Error bounds for small jumps of L\'evy processes

TL;DR

This paper derives error bounds for approximating Levy processes in option pricing, focusing on truncating small jumps or replacing them with Brownian motion, to improve Monte Carlo simulation accuracy.

Contribution

It provides theoretical bounds on errors caused by common approximations of Levy processes in financial modeling.

Findings

Bounds for errors from jump truncation

Bounds for errors from Brownian replacement

Guidelines for approximation accuracy in Monte Carlo methods

Abstract

The pricing of options in exponential Levy models amounts to the computation of expectations of functionals of Levy processes. In many situations, Monte-Carlo methods are used. However, the simulation of a Levy process with infinite Levy measure generally requires either to truncate small jumps or to replace them by a Brownian motion with the same variance. We will derive bounds for the errors generated by these two types of approximation.