zbMATH — the first resource for mathematics

On Cox processes and credit risky securities. (English) Zbl 1274.91459
Summary: A framework is presented for modeling defaultable securities and credit derivatives which allows for dependence between market risk factors and credit risk. The framework reduces the technical issues of modeling credit risk to the same issues faced when modeling the ordinary term structure of interest rates. It is shown how to generalize a model of R. Jarrow, D. Lando and S. Turnbull [“A Markov model for the term structure of credit risk spreads”, Rev. Fin. Stud. 10, No. 2, 481–523 (1997; doi:10.1093/rfs/10.2.481)] to allow for stochastic transition intensities between rating categories and into default. This generalization can handle contracts with payments explicitly linked to ratings. It is also shown how to obtain a term structure model for all different rating categories simultaneously and how to obtain an affine-like structure. An implementation is given in a simple one factor model in which the affine structure gives closed form solutions.

91G40 Credit risk
91G20 Derivative securities (option pricing, hedging, etc.)
Full Text: DOI
[1] Artzner, P. and F. Delbaen. (1995). ?Default Risk Insurance and Incomplete Markets,?Mathematical Finance 5, 187?195. · Zbl 0866.90047
[2] Black, F. and M. Scholes. (1973). ?The Pricing of Options and Corporate Liabilities,?Journal of Political Economy 3, 637?654. · Zbl 1092.91524
[3] Cooper, I. and M. Martin. (1996). ?Default Risk and Derivative Products,?Applied Mathematical Finance 3, 53?74. · Zbl 1097.91531
[4] Duffee, G. (1996). ?Treasury Yields and Corporate Bond Yields Spreads: An Empirical Analysis,? Working Paper, Federal Reserve Board, Washington DC.
[5] Duffie, D. (1992).Dynamic Asset Pricing Theory. Princeton: Princeton University Press. · Zbl 1140.91041
[6] Duffie, D. and M. Huang. (1996). ?Swap Rates and Credit Quality,?Journal of Finance 51(3), 921?949.
[7] Duffie, D. and K. Singleton. (1996). ?Modeling Term Structures of Defaultable Bonds,? Working Paper, Stanford University.
[8] Duffie, D. and K. Singleton. (1997). ?An Econometric Model of the Term Structure of Interest Rate Swap Yields,?Journal of Finance 52(4), 1287?1321.
[9] Duffie, D., M. Schroder, and C. Skiadas. (1996). ?Recursive Valuation of Defaultable Securities and the Timing of Resolution of Uncertainty,?The Annals of Applied Probability 6(4), 1075?1090. · Zbl 0868.90008
[10] Fons, J. and A. Kimball. (1991). ?Corporate Bond Defaults and Default Rates 1970?1990,?The Journal of Fixed Income, 36?47.
[11] Gill, R. and S. Johansen. (1990). ?A Survey of Product-Integration with a View Towards Applications in Survival Analysis,?The Annals of Statistics 18(4), 1501?1555. · Zbl 0718.60087
[12] Grandell, J. (1976). ?Doubly Stochastic Poisson Processes.? Volume 529 ofLecture Notes in Mathematics, New York: Springer. · Zbl 0339.60053
[13] Jarrow, R., D. Lando, and S. Turnbull. (1997). ?A Markov Model for the Term Structure of Credit Risk Spreads,?Review of Financial Studies 10(2), 481?523.
[14] Jarrow, R. and S. Turnbull. (1995). ?Pricing Options on Financial Securities Subject to Credit Risk,?Journal of Finance 50, 53?85.
[15] Karatzas, I. and S. Shreve. (1988).Brownian Motion and Stochastic Calculus. New York: Springer. · Zbl 0638.60065
[16] Lando, D. (1994). ?Three Essays on Contingent Claims Pricing,? PhD Dissertation, Cornell University.
[17] Lando, D. (1997). ?Modelling Bonds and Derivatives with Credit Risk.? In M. Dempster and S. Pliska (eds.),Mathematics of Financial Derivativcs, 369?393. Cambridge University Press.
[18] Longstaff, F. and E. Schwartz. (1995). ?A Simple Approach to Valuing Risky Fixed and Floating Rate Debt,?Journal of Finance 50, 789?819.
[19] Madan, D. and H. Unal. (1995). ?Pricing the Risks of Default,? Working Paper, University of Maryland. · Zbl 1274.91426
[20] Merton, R. C. (1974). ?On the Pricing of Corporate Debt: The Risk Structure of Interest Rates,?Journal of Finance 2, 449?470.
[21] Williams, D. (1991).Probability with Martingales. Cambridge University Press. · Zbl 0722.60001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.