Valuing Credit Derivatives Using an Implied Copula Approach

Abstract

Credit derivatives are among the most important new financial instruments, but also among the most complicated. Each individual issuer is continuously exposed to default risk, and default intensity looking forward is not constant. It typically has a term structure, as revealed in the CDS market. A portfolio of risky bonds, as in a CDO, aggregates the individual risks, and now the correlations among them also become important. CDO tranches then redistribute and split up this aggregate exposure among a set of new securities. Evaluating the resulting tranche exposures requires a model for the individual default risks and their correlations, but even in the industry-standard Gaussian copula model, the problem is computationally intractable without heroic simplifying assumptions. The plainest vanilla model assumes correlations are equal for all pairs of credits. Then, analogous to the way an implied volatility can be extracted from an option's market price, the implied correlation can be extracted from a CDO tranche price. But, as with implied volatility, the resulting tranche correlations differ widely for different tranches, leading to the use of „base correlation,” a different implied correlation concept. Base correlation is still inconsistent with the model it is derived from, but it is not quite as badly behaved as tranche correlations. In this article, Hull and White offer an alternative approach that considerably reduces the inconsistencies in calibrating a copula to a set of CDO tranche prices. The secret is to make default intensities and recovery rates stochastic, rather than requiring a single value. By imposing the restrictions that the single-name CDS and the CDO tranches must all be priced by the model just as they are in the market, and that the probabilities for the set of possible individual default intensities must sum to 1 and exhibit maximum smoothness, Hull and White are able to imply tranche correlations that are much better behaved than the standard approach. The last part of the article extends their procedure in a number of directions, to nonstandard attachment points, bespoke portfolios, and CDO-squared securities.