Optimal importance sampling for Lévy processes

We develop importance sampling estimators for Monte Carlo pricing of European and path-dependent options in models driven by Lévy processes. Using results from the theory of large deviations for processes with independent increments, we compute an explicit asymptotic approximation for the variance of the pay-off under a time-dependent Esscher-style change of measure. Minimizing this asymptotic variance using convex duality, we then obtain an importance sampling estimator of the option price. We show that our estimator is logarithmically optimal among all importance sampling estimators. Numerical tests in the variance gamma model show consistent variance reduction with a small computational overhead.