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Pricing the Zero-Coupon Bond and its Fair Premium Under a Structural Credit Risk Model with Jumps

Part of: Markov processes Integral transforms, operational calculus

Published online by Cambridge University Press:  14 July 2016

Yinghui Dong ,
Guojing Wang  and
Rong Wu
Yinghui Dong*
Affiliation:
Suzhou University and Suzhou University of Science and Technology
Guojing Wang*
Affiliation:
Suzhou University
Rong Wu*
Affiliation:
Nankai University
*
∗Postal address: Department of Mathematics and Center for Financial Engineering, Suzhou University, Suzhou 215006, P. R. China.
∗Postal address: Department of Mathematics and Center for Financial Engineering, Suzhou University, Suzhou 215006, P. R. China.
∗∗∗Postal address: Department of Mathematics and LPMC, Nankai University, Tianjin 300071, P. R. China.
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Abstract

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In this paper we consider a structural form credit risk model with jumps. We investigate the credit spread, the price, and the fair premium of the zero-coupon bond for the proposed model. The price and the fair premium of the bond are associated with the Laplace transform of default time and the firm's expected present market value at default. We give sufficient conditions under which the Laplace transform and the expected present market value of a firm at default are twice continuously differentiable. We derive closed-form expressions for them when the jumps have a hyperexponential distribution. Using the closed-form expressions, we obtain numerical solutions for the default probability, the credit spread, and the fair premium of the bond.

Keywords

Credit spreaddefault probabilityhyperexponential distributionfair premium ratestructural credit risk modelzero-coupon bond

MSC classification

Secondary: 60J75: Jump processes 44A10: Laplace transform
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 2 , June 2011 , pp. 404 - 419
Copyright
Copyright © Applied Probability Trust 2011 

References

Black, F. and Cox, J. (1976). Valuing corporate securities liabilities: some effects of bond indenture provisions. J. Finance 31, 351–367.CrossRefGoogle Scholar
Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. J. Political Economy 81, 637–654.CrossRefGoogle Scholar
Blumenthal, R. M. and Getoor, R. K. (1968). Markov Processes and Potential Theory. Academic Press, New York.Google Scholar
Chen, C.-J. and Panjer, H. (2009). A bridge from ruin theory to credit risk. Rev. Quant. Finance Accounting 32, 373–403.CrossRefGoogle Scholar
Chiu, S. N. and Yin, C. C. (2003). The time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process perturbed by diffusion. Insurance Math. Econom. 33, 59–66.CrossRefGoogle Scholar
Collin-Dufresne, P. and Goldstein, R. S. (2001). Do credit spreads reflect stationary leverage ratios? J. Finance 56, 1929–1929.CrossRefGoogle Scholar
Dufresne, F. and Gerber, H. U. (1991). Risk theory for the compound Poisson process that is perturbed by diffusion. Insurance Math. Econom. 10, 51–59.CrossRefGoogle Scholar
Dynkin, E. B. (1965). Markov Processes, Vols. I, II. Springer, Berlin.Google Scholar
Furrer, H. J. and Schmidli, H. (1994). Exponential inequalities for ruin probabilities of risk processes perturbed by diffusion. Insurance Math. Econom. 15, 23–36.CrossRefGoogle Scholar
Gerber, H. U. (1970). An extension of the renewal equation and its application in the collective theory of risk. Skand. Aktuarietidskrift 1970, 205–210.Google Scholar
Gerber, H. U. and Shiu, E. S. W. (1998). On the time value of ruin. N. Amer. Actuarial J. 2, 48–78.CrossRefGoogle Scholar
Gerber, H. U. and Shiu, E. S. W. (2005). The time value of ruin in a Sparre Andersen model. N. Amer. Actuarial J. 9, 49–84.CrossRefGoogle Scholar
Hilberink, B. and Rogers, L. C. G. (2002). Optimal capital structure and endogenous default. Finance Stoch. 6, 237–263.CrossRefGoogle Scholar
Huang, J.-Z. and Huang, M. (2003). How much of the corporate-treasury yield spread is due to credit risk? Working paper. Available at http://ssrn.com/abstract=307360.Google Scholar
Huang, J.-Z. and Zhou, H. (2008). Specification analysis of structural credit risk models. Working paper. Available at http://ssrn.com/paper=1105640.Google Scholar
Kou, S. G. (2002). A Jump-diffusion model for option pricing. Manag. Sci. 48, 1086–1101.CrossRefGoogle Scholar
Kou, S. G. and Wang, H. (2003). First passage times of a Jump diffusion process. Adv. Appl. Prob. 35, 504–531.CrossRefGoogle Scholar
Kou, S. G. and Wang, H. (2004). Option pricing under a double exponential Jump diffusion model. Manag. Sci. 50, 1178–1192.CrossRefGoogle Scholar
Le Courtois, O. and Quittard-Pinon, F. (2006). Risk-neutral and default probabilities with an endogenous bankruptcy Jump-diffusion model. Asia-Pacific Financial Markets 13, 11–39.CrossRefGoogle Scholar
Lindskog, F. and McNeil, A. J. (2003). Common Poisson shock models: applications to insurance and credit risk modelling. ASTIN Bull. 33, 209–238.CrossRefGoogle Scholar
Longstaff, F. A. and Schwartz, E. S. (1995). A simple approach to valuing risky and floating rate debt. J. Finance 50, 789–819.CrossRefGoogle Scholar
Merton, R. C. (1974). On the pricing of corporate debt: the risk structure of interest rates. J. Finance 29, 449–470.Google Scholar
Ramezani, C. and Zeng, Y. (2007). Maximum likelihood estimation of the double exponential Jump-diffusion process. Ann. Finance 3, 487–507.CrossRefGoogle Scholar
Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin.CrossRefGoogle Scholar
Tsai, C. C.-L. and Willmot, G. E. (2002). On the moments of the surplus process perturbed by diffusion. Insurance Math. Econom. 31, 327–350.CrossRefGoogle Scholar
Wang, G. and Wu, R. (2000). Some distributions for classical risk process that is perturbed by diffusion. Insurance Math. Econom. 26, 15–24.CrossRefGoogle Scholar
Wang, G. and Wu, R. (2008). The expected discounted penalty function for the perturbed compound Poisson risk process with constant interest. Insurance Math. Econom. 42, 59–64.CrossRefGoogle Scholar
Yuen, K. C., Lu, L. and Wu, R. (2009). The compound Poisson process perturbed by a diffusion with a threshold dividend strategy. Appl. Stoch. Models Business Industry 25, 73–93.CrossRefGoogle Scholar
Zhou, C. (2001). The term structure of credit spreads with Jump risk. J. Banking Finance 25, 2015–2040.CrossRefGoogle Scholar