| We develop numerical methods for measuring credit risks and pricing credit basket derivatives, such as the increasingly popular collateralized debt obligations (CDOs), which offer a way to create new classes of securities based on multi-name portfolios. A key issue in the modelling of credit portfolios is how to capture the dependence among the defaults of obligors in the portfolios. In practice, the correlation among obligors are often modelled by factor structures, with the single-factor structure being the most convenient, albeit unrealistic and too simplistic to explain the skew in the implied correlation curves. Once we venture into the more realistic case of multifactor models, the difficulty becomes the computation of model outputs, as traditional methods often require computing time that is exponential in the number of factors. The purpose of this work is to develop pricing methods that are non-simulation based, and are less sensitive to the number of factors. We work within the Normal Copula model, which is the industry standard, and propose two methods for approximating CDO prices and portfolio loss distributions. The first approach is based on a power series expansion in a parameter that scales the correlation among obligors; we express the CDO price in a multifactor model as a series of prices in independent-obligor models. Thus, pricing in a model with correlated defaults is reduced to calculations involving only independent defaults. The second approach is based on the Laplace inversion method, which has become a popular tool for pricing credit derivatives. Due to the lack of explicit formulae for the Laplace transform of portfolios with dependent obligors, one faces the challenge of approximating said transforms efficiently and accurately. We develop a closed-form approximant for the Laplace transforms that is robust to the number of factors; then, we use the approximant along with the readily available Laplace inversion formulae to approximate the loss distribution, or the derivative prices. We also prove some results relating to the speed and accuracy of both approaches. |
