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AN ANALYTICAL APPROXIMATION FORMULA FOR THE PRICING OF CREDIT DEFAULT SWAPS WITH REGIME SWITCHING

Part of: Mathematical finance

Published online by Cambridge University Press:  02 September 2021

XIN-JIANG HE Open the ORCID record for XIN-JIANG HE [Opens in a new window]  and
SHA LIN Open the ORCID record for SHA LIN [Opens in a new window]
XIN-JIANG HE
Affiliation:
School of Economics, Zhejiang University of Technology, Hangzhou, China; e-mail: xinjiang@zjut.edu.cn.
SHA LIN*
Affiliation:
School of Finance, Zhejiang Gongshang University, Hangzhou, China
*
e-mail: linsha@mail.zjgsu.edu.cn.
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Abstract

We derive an analytical approximation for the price of a credit default swap (CDS) contract under a regime-switching Black–Scholes model. To achieve this, we first derive a general formula for the CDS price, and establish the relationship between the unknown no-default probability and the price of a down-and-out binary option written on the same reference asset. Then we present a two-step procedure: the first step assumes that all the future information of the Markov chain is known at the current time and presents an approximation for the conditional price under a time-dependent Black–Scholes model, based on which the second step derives the target option pricing formula written in a Fourier cosine series. The efficiency and accuracy of the newly derived formula are demonstrated through numerical experiments.

Keywords

analytical approximationcredit default swapregime switchingFourier cosine series

MSC classification

Primary: 91G20: Derivative securities
Type
Research Article
Information
The ANZIAM Journal , Volume 63 , Issue 2 , April 2021 , pp. 143 - 162
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Copyright
© Australian Mathematical Society 2021

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References

Artzner, P. and Delbaen, F., “‘Finem Lauda’ or the risks in swaps”, Insur. Math. Econ. 9 (1990) 295–303; doi:10.1016/0167-6687(90)90008-2.CrossRefGoogle Scholar
Artzner, P. and Delbaen, F., “Default risk insurance and incomplete markets 1”, Math. Finance 5 (1995) 187–195; doi:10.1111/j.1467-9965.1995.tb00064.x.CrossRefGoogle Scholar
Boyd, J. P., Chebyshev and Fourier spectral methods, 2nd edn, (Dover Publications, Mineola, NY, 2001).Google Scholar
Buffington, J. and Elliott, R. J., “American options with regime switching”, Int. J. Theor. Appl. Finance 5 (2002) 497–514; doi:10.1142/S0219024902001523.CrossRefGoogle Scholar
Cariboni, J. and Schoutens, W., “Pricing credit default swaps under Lévy models”, J. Compu. Finance 10 (2007) 71–91; doi:10.21314/JCF.2007.172.CrossRefGoogle Scholar
Chen, W. and He, X.-J., “Pricing credit default swaps under a multi-scale stochastic volatility model”, Physica A 468 (2017) 425–433; doi:10.1016/j.physa.2016.10.082.CrossRefGoogle Scholar
De Malherbe, E., “A simple probabilistic approach to the pricing of credit default swap covenants”, J. Risk 8 (2006) 85–113; doi:10.21314/JOR.2006.130 CrossRefGoogle Scholar
Elliott, R. J., Aggoun, L. and Moore, J. B., Hidden Markov models: estimation and control (Springer, New York, 2008).Google Scholar
Elliott, R. J. and Lian, G.-H., “Pricing variance and volatility swaps in a stochastic volatility model with regime switching: discrete observations case”, Quant. Finance 13 (2013) 687–698; doi:10.1080/14697688.2012.676208.CrossRefGoogle Scholar
Eraker, B., “Do stock prices and volatility jump? Reconciling evidence from spot and option prices”, J. Finance 59 (2004) 1367–1403; doi:10.1111/j.1540-6261.2004.00666.x.CrossRefGoogle Scholar
Guo, X., “Information and option pricings”, Quant. Finance 1 (2001) 38–44; doi:10.1080/713665550.CrossRefGoogle Scholar
Hamilton, J. D., “Analysis of time series subject to changes in regime”, J. Econometrics 45 (1990) 39–70; doi:10.1016/0304-4076(90)90093-9.CrossRefGoogle Scholar
He, X.-J. and Chen, W., “The pricing of credit default swaps under a generalized mixed fractional Brownian motion”, Phys. A Stat. Mech. Appl. 404 (2014) 26–33; doi:10.1016/j.physa.2014.02.046.CrossRefGoogle Scholar
He, X.-J. and Chen, W., “A Monte-Carlo based approach for pricing credit default swaps with regime switching”, Comput. Math. Appl. 76 (2018) 1758–1766; doi:10.1016/j.camwa.2018.07.027.CrossRefGoogle Scholar
He, X.-J. and Chen, W., “Pricing foreign exchange options under a hybrid Heston–Cox–Ingersoll–Ross model with regime switching”, IMA J. Manag. Math. (2021) dpab013; doi:10.1093/imaman/dpab013.CrossRefGoogle Scholar
Jarrow, R. A. and Turnbull, S. M., “Pricing derivatives on financial securities subject to credit risk”, J. Finance 50 (1995) 53–85; doi:10.1111/j.1540-6261.1995.tb05167.x.CrossRefGoogle Scholar
Lando, D., “On Cox processes and credit risky securities”, Rev. Deriv. Res. 2 (1998) 99–120; doi:10.1007/BF01531332.CrossRefGoogle Scholar
Lin, S. and He, X.-J., “Analytically pricing European options under a new two-factor Heston model with regime switching”, Comput. Econ. (2021) Lin2021; doi:10.1007/s10614-021-10117-6.CrossRefGoogle Scholar
Lin, S. and He, X.-J., “A closed-form pricing formula for forward start options under a regime-switching stochastic volatility model”, Chaos Soliton. Fract. 144 (2021) 110644; doi:10.1016/j.chaos.2020.110644.CrossRefGoogle Scholar
Lo, C.-F., Lee, H. C. and Hui, C.-H., “A simple approach for pricing barrier options with time-dependent parameters”, Quant. Finance 3 (2003) 98–107; doi:10.1088/1469-7688/3/2/304.CrossRefGoogle Scholar
Longstaff, F. and Schwartz, E., “Valuing credit derivatives”, Journal of Fixed Income 5 (1995) 6–12; doi:10.3905/jfi.1995.408138.CrossRefGoogle Scholar
Madan, D. B. and Unal, H., “Pricing the risks of default”, Rev. Deriv. Res. 2 (1998); doi:10.1007/BF01531333.CrossRefGoogle Scholar
Merton, R. C., “On the pricing of corporate debt: The risk structure of interest rates”, J. Finance 2 (1974) 449–470; doi:10.2307/2978814. Google Scholar
Peiro, A., “Skewness in financial returns”, J. Bank. Finance 23 (1999) 84–862; doi:10.1016/S0378-4266(98)00119-8. CrossRefGoogle Scholar
Tarashev, N. A., “An empirical evaluation of structural credit risk models”, BIS Working Paper, available at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=846304.Google Scholar
Zhou, C., “The term structure of credit spreads with jump risk”, J. Bank. Finance 25 (2001) 201–2040; doi:10.1016/S0378-4266(00)00168-0.CrossRefGoogle Scholar
Zhu, S.-P., Badran, A. and Lu, X., “A new exact solution for pricing European options in a two-state regime-switching economy”, Comput. Math. Appl. 64 (2012) 2744–2755; doi:10.1016/j.camwa.2012.08.005.CrossRefGoogle Scholar