Importance Sampling Applied to Greeks for Jump–Diffusion Models with Stochastic Volatility

Abstract

We develop a variance reduction technique, based on importance sampling in conjunction with the stochastic Robbins–Monro algorithm, for option prices of jump–diffusion models with stochastic volatility. This is done by combining the work developed by Arouna for pricing diffusion models, and extended by Kawai for Lévy processes without a Brownian component. We apply this technique to improve the numerical computation of derivative price sensitivities for general Lévy processes, allowing both Brownian and jump parts. Numerical examples are performed for both the Black–Scholes and Heston models with jumps and for the Barndorff–Nielsen–Shephard model to illustrate the efficiency of this numerical technique. The numerical results support that the proposed methodology improves the efficiency of the usual Monte Carlo procedures.

Keywords: option pricing, Esscher transform, Malliavin calculus, Robbins–Monro algorithm