CDO Pricing: Copula Implied by Risk Neutral Dynamics

Abstract

When dealing with multi-issuer credit derivatives such as CDO, it is customary to refer the reader to either of two approaches: “static models” which focus on the copula between the variables of interest, and “dynamic models” where the diusion of the underlying variables is described directly. While the former is widely used due to its simplicity, it is not clear that there is a well behaved dynamic model consistent with a given static approach. For this reason, it is often argued that an understanding of the dynamics used in model for CDO is required to bring it to par with derivative models used for other asset classes, such as the risk neutral diusion models used for equity, currency and commodity options derived from Black and Scholes, or the characterization of arbitrage free term structure of interest rates obtained by HJM. Clearly, a “dynamic model” implies a certain copula between the random variables of interest. The goal of this article is to develop a unified view compatible with both approaches, and reach a better understanding of the properties that a good “dynamic model” used for pricing and hedging would have when seen as a static model. We focus on credit models where large homogeneous pool portfolio are mathematically possible, a common assumption among practitioners. In a general credit term structure dynamics framework similar to HJM, we identify a “systemic loss” process linked to the survival dynamics that allows to identify the density of loss for large portfolios, and to explicit the default copula between the issuers. We then apply these results to dierent classes of CDO models that have been put forward for their tractability, to see what copula with implied by given a popular dynamic model, and what dynamic models could give rise to some popular copula model. The three classes we review are the one factor copula models, the markovian loss intensity models, and the systemic intensity jump diusion models.