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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf,len=1088
+page_content='A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A  CREDIT INDEX  YOSHIHIRO SHIRAI  Department of Mathematics, University of Maryland, College Park  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' A two dimensional pure jump process is proposed to model the evolution of the risk  free rate and default intensities for the purpose of evaluating option contracts on a credit index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Time evolution in credit markets is assumed to follow a gamma process evaluated at calendar time  in order to reflect different levels of business activity in the credit and Treasury markets, which  ultimately reflect differences in preferences and incentives of credit products investors, as well as  the structure of the credit market itself, with those of their respective counterparts in the Treasury  market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Formulas for the characteristic function, zero coupon bonds and moments of the process  are derived, and its parameters calibrated to market prices of options on a credit index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Model and  market implied credit spreads moments are estimated and compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Introduction  This paper proposes a new valuation method for credit index swaptions (henceforth, CDXOs),  which are options to enter at a predetermined date a credit index swap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' The current literature  (see Brigo & Morini (2011) and Armstrong & Rutkowski (2009), as well as Pedersen (2003) and  Doctor & Goulden (2007)) focuses on developing a Black-type formula for the purpose of retrieving  the CDXO price from its quotation, which is expressed in terms of the underlying spread, and,  particularly, on the issue of including the so called front end protection into the CDXO payoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='1  Apart from this formulation and to the best of the author’s knowledge, there are no generally  accepted and/or standard valuation methods for the pricing of credit index derivatives that also  match the statistical features of credit spreads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' The main contribution of this paper is then to  specify an underlying Markov process X that ultimately defines both short rate and credit spread  dynamics and is such that:  (i) a reliable and fast numerical method can be implemented to obtain CDX and CDXO prices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  (ii) the model parameters can be calibrated to fit sufficiently well the option price surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' and  (iii) the model implied statistical properties of the credit spread fit those implied by the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  We assume in particular that X is the two dimensional process (r, λ), where r is the short rate, and  λ the default intensity process of each entity in the underlying index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' The default time for entity i  is then modeled as the first time the default intensity integrated process Λ reaches a threshold εi,  where ε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=', εn are independent copies of an exponential random variable and n is the numbers of  entities in the index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  E-mail address: yshirai@umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Date: January 16, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' 60G18, 60G51, 91G20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Multiple Gamma Processes, Credit Index Options, Credit Spreads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  1Applying the conversion formula requires several inputs, such as the CDXO annuity (also referred to as the  hypothetical bond of the CDXO), which are typically unavailable to outsiders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Testing and calibration of the model  here proposed with real market data is made possible thanks to time series of CDXO prices provided by Morgan  Stanley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='05332v1  [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='PR]  12 Jan 2023  2  A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  We take mean reverting processes for r and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Randomness of the rate r is represented by a  gamma process gr, whereas for λ it is the sum of a double gamma process gλ ◦ gτ and a scalar  multiple ρ of gr itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Multiple gamma processes were first investigated in Madan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' (2020),  for the purpose of randomizing the speed at which jumps occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' To our knowledge, ours is the  first application of a pure jump process with infinite arrival rate in credit risk modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Our focus  on pure-jump models is also motivated by the possibility that such framework offers to apply the  theory of dynamic spectral risk measures (see Madan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' (2017)), thus introducing nonlinearity  in the valuation of credit index products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Because of this, and although the exploration of the  applications of nonlinear valuations of credit index derivatives is left to future research, we ignore  here the relatively small accounting issues related to the front end protection, and assume that no  defaults can occur before the time T0 at which the forward/swaption contract expires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Default times as above, known as doubly stochastic random times, are commonly used in credit  risk modeling (see Bielecki & Rutkowski (2002) and McNeil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' (2005)) and their development  goes back to Duffie & Singleton (1999), Lando (1998), Jarrow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' (1997)) and Madan & Unal  (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Common specifications for rate and intensity processes are the affine models developed by  Duffee & Kan (6) for diffusion models and Duffie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' (2000)) and Duffie and Garlenau (Duffie  & Garleanu (2001) for basic affine jump-diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' We mention that reduced form model  with non doubly stochastic random times are also possible, although such a direction was not  investigated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' For their development see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=', Kusuoka (1999) and Elliot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  We derive the Levy measure of the process (r, λ), based on which prices of zero coupon bonds  can be computed analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Moments, stationary distribution and characteristic exponent of the  random vector (rt, λt) for t ≥ 0 are also computed analytically, and level curves of its bivariate  density for different parameters are plotted using a 2D-version of the FFT algorithm (similar to  the one in Hurd & Zhou (2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' We then derive analytical formulas for discounted payoff of credit  index swaps, and the partial integro differential equation (PIDE) for credit index swaptions prices,  together with a finite difference scheme for its solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Calibration is performed for each maturity  to all traded strikes of options on the IG CDX index as of 2 January 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' We do not perform  a stability analysis, but we show that the numerical error for a given set of parameters (obtained  from calibration) is close enough to the prices obtained via Montecarlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Finally, we compare market and model implied summary statistics of credit spreads for a specific  maturity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' As shown in Carr & Madan (2001), variance, skewness and kurtosis of an equity position  under the risk neutral measure can be replicated with a continuum of option contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Here it is  shown that variance, skewness and kurtosis of the spread of a credit index can be replicated with a  continuum of credit index swaptions under the measure QA corresponding to choosing as numeraire  the annuity of the index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Our model is then calibrated to market prices for all strikes and for a  specific maturity for the period between 2 January 2020 through 5 June 2020, and market and  model implied variance, skewness and kurtosis of the credit spreads are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' The closer these  are, the better the model approximates the market implied densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' The results of our analysis  show that our model is generally able to capture positive skewness and leptokurtic features of  CDX spreads under the measure QA, and the model and market implied moments are of the same  magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' We observe, in particular, that the replication of credits spreads with option contracts  under the measure QA is a novel way to extract model-free statistical properties of credit spreads  from market prices of options, allowing the validation of any model of credit spreads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' In section 2 we review the basics of credit index  derivatives and their market, and in section 3 the fundamental mathematical framework is intro-  duced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' In section 4 we specify the pure-jump dynamics of short rate and default intensity, derive  the characteristic exponent of the underlying Markov process and the valuation PIDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' A simple  finite difference scheme is tested in section 5, and a comparison of model and market results is  shown in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Section 7 concludes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Credit Index Derivatives and their Market  The last few decades saw a spectacular rise in trading volumes of credit derivatives, such as  credit default swaps, credit index swaps, single tranche credit default obligations, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' One reason  for this is that the main contract’s features of credit default swaps, which form the basic asset  class in credit markets, have been standardized,2, thus allowing a relatively easy implementation  of hedging and speculative strategies and making the credit default swaps market more liquid than  that of corporate bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' However, the details of credit derivatives contracts remain complex and  satisfactory valuation methods for credit index forwards and swaptions are yet to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  To introduce the mathematical problem, recall that a credit default swap (CDS) is an over the  counter contract between two counterparties - the protection buyer and seller - in which protection  against the risk of default of an underlying entity (usually a company issuing bonds in the debt  market) is provided by the seller to the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' The latter pays the former a predetermined premium  K (defined as a credit spread multiplied by the contract’s notional) at regular intervals until the  contract expires and obtains a contingent payment from the seller triggered by any credit event  (such as default, restructuring, downgrade, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=') concerning the underlying entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  A credit index swap (CDX) can be thought of as a portfolio of credit default swaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' There are  two families of credit indices, the CDX, which refers to American companies, and the iTraxx, which  refers to European or to Asian and Australian ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Each family is composed of different indices,  each of which representing a different class of credit quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' A summary of the main credit indices  is shown in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' It is important to observe that, in order to reflect changes in the credit quality  of the constituents, the composition of most credit indices changes every six months on March 20  and September 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Each series of an index corresponds to a specific roll date, and older series  continue to trade, but their market is far less liquid (see McNeil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' (2005)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Name  Pool size  Region  Credit Quality  CDX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='IG  125  North America  Investment Grade  CDX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='IG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='HVOL  30  North America  Low-quality Investment Grade  CDX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='HY  100  North America  Speculative Grade  iTraxx Europe  125  Europe  Investment Grade  iTraxx Europe  30  Europe  Low-quality Investment Grade  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Major credit indices and their characteristics (source: McNeil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' (2005)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Similarly to a CDS, the cash flow associated to a credit index swap consists again of a premium  payment leg (with payments made by the protection buyer) and a default payment leg (with  payments made by the protection seller).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Premium payments, which are defined as a credit spread  multiplied by the index annuity (a measure of the number of underlying issuers for which a credit  event has not occurred yet) are due at deterministic dates T0 < T1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' < TM, where TM is the  maturity of the contract and T0 the inception date (for forward-start contracts T0 > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' A credit  event concerning any of the underlying entities triggers a payment by the seller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Standardized  credit index swaps have quarterly premium payments and maturity at issuance is three, five, seven  and ten years, with five years being the most liquid traded maturity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  There are two main differences between a CDX and a (portfolio of) CDS: (1) the contingent  payment of a CDX is the same for each underlying entity and (2) it does not become an empty  contract after a single credit event occurs, so the expected discounted value of the cumulated losses  before the inception date (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' the above mentioned front end protection) is included in the price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  2For instance, banks and financial institutions typically utilize the ISDA Master Service Agreement, published by  the International Swaps and Derivatives Association, as the framework agreement such that each futures transactions  between the parties of the agreement are mostly defined by it, leaving only specific points of the transaction open to  negotiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  4  A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Review and Assumptions  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Hazard Rates and Doubly Stochastic Random Times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Suppose that:  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' (Ω, F, Q) is a filtered probability space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' {Ft}t≥0 a filtration on (Ω, F, Q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='3  iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' τ : Ω → [0, ∞] is F-measurable and {Ht}t≥0 := σ  �  {11{τ>t}}t≥0  �  , so that τ is an Ht-stopping  time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Λ(t) = log (Q(τ > t|F∞)) is strictly increasing, finite (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Q(τ > t|F∞) > 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' for every  t > 0), Ft-adapted and absolutely continuous, with Λ(t) =  � t  0 λ(s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Then, τ is called a doubly stochastic random time with Ft-conditional hazard rate process λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Since Λ(t) is Ft-adapted, we have Q(τ ≤ t|Ft) = Q(τ ≤ t|F∞) ∀t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Suppose X is a standard exponentially distributed random variable on (Ω, F, Q)  independent of F∞, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Q(X ≤ t|F∞) = 1 − e−t for every t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Let λ(t) be a positive Ft-  adapted stochastic process such that Λ(t) =  � t  0 λ(s)ds is increasing and finite for every t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Let  τ := inf{t ≥ 0 : Λ(t) ≥ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Then τ is a doubly stochastic random time with hazard process λ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' By definition {τ > t} = {Λ(t) < X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Since Λ(t) is F∞-measurable and X is independent of  F∞, we have Q(τ > t|F∞) = Q(Λ(t) < X|F∞) = e−Λ(t), which proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  □  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='4 (Dellacherie Formulas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Let (Ω, F, Q) be a filtered probability space, τ a doubly  stochastic random time with {Ft}t≥0-conditional hazard rate process λ(t) and {rt}t≥0 an Ft-adapted  random process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Suppose that, for some T > 0, X is FT -measurable, {ν(t)}0≤t≤T and {Z(t)}t≥0  are Ft-adapted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='4 If the random variables  |X|e−  � T  t r(s)ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  � T  t  ν(s)e−  � s  t r(u)duds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  � T  t  |Z(s)λ(s)|e−  � s  t r(u)+λ(u)duds  are all integrable with respect to Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' then  E  �  e−  � T  t r(s)ds11{τ>T}X  ��� Ft ∨ Ht  �  = 11{τ>t}E  �  e−  � T  t r(s)+λ(s)dsX  ��� Ft  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  E  �� T  t  ν(s)e−  � s  t r(u)du11{τ>s}ds  ���� Ft ∨ Ht  �  = 11{τ>t}E  �� T  t  ν(s)e−  � s  t r(u)+λ(u)duds  ���� Ft  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  E  �  e−  � τ  t r(s)ds11{t<τ≤T}Z(τ)  ��� Ft ∨ Ht  �  = 11{τ>t}E  �� T  t  Z(s)λ(s)e−  � s  t r(u)+λ(u)duds  ���� Ft  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  where Ht = σ  �  {11{τ>t}}  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' See McNeil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' (2005), proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Basics of Forward CDS and CDX Contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Consider a forward-start CDS with in-  ception date T0, tenor structure T0 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' < TM, CDS spread c and for a notional of 1 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' dollar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Let (Ω, F, Q) be a probability space, {Ft}t≥0 a filtration on it, {r(t)}t≥0 an Ft-adapted random  process, and τ : Ω → [0, ∞] a doubly stochastic random time with hazard rate λ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Assuming that  τ represents the time of the credit event, the payments made by the protection seller (protection  leg) discounted at time t ≤ T0 are given by  Φ(t) = δ(τ)e−  � τ  t r(s)ds11{T0<τ≤TM},  3In credit risk modelling, {Ft}t≥0 is typically generated by some random process Ψ representing some measure  of economic activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  4Typically, X is a survival claim, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' a promised payment if there is no default, ν is a risky stream of payments  that stops when default occurs, and Z is a payment made at default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  5  where δ(t) is the Ft-adapted process representing loss given default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' the premium leg is  given by  Ψ(t) = c  M  �  j=1  e−  � Tj  t  r(s)ds11{τ>Tj}[Tj − Tj−1]  Using Dellacherie formulas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' we have  EQ[Φ(t)|Ft ∨ Ht] = EQ �  δ(s)e−  � τ  t r(s)ds �  11{t<τ≤TM} − 11{t<τ<T0}  �  |Ft ∨ Ht  �  = 11{τ>t}EQ  �� TM  T0  λ(s)δ(s)e−  � s  t r(u)+λ(u)duds)|Ft  �  The present value of the protection buyer’s cash flow is then given by  EQ [Φ(t) − Ψ(t)|Ft] = 11{τ>t}EQ  �� TM  T0  λ(s)δ(s)e−  � s  t r(u)+λ(u)duds)  �  − 11{τ>t}c  M  �  j=1  (Tj − Tj−1)EQ  �  e−  � Tj  t  r(u)+λ(u)du  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Since the CDS spread c(t, T0, TM) is chosen such that the current value of the contract is zero, we  then have  c(t, T0, TM) =  EQ �� TM  T0  λ(s)δ(s)e−  � s  t r(u)+λ(u)duds)  �  �M  j=1(Tj − Tj−1)EQ  �  e−  � Tj  t  r(u)+λ(u)du�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  We next provide the relevant definitions for forward contracts on a credit index (see Brigo &  Morini (2011) for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Suppose that the premium payments occur at T0 < T1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' < TM, where  TM is the maturity of the contract and T0 is the inception date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Define the following quantities:  (i) Cumulated losses: L(t) =  δ  n  �n  j=1 11{τj<t}, where δ is the loss given default (typically  common for each name and nonrandom) and τj is the time of default of entity j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  (ii) Outstanding notional: N(t) = 1 − L(t)  δ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  (iii) Index Annuity:  A(t) =  M  �  j=1  e−  � Tj  t  r(u)du  � Tj  Tj−1  N(s)ds ≈  M  �  j=1  e−  � Tj  t  r(u)duN(Tj)(Tj − Tj−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  (iv) Premium leg: Ψ(t, c) = cA(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  (v) Protection leg:  Φ(t) =  � TM  T0  e−  � s  t r(u)dudL(s) ≈  M  �  j=1  e−  � Tj  t  r(u)du [L(Tj) − L(Tj−1)]  (vi) Front End Protection: F(t) = e−  � T0  t  r(s)dsL(T0), t ≤ T0, is the discounted value at time  t of cumulated losses at time T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  The discounted payoff of a credit index swap is then given by  e−  � T0  t  r(s)ds [Φ(T0) − Ψ(T0) + F(T0)] = Φ(t) − Ψ(t) + F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='1)  Thus, any time a default event is triggered for any of the names composing the index, the name  that defaulted is removed from the index and a payment of size δ/n is made by the protection  seller, provided the default event happens after the inception date of the credit index swap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' If the  6  A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  event happens before the inception date, then the name that defaulted is again removed from the  index, and the loss is paid at the inception of the swap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  As mentioned, we avoid technical complications related to the front end protection, and assume  no defaults occur before inception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' The discounted payoff at 0 ≤ t ≤ T0 is then  e−  � T0  t  r(s)ds [Φ(T0) − Ψ(T0)] = Φ(t) − Ψ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='2)  To model different default rates, we consider a common intensity process {λ(t)}t≥0 for each under-  lying name (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' the pool is “homogeneous”), and, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=', n, we define the default time  τ i = inf {t > 0 : Λ(t) > εi} ,  where ε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=', εn are independent exponential random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Then (McNeil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' (2005), lemma  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='33), the default times are conditionally independent doubly stochastic random times, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' each τ i  is a doubly stochastic random time with respect to Ft and  Q(τ 1 > t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=', τ n > t|F∞) =  n  �  i=1  Q(τ i > t|F∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  In this case, we obtain for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=', n,  EQ  �  e−  � Tℓ  T0 r(u)du11{τ i>Tℓ−1}|FT0 ∨ HT0  �  = EQ  �  e−  � Tℓ  T0 r(u)du11{τ i>Tℓ−1}|FT0 ∨ Hi  T0  �  ,  where for every T ≥ t ≥ 0, Ht = ∨n  i=1Hi  t and Hi  t = σ({11{τ i>t}, t ≥ 0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Hence, if 1 ≤ i ≤ n, 1 ≤ ℓ ≤ M, and using the Dellacherie formulas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='4 and the tower property  of conditional expectation, there is a function gℓ : R3  + → R such that  EQ  �  e−  � Tℓ  T0 r(u)du11{τ i>Tℓ−1}|FT0 ∨ HT0  �  = EQ  �  e−  � Tℓ−1  T0  r(u)du11{τ i>Tℓ−1}P(Tℓ−1, Tℓ)|FT0 ∨ HT0  �  = 11{τ i>T0}EQ  �  e−  � Tℓ−1  T0  r(u)+λ(u)duP(Tℓ−1, Tℓ)|FT0  �  = 11{τ i>T0}gℓ(T0, r(T0), λ(T0)),  where, for every T ≥ t ≥ 0,  P(t, T) = EQ �  e−  � T  t r(u)du|Ft  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Similarly, there is a function hℓ : R3  + → R such that  EQ  �  e−  � Tℓ  T0 r(u)du11{τ i>Tℓ}|FT0 ∨ HT0  �  = 11{τ i>T0}EQ  �  e−  � Tℓ  T0 r(u)+λ(u)du|FT0  �  = 11{τ i>T0}hℓ(T0, r(T0), λ(T0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Therefore, setting g = �M  ℓ=1 gℓ, h = �M  ℓ=1 hℓ, and using the tower property of conditional expecta-  tion, we have  EQ [Φ(T0)|FT0 ∨ HT0] =  M  �  ℓ=1  δ  n  n  �  i=1  EQ  �  e−  � Tℓ  T0 r(u)du(11{τ i>Tℓ−1} − 11{τ i>Tℓ})|FT0 ∨ HT0  �  = δ  n  n  �  i=1  11{τ i>T0} [(g(T0, r(T0), λ(T0)) − h(T0, r(T0), λ(T0)))] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  7  As for the premium leg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' similar calculations give  EQ [A(T0)|FT0 ∨ HT0] =  M  �  ℓ=1  (Tℓ − Tℓ−1) 1  n  n  �  i=1  EQ  �  e−  � Tℓ  T0 r(u)du11{τ i>Tℓ}|FT0 ∨ HT0  �  =  M  �  ℓ=1  (Tℓ − Tℓ−1) 1  n  n  �  i=1  11{τ i>T0}hℓ(T0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' r(T0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' λ(T0))  The payoff π(T0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' r(T0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' λ(T0)) of a receiver credit index swaption5 is then  π(T0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' r(T0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' λ(T0))  =  �  κ  M  �  ℓ=1  (Tℓ − Tℓ−1)hℓ(T0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' r(T0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' λ(T0)) − δ (g(T0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' r(T0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' λ(T0)) − h(T0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' r(T0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' λ(T0)))  �+  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  This allows one to define a PIDE for the price of a credit index swaption at time t in terms of the  vector (t, r(t), λ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' 6  For the approximated payoff π of the credit index swaption, and given the short rate r(t) at time  t, the spread c(t, T0, TM) is a function of λ(t) only:  c(t, λ(t), T0, TM) := δEQ [Φ(t)|Ft]  EQ [A(t)Ft] = δ g(t, r(t), λ(t)) − h(t, r(t), λ(t))  �M  ℓ=1(Tℓ − Tℓ−1)hℓ(t, r(t), λ(t))  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='3)  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' For the case of semiannual payments, the credit spread is given by  κ(t, λ(t), T0, TM) = δEQ [Φ(t)|Ft]  EQ [A(t)Ft] = 2δ  �g(t, r(t), λ(t))  h(t, r(t), λ(t)) − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  If we assume that there is only one payment, then we obtain the following familiar approximated  relationship between credit spread and hazard rate:  κ(t, λ(t), T0, TM) = 2δ  �  K1(t, T0)e  λ(t)  �  1−e−θλ/2  θλ  �  − 1  �  ≈ δK1(t, T0)λ(t) + K2(t, T0),  where K1(t, T0) and K2(t, T0) are constants that do not depend on λ(t) nor r(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' In particular, the  credit spread is, in first order approximation, an affine linear function of the hazard rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' A Model for Stochastic Rate and Default Intensity  In this section, we propose a specification for the dynamics of the vector process X = (r, λ) for  the purpose of pricing credit index derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Inspired by Eberlein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' (2013), a simple choice  for the dynamics of X is  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='1)  �  dr(t) = θr(µr − r(t))dt + dgr(t)  dλ(t) = θλ(µλ − λ(t))dt + (dgλ(t) + ρdgr(t)) ,  where gr and gλ are two independent gamma processes with scale parameters cr and cλ, and shape  parameters γr and γλ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='7 Parameters θr and θλ are positive and measure the speed of  mean reversion toward the long term average µr and µλ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' The magnitude of the impact  5An option contract on a CDX index is of receiver type if the holder has the right, not the obligation, to sell  protection, and it is of payer type if the holder has the right, not the obligation, to buy protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  6The value of λ(t) is not directly observable, but can be retrieved by equating to zero the value of the forward  contract at the strike for which put-call parity holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  7Equivalently, for instance, gr(1) is a gamma random variable with mean γr/cr and variance γr/c2  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Furthermore,  log  �  EQ �  eiugr(t)��  = γrt log  �  cr  cr−iu  �  , and its Levy density ϕr is given by ϕr(x) = γr e−crx  x  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  8  A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  of jumps in the short rate on the default intensity is modeled by the variable ρ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' To better  fit the option price surface,8 and recognizing that economic activity in the highly liquid Treasury  market often evolves at a different pace than in the debt markets, we further subordinate gλ to a  gamma time change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Specifically, given a third gamma process gτ with parameters cτ and γτ, we  set ˜gλ(t) = gλ ◦ gτ(t), and consider  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='2)  �  dr(t) = θr[µr − r(t)]dt + dgr(t)  dλ(t) = θλ[µλ − λ(t)]dt + (d˜gλ(t) + ρdgr(t))) ,  Standard calculations (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Barndorff-Nielsen (1998)) give the following solution for the system  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='2 given the initial condition (r(0), λ(0)):  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='3)  �  r(t) = r(0)e−θrt + µr(1 − e−θrt) +  � t  0 e−θr(t−u)dgr(u)  λ(t) = λ(0)e−θλt + µλ(1 − e−θλt) +  � t  0 e−θλ(t−u)d(ρgr(u) + ˜gλ(u)]  In general,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' given a gamma process g with parameters c and γ and conditioning on gτ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' the  characteristic function of the process ˜g = g ◦ gτ is  E  �  eiθ˜g(t)�  = E  �  ψg(θ)gτ(t)�  = E  �  egτ(t) log(ψg(θ))�  = exp  �  γτt  � ∞  0  �  ex log(ψg(θ)) − 1  �  ϕτ(x)dx  �  = exp  �  γτt  � ∞  0  (ψg(θ)x − 1) ϕτ(x)dx  �  = exp  �  γτt  � ∞  0  �� ∞  0  eiθypx(y)dy − 1  �  ϕτ(x)dx  �  = exp  �  γτt  � ∞  0  � ∞  0  �  eiθy − 1  �  px(y)ϕτ(x)dydx  �  = exp  �  γτt  � ∞  0  �  eiθy − 1  � �� ∞  0  px(y)ϕτ(x)dx  �  dy  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  where ψg(θ) := E  �  eiθg(1)�  and where px is the density of a gamma distribution with parameters c  and γx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Therefore, the Levy density of the process ˜g is the weighted gamma Levy density:  ϕ˜g(y) =  � ∞  0  px(y)ϕτ(x)dx = γτ  e−cτy  y  � ∞  0  (cy)γx  Γ(γx)  e−cx  x dx, y > 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='4)  where Γ is the gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Note that the use of Fubini-Tonelli in the above derivation follows  from the fact that the Levy density ϕ˜g is well defined, which follows from Stirling approximation  and an integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Finally, from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='4, ˜g has infinite arrival rate, and finite variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Characteristic Exponent, Zero Coupon Bond Prices and Valuation PIDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' In this  section we compute the joint characteristic exponent of X and its integrated process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' To simplify  notation, we henceforth drop the tilde and thus assume that the process gλ is subordinated to gτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  8Although not reported here, we experimented with real market data and found that model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='1 is not rich enough  to fit, in particular, the prices of out of the money options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Note also that other modeling choices for λ that are not  investigated here are possible, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' one could consider an integrated truncated bilateral gamma process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  9  Note that the integrated Ornstein-Uhlenbeck (“OU”) process Yr(t) associated to r(t) is given by  Yr(t) =  � t  0  r(s)ds =  � t  0  r(0)e−θrs + µr  �  1 − e−θrs�  ds +  � t  0  � s  0  e−θr(s−u)dgr(u)ds  = r(0)1 − e−θrt  θr  + µr  �  t − 1 − e−θrt  θr  �  +  � t  0  � t  u  e−θr(s−u)dsdgr(u)  = µrt + (r(0) − µr)  �1 − e−θrt  θr  �  +  � t  0  1 − e−θr(t−u)  θr  dgr(u)  Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' the integrated OU process Yλ(t) associated to λ(t) is given by  Yλ(t) = µλt + (λ(0) − µλ)  �1 − e−θλt  θλ  �  +  � t  0  1 − e−θλ(t−u)  θλ  d(ρgr + gλ)(u)  Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' for every α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' α2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' α3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' α4 ∈ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' we have  α1Yr(t) + α2Yλ(t) + α3r(t) + α4λ(t)  = α1  �  µrt + (r(0) − µr)  �1 − e−θrt  θr  ��  + α2  �  µλt + (λ(0) − µλ)  �1 − e−θλt  θλ  ��  + α3  �  r(0)e−θrt + µr(1 − e−θrt)  �  + α4  �  λ(0)e−θλt + µλ(1 − e−θλt)  �  +  � t  0  �  α1  �  1 − e−θr(t−u)  θr  �  + α2ρ  �  1 − e−θλ(t−u)  θλ  �  + α3  �  e−θr(t−u)�  + α4ρ  �  e−θλ(t−u)��  dgr(u)  +  � t  0  �  α2  �  1 − e−θλ(t−u)  θλ  �  + α4  �  e−θλ(t−u)��  dgλ(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Next, set  ξr(t, r, α1, α3) = α1  �  µrt + (r − µr)  �1 − e−θrt  θr  ��  + α3  �  re−θrt + µr(1 − e−θrt)  �  ξλ(t, λ, α2, α4) = α2  �  µλt + (λ − µλ)  �1 − e−θλt  θλ  ��  + α4  �  λe−θλt + µλ(1 − e−θλt)  �  ψr(t, u, α1, α2, α3, α4) = α1  �  1 − e−θr(t−u)  θr  �  + α2ρ  �  1 − e−θλ(t−u)  θλ  �  + α3  �  e−θr(t−u)�  + α4ρe−θλ(t−u)  ψλ(t, u, α2, α4) = α2  �  1 − e−θλ(t−u)  θλ  �  + α4  �  e−θλ(t−u)�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Since |ψr(t, u, α1, α2, α3, α4)| ≤ |α1| + |ρα2| + |α3| + ρ|α4|, we have  EQ  0  �  exp  �� t  0  iψr(t, u, α1, α2, α3, α4)dgr(u)  ��  = lim  n→∞ EQ  0  �  exp  �n−1  �  k=0  iψr(t, kt/n, α1, α2, α3, α4) (gr((k + 1)t/n) − gr(kt/n))  ��  ,  10  A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  where Ft is the filtration generated by X and EQ  0 [·] denotes expectation conditional on F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' By the  stationariness and independence of the increments of the process gr, we have  lim  n→∞ EQ  0  �  exp  �n−1  �  k=0  iψr(t, kt/n, α1, α2, α3, α4) (gr((k + 1)t/n) − gr(kt/n))  ��  = lim  n→∞  n−1  �  k=0  EQ  0 [exp (iψr(t, kt/n, α1, α2, α3, α4)gr(t/n))] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  By the properties of the Levy density ϕr of the process gr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' for every ψ ∈ C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  1  t log  �  EQ  0  �  eiψg(t)��  =  �  R  �  eiψy − 1  �  ϕr(y)dy = −γr log  �  1 − iψ  cr  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  ⇒ lim  n→∞  n−1  �  k=0  EQ  0 [exp (iψr(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' kt/n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' α2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' α3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' α4)gr(t/n))]  = lim  n→∞  n−1  �  k=0  exp  � t  n  � ∞  0  �  eiψr(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='kt/n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='α2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='α3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='α4)y − 1  �  ϕr(y)dy  �  = lim  n→∞ exp  �n−1  �  k=0  t  n  � ∞  0  �  eiψr(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='kt/n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='α2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='α3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='α4)y − 1  �  ϕr(y)dy  �  = exp  �� t  0  � ∞  0  �  eiψr(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='α2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='α3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='α4)y − 1  �  ϕr(y)dydu  �  = exp  �  −γr  � t  0  log  �  1 − iψr(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' α2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' α3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' α4)  cr  �  du  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Similar calculations can be done with respect to ψλ(t, u, α2, α4), yielding  EQ  0  �  exp  �� t  0  iψλ(t, u, α2, α4)dgλ(u)  ��  = exp  �� t  0  � ∞  0  �  eiψλ(t,u,α2,α4)y − 1  �  ϕλ(y)dydu  �  = exp  �  −γτ  � t  0  log  �  1 + γλ  cτ  log  �  1 − i  cλ  ψλ(t, u, α2, α4)  ��  du  �  Therefore, the characteristic exponent of the vector process (Yr(t), Yλ(t), r(t), λ(t)) is given by  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5)  log  �  EQ �  eiα1Yr(T)+iα2Yλ(T)+iα3r(T)+iα4λ(T)|Ft  ��  =  � T  t  � ∞  0  �  eiψr(T,u,α1,α2,α3,α4)y − 1  �  ϕr(y)dydu +  � T  t  � ∞  0  �  eiψλ(T,u,α2,α4)y − 1  �  ϕλ(y)dydu  + iξr(T − t, r(t), α1, α3) + iξλ(T − t, λ(t), α2, α4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  This implies immediately that the risk neutral price P(t, T) at time t of a zero coupon bond with  maturity T and no default risk is given by  P(t, T) = exp  �  −µr(T − t) − (r(t) − µr)  �  1 − e−θr(T−t)  θr  �  +  � T  t  � ∞  0  �  e− 1−e−θr(T −u)  θr  y − 1  �  ϕr(y)dydu  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  11  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Density, Summary Statistics and Stationary Distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' To gain further insights into the  dynamics of the vector (r(t), λ(t)), we will now derive its density fr,λ conditional on (r(0), λ(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Denoting by φ the (joint) Fourier transform,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' and assuming µr = µλ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' for every  α = (α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' α2) ∈ R2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  log (φ(α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' α2)) = log  ��  R2  e−2πi(α1r+α2λ)fr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='λ(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' λ)drdλ  �  = log  �  EQ  0  �  e−2πi(α1r(t)+α2λ(t))��  = −2πiα1r(0)e−θrt − 2πiα2λ(0)e−θλt  − γr  � t  0  log  �  1 + 2πi  cr  �  α1e−θr(t−u) + ρα2e−θλ(t−u)��  du  − γτ  � t  0  log  �  1 + γλ  cτ  log  �  1 + 2πi  cλ  α2e−θλ(t−u)  ��  du  By Fourier inversion and a change of variable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' the joint density of (r(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' λ(t)) is then given by  fr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='λ(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' λ) =  1  4π2  �  R2  ei(α1r+α2λ)φ  �α1  2π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' α2  2π  �  dα1dα2  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='6)  As shown in Hurd & Zhou (2010), this double integral can be computed using a two dimensional  fast Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Specifically, we set N = 213, B = 106, η = 2B  N , λ = 2π  Nη = π  B, b = Nλ  2 = π  η ,  and approximate 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='6 by a double sum over the grid in the frequency domain  F =  �  αk = (αk1, αk2) : k = (k1, k2) ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=', N − 1}2�  , αki = −B + kiη, i = 1, 2  with corresponding grid in the space domain given by  S =  �  xℓ = (xℓ1, xℓ2) : ℓ = (ℓ1, ℓ2) ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=', N − 1}2�  , xℓi = −b + ℓiη, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Thus, we have the approximation  fr,λ(r, λ) ≈ η2  4π2  N−1  �  k1,k2=0  eiαkx′  ℓφ  �αk1  2π , αk2  2π  �  = (−1)ℓ1+ℓ2  �ηN  2π  �2 1  N2  N−1  �  k1,k2=0  e2πikℓ′/N(−1)k1+k2φ  �αk1  2π , αk2  2π  �  ,  where the last double sum can be computed for instance in Matlab with the command ifft2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Figures 1 and 2 show the bivariate density and the marginals of the vector (r(t), λ(t) for t = 1  year, and for r(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0146, θr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5500 cr = 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0005, γr = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='9475, ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='1548, λ(0) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' θλ =  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='3533, cλ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='3178, γλ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0617, cτ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5298, γτ = 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  The characteristic function allows one to compute the basic summary statistics for r(t) and λ(t):  EQ[r(t)] = r(0)e−θrt + 1  i  ∂  ∂α  �  −γr  � t  0  log  �  1 − iαe−θr(t−u)  cr  �  du  �  α=0  = r(0)e−θrt + γr  cr  1 − e−θrt  θr  ,  VQ[r(t)] = − ∂2  ∂α2  �  −γr  � t  0  log  �  1 − iαe−θr(t−u)  cr  �  du  �  α=0  = γr  c2r  1 − e−2θrt  2θr  EQ  0 [λ(t)] = λ(0)e−θλt +  �ργr  cr  + γτ  cτ  γλ  cλ  � 1 − e−θλt  θλ  VQ  0 [λ(t)] =  �ρ2γr  c2r  + γτγλ  c2τc2  λ  (γλ + cτ)  � 1 − e−2θλt  2θλ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  12  A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  (a)  Bivariate  density  of  the  random  vector  (r(t), λ(t)) for t = 1 year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='014  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='016  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='018  r  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='6  10-3  1  2  3  4  5  6  7  8  9  10  11  105  (b) Bivariate density contour of the random vector  (r(t), λ(t)) for t = 1 year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Figure 1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='03  0  50  100  150  200  250  (a) Marginal densities for the short rate for θr ∈  {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='16, 1, 5} and t = 1 year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='007  0  500  1000  1500  2000  2500  (b)  Default  intensity  marginal  for  θλ  ∈  {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='06, 4} and t = 1 year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Figure 2  (a) Stationary/limiting bivariate density of the  random vector (r(t), λ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='014  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='016  r  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='8  10-3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5  105  (b) Stationary/limiting bivariate density contour  of the random vector (r(t), λ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Figure 3  ×105  12  10-  8 -  9  4 -  2 -  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5  ×10-3  2  入0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='02  r×105  5 -  4-  3 -  2 -  1  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5  ×10-3  入  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='02  rA PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  13  These calculations show that higher values of the parameters θr and θλ imply smaller short rate  and default intensity, whereas the smaller θr and θλ, the higher the variance and expected value of  r(t) and λ(t) respectively (see also figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' we observe that the change of variable −(t − u) = v gives  r(t) = r(0)e−θrt +  � t  0  e−θr(t−u)dgr(u)  = r(0)e−θrt +  � 0  −t  eθrvdgr(v) −→  � 0  −∞  eθrvdgr(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  λ(t) = λ(0)e−θλt +  � t  0  e−θλ(t−u)(dgλ(u) + ρdgr(u))  = λ(0)e−θλt +  � 0  −t  eθλv(dgλ(u) + ρdgr(u)) −→  � 0  −∞  e−θλv(dgλ(v) + ρdgr(v))  and also that if the initial condition satisfies  (r(0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' λ(0)) =  �� 0  −∞  eθrvdgr(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  � 0  −∞  eθλv(dgλ(v) + ρdr(v))  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  then  r(t) =  � t  −∞  e−θr(t−u)dgr(u) =  � 0  −∞  eθrvdgr(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  λ(t) =  � t  −∞  eθ−λ(t−u)(dgλ(u) + ρdgr(u)) =  � 0  −∞  eθλv(dgλ(u) + ρdgr(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Thus, the limiting/stationary distribution of the process (r(t), λ(t)) is that of the random vector  �� 0  −∞  eθrvdgr(v),  � 0  −∞  eθλv(dgλ(v) + ρdr(v))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Similar calculations as above give the Fourier transform of the stationary bivariate density:  φ(α1, α2) = exp  �  −γr  � 0  −∞  log  �  1 + 2πi  cr  �  α1eθrv + ρα2eθλv��  dv  −γτ  � 0  −∞  log  �  1 + γλ  cτ  log  �  1 + 2πi  cλ  α2eθλv  ��  dv  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='7)  Figure 3 show the limiting density of the bivariate process (r(t), λ(t)), obtained by Fourier inversion  and approximating the improper integrals in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='7 with their proper version on [−100, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Linear PIDE for CDX Swaption prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' By the first fundamental theorem of asset pricing,  we know that the price u(t, r, λ) of a CDX swpation is given by  u(t, r(t), λ(t)) = EQ �  e−  � T0  t  r(u)duπ(r(T0), λ(T0))  ��� Ft  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Therefore, setting for every t ∈ [0, T]  M(t) = e−  � t  0 r(u)duu(t, r(t), λ(t))  we have  M(t) = EQ [M(T)| Ft]  14  A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' M(t) is a martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Furthermore, by Ito’s lemma for semimartingales, we have  M(t) = M(0) −  � t  0  e−  � s  0 r(u)dur(s)u(s, r(s), λ(s))ds  +  � t  0  e−  � s  0 r(u)du [ut + ur (θr(µr − r(s)) + uλ (θλ(µλ − λ(s))] ds  +  �  (0,t]×R2  +\\{0}  e−  � s  0 r(u)duDt,r,λ  u  (y)N(ds, dy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  where N is the Poisson random measure associated to the process (gr(t), gλ(t)+ρgr(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='9 Therefore,  if we denote by ϕ the Levy density of (gr(t), gλ(t)+ρgr(t)), and we add and subtract the compensator  to M(t), we obtain, after reversing time, the following PIDE for u(t, r(t), λ(t)):  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='8)  �  �  �  �  �  ut + ru − ∇u · α(r(t), λ(t)) −  �  R2  +\\{0} Dt,r,λ  u  (y)ϕ(y)dy = 0  u(0, r, λ) = π(T0, r, λ)  + boundary conditions  where ∇u is the partial gradient of u with respect to r and λ (a two dimensional row vector) and  α(r, λ) := (θr(µr − r), θλ(µλ − λ))T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Finally, it is easy to see that, for every (ur, uλ) ∈ R2,  EQ �  eiurgr(t)+iuλ(ρgr(t)+gλ(t)�  = EQ �  ei(ur+ρuλ)gr(t)�  EQ �  eiuλgλ(t)�  = e  �  {yλ=ρyr}(eiuryr+uλyλ−1)ϕr(yr)dyrdyλe  � ∞  0 (eiuλyλ−1)ϕλ(yλ)dyλ,  which implies  �  R2  +\\{0}  Dt,r,λ  u  (y)ϕ(y)dy =  � ∞  0  Dt,r,λ  u  (yr, ρyr)ϕr(yr)dyr +  � ∞  0  Dt,r,λ  u  (0, yλ)ϕλ(yλ)dyλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Numerical Results  We implemented a finite difference scheme for the valuation PIDE, whose construction is reported  in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' The scheme was then tested taking as final condition the payoff of a forward  start CDX swap, whose current value admits an integral representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' The proof is based on  9It is easy to see that jumps in r and λ correspond to jumps in gr and gr + ρgλ, and their magnitude is also the  same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' In fact,  lim  s→0 r(t + s) − r(t) = lim  s→0  � t+s  t  e−θr(t−u)dgr(u) = lim  s→0  � s  0  eθrudgr(u + t)  = lim  s→0 lim  n→∞  n  �  k=0  eθr ks  n [gr(t + (k + 1)s/n) − gr(t + ks/n)]  ≤ lim  s→0 lim  n→0  n  �  k=0  eθrs[gr(t + (k + 1)s/n) − gr(t + ks/n)]  = lim  s→0 eθrs[gr(t + s) − gr(t)] = lim  s→0[gr(t + s) − gr(t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  On the other hand, since θr > 0,  lim  s→0[gr(t + s) − gr(t)] = lim  s→0  � s  0  dgr(u + t) ≤ lim  s→0  � s  0  eθrudgr(u + t) = lim  s→0 r(t + s) − r(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Similar considerations hold for jumps in the default intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Therefore, the Poisson random measure associated to  (r(t), λ(t)) must be the same as the one associated to (gr(t), gλ(t) + ρgr(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  15  calculations that are similar to those performed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' In particular, we obtain that, for  every t ∈ [0, T0], ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  EQ  �  e−  � T0  t  r(u)duEQ  �  e−  � Tℓ  T0 r(u)+λ(u)du|FT0  ����� Ft  �  = EQ �  e−  � Tℓ  t  r(u)dueξr(Tℓ−T0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='r(T0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0)+ξλ(Tℓ−T0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='λ(T0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0)|Ft  �  × e  � Tℓ  T0  � ∞  0 (eψr(Tℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0)y−1)ϕr(y)dydu+  � Tℓ  T0  � ∞  0 (eψλ(Tℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0)y−1)ϕλ(y)dydu  = eξr(T0−t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='r(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='b3)+ξλ(T0−t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='λ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='b4)  × e  � T0  t  � ∞  0 (eψr(T0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='b3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='b4)y−1)ϕr(y)dydu+  � T0  t  � ∞  0 (eψλ(T0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='b4)y−1)ϕλ(y)dydu  × e  � Tℓ  T0  � ∞  0 (eψr(Tℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0)y−1)ϕr(y)dydu+  � Tℓ  T0  � ∞  0 (eψλ(Tℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0)y−1)ϕλ(y)dydu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  EQ  �  e−  � T0  t  r(u)duEQ  �  e−  � Tℓ−1  T0  r(u)+λ(u)duP(Tℓ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Tℓ)|FT0  ����� Ft  �  = EQ  �  e−  � T0  t  r(u)duEQ  �  e−  � Tℓ−1  T0  r(u)+λ(u)duer(Tℓ−1)α3|FT0  ����� Ft  �  × e  � Tℓ  Tℓ−1  � ∞  0  �  e  − 1−e−θr(Tℓ−u)  θr  y−1  �  ϕr(y)dydu  = EQ �  e−  � T0  t  r(u)dueξr(Tℓ−1−T0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='r(T0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='α3)+ξλ(Tℓ−1−T0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='λ(T0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0)��� Ft  �  × e  � Tℓ−1  T0  � ∞  0  �  eψr(Tℓ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='α3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0)y−1  �  ϕr(y)dydu+  � Tℓ−1  T0  � ∞  0  �  eψλ(Tℓ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0)y−1  �  ϕλ(y)dydu  × e  � Tℓ  Tℓ−1  � ∞  0  �  e  − 1−e−θr(Tℓ−u)  θr  y−1  �  ϕr(y)dydu  = eξr(T0−t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='r(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='a3)+ξλ(T0−t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='λ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='a4)  × e  � T0  t  � ∞  0 (eψr(T0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='a3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='a4)y−1)ϕr(y)dydu+  � T0  t  � ∞  0 (eψλ(T0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='a4)y−1)ϕλ(y)dydu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  × e  � Tℓ−1  T0  � ∞  0  �  eψr(Tℓ−1,u,−1,−1,α3,0)y−1  �  ϕr(y)dydu+  � Tℓ−1  T0  � ∞  0  �  eψλ(Tℓ−1,u,−1,0)y−1  �  ϕλ(y)dydu  × e  � Tℓ  Tℓ−1  � ∞  0  �  e  − 1−e−θr(Tℓ−u)  θr  y−1  �  ϕr(y)dydu  ,  where  α3 := −1 − e−θr(Tℓ−Tℓ−1)  θr  , a3 := −1 − e−θr(Tℓ−1−T0)  θr  + α3e−θr(Tℓ−1−T0),  a4 := −1 − e−θλ(Tℓ−1−T0)  θλ  , b3 := −1 − e−θr(Tℓ−T0)  θr  , b4 := −1 − e−θλ(Tℓ−T0)  θλ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  We considered as before the following set of parameters,  r(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0146, θr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5500 cr = 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0005, γr = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='9475, ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='1548;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  λ(0) = 0, θλ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='3533, cλ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='3178, γλ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0617, cτ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5298, γτ = 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0001,  and we also assumed that the forward contract matures in 15 days, while the underlying asset  is a 5-year receiver swap with recovery rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='4, strike κ = 60 bps and semiannual payments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Figure 4 shows the price surface for the forward start credit index swap generated by solving 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='8  16  A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  assuming N = 50 (left) and N = 100 (right), and with initial condition given by the payoff of the  swap at maturity of the forward contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' The ℓ∞ absolute error is plotted in figure 5(a) for strikes  κ = 50 : 10 : 100 bps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Note that even to compute the analytical solutions certain integrations  were performed numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Figure 5 (b) shows the ℓ∞ absolute error of the solution computed via  Montecarlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' For both the PIDE and Montecarlo cases the error is relatively high, and  although one could reduce the error for instance by more accurately computing the gamma time  changed gamma Levy density 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='4 (at the price of higher computational costs), we observe that the  error is less than or at least comparable with the bid-ask spread observed in the option market,  which, in the period considered, is at least 2 bps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  In the case of a forward contract, the actual price of the contract, including the front end  protection, can be analytically computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' we have  EQ �  e−  � Tℓ  t  r(u)du11{τ i>Tℓ−1}|Ft ∨ Ht  �  = 11{τ i>t}EQ  �  e−  � Tℓ−1  t  r(u)+λ(u)duP(Tℓ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Tℓ)|Ft  �  = 11{τ i>t}EQ  �  e−  � Tℓ−1  t  r(u)+λ(u)duer(Tℓ−1)α3|Ft  �  e  � Tℓ  Tℓ−1  � ∞  0  �  e  − 1−e−θr(Tℓ−u)  θr  y−1  �  ϕr(y)dydu  = 11{τ i>t}eξr(Tℓ−1−t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='r(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='α3)+ξλ(Tℓ−1−t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='λ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0)  × e  � Tℓ−1  t  � ∞  0  �  eψr(Tℓ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='α3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0)y−1  �  ϕr(y)dydu+  � Tℓ−1  t  � ∞  0  �  eψλ(Tℓ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0)y−1  �  ϕλ(y)dydu  × e  � Tℓ  Tℓ−1  � ∞  0  �  e  − 1−e−θr(Tℓ−u)  θr  y−1  �  ϕr(y)dydu  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  and  EQ �  e−  � Tℓ  t  r(u)du11{τ i>Tℓ}|Ft  �  = 11{τ i>t}EQ �  e−  � Tℓ  t  r(u)+λ(u)du|Ft ∨ Ht  �  = 11{τ i>t}eξr(Tℓ−t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='r(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0)+ξλ(Tℓ−t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='λ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0)  × e  � Tℓ  t  � ∞  0 (eψr(Tℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0)y−1)ϕr(y)dydu+  � Tℓ  t  � ∞  0 (eψλ(Tℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0)y−1)ϕλ(y)dydu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' as before,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  α3 = −1 − e−θr(Tℓ−Tℓ−1)  θr  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Figures 6 shows that the value of the front end protection is relatively small for this set of  parameters, although, as noted for instance in Brigo & Morini (2011), its value can be substantial  for higher values of λ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  We now turn our attention to the option contracts on a CDX index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' The numerical price is  shown in figure 7 (a), while the ℓ∞ absolute error with respect to the Montecarlo generated surface  for κ = 50 : 10 : 100 and for r(0) = 146 bps is shown in figure 7 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Finally, the question of convergence of the numerical method for the case of an option payoff is  addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' For r(0) = 146 bps, consider CDX spreads κ in the range 60 to 50 bps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' The resulting  prices for various values of N, reported in table 2, show that convergence up to the second decimal  (in bps) is obtained already for N = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  17  (a)  (b)  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Numerical price surface (in bps) for a forward-start credit index swap, assuming  N = 50 (a) and N = 100 (b), and for M = 100 and Nsim = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5  6  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5  7  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5  8  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5  9  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5  10  10-3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='485  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='49  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='495  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='505  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='51  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='515  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='52  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='525  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='53  (a)  5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5  6  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5  7  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5  8  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5  9  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5  10  10-3  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='62  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='64  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='66  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='68  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='72  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='74  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='76  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='78  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='8  (b)  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' ℓ∞ absolute error (a) and ℓ∞ difference (in bps) with Montecarlo generated  price surface (b) for strikes κ = 50 : 10 : 100 bps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  (a)  (b)  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Price surface (in bps) of a forward-start CDX including FEP on a 50 × 50 grid  (a) and comparison with the numerical solution (excluding FEP) of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='1 for strike κ = 100  bps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  54  52  50  48  46  44  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
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+page_content='04  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='06  r8  9  4 -  2  0  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
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+page_content='06  r5550  45  40  35  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
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+page_content='50  45  40、  35  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
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+page_content='08  r0  0218  A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  (a)  50  55  60  65  70  75  80  85  90  95  100  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
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+page_content='5  (b)  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Receiver CDX option price surface for κ = 60 bps and N = 50 (left) and ℓ∞  absolute difference with the Montecarlo generated price surface for κ = 50 : 10 : 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  N/κ  60 bps  50 bps  Cpu time  50  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
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+page_content='205160e+01  250  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='98665  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='06030  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='590890e+01  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' CDXO price (in bps) and cpu time for different values of N and strike price κ  bps and for M = Nsim = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Comparison with Market Data  We calibrated the model to the Treasury yield curve (from www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='treasury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='gov) and to CDX option  prices (provided by Morgan Stanley) as of 2 January 2020 across traded strikes and for each traded  maturity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Strike prices are expressed in terms of CDX spreads, and they range from 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5 bps up to  120 bps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Traded maturities are 13, 43, 76, 104, 139 and 167 business days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' The spot CDX spread as  of 2 January 2020 was 44 bps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' We considered strikes that are up to 30% out of the money (OTM)  for receiver and payer contracts for each available maturity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='10 Calibrated parameters for the short  rate are the same as those considered above, while those for the default intensity are reported in  table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Figure 8 compares the corresponding OTM model and market price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Term (years)  θλ  ρ  cλ  γλ  cτ  γτ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
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+page_content='2892  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='0241  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='8397  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='4584  Table 3  10A receiver option, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' an option to sell protection, is OTM if the spot spread is higher than the strike spread,  while a payer option is OTM if the spot spread is lower than the strike spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  6040  20  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='04  0  入  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
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+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='4  r0A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  19  1  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' OTM mid price (asterisk) and model price (circle) for maturities and strikes  traded on 2 January 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Each color corresponds to one of the following strikes: 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5, 45,  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5, 50, 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5, 55, 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5 (in bps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Maturities are as reported in table 3 and model prices are  computed using the corresponding parameters also reported in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  It is also possible to compare market and model implied summary statistics (variance, skewness,  kurtosis) of credit spreads for a specific maturity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' In particular, as shown above, the payoff of a  receiver CDX option maturing at time T0 with strike spread c is given by  π(T0) =  �  cEQ[A(T0)|FT0] − δEQ[Φ(T0)|FT0]  �+  ,  while the spot credit spread c(T0) satisfies  c(T0)EQ [A(T0)|FT0] = δEQ [Φ(T0)|FT0] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Therefore, we have  π(T0) = EQ[A(T0)|FT0] (c − c(T0))+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Taking the index annuity as numeraire,11 and letting QA denote the associated probability measure,  the price ur(t, c) and up(t, c) of a receiver/payer option at time t is given by  ur(t, c) = EQ[A(t)|Ft]EQA[(c − c(T0))+ |Ft], up(t, c) = EQ[A(t)|Ft]EQA[(c(T0) − c)+ |Ft].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  We can then use a result due to Madan and Carr (see Carr & Madan (2001)), according to which  a twice continuously differentiable payoff function H(c) ∈ C2(R) can be written as  H(c) = H(ˆc) + (c − ˆc)H′(ˆc) +  � ∞  ˆc  H′′(c)(c − c)+dc +  � ˆc  0  H′′(c)(c − c)+dc,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='1)  where ˆc ≥ 0 is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' This allows one to compute model-free summary statistics of the spot  credit spread c(T0) under QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  To do so, define the volatility, cubic and quartic contracts as  (c − cf)2, (c − cf)3, (c − cf)4, where cf = EQA[c(T0)] is the forward credit spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Setting ˆc = cf,  11Technically, the index annuity may be null on a set of positive measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  This happens in the case of an  armageddon event, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' all the entities in the index default prior to the option expiration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Our assumption that such  event has approximately zero probability (which is more likely the shorter the maturity of the option) ensures that  the approximation error in taking the index annuity as numeraire is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  0  ×10-34  0  米  米  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5 -  米  0  米  3 -  米  米  米  0  米  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5 -  米  0  2 -  米  米  米  0 0  米  0  米  果  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5 -  0  D  米  米  米  1 -  果  米  米  来  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5-  米  0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='8  5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='2  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='4  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='6  ×10-3  Strike米  8  米  0  米  米  0米  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5  0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='8  Term20  A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='1 implies  EQ  0 [A(0)]EQA  0 [(c(T0) − cf)2]  = 2EQ  0 [A(0)]EQA  0  �� ∞  cf  (c(T0) − c)+dc +  � cf  0  (c − c(T0))+dc  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  EQ  0 [A(0)]EQA  0 [(c(T0) − cf)3]  = 6EQ  0 [A(0)]EQA  0  �� ∞  cf  (c − cf)(c(T0) − c)+dc +  � cf  0  (c − cf)(c − c(T0))+dc  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  EQ  0 [A(0)]EQA  0 [(c(T0) − cf)4]  = 12EQ  0 [A(0)]EQA  0  �� ∞  cf  (c − cf)2(c(T0) − c)+dc +  � cf  0  (c − cf)2(c − c(T0))+dc  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  which imply,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' under reasonable assumptions on c(T0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  EQ  0 [A(0)]EQA  0 [(c(T0) − cf)2] = 2  � ∞  cf  up(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' c)dc + 2  � cf  0  ur(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' c)dc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='2)  EQ  0 [A(0)]EQA  0 [(c(T0) − cf)3] = 6  � ∞  cf  (c − cf)up(0, c)dc + 6  � cf  0  (c − cf)ur(0, c)dc,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='3)  EQ  0 [A(0)]EQA  0 [(c(T0) − cf)4] = 12  � ∞  cf  (c − cf)2up(0, c)dc + 12  � cf  0  (c − cf)2ur(0, c)dc,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='4)  Assuming that non traded deep OTM otions have zero price, one can think of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='2, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='4 as  reasonable approximations of the first three moments of c(T0), multiplied by the current value of  the index annuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Note also that calculation of cf is straightforward since, following the standard  put-call parity argument, the price fp(t, c) at time t of a payer forward with spread c is  fp(T0, cf) =  �  δEQ[Φ(T0)|FT0] − cfEQ[A(T0)|FT0]  �+  −  �  cfEQ[A(T0)|FT0] − δEQ[Φ(T0)|FT0]  �+  = up(T0, cf) − ur(T0, cf)  and so fp(0, cf) = up(0, cf) − ur(0, cf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Since fp(0, 0) = EQ  0 [A(0)]EQA  0 [c(T0)] = EQ  0 [A(0)]cf, the no  arbitrage model-free value of the annuity is  EQ  0 [A(0)] = fp(0, 0)  cf  ≈ up(0, 0)  cf  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5)  A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  21  Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' the market implied spread’s variance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' µ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' skewness µ3 and kurtosis µ4 under QA are:  µ2 := EQA �  (c(T0) − cf)2�  ≈  2cf  fp(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' 0)  �� ∞  cf  up(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' c)dc +  � cf  0  ur(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' c)dc  �  µ3 :=  EQA[(c(T0) − cf)3]  EQA [(c(T0) − cf)2]3/2  ≈  6cf  µ3/2  2  fp(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' 0)  �� ∞  cf  (c − cf)up(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' c)dc +  � cf  0  (c − cf)ur(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' c)dc  �  µ4 := EQA[(c(T0) − cf)4]  EQA [(c(T0) − cf)2]4  ≈  12cf  µ4  2fp(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' 0)  �� ∞  cf  (c − cf)2up(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' c)dc + 12  � cf  0  (c − cf)2ur(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' c)dc  �  As shown in table 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' model and market implied spread statistics under QA are relatively close,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  evidencing accuracy of model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='2 in explaining short rate and default intensity dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Note in  particular that model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='2 is able to capture the positive skewness and leptokurtic feature of CDX  spreads under the measure QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
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+page_content='29  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='566200e+01  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='788540e+01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='39  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='458586e+01  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='487553e+01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='46  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='212721e+01  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='028359e+01  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Market and model implied CDX spread statistics for the maturities traded on 2  January 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  It is also worth noting that, under 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='2,  EQA  0 [H(c(T0))] =  EQ  0  �  e−  � T0  0  r(u)du �M  ℓ=1(Tℓ − Tℓ−1)EQ  �  e−  � Tℓ  T0 r(u)+λ(u)du|FT0  �  H(c(T0))  �  �M  ℓ=1(Tℓ − Tℓ−1)EQ  0  �  e−  � Tℓ  0  r(u)+λ(u)du�  ,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='6)  which can be computed via montecarlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  We performed such computation, but the  resulting model implied variance, skewness and kurtosis are not in line with those computed above,  ultimately because the statistics in table 4 assume that the prices of deep OTM options are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  22  A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  Jan  Feb  Mar  Apr  May  Jun  2020  101  102  103  104  Variance  (a)  Jan  Feb  Mar  Apr  May  Jun  2020  10-2  10-1  100  101  Skewness  (b)  Jan  Feb  Mar  Apr  May  Jun  2020  100  101  Kurtosis  (c)  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Daily market implied and model statistics under QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Table 5 shows model and market implied moments significant correlation between 2 January  2020 and 5 June 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Calibration was performed each day using Nelder-Mead algorithm with  starting point the optimal parameters for the previous day and maximum 100 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Statistics  Correlation Coefficient  Variance  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='8892  Skewness  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='3238  Kurtosis  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='4759  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Correlation coefficient for the time series of market and model implied spread  statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Finally, figure 10 shows the daily realized short rate, default intensity and the parameter ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Conclusions  We introduced a pure-jump dynamics for the simultaneous modelling of the short rate and default  intensity of a pool of entities with similar credit qualities, with the former being a gamma process  and the latter also a gamma process but subordinated to another (independent) gamma process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' We  tested a simple finite difference scheme for the valuation PIDE for forward and option contracts on  A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  23  Jan  Feb  Mar  Apr  May  Jun  2020  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='014  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='016  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='018  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='02  (a)  Jan  Feb  Mar  Apr  May  Jun  2020  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='9  (b)  Jan  Feb  Mar  Apr  May  Jun  2020  0  5  10  15  20  25  (c)  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Daily realized short rate, hazard rate and parameter ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  derivatives determined by short rate and default intensity, and showed that the numerical solution  approximates the exact one or a simulated one with reasonable margin of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' We calibrated the  model to the CDX option price surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' For January 2 2020, the calibration error is generally low,  but it can be substantial and especially as maturity increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Finally, we derived a market implied  formula for variance, skewness and kurtosis of the credit spread under the Annuity measure, and  reported that market and model implied statistics over the year 2020 are of similar magnitude and  positively correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Aknowledgement  This paper is a revised version of a chapter of the author’s doctoral dissertation, conducted under  the supervision of Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Dilip B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Madan at the University of Maryland, College Park.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Appendix: The Finite Difference Scheme for the Valuation PIDE  In our finite difference approximation, we treat the integral term fully explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Consider the  following mesh on the region [0, T] × [0, rmax] × [0, λmax]:  D =  �  �  �  �  �  tj = j∆t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' ∆t = T  M ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=', M  ri = i∆r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' ∆r = rmax  N ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=', N  λk = k∆λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' ∆λ = λmax  L ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=', L  24  A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  We pick λmax = ρrmax and L = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' We denote by (tj, ri, λk) ∈ R3  + the grid points in D, and  let uj  i,k = u(tj, ri, λk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Assuming that the (N + 1)2 values uj  i,k are known for fixed tj, we need  to construct the difference equation for each point (tj+1, ri, λk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Space and time derivatives are  approximated using central and forward differences respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  ur(tj+1, ri, λk) =  uj+1  i+1,k − uj+1  i−1,k  2∆r  + O(∆r2), uλ(tj+1, ri, λk) =  uj+1  i,k+1 − uj+1  i,k−1  2∆λ  + O(∆λ2),  ut(tj+1, ri, λk) =  uj+1  i,k − uj  i,k  ∆t  + O(∆t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  We then obtain the following equation for the point (tj+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' λk):  uj+1  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='k − uj  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='k  ∆t  + riuj+1  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='k − α1  uj+1  i+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='k − uj+1  i−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='k  2∆r  − α2  uj+1  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='k+1 − uj+1  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='k−1  2∆λ  ≈  � ∞  0  Dtj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='λk  u  (yr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' 0)ϕr(yr)dyr +  � ∞  0  Dtj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='λk  u  (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' yλ)ϕλ(yλ)dyλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Equivalently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' we have  uj+1  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='k (1 + ∆tri) − ∆tα1  2∆r  �  uj+1  i+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='k − uj+1  i−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='k  �  − ∆tα2  2∆λ  �  uj+1  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='k+1 − uj+1  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='k−1  �  ≈ uj  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='k +  � ∞  0  Dtj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='λk  u  (yr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' 0)ϕr(yr)dyr +  � ∞  0  Dtj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='λk  u  (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' yλ)ϕλ(yλ)dyλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='1)  The integral terms in 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='1 can be treated easily via montecarlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Specifically, at the grid  point (tj, ri, λk), having generated Nsim exponentially distributed random variables {Y r  s }s=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=',Nsim  with parameter cr, we have  � ∞  0  Dtj,ri,λk  u  (yr, ρyr)ϕr(yr)dyr ≈  1  Nsim  Nsim  �  s=1  Dtj,ri,λk  u  (Y r  s , ρY r  s ) γr  crY rs  ,  and similarly for the second integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' For every s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=', Nsim, we compute Dtj,ri,λk  u  (Y r  s , ρY r  s ) and  Dtj,ri,λk  u  (0, Y λ  s ) by linearly interpolating uj, and obtain the following difference equation  −Si,kuj+1  i,k−1 − Wi,kuj+1  i−1,k + Ci,kuj+1  i,k − Ei,kuj+1  i+1,k − Ni,kuj+1  i,k+1 = uj  i,k + ∆tRj  i,k,  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='2)  where  Si,k = −∆tα2  2∆λ , Wi,k = −∆tα1  2∆r , Ei,k = ∆tα1  2∆r , Ci,k = 1 + ∆tri, Ni,k = ∆tα2  2∆λ ,  Rj  i,k =  1  Nsim  Nsim  �  s=1  Dtj,ri,λk  u  (Y r  s , ρY r  s ) γr  crY rs  +  1  Nsim  Nsim  �  s=1  Dtj,ri,λk  u  (Y λ  s , 0) γλ  cλY λ  s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Implementation of Boundary Conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' We impose homogeneous Neumann boundary conditions  for each time t ∈ [0, T]:  uλλ(t, r, λL) = 0 − uλλ(t, r, 0) = 0, −urr(t, 0, λ) = 0, urr(t, rN, λ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='3)  We thus solve  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='4)  �  ut + ru − ∇u · α =  � ∞  0 Dt,r,λ  u  (yr, ρyr)ϕr(yr)dyr +  � ∞  0 Dt,r,λ  u  (0, yλ)ϕλ(yλ)dyλ  uλλ(t, r, λL) = 0, uλλ(t, r, 0) = 0, urr(t, 0, λ) = 0, urr(t, rN, λ) = 0  We implement 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='3 at the points (tj, r1, λk), (tj, rN−1, λk), (tj, ri, λ1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' we set  uj+1  0,k = 2uj+1  1,k − uj+1  2,k , uj+1  N,k = 2uj+1  N−1,k − uj+1  N−2,k, uj+1  i,L+1 = 2uj+1  i,L − uj+1  i,L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='uj+1  i,0  = 2uj+1  i,1 − uj+1  i,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  A PURE JUMP MODEL FOR THE VALUATION OF OPTIONS ON A CREDIT INDEX  25  References  Armstrong, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
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+page_content=' Valuation of Credit Default Index Swaps and Swaptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  International Journal of Theoretical and Applied Finance, 12(7), 1027–1053.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
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+page_content=' Finance and Stochastics,  2, 41–68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
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+page_content=' Springer  Finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
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+page_content=' Financial  Analysts Journal, 57(1), 41–59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
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+page_content=' Econometrica, 68(6), 1343–1376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
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+page_content=' Review of Financial Studies, 10(2), 481–523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
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+page_content=' Advances in Mathematical Economics, 1,  69–81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
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+page_content=' Review of Derivatives Research,  2(2-3), 99–120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Madan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
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+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' On dynamic spectral risk measures, a limit theorem  and optimal portfolio allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Finance and Stochastics, 21, 1073–1102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Madan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
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+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Bilateral Multiple Gamma Returns: Their Risks  and Rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' International Journal of Financial Engineering, 7(1), 1–27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Madan, Dilip, & Unal, Haluk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Pricing the risks of default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Review of Derivatives Research,  2, 121–160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  McNeil, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=', Frey, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=', & Embrecht, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Quantitative Risk Management-Concepts, Techniques  and Tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Princeton University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content='  Pedersen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Valuation of Portfolio Credit Default Swaptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}
+page_content=' Lehman Brothers Quantitative  Credit Research, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQf6g5Z/content/2301.05332v1.pdf'}