Implied Correlations: Smiles or Smirks?

Abstract

Developing a pricing model for CDOs that can actually be implemented is a very challenging problem. In a credit portfolio with N obligors, there are N individual default probabilities, (N2 - N)/2 correlations, and N recovery rates, all of which are important determinants of the portfolio loss distribution. Fitting the default probabilities alone typically requires specifying probability distributions for the shared and idiosyncratic underlying factors. In light of all this complexity, the Gaussian copula model applied to a homogeneous portfolio has become the industry standard approach. But the shortcomings of this model in explaining market prices for CDO tranches are very visible in the strange, but consistent, v-shaped implied correlation skew. Agca, Agrawal, and Islam consider the many simplifying assumptions of the Gaussian copula model and explore the importance of each by weakening one at a time and studying how the correlation skew changes. Does allowing the common factor or the idiosyncratic factors to come from a fat-tailed distribution, rather than the normal, make a big difference? For the shared factor, yes, but for the idiosyncratic factor, no. How important is the assumption that all correlations are equal versus being distributed over a wide range as is the case when the correlations are estimated from the data? Neither seems to make much difference to the implied correlation skew. In the end, we learn which assumptions make the most difference in CDO pricing—even though the true source of the correlation skew remains elusive.