Copula-Dependent Defaults in Intensity Models

TLDR

The paper proposes a new method to embed dynamic default dependence into intensity‑based default risk models. The authors construct a copula‑dependent intensity model that couples an arbitrary copula of default times with individual intensity processes, deriving survival and spread dynamics that exhibit contagion‑induced jumps, and demonstrate the framework for Archimedean copulae and its calibration to CreditMetrics‑style Gaussian dependence. For Clayton copula the spread jumps are proportional to pre‑default intensity, and omitting other obligors collapses the model to a single‑obligor intensity model, while the framework also allows calibration of default correlation to a CreditMetrics‑style Gaussian structure.

Abstract

In this paper we present a new approach to incorporate dynamic default dependency in intensity-based default risk models. The model uses an arbitrary default dependency structure which is specified by the Copula of the times of default, this is combined with individual intensity-based models for the defaults of the obligors without loss of the calibration of the individual default-intensity models. The dynamics of the survival probabilities and credit spreads of individual obligors are derived and it is shown that in situations with positive dependence, the default of one obligor causes the credit spreads of the other obligors to jump upwards, as it is experienced empirically in situations with credit contagion. For the Clayton copula these jumps are proportional to the pre-default intensity. If information about other obligors is excluded, the model reduces to a standard intensity model for a single obligor, thus greatly facilitating its calibration. To illustrate the results they are also presented for Archimedean copulae in general, and Gumbel and Clayton copulae in particular. Furthermore it is shown how the default correlation can be calibrated to a Gaussian dependency structure of CreditMetrics-type.