Quadratic Transform Approximation for CDO Pricing in Multifactor Models

The multifactor version of copula models has the ability to generate complex correlation structure among defaults that is useful in fitting the base correlation skew. However, multifactor models have often been dismissed for their intractability. Even the semianalytical approach using Laplace transforms is computationally challenging, because although the model is tractable upon conditioning on the factors, unconditioning usually requires high efforts of integrating out the factors. To circumvent this problem, this paper develops a fast, closed-form approximation to the Laplace transform in multifactor models. The method, which approximates the conditional transform in a way that lends itself to closed-form unconditioning in arbitrarily high dimensions, is applicable to a range of models with Gaussian factors, including models that extend the standard Gaussian copula to allow stochastic recovery rates and factor loadings. We analyze the accuracy and convergence properties of the approximation. Numerical examples illustrate the speed and accuracy of the method.