We consider a rating-based model for the term structure of credit risk spreads wherein the credit-worthiness of the issuer is represented as a finite-state continuous time Markov process. This approach entails a progressive drift in credit quality towards default. A model of the economy is presented featuring stochastic transition probabilities; credit instruments are valued via an ultra parabolic Hamilton-Jacobi system of equations discretized utilizing the method-of-lines finite difference method. Computations for a callable bond are presented demonstrating the efficiency of the method.
Cite this paper
|||Bielicki, T. and Rutkowski, M. (2002) Credit Risk: Modeling, Valuation and Hedging. Springer-Verlag, Berlin.|
|||Lando, D. (1998) On Cox Processes and Credit Risky Securities. Review of Derivative Research, 2, 99-120.
|||Duffie, D. and Singleton, K. (1999) Modeling Term Structures of Defaultable Bonds. Review of Financial Studies, 12, 687-720.
|||Longstaff, F. and Schwartz, E. (1995) Valuing Credit Derivatives. The Journal of Fixed Income, 5, 6-12.
|||Jobst, N. and Zenios, S.A. (2005) On the Simulation of Interest Rate and Credit Risk Sensitive Securities. European Journal of Operational Research, 161, 298-324.
|||Jarrow, R., Lando, D. and Turnbull, S. (1997) A Markov Model for the Term Structure of Credit Risk Spreads. Review of Financial Studies, 10, 481-523.
|||Das, S. and Tufano, P. (1996) Pricing Credit Sensitive Debt When Interest Rates, Credit Ratings and Credit Spreads Are Stochastic. Journal of Financial Engineering, 5, 161-198.|
|||Arvantis, A., Gregory, J. and Laurent, J.-P. (1999) Building Models for Credit Spreads. The Journal of Derivatives, 6, 27-43.|
|||Elliott, R.J. and Mamon R.S. (2002) An Interest Rate Model with a Markovian Mean Reverting Level. Quantitative Finance, 2, 454-458.
|||Wu, S. and Zeng, Y. (2005) A General Equilibrium Model of the Term Structure of Interest Rates under Regime-Switching Risk. International Journal of Theoretical and Applied Finance, 8, 1-31.|
|||Harrison, J.M. and Pliska, S. (1981) Martingales and Stochastic Integrals in the Theory of Continuous Trading. Stochastic Processes and Their Applications, 11, 215-260.
|||Jarrow, R. and Turnbull, S. (1995) Pricing Derivatives on Financial Securities Subject to Credit Risk. Journal of Finance, 50, 53-85.
|||Karlin, S. and Taylor, H. (1975) A First Course in Stochastic Processes. Academic Press, New York.|
|||Elliott, R.J., Aggoun, L. and Moore, J.B. (1994) Hidden Markov Models: Estimation and Control. Springer-Verlag, New York.|
|||Brémaud, P. (1998) Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues. Springer-Verlag, New York.|
|||Marcozzi, M.D. (2001) On the Approximation of Optimal Stopping Problems with Application to Financial Mathematics. SIAM Journal on Scientific Computing, 22, 1865-1884.
|||Marcozzi, M.D. (2015) Optimal Control of Ultradiffusion Processes with Application to Mathematical Finance. International Journal of Computer Mathematics, 92, 296-318.