Unifying discrete structural models and reduced-form models in credit risk using a jump-diffusion process

Merton [The Journal of Finance 29 (1974) 449] 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. [Journal of Finance 39 (1984) 611] 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 Merton [Journal of Financial Economics 3 (1976) 125] 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.