Contagion Models in Credit Risk

This chapter aims to give an account of mathematical techniques for credit risk models where there is contagion between the obligors, i.e. default of one party either directly causes default of other parties or (more commonly) changes other parties’ risk of default. While various approaches are possible, the treatment here concentrates on ‘reduced-form’ models based on Markov chains. We argue that such models provide a flexible and computationally efficient framework. Subsidiary but important themes of the chapter are the role of information (i.e., whether various factors influencing default risk are observable or ‘latent’), and changes of measure, either from ‘physical’ to ‘risk neutral’ and vice versa or, in the context of econometric studies, computation of likelihood functions for parameter estimation. We start in Section 2 with a general discussion of joint distributions and copulas, mainly to point out that ‘contagion’ is in some sense already built into the copula concept. Section 3 gives a general formulation of the reduced-form model and a taxonomy of models distinguishing between factor, frailty and contagion models. Section 4 gives some background information about Markov processes, Markov chains and phase-type distributions as required for the subsequent sections. We then discuss, in Section 5, four simple but effective Markov chain-based models with applications in counterparty risk and credit risk for inhomogeneous and homogeneous portfolios1. The following two sections, §6 and §7 develop the ‘subsidiary themes’ mentioned above, before we return in Section 8 to further development of the Enhanced Risk homogeneous portfolio model, introduced in Section 5.4, in the light of these these themes. Finally, following the concluding Section 9, Appendix A summarizes information about Piecewise-Deterministic Markov Processes, which play an essential role in our discussion.