Importance sampling for multimodal functions and application to pricing exotic options

We consider importance sampling (IS) to increase the efficiency of Monte Carlo integration, especially for pricing exotic options where the random input is multivariate normal. When the importance function (the product of integrand and original density) is multimodal, determining a good IS density is a difficult task. We propose an automated importance sampling density selection procedure (AISDE). AISDE selects an IS density as a mixture of multivariate normal densities with modes at certain local maxima of the importance function. When the simulation input is multivariate normal, we use principal component analysis to obtain a reduced-dimension, approximate importance function, which allows efficient identification of a good IS density via AISDE in original problem dimensions over 100. We present Monte Carlo experimental results on randomly generated option-pricing problems (including path-dependent options), demonstrating large and consistent efficiency improvement.