Multilevel Monte Carlo For Exponential L\'{e}vy Models

TL;DR

This paper demonstrates the effectiveness of multilevel Monte Carlo methods for pricing options under exponential Lévy models, providing convergence analysis and numerical results for various processes.

Contribution

It introduces convergence rate estimates for discrete supremum monitoring in Lévy processes and applies multilevel Monte Carlo to improve option pricing efficiency.

Findings

Multilevel Monte Carlo reduces computational cost for Lévy process options.

Convergence rate bounds are derived for Variance Gamma, NIG, and α-stable processes.

Numerical experiments confirm the efficiency and accuracy of the proposed methods.

Abstract

We apply multilevel Monte Carlo for option pricing problems using exponential L\'{e}vy models with a uniform timestep discretisation to monitor the running maximum required for lookback and barrier options. The numerical results demonstrate the computational efficiency of this approach. We derive estimates of the convergence rate for the error introduced by the discrete monitoring of the running supremum of a broad class of L\'{e}vy processes. We use these to obtain upper bounds on the multilevel Monte Carlo variance convergence rate for the Variance Gamma, NIG and $\alpha$-stable processes used in the numerical experiments. We also show numerical results and analysis of a trapezoidal approximation for Asian options.