Optimal importance sampling for L\'evy Processes

TL;DR

This paper introduces a new importance sampling method for Monte Carlo valuation of options in Le9vy process models, achieving efficient variance reduction with minimal computational cost.

Contribution

It extends importance sampling techniques to Le9vy-driven models using large deviations theory and convex duality for explicit variance minimization.

Findings

Significant variance reduction in European basket options

Effective variance reduction for Asian options in variance gamma model

Low additional computational overhead

Abstract

We develop generic and efficient importance sampling estimators for Monte Carlo evaluation of prices of single- and multi-asset European and path-dependent options in asset price models driven by L\'evy processes, extending earlier works which focused on the Black-Scholes and continuous stochastic volatility models. Using recent results from the theory of large deviations on the path space for processes with independent increments, we compute an explicit asymptotic approximation for the variance of the pay-off under an Esscher-style change of measure. Minimizing this asymptotic variance using convex duality, we then obtain an easy to compite asymptotically efficient importance sampling estimator of the option price. Numerical tests for European baskets and for Asian options in the variance gamma model show consistent variance reduction with a very small computational overhead.