Asymptotically Optimal Importance Sampling and Stratification for Pricing Path-Dependent Options

This paper develops a variance reduction technique for Monte Carlo simulations of path-dependent options driven by high-dimensional Gaussian vectors. The method combines {\it importance sampling} based on a change of drift with {\it stratified sampling} along a small number of key dimensions.  The change of drift is selected through a large deviations analysis and is shown to be optimal in an asymptotic sense. The drift selected has an interpretation as the path of the underlying state variables that maximizes the product of probability and payoff --- the most important path.  The directions used for stratified sampling are optimal for a quadratic approximation to the integrand or payoff function.  Indeed, under differentiability assumptions our importance sampling 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 method with path-dependent options, a stochastic volatility model, and interest rate derivatives. The method reveals novel features of the structure of their payoffs.