A Simple Model for Valuing Default Swaps when both Market and Credit Risk are Correlated

This paper provides a simple analytic formula for valuing default swaps with correlated market and credit risk. We illustrate the numerical implementation of this model by inferring the default probability parameters implicit in default swap quotes for twenty two companies over the time period 8/21/00 to 10/31/00. For comparison, with also provide implicit estimates for the standard model (a special case of our approach) where market and credit risk are statistically independent. A Simple Model for Valuing Default Swaps when both Market and Credit Risk are Correlated A recent Risk magazine [2000] survey showed that the market for default swaps, as measured by the notional amounts of contracts traded per year, has grown from about 50 billion in 1998 to over 400 billion dollars in 2000. This exponential growth has generated significant interest in the fair valuation of default swaps by both the academic and practitioner communities. The existing literature investigating the valuation of default swaps (see Hull and White [2000, 2001], Martin, Thompson and Browne [2000], Wei [2001], and the survey paper by Cheng [2001]) gives the impression that simple models for pricing default swaps are only available when credit and market risk are statistically independent. Indeed, it is commonly believed that models incorporating correlated market and credit risk are quite complex, necessarily requiring burdensome recursive numerical procedures. For example, from Hull and White [2000, p. 30] “Like most other approaches, ours assumes that default probabilities, interest rates, and recovery rates are mutually independent. Unfortunately, it does not seem to be possible to relax these assumptions without a considerably more complex model.” In contradiction with this commonly held belief, this paper provides a simple analytic formula for the valuation of default swaps when market and credit risk are correlated. This analytic formula is easy to understand and to compute. This formula is derived in the context of the reduced-form credit risk model from Jarrow [2001] where correlated defaults arise due to the fact that a firm’s default intensities depend on common macro-factors. The common macro-factor used in our paper is the spot rate of interest. The spot rate of interest is assumed to follow an extended Vasicek model in the HJM framework. We illustrate the numerical implementation of this model by inferring the default probability parameters implicit in the term structure of default swap quotes for twenty two different companies over the time period 8/21/00 to 10/31/00. The data used for this investigation was downloaded from Enron’s web site. The twenty-two different firms were chosen to stratify various industry groupings: financial, food and beverages,