Unifying discrete structural models and reduced-form models in credit risk using a jump-diffusion process. (English) Zbl 1103.91356

Summary: R. C. Merton [ J. Finance 29, 449 (1974)] pioneered the structural model using a diffusion process to model the firm value evolution. Since a sudden drop of firm value is impossible, Jones et al. [J. Finance 39, 611 (1984)] argue that the short-term yield spread and the default probability are too small. Zhou [A jump-diffusion approach to modelling credit risk and valuing defaultable securities, Federal Reserve Board, Washington (1997)] uses a jump-diffusion process that is originally proposed by R. C. Merton [J. Financial Econ. 3, 125–144 (1976)] to model the firm value process. However, a method for finding the jump distribution is not developed. In a reduced-form model, the default probability (or intensity of default) and the mean recovery rate are obtained from the market spread by using model-specific assumptions. However, the capital structure that triggers the default usually is not used. In this paper, we propose methods to remove the discrepancy of yield spreads between structural models and reduced-form models and unify these two models. We first show the equivalence of yield spreads between structural models and reduced-form models and then find the implied jump distribution based on the market spread. The mean recovery rate for multiple seniorities and the mean recovery rate are thus obtained.


91B30 Risk theory, insurance (MSC2010)
91B28 Finance etc. (MSC2000)
60J75 Jump processes (MSC2010)
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[1] Artzner, P., Delbaen, 1995. Default risk insurance and incomplete markets. Mathematical Finance 5, 187-195. · Zbl 0866.90047
[2] Black, F.; Scholes, M., The pricing of options and corporate liabilities, Journal of political economy, 81, 637-654, (1973) · Zbl 1092.91524
[3] Carty, L.V., Lieberman, D., 1996. Corporate bond defaults and default rates 1938-1995. Moody’s Investors Service, Global Credit Research.
[4] Duffie, D., Singleton, K.J., 1995. Modelling term structures of defaultable bonds. Working Paper. Stanford University (revised as Review of Financial Studies 12 (1999) 687-720).
[5] Jarrow, R.A.; Turnbull, S.M., Pricing derivatives on financial securities subject to credit risk, The journal of finance, 50, 53-85, (1995)
[6] Jarrow, R.A.; Lando, D.; Turnbull, S.M., A Markov model for the term structure of credit risk spreads, The review of financial studies, 10, 481-523, (1997)
[7] Jones, E.P.; Mason, S.P.; Rosenfeld, E., Contingent claims analysis of corporate capital structures: an empirical investigation, Journal of finance, 39, 611-627, (1984)
[8] Lando, D., 1997. Modelling bonds and derivatives with default risk. In: Dempster, M., Pliska, S. (Eds.), Mathematics of Derivative Securities. Cambridge University Press, Cambridge, pp. 369-393. · Zbl 0912.90053
[9] Li, D., 1998. Constructing a credit curve. Credit Risk: Risk Special Report, November, pp. 40-44.
[10] Madan, D.B., Unal, H., 1996. Pricing the risks of default. Working Paper. · Zbl 1274.91426
[11] Merton, R.C., On the pricing of corporate debt: the risk structure of interest rates, The journal of finance, 29, 449-470, (1974)
[12] Merton, R.C., Option pricing when underlying stock returns are discontinuous, Journal of financial economics, 3, 125-144, (1976) · Zbl 1131.91344
[13] Zhou, C., 1997. A Jump-diffusion Approach to Modelling Credit Risk and Valuing Defaultable Securities. Federal Reserve Board, Washington.
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