Sébastien Hitier, BNP Paribas

Title: Default Copula implied by Dynamic Credit Models

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 
diffusion 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 diffusion 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 different classes of CDO models that 
have been put forward for their tractability, to see what copula is implied by 
given a 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 diffusion 
models.