CDO Pricing: Copula Implied by Risk Neutral Dynamics

When dealing with multi-issuer credit derivatives such as CDO, it is customary to referthe reader to either of two approaches: “static models” which focus on the copula between thevariables of interest, and “dynamic models” where the diffusion of the underlying variablesis described directly. While the former is widely used due to its simplicity, it is not clearthat there is a well behaved dynamic model consistent with a given static approach. For thisreason, it is often argued that an understanding of the dynamics used in model for CDO isrequired to bring it to par with derivative models used for other asset classes, such as therisk neutral diffusion models used for equity, currency and commodity options derived fromBlack and Scholes, or the characterization of arbitrage free term structure of interest ratesobtained by HJM.Clearly, a “dynamic model” implies a certain copula between the random variables ofinterest. 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 forpricing and hedging would have when seen as a static model.We focus on credit models where large homogeneous pool portfolio are mathematicallypossible, a common assumption among practitioners. In a general credit term structuredynamics framework similar to HJM, we identify a “systemic loss” process linked to thesurvival dynamics that allows to identify the density of loss for large portfolios, and toexplicit the default copula between the issuers. We then apply these results to differentclasses of CDO models that have been put forward for their tractability, to see what copulawith implied by given a popular dynamic model, and what dynamic models could give rise tosome popular copula model. The three classes we review are the one factor copula models,the markovian loss intensity models, and the systemic intensity jump diffusion models.