Valuation of Credit Default Swap with Counterparty Default Risk by Structural Model

ABSTRACT

This paper provides a methodology for valuing a credit default swap (CDS) with considering a counterparty default risk. Using a structural framework, we study the correlation of the reference entity and the counterparty through the joint distribution of them. The default event discussed in our model is associated to whether the minimum value of the companies in stochastic processes has reached their thresholds (default barriers). The joint probability of minimums of correlated Brownian motions solves the backward Kolmogorov equation, which is a two dimensional partial differential equation. A closed pricing formula is obtained. Numerical methodology, parameter analysis and calculation examples are implemented.

This paper provides a methodology for valuing a credit default swap (CDS) with considering a counterparty default risk. Using a structural framework, we study the correlation of the reference entity and the counterparty through the joint distribution of them. The default event discussed in our model is associated to whether the minimum value of the companies in stochastic processes has reached their thresholds (default barriers). The joint probability of minimums of correlated Brownian motions solves the backward Kolmogorov equation, which is a two dimensional partial differential equation. A closed pricing formula is obtained. Numerical methodology, parameter analysis and calculation examples are implemented.

KEYWORDS

CDS Spread, Counterparty Default Risk, Structural Model, PDE Method, Monte Carlo Calculation

CDS Spread, Counterparty Default Risk, Structural Model, PDE Method, Monte Carlo Calculation

Cite this paper

nullJ. Liang, P. Zhou, Y. Zhou and J. Ma, "Valuation of Credit Default Swap with Counterparty Default Risk by Structural Model,"*Applied Mathematics*, Vol. 2 No. 1, 2011, pp. 106-117. doi: 10.4236/am.2011.21012.

nullJ. Liang, P. Zhou, Y. Zhou and J. Ma, "Valuation of Credit Default Swap with Counterparty Default Risk by Structural Model,"

References

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[3] F. Longstaff and E. Schwartz, “A Simple Approach to Valuing Risky Fixed and Floating Rate Debt,” Journal of Finance, Vol. 50, No. 3, 1995, pp. 789-819. doi:10.2307/ 2329288

[4] C. Zhou, “A Jump-Diffusion Approach to Modeling Credit Risk and Valuing Defaultable Securities,” Finance and Economics Discussion Series, Working Paper, Board of Governors of the Federal Reserve System, Washington DC, 1997.

[5] C. Zhou, “An Analysis of Default Correlation and Multiple Defaults,” Review of Finance Studies, Vol. 14, No. 2, 2001, pp. 555-576. doi:10.1093/rfs/14.2.555

[6] C. Zhou, “The Term Structure of Credit Spreads with Jump Risk,” Journal of Banking & Finance, Vol. 25, No. 11, 2001, pp. 2015-2040. doi:10.1016/S0378-4266(00)00 168-0

[7] D. Duffie and K. J. Singleton, “Modeling Term Structures of Defaultable Bonds,” Review of Financial Studies, Vol. 12, No. 4, 1999, pp. 687-720. doi:10.1093/rfs/12.4. 687

[8] D. Duffie and K. J. Singleton, “Credit Risk,” Princeton University Press, Princeton, 2003.

[9] D. Lando, “On Cox Processes and Credit Risky Securities,” Review of Derivatives Research, Vol. 2, No. 2-3, 1998, pp. 99-120. doi:10.1007/BF01531332

[10] J. Hull and A. White, “Valuing Credit Default Swaps I: No Counterparty Default Risk,” Journal of Derivatives, Vol. 8, No. 1, 2000, pp. 29-40. doi:10.3905/jod.2000. 319115

[11] J. Hull and A. White, “Valuing Credit Default Swaps II: Modeling Default Correlations,” Journal of Derivatives, Vol. 8, No. 3, 2001, pp. 12-22. doi:10.3905/jod.2001. 319153

[12] R. Jarrow and Y. Yildirim, “A Simple Model for Valuing Default Swaps when Both Market and Credit Risk are Correlated,” Journal of Fixed Income, Vol. 11, No. 4, 2002, pp. 7-19. doi:10.3905/jfi.2002.319308

[13] R. Jarrow and F. Yu, “Counterparty Risk and the Pricing of Defaultable Securities,” Journal of Finance, Vol. 56, No. 5, 2001, pp. 1765-1799. doi:10.1111/0022-1082.003 89

[14] F. Yu, “Correlated Defaults and the Valuation of Defaultable Securities,” Proceedings of 2nd International Conference on Credit Risk, Montréal, 15-16 April 2004, pp. 1-30.

[15] S. Y. Leung and Y. K. Kwok, “Credit Default Swap Valuation with Counterparty Risk,” The Kyoto Economic Review, Vol. 74, No. 1, 2005, pp. 25-45.

[16] R. Jarrow and S. M. Turnbull, “Pricing Derivatives on Financial Securities Subject to Credit Risk,” Journal of Finance, Vol. 50, No. 1, 1995, pp. 53-86. doi:10.2307/ 2329239

[17] J. Hull and A. White, “Valuation of a CDO and nth to Default CDS without Monte Carlo Simulation,” Journal of Derivatives, Vol. 12, No. 2, 2004, pp. 8-23. doi:10. 3905/jod.2004.450964

[18] J. Hull, M. Predescu and A. White, “The Relationship between Credit Default Swap Spreads, Bond Yields, and Credit Rating Announcements,” Journal of Banking & Finance, Vol. 28, No. 11, 2004, pp. 2789-2811. doi:10. 1016/j.jbankfin.2004.06.010

[19] J. Hull, M. Predescu and A. White, “The Valuation of Correlation-Dependent Credit Derivatives Using a Structural Model,” Journal of Credit Risk, Vol. 6, No. 3, 2010, pp. 99-132.

[20] M. Kijima and Y. Muromachi, “Credit Events and the Valuation of Credit Derivatives of Basket Type,” Review of Derivatives Research, Vol. 4, No. 1, 2000, pp. 55-79. doi:10.1023/A:1009676412322

[21] M. Kijima and Y. Muromachi, “Valuation of a Credit Swap of the Basket Type,” Review of Derivatives Research, Vol. 4, No. 1, 2000, pp. 81-97. doi:10.1023/A: 1009628513231

[22] M. Wise and V. Bhansali, “Correlated Random Walks and the Joint Survival Probability,” Working Paper, Caltech and PIMCO, pp. 1-13.

[23] P. Zhou and J. Liang, “Analysis of Credit Default Swap,” Applied Mathematics-JCU, Vol. 22, 2007, pp. 311-314.

[24] K. Giesecke and L. R. Goldberg, “The Market Price of Credit Risk,” Working Paper, Cornell University, New York, 2003, pp. 1-29.

[25] J. Hull, “Options, Futures and Other Derivatives,” 7th Edition, Prentice Education, Upper Saddle River, 2009.

[26] H. He, P. W. Keirstead and J. Rebholz, “Double Lookbacks,” Mathematical Finance, Vol. 8, No. 3, 1998, pp. 201-228. doi:10.1111/1467-9965.00053

[27] I. S. Gradshteyn and I. M. Ryzhik, “Table of Integrals, Series, and Products,” Academic Press, New York, 1980

[1] R. Merton, “On the Valuing of Corporate Debt: The Risk Structure of Interest Rates,” Journal of Finance, Vol. 29, No. 2, 1974, pp. 449-470. doi:10.2307/2978814

[2] F. Black and J. Cox, “Valuing Corporate Securities: Some Effects of Bond Indenture Provisions,” Journal of Finance, Vol. 31, No. 2, 1976, pp. 351-367. doi:10.2307/ 2326607

[3] F. Longstaff and E. Schwartz, “A Simple Approach to Valuing Risky Fixed and Floating Rate Debt,” Journal of Finance, Vol. 50, No. 3, 1995, pp. 789-819. doi:10.2307/ 2329288

[4] C. Zhou, “A Jump-Diffusion Approach to Modeling Credit Risk and Valuing Defaultable Securities,” Finance and Economics Discussion Series, Working Paper, Board of Governors of the Federal Reserve System, Washington DC, 1997.

[5] C. Zhou, “An Analysis of Default Correlation and Multiple Defaults,” Review of Finance Studies, Vol. 14, No. 2, 2001, pp. 555-576. doi:10.1093/rfs/14.2.555

[6] C. Zhou, “The Term Structure of Credit Spreads with Jump Risk,” Journal of Banking & Finance, Vol. 25, No. 11, 2001, pp. 2015-2040. doi:10.1016/S0378-4266(00)00 168-0

[7] D. Duffie and K. J. Singleton, “Modeling Term Structures of Defaultable Bonds,” Review of Financial Studies, Vol. 12, No. 4, 1999, pp. 687-720. doi:10.1093/rfs/12.4. 687

[8] D. Duffie and K. J. Singleton, “Credit Risk,” Princeton University Press, Princeton, 2003.

[9] D. Lando, “On Cox Processes and Credit Risky Securities,” Review of Derivatives Research, Vol. 2, No. 2-3, 1998, pp. 99-120. doi:10.1007/BF01531332

[10] J. Hull and A. White, “Valuing Credit Default Swaps I: No Counterparty Default Risk,” Journal of Derivatives, Vol. 8, No. 1, 2000, pp. 29-40. doi:10.3905/jod.2000. 319115

[11] J. Hull and A. White, “Valuing Credit Default Swaps II: Modeling Default Correlations,” Journal of Derivatives, Vol. 8, No. 3, 2001, pp. 12-22. doi:10.3905/jod.2001. 319153

[12] R. Jarrow and Y. Yildirim, “A Simple Model for Valuing Default Swaps when Both Market and Credit Risk are Correlated,” Journal of Fixed Income, Vol. 11, No. 4, 2002, pp. 7-19. doi:10.3905/jfi.2002.319308

[13] R. Jarrow and F. Yu, “Counterparty Risk and the Pricing of Defaultable Securities,” Journal of Finance, Vol. 56, No. 5, 2001, pp. 1765-1799. doi:10.1111/0022-1082.003 89

[14] F. Yu, “Correlated Defaults and the Valuation of Defaultable Securities,” Proceedings of 2nd International Conference on Credit Risk, Montréal, 15-16 April 2004, pp. 1-30.

[15] S. Y. Leung and Y. K. Kwok, “Credit Default Swap Valuation with Counterparty Risk,” The Kyoto Economic Review, Vol. 74, No. 1, 2005, pp. 25-45.

[16] R. Jarrow and S. M. Turnbull, “Pricing Derivatives on Financial Securities Subject to Credit Risk,” Journal of Finance, Vol. 50, No. 1, 1995, pp. 53-86. doi:10.2307/ 2329239

[17] J. Hull and A. White, “Valuation of a CDO and nth to Default CDS without Monte Carlo Simulation,” Journal of Derivatives, Vol. 12, No. 2, 2004, pp. 8-23. doi:10. 3905/jod.2004.450964

[18] J. Hull, M. Predescu and A. White, “The Relationship between Credit Default Swap Spreads, Bond Yields, and Credit Rating Announcements,” Journal of Banking & Finance, Vol. 28, No. 11, 2004, pp. 2789-2811. doi:10. 1016/j.jbankfin.2004.06.010

[19] J. Hull, M. Predescu and A. White, “The Valuation of Correlation-Dependent Credit Derivatives Using a Structural Model,” Journal of Credit Risk, Vol. 6, No. 3, 2010, pp. 99-132.

[20] M. Kijima and Y. Muromachi, “Credit Events and the Valuation of Credit Derivatives of Basket Type,” Review of Derivatives Research, Vol. 4, No. 1, 2000, pp. 55-79. doi:10.1023/A:1009676412322

[21] M. Kijima and Y. Muromachi, “Valuation of a Credit Swap of the Basket Type,” Review of Derivatives Research, Vol. 4, No. 1, 2000, pp. 81-97. doi:10.1023/A: 1009628513231

[22] M. Wise and V. Bhansali, “Correlated Random Walks and the Joint Survival Probability,” Working Paper, Caltech and PIMCO, pp. 1-13.

[23] P. Zhou and J. Liang, “Analysis of Credit Default Swap,” Applied Mathematics-JCU, Vol. 22, 2007, pp. 311-314.

[24] K. Giesecke and L. R. Goldberg, “The Market Price of Credit Risk,” Working Paper, Cornell University, New York, 2003, pp. 1-29.

[25] J. Hull, “Options, Futures and Other Derivatives,” 7th Edition, Prentice Education, Upper Saddle River, 2009.

[26] H. He, P. W. Keirstead and J. Rebholz, “Double Lookbacks,” Mathematical Finance, Vol. 8, No. 3, 1998, pp. 201-228. doi:10.1111/1467-9965.00053

[27] I. S. Gradshteyn and I. M. Ryzhik, “Table of Integrals, Series, and Products,” Academic Press, New York, 1980