- Email: [email protected]
Valuing credit derivatives in a jump-diffusion model
Applied Mathematics and Computation 190 (2007) 627–632 www.elsevier.com/locate/amc
Valuing credit derivatives in a jump-diffusion model Xinhua Hu *, Zhongxing Ye Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China
Abstract This paper presents a simple framework for valuing single-name credit derivatives in jump-diffusion models. The Gaver–Stehfest algorithm is used to calculate the CDS spread when the value of the reference entity is assumed to the double exponential jump-diffusion process. We model directly the credit spread using a geometric Ornstein–Uhlenbeck process with a jump and derive the pricing formula for credit spread option. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Credit derivatives; Jump-diffusion model; Gaver–Stehfest algorithm
1. Introduction Credit derivatives are financial contracts whose payoffs are contingent on the creditworthiness of some financial entities. Credit derivatives are means of transferring credit risk (on a reference entity) between two parties by means of bilateral agreements. They can refer to a single credit instrument or a basket of instruments. In the last decade, credit derivatives have become increasingly popular. According to the International Swaps and Derivatives Association (ISDA), by the end of 2006 the size of the global credit derivatives market will be $20 trillion. There are some common credit derivatives, such as credit default swaps, credit linked note, credit spread option, etc. The literature on credit derivatives is growing rapidly. There are two main approaches to modeling credit derivatives: the asset based method and intensity based method. Important theoretical work in the area includes Das [2], Das and Sundaram [3,4], Duffie [5], Duffie and Singleton [6], Hull and White [8–11], Jamshidian [12], Jarrow and Turnbull [13,14], Jarrow and Yildirim [15], Lando [18], Longstaff and Schwartz [19], Schonbucher [20], and many others. Our approaches to valuing derivatives belong to structural methods in this article. The remainder of this paper is structured as follows. Section 2 studies the pricing of CDS in a jump-diffusion model. In Section 3 we derive a formula for credit spread option pricing. Section 4 discusses the pricing of other credit derivatives in our framework. Section 5 concludes.
*
Corresponding author. E-mail address: [email protected] (X. Hu).
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.01.088
628
X. Hu, Z. Ye / Applied Mathematics and Computation 190 (2007) 627–632
2. CDS valuation Credit Default Swaps (CDS) are the most important and widely used single-name credit derivatives. A typical CDS contract has a life of 5 years during which the buyer of credit protection pays a periodic fee s, known as the swap premium, times the outstanding notional to the seller of protection in return for protection against a potential credit event of a given firm known as the underlying reference entity. Credit events in practice are associated with credit-rating downgrading, firm restructuring, and default, among others. In this paper, the credit event refers only to the default of the reference entity. When default occurs CDS reduce the outstanding notional principal and trigger two events. First there is an accrual payment to bring the periodic payments up to date. Second the seller of protection makes a payment equal to the loss to the buyer of protection. The loss is the reduction in the notional principal times one less the recovery rate, R. We follow Hull and White [11] and assume that defaults occur in the middle of a period for simplicity. The value of a CDS is the present value of the expected cash flows. The present values of the regular payments at a rate of 100% per year, P, can be written as P ¼s
N X ðti ti1 ÞE½F ðti Þerti :
ð1Þ
i¼1
The present values of the accrual payment that occurs when defaults reduce the notional principal, A, can be written as A¼s
N X 1 ðti ti1 ÞfE½F ðti1 Þ F ðti Þgerðti1 þti Þ=2 : 2 i¼1
ð2Þ
And the present values of the expected loss due to default, C, can be written as C ¼ ð1 RÞ
N X
fE½F ðti1 Þ F ðti Þgerðti1 þti Þ=2 ;
ð3Þ
i¼1
where ti (i ¼ 1; . . . ; N Þ is the premium payment date and F ðti Þ is the notional principal outstanding at ti; tN ¼ T is the expiration date, r is the risk-free rate of interest, and expectations are taken over a risk-neutral density. The fair spread s makes the mark-to-market equal to zero, i.e., P þ A ¼ C. If we assume the notional principal of CDS is one, then we have E½F ðti Þ ¼ 1 Qðti Þ, where Q(t) is the risk-neutral probability that the reference entity has defaulted before time t. From (1)–(3), we can obtain the spread PN ð1 RÞ i¼1 ðQðti Þ Qðti1 ÞÞerðti1 þti Þ=2 s ¼ PN : ð4Þ rti þ 0:5ðQðt Þ Qðt rðti1 þti Þ=2 i i1 ÞÞe i¼1 ðt i t i1 Þ½ð1 Qðt i ÞÞe It is clear that the key element in pricing CDS is the determination of the default probability of the reference entity Q(t). Next we will solve the cumulative default rate Q(t) in a structural model. We assume that the value of the reference entity’s assets V ¼ fV t gtP0 satisfies V t ¼ V 0 expðX t Þ;
ð5Þ
V 0 > 0;
where ðX t ÞtP0 is a jump-diffusion process given by X t ¼ l dt þ r dW t þ
Nt X
ð6Þ
Y i:
i¼1
The sequence of jump sizes fY i giP1 is i:i:d: with an asymmetric double exponential density first proposed by Kou [16]: f ðyÞ ¼ pg1 eg1 y 1fyP0g þ ð1 pÞg2 eg2 y 1fy<0g ;
g1 > 0; g2 > 0;
ð7Þ
where p represents the probabilities of upward jumps, the standard Brownian motion fW t gtP0 , Poisson process fN t gtP0 and jump sizes fY i giP1 are assumed to be independent under risk-neutral measure Q. The double exponential distribution, which performs two-sided jumps, has the leptokurtic feature of the jump size that
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provides the peak and tails of the return distribution found in reality. Moreover, it has a memoryless feature which makes it easier to calculate expected means and variance terms. The process fX t gtP0 is a Levy process, i.e., a process with stationary and independent increments. The Laplace exponent of Xt can be obtained as EQ ½ebX t ¼ eGðbÞt , where 1 2 2 pg1 ð1 pÞg2 þ 1 : ð8Þ GðbÞ ¼ lb þ r b þ k 2 g1 b g2 þ b For a > 0, the equation GðxÞ ¼ a has exactly four real roots: b1;a ; b2;a ; b3;a ; b4;a , where 1 < b4;a < g2 < b3;a < 0 < b1;a < g1 < b2;a < 1. The reference entity defaults whenever the value of its assets falls below a fall below the default threshold H. We define the first-passage time s ¼ infft P 0; V t 6 H g ¼ infft P 0; X t 6 bg;
b < 0;
ð9Þ
where b ¼ logðH =V 0 Þ, and we have QðtÞ Qðs 6 tÞ ¼ Qð inf X s 6 bÞ 06s6t
ð10Þ
Using the results of Kou and Wang [17], we can deduce the Laplace transforms of s for any a > 0, g2 b3;a b4;a b4;a g2 b3;a ebb3;a þ ebb4;a : ð11Þ g2 b4;a b3;a g2 b4;a b3;a R1 R1 b Let QðaÞ ¼ 0 eat Qðs 6 tÞdt ¼ 1a 0 eat dQðs 6 tÞ ¼ 1a EQ ½eas be the Laplace transform of Q(t), from the Gaver–Stehfest algorithm for Laplace inversion [1,7], we obtain EQ ½eas ¼
e n ðtÞ; QðtÞ ¼ lim Q n!1
ð12Þ
where n X k n e n ðtÞ ¼ log 2 ð2nÞ! b ðn þ kÞ log 2 : Q ð1Þ Q t n!ðn 1Þ! k¼0 t k
ð13Þ
To accelerate convergence, a linear combination of the terms can be used as the approximant of Q(t) [1,21], that is, for large n, n X kn e k ðtÞ; ð1Þnk ð14Þ Q QðtÞ k!ðn kÞ! k¼1 P kn e k ðtÞ QðtÞ ¼ oðnk Þ as n ! 1 for all k. Now we can use the Gaver–Stehfest Q moreover, nk¼1 ð1Þnk k!ðnkÞ! algorithm to compute Qðti Þ at fixed time ti ði ¼ 1; . . . ; N Þ. Then we can calculate the fair spreads s using formula (4) in a double exponential jump-diffusion model. As a numerical illustration, we will present default probability Qðs 6 tÞ of the reference entity and fair CDS spreads s with different expiration dates. For all the computations, we take the values of the parameters for the double exponential jump diffusion as follows: the drift coefficient l = 0.1, the volatility r = 0.2, the probability of upward jumps p = 0.4, the mean of upward (downward) jumps 1=g1 ¼ 0:02ð1=g2 ¼ 0:03Þ and the intensity of the jump process k = 3, the default barrier b = 2. For the parameter of the CDS maturity, we take T as equal to 1 year, 2 years, . . . , 10 years. For simplicity we assume that the CDS payments are made annually. Other parameters are the risk-free interest rate r = 5% and recovery rate R = 40%. The results are presented in Fig. 1. 3. Credit spread option valuation Spread options are derivative instruments whose payoff depends on the spread between two different assets or variables. Some of the important spread options are credit spread options. Credit spread option offer investors whose portfolio values are highly sensitive to shifts in the spread between defaultable and non-defaultable yields an important tool for managing and hedging their exposure to this type of risk. The pricing of credit
X. Hu, Z. Ye / Applied Mathematics and Computation 190 (2007) 627–632
spread (basis points)
630
16.00%
100 90 80 70 60 50 40 30 20 10 0
14.00% 12.00% 10.00% 8.00% 6.00% 4.00% 2.00% 0.00% 1
2
3
4
5
6
7
8
9
10
Time - Years CDS Spreads
Default Probability
Fig. 1. CDS spreads and cumulative default probability.
spread options has been addressed in a number of previous publications. In many intensity-based credit risk models, credit spread options can be priced by transfer of techniques from the pricing of default-free bond options, see e.g. Duffie and Singleton [6], Jamshidian [12] or Schonbucher [20]. Das and Sundaram [4] give a discrete-time HJM-approach to credit spreads. Longstaff and Schwartz [19] have developed a valuation model for credit spread option and derived a closed-form solution. They capture the mean-reverting property of credit spreads and model the logarithm of the spread by Ornstein–Uhlenbeck process. However, when the rare catastrophic shocks occur or the regime shifts in the economy or industry, the credit spreads may have jumps. We will take the proposition one step further by modeling directly the spread using a geometric Ornstein–Uhlenbeck process with a jump. We assume that the risk-neutral dynamics of the credit spread s(t) are given by stochastic differential equations: Z 1 dsðtÞ ¼ jðl log sðtÞÞdt þ r dW 1 ðtÞ þ xðJ ðdx; dtÞ kf ðdxÞdtÞ; ð15Þ sðtÞ 1 where the mean reversion coefficient j and the volatility r are positive constants and l is real constant. The jump measure J ðdx; dtÞ is a Poisson random measure on R ½0; T with intensity measure kf ðdxÞdt, that is, RR PN t xJ ðds; dxÞ ¼ i¼1 Y i is a compound Poisson process with intensity k > 0 and jump size distribution ½0;tR f. The jumps sizes Yi are i.i.d. with distribution f. ðN t ÞtP0 is a Poisson process with intensity k and W 1 ðtÞ is a standard Brownian motion. ðY i ÞiP1 ; N t and W 1 ðtÞ are mutually independent. Eq. (15) can be best understood after a simple transformation leading to the dynamics of the logarithms of the credit spread. Setting X ðtÞ ¼ log sðtÞ, a simple application of Itoˆ’s formula gives Z 1 dX ðtÞ ¼ ðc jX ðtÞÞdt þ r dW 1 ðtÞ þ logð1 þ xÞJ ðdx; dtÞ; ð16Þ 1
which shows that the logarithm of the credit R 1 spread is nothing but classical Ornstein–Uhlenbeck process with Poisson jump. Where c ¼ jl 12 r2 k 1 xf ðdxÞ. In particular, Eq. (16) is consistent with Longstaff and Schwartz’s assumption if k = 0. Eq. (16) can be explicitly solved, giving Z t Z Z c X ðtÞ ¼ ejt log sð0Þ þ ð1 ejt Þ þ r ejðtsÞ dW ðsÞ þ ejðtsÞ logð1 þ xÞJ ðdx; dsÞ: ð17Þ j ½0;t½1;1Þ 0 Rt Setting ZðtÞ ¼ r 0 ejðtsÞ dW 1 ðtÞ, then Z(t) is normally distributed with mean 0 and variance r2 ðtÞ, where r2 ðtÞ ¼ r2 ð1 e2jt Þ=ð2jÞ. Hence sðtÞ ¼ expflðtÞ þ ZðtÞg
Nt Y
ð1 þ Y i Þ
expðjðtsi ÞÞ
;
i¼1
where lðtÞ ¼ ejt log sð0Þ þ cð1 ejt Þ=c, ðsi ; Y i Þ representing the ith jump time and the jump size.
ð18Þ
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To allow for random interests in the model, we assume that the risk-neutral dynamics of the short-term interest rate r follow the square-root process (CIR model) pffiffiffiffiffiffiffiffi drðtÞ ¼ bðr rðtÞÞdt þ d rðtÞdW 2 ðtÞ; ð19Þ where b, r and d are positive constants and satisfy 2br > d2 . W 2 ðtÞ is also a standard Brownian motion independent from ðY i ÞiP1 ; ðN t ÞtP0 and W 1 ðtÞ. Given a short-term interest rate process r and a risk-neutral measure Q, the price of a riskless zero-coupon bond with time to maturity T is Bðt; r; T Þ ¼ AðT ÞeBðT ÞrðtÞ ;
ð20Þ
where
2heðbþhÞT =2 AðT Þ ¼ 2h þ ðb þ hÞðehT 1Þ
2br=d2 ;
BðT Þ ¼
2ðehT 1Þ 2h þ ðb þ hÞðehT 1Þ
and
h¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 þ 2d2 :
Let Cðs; r; T Þ denote the value of a European call option on the level of credit spread with payoff function H ðsðT ÞÞ ¼ maxfsðT Þ K; 0g, where the strike price of option is K and time to expiration is T. Applying the martingale method of derivative pricing, Cðs; r; T Þ ¼ Bðr; T ÞEQ ½H ðsðT ÞÞ, we obtain " # 1 n Y X ðkT Þn ekT expðjðtsi ÞÞ lðT Þþ1r2 ðT Þ 2 En Cðs; r; T Þ ¼ Bðr; T Þ ð1 þ Y i Þ e N ðd 1 Þ KN ðd 2 Þ ; ð21Þ n! n¼0 i¼1 where N(Æ) is the cumulative standard normal distribution function, Bðr; T Þ Bð0; r; T Þ, and Qn expðjðtsi ÞÞ lðT Þ log K þ log i¼1 ð1 þ Y i Þ d 1 ¼ d 2 þ rðT Þ; d 2 ¼ ; rðT Þ Q En is the expectation operator over the random variable ni¼1 ð1 þ Y i Þexpðjðtsi ÞÞ . Similarly, we can derive the pricing formula of a European put option on the credit spread with payoff function H ðsðT ÞÞ ¼ maxfK sðT Þ; 0g, " # 1 n X Y ðkT Þn ekT expðjðtsi ÞÞ lðT Þþ1r2 ðT Þ 2 P ðs; r; T Þ ¼ Bðr; T Þ En KN ðd 2 Þ ð1 þ Y i Þ e N ðd 1 Þ : ð22Þ n! n¼0 i¼1 4. Other credit derivatives valuation We can also price other single-name credit derivatives such as credit-linked notes and total return swap in a jump-diffusion model. A credit-linked note is a security with an embedded credit default swap allowing the issuer to transfer a specific credit risk to credit investors. It combines a regular coupon-paying note with some credit risk feature. Under this structure, the coupon or price of the note is linked to the performance of a reference asset. Similarly, if we model the value of the reference entity by the double exponential jump-diffusion process, the coupon rate of the credit-linked note can be derived. A total rate of return swap is swap under which two parties periodically pay each other the total return on one or two reference assets or a basket of assets they do not necessarily hold. The protection buyer makes payments linked to the total return on a portfolio of assets or loans. The protection seller makes payments tied to a reference rate, such as the yield on an equivalent treasury issue (or LIBOR) plus a spread. A total return swap removes all risk of the underlying assets without selling the assets. Unlike a CDS, a total return swap also has some exposure to market risk. The pricing of total return swap is similar to the valuation of CDS in our framework. 5. Conclusions We have proposed a jump-diffusion model for valuing credit derivatives. The CDS spreads are easy to calculate by the Gaver–Stehfest algorithm for Lapalace inversion when the value of the reference entity follows the double exponential jump-diffusion process. The numerical results match empirical observations. We have
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presented closed-form valuation expression for European calls and puts on credit spread by modeling directly the credit spread using a geometric Ornstein–Uhlenbeck process with a jump. Our structural approaches in a jump-diffusion model can be applied to price other credit derivatives. Acknowledgement This research is supported by the National Natural Science Foundation of China under Grant number 70671069. References [1] J. Abate, W. Whitt, The Fourier-series method for inverting transforms of probability distributions, Queueing Systems 10 (1992) 5– 88. [2] S.R. Das, Credit risk derivatives, Journal of Derivatives 2 (1995) 7–21. [3] S.R. Das, R.K. Sundaram, A direct approach to arbitrage-free pricing of credit derivatives, Working Paper 6635, National Bureau of Economic Research, 1998. [4] S.R. Das, R.K. Sundaram, A discrete-time approach to arbitrage-free pricing of credit derivatives, Management Science 46 (2000) 46– 62. [5] D. Duffie, Credit swap valuation, Financial Analysts Journal 55 (1999) 73–87. [6] D. Duffie, K.J. Singleton, Modeling term structures of defaultable bonds, Review of Financial Studies 12 (1999) 687–720. [7] D.P. Gaver, Observing stochastic processes and approximate transform inversion, Operations Research 14 (1966) 444–459. [8] J. Hull, A. White, Valuing Credit Default Swaps I: no counterparty default risk, The Journal of Derivatives 8 (1) (2000) 29–40. [9] J. Hull, A. White, Valuing credit default swaps II: modeling default correlations, The Journal of Derivatives 8 (3) (2001) 12–22. [10] J. Hull, A. White, The valuation of credit default swap options, The Journal of Derivatives 10 (3) (2003) 40–50. [11] J. Hull, A. White, Valuing credit derivatives using an implied copula approach, Journal of Derivatives 14 (2) (2006) 8–28. [12] F. Jamshidian, Valuation of credit default swap and swaptions, Working paper, NIB Capital Bank, October 2002. [13] R.A. Jarrow, S.M. Turnbull, Pricing derivatives on financial securities subject to credit risk, Journal of Finance 50 (1995) 53–85. [14] R.A. Jarrow, S.M. Turnbull, The intersection of market and credit risk, Journal of Banking and Finance 24 (2000) 271–299. [15] R.A. Jarrow, Y. Yildirim, Valuing default swaps under market and credit risk correlation, Journal of Fixed Income 11 (2002) 7–19. [16] S. Kou, A jump-diffusion model for option pricing, Management Science 48 (2002) 1086–1101. [17] S. Kou, H. Wang, First passage times of a jump diffusion process, Advances in Applied Probability 35 (2003) 504–531. [18] D. Lando, On Cox processes and credit-risky securities, Review of Derivatives Research 2 (1998) 99–120. [19] F.A. Longstaff, E.S. Schwartz, Valuing credit derivatives, The Journal of Fixed Income 5 (1995) 6–12. [20] P.J. Schonbucher, A tree implementation of a credit spread model for credit derivatives, The Journal of Computational Finance 6 (2) (2002). [21] H. Stehfest, Algorithm 368. Numerical inversion of Laplace transforms, Communications of the ACM 13 (1970) 47–49 and 13(10) 624.
Valuing credit derivatives in a jump-diffusion model Xinhua Hu *, Zhongxing Ye Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China
Abstract This paper presents a simple framework for valuing single-name credit derivatives in jump-diffusion models. The Gaver–Stehfest algorithm is used to calculate the CDS spread when the value of the reference entity is assumed to the double exponential jump-diffusion process. We model directly the credit spread using a geometric Ornstein–Uhlenbeck process with a jump and derive the pricing formula for credit spread option. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Credit derivatives; Jump-diffusion model; Gaver–Stehfest algorithm
1. Introduction Credit derivatives are financial contracts whose payoffs are contingent on the creditworthiness of some financial entities. Credit derivatives are means of transferring credit risk (on a reference entity) between two parties by means of bilateral agreements. They can refer to a single credit instrument or a basket of instruments. In the last decade, credit derivatives have become increasingly popular. According to the International Swaps and Derivatives Association (ISDA), by the end of 2006 the size of the global credit derivatives market will be $20 trillion. There are some common credit derivatives, such as credit default swaps, credit linked note, credit spread option, etc. The literature on credit derivatives is growing rapidly. There are two main approaches to modeling credit derivatives: the asset based method and intensity based method. Important theoretical work in the area includes Das [2], Das and Sundaram [3,4], Duffie [5], Duffie and Singleton [6], Hull and White [8–11], Jamshidian [12], Jarrow and Turnbull [13,14], Jarrow and Yildirim [15], Lando [18], Longstaff and Schwartz [19], Schonbucher [20], and many others. Our approaches to valuing derivatives belong to structural methods in this article. The remainder of this paper is structured as follows. Section 2 studies the pricing of CDS in a jump-diffusion model. In Section 3 we derive a formula for credit spread option pricing. Section 4 discusses the pricing of other credit derivatives in our framework. Section 5 concludes.
*
Corresponding author. E-mail address: [email protected] (X. Hu).
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.01.088
628
X. Hu, Z. Ye / Applied Mathematics and Computation 190 (2007) 627–632
2. CDS valuation Credit Default Swaps (CDS) are the most important and widely used single-name credit derivatives. A typical CDS contract has a life of 5 years during which the buyer of credit protection pays a periodic fee s, known as the swap premium, times the outstanding notional to the seller of protection in return for protection against a potential credit event of a given firm known as the underlying reference entity. Credit events in practice are associated with credit-rating downgrading, firm restructuring, and default, among others. In this paper, the credit event refers only to the default of the reference entity. When default occurs CDS reduce the outstanding notional principal and trigger two events. First there is an accrual payment to bring the periodic payments up to date. Second the seller of protection makes a payment equal to the loss to the buyer of protection. The loss is the reduction in the notional principal times one less the recovery rate, R. We follow Hull and White [11] and assume that defaults occur in the middle of a period for simplicity. The value of a CDS is the present value of the expected cash flows. The present values of the regular payments at a rate of 100% per year, P, can be written as P ¼s
N X ðti ti1 ÞE½F ðti Þerti :
ð1Þ
i¼1
The present values of the accrual payment that occurs when defaults reduce the notional principal, A, can be written as A¼s
N X 1 ðti ti1 ÞfE½F ðti1 Þ F ðti Þgerðti1 þti Þ=2 : 2 i¼1
ð2Þ
And the present values of the expected loss due to default, C, can be written as C ¼ ð1 RÞ
N X
fE½F ðti1 Þ F ðti Þgerðti1 þti Þ=2 ;
ð3Þ
i¼1
where ti (i ¼ 1; . . . ; N Þ is the premium payment date and F ðti Þ is the notional principal outstanding at ti; tN ¼ T is the expiration date, r is the risk-free rate of interest, and expectations are taken over a risk-neutral density. The fair spread s makes the mark-to-market equal to zero, i.e., P þ A ¼ C. If we assume the notional principal of CDS is one, then we have E½F ðti Þ ¼ 1 Qðti Þ, where Q(t) is the risk-neutral probability that the reference entity has defaulted before time t. From (1)–(3), we can obtain the spread PN ð1 RÞ i¼1 ðQðti Þ Qðti1 ÞÞerðti1 þti Þ=2 s ¼ PN : ð4Þ rti þ 0:5ðQðt Þ Qðt rðti1 þti Þ=2 i i1 ÞÞe i¼1 ðt i t i1 Þ½ð1 Qðt i ÞÞe It is clear that the key element in pricing CDS is the determination of the default probability of the reference entity Q(t). Next we will solve the cumulative default rate Q(t) in a structural model. We assume that the value of the reference entity’s assets V ¼ fV t gtP0 satisfies V t ¼ V 0 expðX t Þ;
ð5Þ
V 0 > 0;
where ðX t ÞtP0 is a jump-diffusion process given by X t ¼ l dt þ r dW t þ
Nt X
ð6Þ
Y i:
i¼1
The sequence of jump sizes fY i giP1 is i:i:d: with an asymmetric double exponential density first proposed by Kou [16]: f ðyÞ ¼ pg1 eg1 y 1fyP0g þ ð1 pÞg2 eg2 y 1fy<0g ;
g1 > 0; g2 > 0;
ð7Þ
where p represents the probabilities of upward jumps, the standard Brownian motion fW t gtP0 , Poisson process fN t gtP0 and jump sizes fY i giP1 are assumed to be independent under risk-neutral measure Q. The double exponential distribution, which performs two-sided jumps, has the leptokurtic feature of the jump size that
X. Hu, Z. Ye / Applied Mathematics and Computation 190 (2007) 627–632
629
provides the peak and tails of the return distribution found in reality. Moreover, it has a memoryless feature which makes it easier to calculate expected means and variance terms. The process fX t gtP0 is a Levy process, i.e., a process with stationary and independent increments. The Laplace exponent of Xt can be obtained as EQ ½ebX t ¼ eGðbÞt , where 1 2 2 pg1 ð1 pÞg2 þ 1 : ð8Þ GðbÞ ¼ lb þ r b þ k 2 g1 b g2 þ b For a > 0, the equation GðxÞ ¼ a has exactly four real roots: b1;a ; b2;a ; b3;a ; b4;a , where 1 < b4;a < g2 < b3;a < 0 < b1;a < g1 < b2;a < 1. The reference entity defaults whenever the value of its assets falls below a fall below the default threshold H. We define the first-passage time s ¼ infft P 0; V t 6 H g ¼ infft P 0; X t 6 bg;
b < 0;
ð9Þ
where b ¼ logðH =V 0 Þ, and we have QðtÞ Qðs 6 tÞ ¼ Qð inf X s 6 bÞ 06s6t
ð10Þ
Using the results of Kou and Wang [17], we can deduce the Laplace transforms of s for any a > 0, g2 b3;a b4;a b4;a g2 b3;a ebb3;a þ ebb4;a : ð11Þ g2 b4;a b3;a g2 b4;a b3;a R1 R1 b Let QðaÞ ¼ 0 eat Qðs 6 tÞdt ¼ 1a 0 eat dQðs 6 tÞ ¼ 1a EQ ½eas be the Laplace transform of Q(t), from the Gaver–Stehfest algorithm for Laplace inversion [1,7], we obtain EQ ½eas ¼
e n ðtÞ; QðtÞ ¼ lim Q n!1
ð12Þ
where n X k n e n ðtÞ ¼ log 2 ð2nÞ! b ðn þ kÞ log 2 : Q ð1Þ Q t n!ðn 1Þ! k¼0 t k
ð13Þ
To accelerate convergence, a linear combination of the terms can be used as the approximant of Q(t) [1,21], that is, for large n, n X kn e k ðtÞ; ð1Þnk ð14Þ Q QðtÞ k!ðn kÞ! k¼1 P kn e k ðtÞ QðtÞ ¼ oðnk Þ as n ! 1 for all k. Now we can use the Gaver–Stehfest Q moreover, nk¼1 ð1Þnk k!ðnkÞ! algorithm to compute Qðti Þ at fixed time ti ði ¼ 1; . . . ; N Þ. Then we can calculate the fair spreads s using formula (4) in a double exponential jump-diffusion model. As a numerical illustration, we will present default probability Qðs 6 tÞ of the reference entity and fair CDS spreads s with different expiration dates. For all the computations, we take the values of the parameters for the double exponential jump diffusion as follows: the drift coefficient l = 0.1, the volatility r = 0.2, the probability of upward jumps p = 0.4, the mean of upward (downward) jumps 1=g1 ¼ 0:02ð1=g2 ¼ 0:03Þ and the intensity of the jump process k = 3, the default barrier b = 2. For the parameter of the CDS maturity, we take T as equal to 1 year, 2 years, . . . , 10 years. For simplicity we assume that the CDS payments are made annually. Other parameters are the risk-free interest rate r = 5% and recovery rate R = 40%. The results are presented in Fig. 1. 3. Credit spread option valuation Spread options are derivative instruments whose payoff depends on the spread between two different assets or variables. Some of the important spread options are credit spread options. Credit spread option offer investors whose portfolio values are highly sensitive to shifts in the spread between defaultable and non-defaultable yields an important tool for managing and hedging their exposure to this type of risk. The pricing of credit
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spread (basis points)
630
16.00%
100 90 80 70 60 50 40 30 20 10 0
14.00% 12.00% 10.00% 8.00% 6.00% 4.00% 2.00% 0.00% 1
2
3
4
5
6
7
8
9
10
Time - Years CDS Spreads
Default Probability
Fig. 1. CDS spreads and cumulative default probability.
spread options has been addressed in a number of previous publications. In many intensity-based credit risk models, credit spread options can be priced by transfer of techniques from the pricing of default-free bond options, see e.g. Duffie and Singleton [6], Jamshidian [12] or Schonbucher [20]. Das and Sundaram [4] give a discrete-time HJM-approach to credit spreads. Longstaff and Schwartz [19] have developed a valuation model for credit spread option and derived a closed-form solution. They capture the mean-reverting property of credit spreads and model the logarithm of the spread by Ornstein–Uhlenbeck process. However, when the rare catastrophic shocks occur or the regime shifts in the economy or industry, the credit spreads may have jumps. We will take the proposition one step further by modeling directly the spread using a geometric Ornstein–Uhlenbeck process with a jump. We assume that the risk-neutral dynamics of the credit spread s(t) are given by stochastic differential equations: Z 1 dsðtÞ ¼ jðl log sðtÞÞdt þ r dW 1 ðtÞ þ xðJ ðdx; dtÞ kf ðdxÞdtÞ; ð15Þ sðtÞ 1 where the mean reversion coefficient j and the volatility r are positive constants and l is real constant. The jump measure J ðdx; dtÞ is a Poisson random measure on R ½0; T with intensity measure kf ðdxÞdt, that is, RR PN t xJ ðds; dxÞ ¼ i¼1 Y i is a compound Poisson process with intensity k > 0 and jump size distribution ½0;tR f. The jumps sizes Yi are i.i.d. with distribution f. ðN t ÞtP0 is a Poisson process with intensity k and W 1 ðtÞ is a standard Brownian motion. ðY i ÞiP1 ; N t and W 1 ðtÞ are mutually independent. Eq. (15) can be best understood after a simple transformation leading to the dynamics of the logarithms of the credit spread. Setting X ðtÞ ¼ log sðtÞ, a simple application of Itoˆ’s formula gives Z 1 dX ðtÞ ¼ ðc jX ðtÞÞdt þ r dW 1 ðtÞ þ logð1 þ xÞJ ðdx; dtÞ; ð16Þ 1
which shows that the logarithm of the credit R 1 spread is nothing but classical Ornstein–Uhlenbeck process with Poisson jump. Where c ¼ jl 12 r2 k 1 xf ðdxÞ. In particular, Eq. (16) is consistent with Longstaff and Schwartz’s assumption if k = 0. Eq. (16) can be explicitly solved, giving Z t Z Z c X ðtÞ ¼ ejt log sð0Þ þ ð1 ejt Þ þ r ejðtsÞ dW ðsÞ þ ejðtsÞ logð1 þ xÞJ ðdx; dsÞ: ð17Þ j ½0;t½1;1Þ 0 Rt Setting ZðtÞ ¼ r 0 ejðtsÞ dW 1 ðtÞ, then Z(t) is normally distributed with mean 0 and variance r2 ðtÞ, where r2 ðtÞ ¼ r2 ð1 e2jt Þ=ð2jÞ. Hence sðtÞ ¼ expflðtÞ þ ZðtÞg
Nt Y
ð1 þ Y i Þ
expðjðtsi ÞÞ
;
i¼1
where lðtÞ ¼ ejt log sð0Þ þ cð1 ejt Þ=c, ðsi ; Y i Þ representing the ith jump time and the jump size.
ð18Þ
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631
To allow for random interests in the model, we assume that the risk-neutral dynamics of the short-term interest rate r follow the square-root process (CIR model) pffiffiffiffiffiffiffiffi drðtÞ ¼ bðr rðtÞÞdt þ d rðtÞdW 2 ðtÞ; ð19Þ where b, r and d are positive constants and satisfy 2br > d2 . W 2 ðtÞ is also a standard Brownian motion independent from ðY i ÞiP1 ; ðN t ÞtP0 and W 1 ðtÞ. Given a short-term interest rate process r and a risk-neutral measure Q, the price of a riskless zero-coupon bond with time to maturity T is Bðt; r; T Þ ¼ AðT ÞeBðT ÞrðtÞ ;
ð20Þ
where
2heðbþhÞT =2 AðT Þ ¼ 2h þ ðb þ hÞðehT 1Þ
2br=d2 ;
BðT Þ ¼
2ðehT 1Þ 2h þ ðb þ hÞðehT 1Þ
and
h¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 þ 2d2 :
Let Cðs; r; T Þ denote the value of a European call option on the level of credit spread with payoff function H ðsðT ÞÞ ¼ maxfsðT Þ K; 0g, where the strike price of option is K and time to expiration is T. Applying the martingale method of derivative pricing, Cðs; r; T Þ ¼ Bðr; T ÞEQ ½H ðsðT ÞÞ, we obtain " # 1 n Y X ðkT Þn ekT expðjðtsi ÞÞ lðT Þþ1r2 ðT Þ 2 En Cðs; r; T Þ ¼ Bðr; T Þ ð1 þ Y i Þ e N ðd 1 Þ KN ðd 2 Þ ; ð21Þ n! n¼0 i¼1 where N(Æ) is the cumulative standard normal distribution function, Bðr; T Þ Bð0; r; T Þ, and Qn expðjðtsi ÞÞ lðT Þ log K þ log i¼1 ð1 þ Y i Þ d 1 ¼ d 2 þ rðT Þ; d 2 ¼ ; rðT Þ Q En is the expectation operator over the random variable ni¼1 ð1 þ Y i Þexpðjðtsi ÞÞ . Similarly, we can derive the pricing formula of a European put option on the credit spread with payoff function H ðsðT ÞÞ ¼ maxfK sðT Þ; 0g, " # 1 n X Y ðkT Þn ekT expðjðtsi ÞÞ lðT Þþ1r2 ðT Þ 2 P ðs; r; T Þ ¼ Bðr; T Þ En KN ðd 2 Þ ð1 þ Y i Þ e N ðd 1 Þ : ð22Þ n! n¼0 i¼1 4. Other credit derivatives valuation We can also price other single-name credit derivatives such as credit-linked notes and total return swap in a jump-diffusion model. A credit-linked note is a security with an embedded credit default swap allowing the issuer to transfer a specific credit risk to credit investors. It combines a regular coupon-paying note with some credit risk feature. Under this structure, the coupon or price of the note is linked to the performance of a reference asset. Similarly, if we model the value of the reference entity by the double exponential jump-diffusion process, the coupon rate of the credit-linked note can be derived. A total rate of return swap is swap under which two parties periodically pay each other the total return on one or two reference assets or a basket of assets they do not necessarily hold. The protection buyer makes payments linked to the total return on a portfolio of assets or loans. The protection seller makes payments tied to a reference rate, such as the yield on an equivalent treasury issue (or LIBOR) plus a spread. A total return swap removes all risk of the underlying assets without selling the assets. Unlike a CDS, a total return swap also has some exposure to market risk. The pricing of total return swap is similar to the valuation of CDS in our framework. 5. Conclusions We have proposed a jump-diffusion model for valuing credit derivatives. The CDS spreads are easy to calculate by the Gaver–Stehfest algorithm for Lapalace inversion when the value of the reference entity follows the double exponential jump-diffusion process. The numerical results match empirical observations. We have
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presented closed-form valuation expression for European calls and puts on credit spread by modeling directly the credit spread using a geometric Ornstein–Uhlenbeck process with a jump. Our structural approaches in a jump-diffusion model can be applied to price other credit derivatives. Acknowledgement This research is supported by the National Natural Science Foundation of China under Grant number 70671069. References [1] J. Abate, W. Whitt, The Fourier-series method for inverting transforms of probability distributions, Queueing Systems 10 (1992) 5– 88. [2] S.R. Das, Credit risk derivatives, Journal of Derivatives 2 (1995) 7–21. [3] S.R. Das, R.K. Sundaram, A direct approach to arbitrage-free pricing of credit derivatives, Working Paper 6635, National Bureau of Economic Research, 1998. [4] S.R. Das, R.K. Sundaram, A discrete-time approach to arbitrage-free pricing of credit derivatives, Management Science 46 (2000) 46– 62. [5] D. Duffie, Credit swap valuation, Financial Analysts Journal 55 (1999) 73–87. [6] D. Duffie, K.J. Singleton, Modeling term structures of defaultable bonds, Review of Financial Studies 12 (1999) 687–720. [7] D.P. Gaver, Observing stochastic processes and approximate transform inversion, Operations Research 14 (1966) 444–459. [8] J. Hull, A. White, Valuing Credit Default Swaps I: no counterparty default risk, The Journal of Derivatives 8 (1) (2000) 29–40. [9] J. Hull, A. White, Valuing credit default swaps II: modeling default correlations, The Journal of Derivatives 8 (3) (2001) 12–22. [10] J. Hull, A. White, The valuation of credit default swap options, The Journal of Derivatives 10 (3) (2003) 40–50. [11] J. Hull, A. White, Valuing credit derivatives using an implied copula approach, Journal of Derivatives 14 (2) (2006) 8–28. [12] F. Jamshidian, Valuation of credit default swap and swaptions, Working paper, NIB Capital Bank, October 2002. [13] R.A. Jarrow, S.M. Turnbull, Pricing derivatives on financial securities subject to credit risk, Journal of Finance 50 (1995) 53–85. [14] R.A. Jarrow, S.M. Turnbull, The intersection of market and credit risk, Journal of Banking and Finance 24 (2000) 271–299. [15] R.A. Jarrow, Y. Yildirim, Valuing default swaps under market and credit risk correlation, Journal of Fixed Income 11 (2002) 7–19. [16] S. Kou, A jump-diffusion model for option pricing, Management Science 48 (2002) 1086–1101. [17] S. Kou, H. Wang, First passage times of a jump diffusion process, Advances in Applied Probability 35 (2003) 504–531. [18] D. Lando, On Cox processes and credit-risky securities, Review of Derivatives Research 2 (1998) 99–120. [19] F.A. Longstaff, E.S. Schwartz, Valuing credit derivatives, The Journal of Fixed Income 5 (1995) 6–12. [20] P.J. Schonbucher, A tree implementation of a credit spread model for credit derivatives, The Journal of Computational Finance 6 (2) (2002). [21] H. Stehfest, Algorithm 368. Numerical inversion of Laplace transforms, Communications of the ACM 13 (1970) 47–49 and 13(10) 624.