Small noise expansion and importance sampling

Abstract

Consider the second-order differential operators L 0 =−∂ t −a 1 (y)∂ x −a 2 (y)∂ xx and $\tilde{L}$ =−b 1 (y)∂ y −b 2 (y)∂ yy −c(y)∂ xy and let u ε (t,x,y) be the solution of the parabolic problem L 0 +ε $\tilde{L}$ u=0 on [0,T)×R 2 with terminal condition u ε (T,x,y)=ϕ(x), for given ε∈R. We provide an explicit asymptotic expansion of the solution u ε around the value ε=0. The expansion coefficients of any order are determined by an explicit induction scheme involving the derivatives of u 0 with respect to x. The results are applied for the computation of European contingent claim prices by Monte Carlo simulations in stochastic volatility models, which are popular in the financial literature. The asymptotic expansion is used as accelerator in an importance sampling variance reduction procedure.