Modeling Default Dependence with Threshold Models

Abstract

Default risk is one of the most important and fastest growing areas in derivatives. Single-name credit default swaps are now well established, and markets for contracts based on the default experience of a credit portfolio, like collateralized debt obligations (CDOs) and basket default swaps, are developing rapidly. Portfolio credit risk requires default correlation as a critical input. But defaults are rare, so default correlation must be estimated in another way, typically from correlations in stock returns. In a „structural” credit risk model, a credit event occurs when the net asset value of the firm falls below a critical threshold. Transitions among credit rating classes can be modeled with a set of thresholds corresponding to the different ratings. The widely used „normal copula” model treats the default probability for a single issuer, and default correlations among pairs of issuers, as arising from exposure to a set of common stochastic risk factors plus an idiosyncratic term. The resulting models are intuitive, but calibration to (implied) correlations embedded in market CDO prices can be difficult, especially as the number of factors increases. In this article, Overbeck and Schmidt offer an alternative structural approach that produces similar values to the normal copula model, with considerably less difficulty. Their trick is to alter the time scales of the Wiener processes driving the underlying asset values, which is equivalent to allowing time-varying volatilities, and permits a much easier calibration of the correlations.