Consider the second-order differential operators ℒ0 = -∂t - a1(y)∂x - a2(y)∂xx and ℒ̃ = -b1(y)∂y - b2(y)∂yy - c(y)∂xy and let uε(t, x, y) be the solution of the parabolic problem ℒ0 + εℒ̃u = 0 on [0, T) × ℝ2 with terminal condition uε(T, x, y) = φ(x), for given ε ∈ ℝ. We provide an explicit asymptotic expansion of the solution uε around the value ε = 0. The expansion coefficients of any order are determined by an explicit induction scheme involving the derivatives of u0 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.
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