Computation of copulas by Fourier methods

TL;DR

This paper introduces a Fourier-based integral representation for copulas of dependent variables, enabling computation in complex financial models like Lévy and affine processes, with an application to the NIG Lévy process.

Contribution

It provides a novel Fourier integral representation for copulas, extending their computation to a broad class of financial models including Lévy and affine processes.

Findings

Derived an integral representation for copulas using Fourier methods.

Applied the method to compute the implied copula of the NIG Lévy process.

Demonstrated the approach's effectiveness for models with time-dependent dependence.

Abstract

We provide an integral representation for the (implied) copulas of dependent random variables in terms of their moment generating functions. The proof uses ideas from Fourier methods for option pricing. This representation can be used for a large class of models from mathematical finance, including L\'evy and affine processes. As an application, we compute the implied copula of the NIG L\'evy process which exhibits notable time-dependence.