Sampling nested Archimedean copulas with applications to CDO pricing

Abstract

Copulas are distribution functions with standard uniform univariate margins. One particular parametric class of copulas is the class Archimedean copulas. These copulas are explicit and can be expressed in terms of a one-dimensional function called the generator of the Archimedean copula. Archimedean copulas are permutation symmetric in their arguments. To circumvent this rather strong assumption in large dimensions, a new class of copulas has recently been suggested, the class of nested Archimedean copulas. Since members of this flexible class are built by nesting Archimedean copulas at different levels, nested Archimedean copulas allow one to precisely capture hierarchical structures often inherent in practical applications. Sampling Archimedean and nested Archimedean copulas can be achieved with an algorithm based on Laplace-Stieltjes transforms, assuming the Laplace-Stieltjes inverses corresponding to the generators are easy to sample. Finding efficient sampling strategies for these distribution functions on the positive real line is the main goal of this dissertation. As a general-purpose approach, we first investigate several algorithms for numerically inverting Laplace transforms to access these distributions. We then develop fast and explicit sampling algorithms for both Archimedean and nested Archimedean copulas constructed with commonly used generators. On our way through the Archimedean world, we obtain several general results concerning the construction and sampling of nested Archimedean copulas. Our findings lead to multiple byproducts, including an efficient sampling algorithm for exponentially tilted distributions. As an application, a pricing model for collateralized debt obligations is developed.