A Flexible Galerkin Scheme for Option Pricing in L\'evy Models

TL;DR

This paper introduces a flexible finite element Galerkin method for solving the partial integro differential equations in Le9vy models used for option pricing, leveraging Fourier representations for efficiency and applicability.

Contribution

It develops a novel, flexible Galerkin scheme that utilizes Fourier symbols to efficiently solve option pricing equations across various Le9vy models.

Findings

Method is numerically feasible for Merton, NIG, and CGMY models.

Provides a flexible approach adaptable to multiple Le9vy models.

Empirical results confirm the method's effectiveness.

Abstract

One popular approach to option pricing in L\'evy models is through solving the related partial integro differential equation (PIDE). For the numerical solution of such equations powerful Galerkin methods have been put forward e.g. by Hilber et al. (2013). As in practice large classes of models are maintained simultaneously, flexibility in the driving L\'evy model is crucial for the implementation of these powerful tools. In this article we provide such a flexible finite element Galerkin method. To this end we exploit the Fourier representation of the infinitesimal generator, i.e. the related symbol, which is explicitly available for the most relevant L\'evy models. Empirical studies for the Merton, NIG and CGMY model confirm the numerical feasibility of the method.