Correlation Smile Matching for CDO Tranches with α Stable Distributions and Fitted Archimedan Copulas

Abstract

As an extension of the standard Gaussian copula model to price CDO tranche swaps we present a generalization of a one-factor copula model based on stable distributions. For special parameter values these distributions coincide with Gaussian or Cauchy distributions, but changing the parameters allows a continuous deformation away from the Gaussian copula. All these factor copulas are embedded into a framework of stochastic correlations. We furthermore generalize the linear dependency in the usual factor approach to a more general Archimedean copula dependency between the individual trigger variable and the common latent factor. Our analysis is carried out on a non-homogeneous correlation structure of the underlying portfolio. CDO tranche market premia, even through the correlation crisis in May 2005, can be reproduced by certain models. From a numerical perspective all these models are simple since calculations can be reduced to one dimensional numerical integrals.