A Jump-Diffusion Approach to Modeling Credit Risk and Valuing Defaultable Securities

by Chunsheng Zhou of the Federal Reserve Board

March 1997

Abstract: Since Black and Scholes (1973) and Merton (1974), structural models of credit risk have relied almost exclusively on diffusion processes to model the evolution of firm value.  While a diffusion approach is convenient, in empirical application, it has produced very disappointing results.  Jones, Mason, and Rosenfeld (1984) find that the credit spreads on corporate bonds are too high to be matched by the diffusion approach.  Also, because the instantaneous default probability of a healthy firm is zero under a continuous process, the diffusion approach predicts that the term structure of credit spreads should always start at zero and slope upward for firms that are not currently in financial distress, but the empirical literature shows that the actual credit spread curves are sometimes at or even downward-sloping.

If a diffusion approach cannot capture the basic features of credit risk, what approach can?  This paper develops a new structural approach to valuing default-risky securities by modeling the evolution of firm value as a jump-diffusion process.  Under a jump-diffusion process, a firm can default instantaneously because of a sudden drop in its value. With this characteristic, a jump-diffusion model can match the size of credit spreads on corporate bonds and can generate various shapes of yield spread curves and marginal default rate curves, including upward-sloping, downward-sloping, at, and hump-shaped, even if the firm is currently in good financial standing.  The model also links recovery rates to firm value at default in a natural way so that variation in recovery rates is endogenously generated in the model.  The model is also consistent with many other stylized empirical facts in the credit-risk literature.

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