Review of Derivatives Research

, Volume 2, Issue 2–3, pp 99–120 | Cite as

On cox processes and credit risky securities

  • David Lando


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 Jarrow, Lando and Turnbull (1997) 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.


credit risk Cox process credit derivatives ratings 


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  1. Artzner, P. and F. Delbaen. (1995). “Default Risk Insurance and Incomplete Markets,”Mathematical Finance 5, 187–195.Google Scholar
  2. Black, F. and M. Scholes. (1973). “The Pricing of Options and Corporate Liabilities,”Journal of Political Economy 3, 637–654.Google Scholar
  3. Cooper, I. and M. Martin. (1996). “Default Risk and Derivative Products,”Applied Mathematical Finance 3, 53–74.Google Scholar
  4. Duffee, G. (1996). “Treasury Yields and Corporate Bond Yields Spreads: An Empirical Analysis,” Working Paper, Federal Reserve Board, Washington DC.Google Scholar
  5. Duffie, D. (1992).Dynamic Asset Pricing Theory. Princeton: Princeton University Press.Google Scholar
  6. Duffie, D. and M. Huang. (1996). “Swap Rates and Credit Quality,”Journal of Finance 51(3), 921–949.Google Scholar
  7. Duffie, D. and K. Singleton. (1996). “Modeling Term Structures of Defaultable Bonds,” Working Paper, Stanford University.Google Scholar
  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.Google Scholar
  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.Google Scholar
  10. Fons, J. and A. Kimball. (1991). “Corporate Bond Defaults and Default Rates 1970–1990,”The Journal of Fixed Income, 36–47.Google Scholar
  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.Google Scholar
  12. Grandell, J. (1976). “Doubly Stochastic Poisson Processes.” Volume 529 ofLecture Notes in Mathematics, New York: Springer.Google Scholar
  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.Google Scholar
  14. Jarrow, R. and S. Turnbull. (1995). “Pricing Options on Financial Securities Subject to Credit Risk,”Journal of Finance 50, 53–85.Google Scholar
  15. Karatzas, I. and S. Shreve. (1988).Brownian Motion and Stochastic Calculus. New York: Springer.Google Scholar
  16. Lando, D. (1994). “Three Essays on Contingent Claims Pricing,” PhD Dissertation, Cornell University.Google Scholar
  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.Google Scholar
  18. Longstaff, F. and E. Schwartz. (1995). “A Simple Approach to Valuing Risky Fixed and Floating Rate Debt,”Journal of Finance 50, 789–819.Google Scholar
  19. Madan, D. and H. Unal. (1995). “Pricing the Risks of Default,” Working Paper, University of Maryland.Google Scholar
  20. Merton, R. C. (1974). “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates,”Journal of Finance 2, 449–470.Google Scholar
  21. Williams, D. (1991).Probability with Martingales. Cambridge University Press.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • David Lando
    • 1
  1. 1.Department of Operations ResearchUniversity of CopenhagenCopenhagen ØDenmark

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