Computing Greeks for L\'evy Models: The Fourier Transform Approach

TL;DR

This paper introduces a Fourier transform-based method to compute Greeks for exponential Lévy models, enabling accurate and efficient European option sensitivity calculations with minimal approximation error.

Contribution

It derives exact formulas for Greeks using the Lewis formula, allowing precise Fourier-based approximations and comparison across models like Merton and Variance Gamma.

Findings

Exact formulas for Greeks enable high accuracy.

Fourier transform approach reduces computational error.

Method applicable to various Lévy models.

Abstract

The computation of Greeks for exponential L\'evy models are usually approached by Malliavin Calculus and other methods, as the Likelihood Ratio and the finite difference method. In this paper we obtain exact formulas for Greeks of European options based on the Lewis formula for the option value. Therefore, it is possible to obtain accurate approximations using Fast Fourier Transform. We will present an exhaustive development of Greeks for Call options. The error is shown for all Greeks in the Black-Scholes model, where Greeks can be exactly computed. Other models used in the literature are compared, such as the Merton and Variance Gamma models. The presented formulas can reach desired accuracy because our approach generates error only by approximation of the integral.