European and Asian Greeks for exponential L\'evy processes

TL;DR

This paper provides explicit formulas for European and Asian Greeks in exponential Lévy models with Brownian motion, using Malliavin calculus, and compares numerical methods for different payoff types.

Contribution

It introduces new closed-form expressions for Greeks in exponential Lévy models using Malliavin calculus, extending previous literature and improving numerical computation methods.

Findings

Malliavin Monte Carlo Greeks outperform finite difference for discontinuous payoffs.

Combination of Malliavin Monte Carlo and finite difference improves convergence for continuous payoffs.

Results apply to generalized Asian options, including fixed and floating strike types.

Abstract

In this paper we give easy-to-implement closed-form expressions for European and Asian Greeks for general L2-payoff functions and underlying assets in an exponential L\'evy process model with nonvanishing Brownian motion part. The results are based on Hilbert space valued Malliavin Calculus and extend previous results from the literature. Numerical experiments suggest, that in the case of a continuous payoff function, a combination of Malliavin Monte Carlo Greeks and the finite difference method has a better convergence behavior, whereas in the case of discontinuous payoff functions, the Malliavin Monte Carlo method clearly is the superior method compared to the finite difference approach, for first- and second order Greeks. Reduction arguments from the literature based on measure change imply that the expressions for the Greeks in this paper also hold true for generalized Asian options in particular for fixed and floating strike Asian options.