Increase in time to expiration
+Increases
+Increases Increase in interest rates
+Increases
+Decreases American versus European Options: Variables Relating to Early Exercise
+A primary distinction between American and European options is that an American option can be exercised at any time prior to its expiration, while European options can be exercised only at expiration. The possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. There is one compensating factor that enables the former to be valued using models designed for the latter. In most cases, the time premium associated with the remaining life of an option and transaction costs make early exercise suboptimal. In other words, the holders of in-the-money options generally get much more by selling the options to someone else than by exercising the options.
+OPTION PRICING MODELS
+Option pricing theory has made vast strides since 1972, when Fischer Black and Myron Scholes published their pathbreaking paper that provided a model for valuing dividend-protected European options. Black and Scholes used a “replicating portfolio”—a portfolio composed of the underlying asset and the risk-free asset that had the same cash flows as the option being valued—and the notion of arbitrage to come up with their final formulation. Although their derivation is mathematically complicated, there is a simpler binomial model for valuing options that draws on the same logic.
+Binomial Model
+The binomial option pricing model is based on a simple formulation for the asset price process in which the asset, in any time period, can move to one of two possible prices. The general formulation of a stock price process that follows the binomial path is shown in Figure 5.3. In this figure, S is the current stock price; the price moves up to Su with probability p and down to Sd with probability 1 – p in any time period. Figure 5.3 General Formulation for Binomial Price Path Creating a Replicating Portfolio
+The objective in creating a replicating portfolio is to use a combination of risk-free borrowing/lending and the underlying asset to create the same cash flows as the option being valued. The principles of arbitrage apply then, and the value of the option must be equal to the value of the replicating portfolio. In the case of the general formulation shown in Figure 5.3, where stock prices can move either up to Su or down to Sd in any time period, the replicating portfolio for a call with strike price K will involve borrowing $B and acquiring Δ of the underlying asset, where: where Cu = Value of the call if the stock price is Su
+Cd = Value of the call if the stock price is Sd
+In a multiperiod binomial process, the valuation has to proceed iteratively (i.e., starting with the final time period and moving backward in time until the current point in time). The portfolios replicating the option are created at each step and valued, providing the values for the option in that time period. The final output from the binomial option pricing model is a statement of the value of the option in terms of the replicating portfolio, composed of Δ shares (option delta) of the underlying asset and risk-free borrowing/lending. ILLUSTRATION 5.1: Binomial Option Valuation
+Assume that the objective is to value a call with a strike price of $50, which is expected to expire in two time periods, on an underlying asset whose price currently is $50 and is expected to follow a binomial process: Now assume that the interest rate is 11%. In addition, define: The objective is to combined Δ shares of stock and B dollars of borrowing to replicate the cash flows from the call with a strike price of $50. This can be done iteratively, starting with the last period and working back through the binomial tree.
+Step 1: Start with the end nodes and work backward: Thus, if the stock price is $70 at t = 1, borrowing $45 and buying one share of the stock will give the same cash flows as buying the call. The value of the call at t = 1, if the stock price is $70, is therefore: Considering the other leg of the binomial tree at t = 1, If the stock price is $35 at t = 1, then the call is worth nothing.
+Step 2: Move backward to the earlier time period and create a replicating portfolio that will provide the cash flows the option will provide. In other words, borrowing $22.50 and buying five-sevenths of a share will provide the same cash flows as a call with a strike price of $50 over the call’s lifetime. The value of the call therefore has to be the same as the cost of creating this position. The Determinants of Value
+The binomial model provides insight into the determinants of option value. The value of an option is not determined by the expected price of the asset but by its current price, which, of course, reflects expectations about the future. This is a direct consequence of arbitrage. If the option value deviates from the value of the replicating portfolio, investors can create an arbitrage position (i.e., one that requires no investment, involves no risk, and delivers positive returns). To illustrate, if the portfolio that replicates the call costs more than the call does in the market, an investor could buy the call, sell the replicating portfolio, and be guaranteed the difference as a profit. The cash flows on the two positions will offset each other, leading to no cash flows in subsequent periods. The call option value also increases as the time to expiration is extended, as the price movements (u and d) increase, and with increases in the interest rate.
+While the binomial model provides an intuitive feel for the determinants of option value, it requires a large number of inputs, in terms of expected future prices at each node. As time periods are made shorter in the binomial model, you can make one of two assumptions about asset prices.You can assume that price changes become smaller as periods get shorter; this leads to price changes becoming infinitesimally small as time periods approach zero, leading to a continuous price process. Alternatively, you can assume that price changes stay large even as the period gets shorter; this leads to a jump price process, where prices can jump in any period. This section considers the option pricing models that emerge with each of these assumptions.
+Black-Scholes Model
+When the price process is continuous (i.e., price changes become smaller as time periods get shorter), the binomial model for pricing options converges on the Black-Scholes model. The model, named after its cocreators, Fischer Black and Myron Scholes, allows us to estimate the value of any option using a small number of inputs, and has been shown to be robust in valuing many listed options.
+The Model
+While the derivation of the Black-Scholes model is far too complicated to present here, it is based on the idea of creating a portfolio of the underlying asset and the riskless asset with the same cash flows, and hence the same cost, as the option being valued. The value of a call option in the Black-Scholes model can be written as a function of the five variables: S = Current value of the underlying asset
+K = Strike price of the option
+t = Life to expiration of the option
+r = Riskless interest rate corresponding to the life of the option
+σ2 = Variance in the ln(value) of the underlying asset The value of a call is then: Note that e–rt is the present value factor, and reflects the fact that the exercise price on the call option does not have to be paid until expiration, since the model values European options. N(d1) and N(d2) are probabilities, estimated by using a cumulative standardized normal distribution, and the values of d1 and d2 obtained for an option. The cumulative distribution is shown in Figure 5.4. Figure 5.4 Cumulative Normal Distribution In approximate terms, N(d2) yields the likelihood that an option will generate positive cash flows for its owner at exercise (i.e., that S > K in the case of a call option and that K > S in the case of a put option). The portfolio that replicates the call option is created by buying N(d1) units of the underlying asset, and borrowing Ke–rt N(d2). The portfolio will have the same cash flows as the call option, and thus the same value as the option. N(d1), which is the number of units of the underlying asset that are needed to create the replicating portfolio, is called the option delta. A NOTE ON ESTIMATING THE INPUTS TO THE BLACK-SCHOLES MODEL
+The Black-Scholes model requires inputs that are consistent on time measurement. There are two places where this affects estimates. The first relates to the fact that the model works in continuous time, rather than discrete time. That is why we use the continuous time version of present value (exp–rt) rather than the discrete version, (1 + r)–t. It also means that the inputs such as the riskless rate have to be modified to make them continuous time inputs. For instance, if the one-year Treasury bond rate is 6.2 percent, the risk-free rate that is used in the Black-Scholes model should be: The second relates to the period over which the inputs are estimated. For instance, the preceding rate is an annual rate. The variance that is entered into the model also has to be an annualized variance. The variance, estimated from ln(asset prices), can be annualized easily because variances are linear in time if the serial correlation is zero. Thus, if monthly or weekly prices are used to estimate variance, the variance is annualized by multiplying by 12 or 52, respectively. ILLUSTRATION 5.2: Valuing an Option Using the Black-Scholes Model
+On March 6, 2001, Cisco Systems was trading at $13.62. We will attempt to value a July 2001 call option with a strike price of $15, trading on the CBOE on the same day for $2. The following are the other parameters of the options: The annualized standard deviation in Cisco Systems stock price over the previous year was 81%. This standard deviation is estimated using weekly stock prices over the year, and the resulting number was annualized as follows: The option expiration date is Friday, July 20, 2001. There are 103 days to expiration, and the annualized Treasury bill rate corresponding to this option life is 4.63%. The inputs for the Black-Scholes model are as follows: Current stock price (S) = $13.62
+Strike price on the option = $15
+Option life = 103/365 = 0.2822
+Standard deviation in ln(stock prices) = 81%
+Riskless rate = 4.63% Inputting these numbers into the model, we get: Using the normal distribution, we can estimate the N(d1) and N(d2): The value of the call can now be estimated: Since the call is trading at $2, it is slightly overvalued, assuming that the estimate of standard deviation used is correct. IMPLIED VOLATILITY
+The only input in the Black Scholes on which there can be significant disagreement among investors is the variance. While the variance is often estimated by looking at historical data, the values for options that emerge from using the historical variance can be different from the market prices. For any option, there is some variance at which the estimated value will be equal to the market price. This variance is called an implied variance.
+Consider the Cisco option valued in Illustration 5.2. With a standard deviation of 81 percent, the value of the call option with a strike price of $15 was estimated to be $1.87. Since the market price is higher than the calculated value, we tried higher standard deviations, and at a standard deviation 85.40 percent the value of the option is $2 (which is the market price). This is the implied standard deviation or implied volatility. Model Limitations and Fixes
+The Black-Scholes model was designed to value European options that can be exercised only at maturity and whose underlying assets do not pay dividends. In addition, options are valued based on the assumption that option exercise does not affect the value of the underlying asset. In practice, assets do pay dividends, options sometimes get exercised early, and exercising an option can affect the value of the underlying asset. Adjustments exist that, while not perfect, provide partial corrections to the Black-Scholes model.
+Dividends
+The payment of a dividend reduces the stock price; note that on the ex-dividend day, the stock price generally declines. Consequently, call options become less valuable and put options more valuable as expected dividend payments increase. There are two ways of dealing with dividends in the Black-Scholes model: 1. Short-term options. One approach to dealing with dividends is to estimate the present value of expected dividends that will be paid by the underlying asset during the option life and subtract it from the current value of the asset to use as S in the model. 2. Long-term options. Since it becomes less practical to estimate the present value of dividends the longer the option life, an alternate approach can be used. If the dividend yield (y = Dividends/Current value of the asset) on the underlying asset is expected to remain unchanged during the life of the option, the Black-Scholes model can be modified to take dividends into account. From an intuitive standpoint, the adjustments have two effects. First, the value of the asset is discounted back to the present at the dividend yield to take into account the expected drop in asset value resulting from dividend payments. Second, the interest rate is offset by the dividend yield to reflect the lower carrying cost from holding the asset (in the replicating portfolio). The net effect will be a reduction in the value of calls estimated using this model. ILLUSTRATION 5.3: Valuing a Short-Term Option with Dividend Adjustments—The Black-Scholes Correction
+Assume that it is March 6, 2001, and that AT&T is trading at $20.50 a share. Consider a call option on the stock with a strike price of $20, expiring on July 20, 2001. Using past stock prices, the annualized standard deviation in the log of stock prices for AT&T is estimated at 60%. There is one dividend, amounting to $0.15, and it will be paid in 23 days. The riskless rate is 4.63%. Present value of expected dividend = $0.15/1.046323/365 = $0.15
+Dividend-adjusted stock price = $20.50 – $0.15 = $20.35
+Time to expiration = 103/365 = 0.2822
+Variance in ln(stock prices) = 0.62 = 0.36
+Riskless rate = 4.63% The value from the Black-Scholes model is: The call option was trading at $2.60 on that day. ILLUSTRATION 5.4: Valuing a Long-Term Option with Dividend Adjustments—Primes and Scores
+The CBOE offers longer-term call and put options on some stocks. On March 6, 2001, for instance, you could have purchased an AT&T call expiring on January 17, 2003. The stock price for AT&T is $20.50 (as in the previous example). The following is the valuation of a call option with a strike price of $20. Instead of estimating the present value of dividends over the next two years, assume that AT&T’s dividend yield will remain 2.51% over this period and that the risk-free rate for a two-year Treasury bond is 4.85%. The inputs to the Black-Scholes model are: The value from the Black-Scholes model is: The call was trading at $5.80 on March 8, 2001. optst.xls: This spreadsheet allows you to estimate the value of a short-term option when the expected dividends during the option life can be estimated. optlt.xls: This spreadsheet allows you to estimate the value of an option when the underlying asset has a constant dividend yield. Early Exercise
+The Black-Scholes model was designed to value European options that can be exercised only at expiration. In contrast, most options that we encounter in practice are American options and can be exercised at any time until expiration. As mentioned earlier, the possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. In general, though, with traded options, it is almost always better to sell the option to someone else rather than exercise early, since options have a time premium (i.e., they sell for more than their exercise value). There are two exceptions. One occurs when the underlying asset pays large dividends, thus reducing the expected value of the asset. In this case, call options may be exercised just before an ex-dividend date, if the time premium on the options is less than the expected decline in asset value as a consequence of the dividend payment. The other exception arises when an investor holds both the underlying asset and deep in-the-money puts (i.e., puts with strike prices well above the current price of the underlying asset) on that asset at a time when interest rates are high. In this case, the time premium on the put may be less than the potential gain from exercising the put early and earning interest on the exercise price.
+There are two basic ways of dealing with the possibility of early exercise. One is to continue to use the unadjusted Black-Scholes model and to regard the resulting value as a floor or conservative estimate of the true value. The other is to try to adjust the value of the option for the possibility of early exercise. There are two approaches for doing so. One uses the Black-Scholes model to value the option to each potential exercise date. With options on stocks, this basically requires that the investor values options to each ex-dividend day and chooses the maximum of the estimated call values. The second approach is to use a modified version of the binomial model to consider the possibility of early exercise. In this version, the up and the down movements for asset prices in each period can be estimated from the variance. 2
+Approach 1: Pseudo-American Valuation
+Step 1: Define when dividends will be paid and how much the dividends will be.
+Step 2: Value the call option to each ex-dividend date using the dividend-adjusted approach described earlier, where the stock price is reduced by the present value of expected dividends.
+Step 3: Choose the maximum of the call values estimated for each ex-dividend day. ILLUSTRATION 5.5: Using Pseudo-American Option Valuation to Adjust for Early Exercise
+Consider an option with a strike price of $35 on a stock trading at $40. The variance in the ln(stock prices) is 0.05, and the riskless rate is 4%. The option has a remaining life of eight months, and there are three dividends expected during this period: Expected Dividend
+Ex-Dividend Day $0.80
+In 1 month $0.80
+In 4 months $0.80
+In 7 months The call option is first valued to just before the first ex-dividend date: The value from the Black-Scholes model is: The call option is then valued to before the second ex-dividend date: The value of the call based on these parameters is: The call option is then valued to before the third ex-dividend date: The value of the call based on these parameters is: The call option is then valued to expiration: The value of the call based on these parameters is: Approach 2: Using the Binomial Model
+The binomial model is much more capable of handling early exercise because it considers the cash flows at each time period, rather than just at expiration. The biggest limitation of the binomial model is determining what stock prices will be at the end of each period, but this can be overcome by using a variant that allows us to estimate the up and the down movements in stock prices from the estimated variance. There are four steps involved:
+Step 1: If the variance in ln(stock prices) has been estimated for the Black-Scholes valuation, convert these into inputs for the binomial model: where u and d are the up and the down movements per unit time for the binomial, and dt is the number of periods within each year (or unit time).
+Step 2: Specify the period in which the dividends will be paid and make the assumption that the price will drop by the amount of the dividend in that period.
+Step 3: Value the call at each node of the tree, allowing for the possibility of early exercise just before ex-dividend dates. There will be early exercise if the remaining time premium on the option is less than the expected drop in option value as a consequence of the dividend payment.
+Step 4: Value the call at time 0, using the standard binomial approach. bstobin.xls: This spreadsheet allows you to estimate the parameters for a binomial model from the inputs to a Black-Scholes model. Impact of Exercise on Underlying Asset Value
+The Black-Scholes model is based on the assumption that exercising an option does not affect the value of the underlying asset. This may be true for listed options on stocks, but it is not true for some types of options. For instance, the exercise of warrants increases the number of shares outstanding and brings fresh cash into the firm, both of which will affect the stock price.3 The expected negative impact (dilution) of exercise will decrease the value of warrants, compared to otherwise similar call options. The adjustment for dilution to the stock price is fairly simple in the Black-Scholes valuation. The stock price is adjusted for the expected dilution from the exercise of the options. In the case of warrants, for instance: where S = Current value of the stock
+nw = Number of warrants outstanding
+W = Value of warrants outstanding
+ns = Number of shares outstanding
+When the warrants are exercised, the number of shares outstanding will increase, reducing the stock price. The numerator reflects the market value of equity, including both stocks and warrants outstanding. The reduction in S will reduce the value of the call option.
+There is an element of circularity in this analysis, since the value of the warrant is needed to estimate the dilution-adjusted S and the dilution-adjusted S is needed to estimate the value of the warrant. This problem can be resolved by starting the process off with an assumed value for the warrant (e.g., the exercise value or the current market price of the warrant). This will yield a value for the warrant, and this estimated value can then be used as an input to reestimate the warrant’s value until there is convergence. FROM BLACK-SCHOLES TO BINOMIAL
+The process of converting the continuous variance in a Black-Scholes model to a binomial tree is a fairly simple one. Assume, for instance, that you have an asset that is trading at $30 currently and that you estimate the annualized standard deviation in the asset value to be 40 percent; the annualized riskless rate is 5 percent. For simplicity, let us assume that the option that you are valuing has a four-year life and that each period is a year. To estimate the prices at the end of each of the four years, we begin by first estimating the up and down movements in the binomial: Based on these estimates, we can obtain the prices at the end of the first node of the tree (the end of the first year): Progressing through the rest of the tree, we obtain the following numbers: ILLUSTRATION 5.6: Valuing a Warrant on Avatek Corporation
+Avatek Corporation is a real estate firm with 19.637 million shares outstanding, trading at $0.38 a share. In March 2001 the company had 1.8 million options outstanding, with four years to expiration and with an exercise price of $2.25. The stock paid no dividends, and the standard deviation in ln(stock prices) was 93%. The four-year Treasury bond rate was 4.9%. (The options were trading at $0.12 apiece at the time of this analysis.)
+The inputs to the warrant valuation model are as follows: The results of the Black-Scholes valuation of this option are: The options were trading at $0.12 in March 2001. Since the value was equal to the price, there was no need for further iterations. If there had been a difference, we would have reestimated the adjusted stock price and option value. If the options had been non-traded (as is the case with management options), this calculation would have required an iterative process, where the option value is used to get the adjusted value per share and the value per share to get the option value. warrant.xls: This spreadsheet allows you to estimate the value of an option when there is a potential dilution from exercise. The Black-Scholes Model for Valuing Puts
+The value of a put can be derived from the value of a call with the same strike price and the same expiration date: where C is the value of the call and P is the value of the put. This relationship between the call and put values is called put-call parity, and any deviations from parity can be used by investors to make riskless profits. To see why put-call parity holds, consider selling a call and buying a put with exercise price K and expiration date t, and simultaneously buying the underlying asset at the current price S. The payoff from this position is riskless and always yields K at expiration (t). To see this, assume that the stock price at expiration is S*. The payoff on each of the positions in the portfolio can be written as follows: Position
+Payoffs at t if S* > K
+Payoffs at t if S* < K Sell call
+–(S* – K)
+0 Buy put
+0
+K – S* Buy stock
+S*
+S* Total
+K
+K Since this position yields K with certainty, the cost of creating this position must be equal to the present value of K at the riskless rate (K e–rt). Substituting the Black-Scholes equation for the value of an equivalent call into this equation, we get: Thus, the replicating portfolio for a put is created by selling short [1 – N(d1)] shares of stock and investing K e–rt[1 – N(d2)] in the riskless asset. ILLUSTRATION 5.7: Valuing a Put Using Put-Call Parity: Cisco Systems and AT&T
+Consider the call on Cisco Systems that we valued in Illustration 5.2. The call had a strike price of $15 on the stock, had 103 days left to expiration, and was valued at $1.87. The stock was trading at $13.62, and the riskless rate was 4.63%. The put can be valued as follows: The put was trading at $3.38.
+Also, a long-term call on AT&T was valued in Illustration 5.4. The call had a strike price of $20, 1.8333 years left to expiration, and a value of $6.63. The stock was trading at $20.50 and was expected to maintain a dividend yield of 2.51% over the period. The riskless rate was 4.85%. The put value can be estimated as follows: The put was trading at $3.80. Both the call and the put were trading at different prices from our estimates, which may indicate that we have not correctly estimated the stock’s volatility. Jump Process Option Pricing Models
+If price changes remain larger as the time periods in the binomial model are shortened, it can no longer be assumed that prices change continuously. When price changes remain large, a price process that allows for price jumps is much more realistic. Cox and Ross (1976) valued options when prices follow a pure jump process, where the jumps can only be positive. Thus, in the next interval, the stock price will either have a large positive jump with a specified probability or drift downward at a given rate.
+Merton (1976) considered a distribution where there are price jumps superimposed on a continuous price process. He specified the rate at which jumps occur (λ) and the average jump size (k), measured as a percentage of the stock price. The model derived to value options with this process is called a jump diffusion model. In this model, the value of an option is determined by the five variables specified in the Black-Scholes model, and the parameters of the jump process (λ, k). Unfortunately, the estimates of the jump process parameters are so difficult to make for most firms that they overwhelm any advantages that accrue from using a more realistic model. These models, therefore, have seen limited use in practice.
+EXTENSIONS OF OPTION PRICING
+All the option pricing models described so far—the binomial, the Black-Scholes, and the jump process models—are designed to value options with clearly defined exercise prices and maturities on underlying assets that are traded. However, the options we encounter in investment analysis or valuation are often on real assets rather than financial assets. Categorized as real options, they can take much more complicated forms. This section considers some of these variations.
+Capped and Barrier Options
+With a simple call option, there is no specified upper limit on the profits that can be made by the buyer of the call. Asset prices, at least in theory, can keep going up, and the payoffs increase proportionately. In some call options, though, the buyer is entitled to profits up to a specified price but not above it. For instance, consider a call option with a strike price of K1 on an asset. In an unrestricted call option, the payoff on this option will increase as the underlying asset’s price increases above K1. Assume, however, that if the price reaches K2, the payoff is capped at (K2 – K1). The payoff diagram on this option is shown in Figure 5.5. Figure 5.5 Payoff on Capped Call This option is called a capped call. Notice, also, that once the price reaches K2, there is no longer any time premium associated with the option, and the option will therefore be exercised. Capped calls are part of a family of options called barrier options, where the payoff on and the life of the option are functions of whether the underlying asset price reaches a certain level during a specified period.
+The value of a capped call is always lower than the value of the same call without the payoff limit. A simple approximation of this value can be obtained by valuing the call twice, once with the given exercise price and once with the cap, and taking the difference in the two values. In the preceding example, then, the value of the call with an exercise price of K1 and a cap at K2 can be written as: Barrier options can take many forms. In a knockout option, an option ceases to exist if the underlying asset reaches a certain price. In the case of a call option, this knockout price is usually set below the strike price, and this option is called a down-and-out option. In the case of a put option, the knockout price will be set above the exercise price, and this option is called an up-and-out option. Like the capped call, these options are worth less than their unrestricted counterparts. Many real options have limits on potential upside, or knockout provisions, and ignoring these limits can result in the overstatement of the value of these options.
+Compound Options
+Some options derive their value not from an underlying asset, but from other options. These options are called compound options. Compound options can take any of four forms—a call on a call, a put on a put, a call on a put, or a put on a call. Geske (1979) developed the analytical formulation for valuing compound options by replacing the standard normal distribution used in a simple option model with a bivariate normal distribution in the calculation.
+Consider, for instance, the option to expand a project that is discussed in Chapter 30. While we will value this option using a simple option pricing model, in reality there could be multiple stages in expansion, with each stage representing an option for the following stage. In this case, we will undervalue the option by considering it as a simple rather than a compound option.
+Notwithstanding this discussion, the valuation of compound options becomes progressively more difficult as more options are added to the chain. In this case, rather than wreck the valuation on the shoals of estimation error, it may be better to accept the conservative estimate that is provided with a simple valuation model as a floor on the value.
+Rainbow Options
+In a simple option, the uncertainty is about the price of the underlying asset. Some options are exposed to two or more sources of uncertainty, and these options are rainbow options. Using the simple option pricing model to value such options can lead to biased estimates of value. As an example, consider an undeveloped oil reserve as an option, where the firm that owns the reserve has the right to develop the reserve. Here there are two sources of uncertainty. The first is obviously the price of oil, and the second is the quantity of oil that is in the reserve. To value this undeveloped reserve, we can make the simplifying assumption that we know the quantity of oil in the reserve with certainty. In reality, however, uncertainty about the quantity will affect the value of this option and make the decision to exercise more difficult.4
+CONCLUSION
+An option is an asset with payoffs that are contingent on the value of an underlying asset. A call option provides its holder with the right to buy the underlying asset at a fixed price, whereas a put option provides its holder with the right to sell at a fixed price, at any time before the expiration of the option. The value of an option is determined by six variables—the current value of the underlying asset, the variance in this value, the expected dividends on the asset, the strike price and life of the option, and the riskless interest rate. This is illustrated in both the binomial and the Black-Scholes models, which value options by creating replicating portfolios composed of the underlying asset and riskless lending or borrowing. These models can be used to value assets that have option like characteristics.
+QUESTIONS AND SHORT PROBLEMS
+In the problems following, use an equity risk premium of 5.5 percent if none is specified. 1. The following are prices of options traded on Microsoft Corporation, which pays no dividends. The stock is trading at $83, and the annualized riskless rate is 3.8%. The standard deviation in ln(stock prices) (based on historical data) is 30%. a. Estimate the value of a three-month call with a strike price of $85.
+b. Using the inputs from the Black-Scholes model, specify how you would replicate this call.
+c. What is the implied standard deviation in this call?
+d. Assume now that you buy a call with a strike price of $85 and sell a call with a strike price of $90. Draw the payoff diagram on this position.
+e. Using put-call parity, estimate the value of a three-month put with a strike price of $85. 2. You are trying to value three-month call and put options on Merck with a strike price of $30. The stock is trading at $28.75, and the company expects to pay a quarterly dividend per share of $0.28 in two months. The annualized riskless interest rate is 3.6%, and the standard deviation in log stock prices is 20%. a. Estimate the value of the call and put options, using the Black-Scholes model.
+b. What effect does the expected dividend payment have on call values? On put values? Why? 3. There is the possibility that the options on Merck described in the preceding problem could be exercised early. a. Use the pseudo-American call option technique to determine whether this will affect the value of the call.
+b. Why does the possibility of early exercise exist? What types of options are most likely to be exercised early? 4. You have been provided the following information on a three-month call: a. If you wanted to replicate buying this call, how much money would you need to borrow?
+b. If you wanted to replicate buying this call, how many shares of stock would you need to buy? 5. Go Video, a manufacturer of video recorders, was trading at $4 per share in May 1994. There were 11 million shares outstanding. At the same time, it had 550,000 one-year warrants outstanding, with a strike price of $4.25. The stock has had a standard deviation of 60%. The stock does not pay a dividend. The riskless rate is 5%. a. Estimate the value of the warrants, ignoring dilution.
+b. Estimate the value of the warrants, allowing for dilution.
+c. Why does dilution reduce the value of the warrants? 6. You are trying to value a long-term call option on the NYSE Composite index, expiring in five years, with a strike price of 275. The index is currently at 250, and the annualized standard deviation in stock prices is 15%. The average dividend yield on the index is 3% and is expected to remain unchanged over the next five years. The five-year Treasury bond rate is 5%. a. Estimate the value of the long-term call option.
+b. Estimate the value of a put option with the same parameters.
+c. What are the implicit assumptions you are making when you use the Black-Scholes model to value this option? Which of these assumptions are likely to be violated? What are the consequences for your valuation? 7. A new security on AT&T will entitle the investor to all dividends on AT&T over the next three years, limiting upside potential to 20% but also providing downside protection below 10%. AT&T stock is trading at $50, and three-year call and put options are traded on the exchange at the following prices: How much would you be willing to pay for this security? 1 Note, though, that higher variance can reduce the value of the underlying asset. As a call option becomes more in-the-money, the more it resembles the underlying asset. For very deep in-the-money call options, higher variance can reduce the value of the option.
+2 To illustrate, if σ2 is the variance in ln(stock prices), the up and the down movements in the binomial can be estimated as follows: where u and d are the up and down movements per unit time for the binomial, T is the life of the option, and m is the number of periods within that lifetime.
+3 Warrants are call options issued by firms, either as part of management compensation contracts or to raise equity.
+4 The analogy to a listed option on a stock is the case where you do not know with certainty what the stock price is when you exercise the option. The more uncertain you are about the stock price, the more margin for error you have to give yourself when you exercise the option, to ensure that you are in fact earning a profit.