Efficient simulation of tail probabilities of sums of dependent random variables

Abstract

We study asymptotically optimal simulation algorithms for approximating the tail probability of P(e X 1 +⋯+ e X d >u) as u→∞. The first algorithm proposed is based on conditional Monte Carlo and assumes that (X 1,…,X d ) has an elliptical distribution with very mild assumptions on the radial component. This algorithm is applicable to a large class of models in finance, as we demonstrate with examples. In addition, we propose an importance sampling algorithm for an arbitrary dependence structure that is shown to be asymptotically optimal under mild assumptions on the marginal distributions and, basically, that we can simulate efficiently (X 1,…,X d |X j >b) for large b. Extensions that allow us to handle portfolios of financial options are also discussed.