A Dimension and Variance Reduction Monte-Carlo Method for Option Pricing under Jump-Diffusion Models

Abstract

We develop a highly efficient MC method for computing plain vanilla European option prices and hedging parameters under a very general jump-diffusion option pricing model which includes stochastic variance and multi-factor Gaussian interest short rate(s). The focus of our MC approach is variance reduction via dimension reduction. More specifically, the option price is expressed as an expectation of a unique solution to a conditional Partial Integro-Differential Equation (PIDE), which is then solved using a Fourier transform technique. Important features of our approach are (i) the analytical tractability of the conditional PIDE is fully determined by that of the Black-Scholes-Merton model augmented with the same jump component as in our model, and (ii) the variances associated with all the interest rate factors are completely removed when evaluating the expectation via iterated conditioning applied to only the Brownian motion associated with the variance factor. For certain cases when numerical methods are either needed or preferred, we propose a discrete fast Fourier transform method to numerically solve the conditional PIDE efficiently. Our method can also effectively compute hedging parameters. Numerical results show that the proposed method is highly efficient.

Keywords: conditional Monte Carlo, variance reduction, dimension reduction, partial-integro~differential~equations, jump diffusions, fast Fourier transform, normal, double-exponential