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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

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 & 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.  Google Scholar

[2]

T. Bielecki and M. Rutkowski, "Credit Risk: Modeling, Valuation and Hedging," Springer Finance, Springer-Verlag, Berlin, 2002.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

B. Øksendal, "Stochastic Differential Equations. An Introduction with Applications," fifth edition, Universitext, Springer-Verlag, Berlin, 1998.  Google Scholar

[7]

A. Friedman, "Variational Principles and Free Boundary Problems," Second edition, Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1988.  Google Scholar

[8]

D. Lando, On Cox processes and credit risky securities, Review of Derivatives Research, 2 (1998), 99-120. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

T. Bielecki and M. Rutkowski, "Credit Risk: Modeling, Valuation and Hedging," Springer Finance, Springer-Verlag, Berlin, 2002.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

B. Øksendal, "Stochastic Differential Equations. An Introduction with Applications," fifth edition, Universitext, Springer-Verlag, Berlin, 1998.  Google Scholar

[7]

A. Friedman, "Variational Principles and Free Boundary Problems," Second edition, Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1988.  Google Scholar

[8]

D. Lando, On Cox processes and credit risky securities, Review of Derivatives Research, 2 (1998), 99-120. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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