A fully non-linear PDE problem from pricing CDS with counterparty risk

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September 2012, 17(6): 2001-2016. doi: 10.3934/dcdsb.2012.17.2001

A fully non-linear PDE problem from pricing CDS with counterparty risk

Bei Hu ^1, , Lishang Jiang ^2, , Jin Liang ^2, and Wei Wei ^2,

1.	Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556
2.	Department of Mathematics, Tongji University, Shanghai 200092, China, China, China

Received July 2011 Revised September 2011 Published May 2012

Abstract
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In this study, we establish a financial credit derivative pricing model for a contract which is subject to counterparty risks. The model leads to a fully nonlinear partial differential equation problem. We study this PDE problem and obtained a solution as the limit of a sequence of semi-linear PDE problems which also arise from financial models. Moreover, the problems and methods build a bridge between two main risk frameworks: structure and intensity models. We obtain the uniqueness, regularities and some properties of the solution of this problem.

Keywords: structure model, fully non-linear PDE., intensity model, Credit derivative pricing, counterparty risk.

Mathematics Subject Classification: 35M99, 91G8.

Citation: Bei Hu, Lishang Jiang, Jin Liang, Wei Wei. A fully non-linear PDE problem from pricing CDS with counterparty risk. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2001-2016. doi: 10.3934/dcdsb.2012.17.2001

References:

[1]	F. Black and J. Cox, Valuing corporate securities: Some effects of bond indenture provisions, Journal of Finance, 31 (1976), 351-367. doi: 10.1111/j.1540-6261.1976.tb01891.x.
[2]	T. Bielecki and M. Rutkowski, "Credit Risk: Modeling, Valuation and Hedging," Springer Finance, Springer-Verlag, Berlin, 2002.
[3]	S. Crepey, M. Jeanblanc and B. Zargari, Counterparty risk on a CDS in a Markov Chain Copula model with joint defaults, working paper, 2009.
[4]	J. Cox, J. Ingersoll and S. Ross, A Theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407. doi: 10.2307/1911242.
[5]	D. Duffie and K. J. Singleton, Modeling term structures of defaultable bonds, Review of Financial Studies, 12 (1999), 687-720. doi: 10.1093/rfs/12.4.687.
[6]	B. Øksendal, "Stochastic Differential Equations. An Introduction with Applications," fifth edition, Universitext, Springer-Verlag, Berlin, 1998.
[7]	A. Friedman, "Variational Principles and Free Boundary Problems," Second edition, Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1988.
[8]	D. Lando, On Cox processes and credit risky securities, Review of Derivatives Research, 2 (1998), 99-120.
[9]	F. Longstaff and E. Schwartz, A simple approach to valuing risky fixed and floating rate debt, Journal of Finance, 50 (1995), 789-819. doi: 10.2307/2329288.
[10]	R. Merton, On the valuing of corporate debt: The risk structure of interest rates, Journal of Finance, 29 (1974), 449-470. doi: 10.1111/j.1540-6261.1974.tb03058.x.

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References:

[1]	F. Black and J. Cox, Valuing corporate securities: Some effects of bond indenture provisions, Journal of Finance, 31 (1976), 351-367. doi: 10.1111/j.1540-6261.1976.tb01891.x.
[2]	T. Bielecki and M. Rutkowski, "Credit Risk: Modeling, Valuation and Hedging," Springer Finance, Springer-Verlag, Berlin, 2002.
[3]	S. Crepey, M. Jeanblanc and B. Zargari, Counterparty risk on a CDS in a Markov Chain Copula model with joint defaults, working paper, 2009.
[4]	J. Cox, J. Ingersoll and S. Ross, A Theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407. doi: 10.2307/1911242.
[5]	D. Duffie and K. J. Singleton, Modeling term structures of defaultable bonds, Review of Financial Studies, 12 (1999), 687-720. doi: 10.1093/rfs/12.4.687.
[6]	B. Øksendal, "Stochastic Differential Equations. An Introduction with Applications," fifth edition, Universitext, Springer-Verlag, Berlin, 1998.
[7]	A. Friedman, "Variational Principles and Free Boundary Problems," Second edition, Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1988.
[8]	D. Lando, On Cox processes and credit risky securities, Review of Derivatives Research, 2 (1998), 99-120.
[9]	F. Longstaff and E. Schwartz, A simple approach to valuing risky fixed and floating rate debt, Journal of Finance, 50 (1995), 789-819. doi: 10.2307/2329288.
[10]	R. Merton, On the valuing of corporate debt: The risk structure of interest rates, Journal of Finance, 29 (1974), 449-470. doi: 10.1111/j.1540-6261.1974.tb03058.x.

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