Asymptotically optimal importance sampling and stratification for pricing path-dependent options

This paper develops a variance reduction technique for Monte Carlo simulati ons of path-dependent options driven by high-dimensional Gaussian vectors. The method combines importance sampling based on a change of drift with str atified sampling along a small number of key dimensions. The change of drif t is selected through a large deviations analysis and is shown to be optima l in an asymptotic sense. The drift selected has an interpretation as the p ath of the underlying state variables which maximizes the product of probab ility and payoff-the most important path. The directions used for stratifie d sampling are optimal for a quadratic approximation to the integrand or pa yoff function. Indeed, under differentiability assumptions our importance s ampling method eliminates variability due to the linear part of the payoff function, and stratification eliminates much of the variability due to the quadratic part of the payoff The two parts of the method are Linked because the asymptotically optimal drift vector frequently provides a particularly effective direction for stratification. We illustrate the use of the metho d with path-dependent options, a stochastic volatility model, and interest rate derivatives. The method reveals novel features of the structure of the ir payoffs.