Small noise expansion and importance sampling

Article type: Research Article

Authors: Fournie, Eric | Lebuchoux, Jérôme | Touzi, Nizar

Affiliations: Caisse Autonome de Refinancement, 2 square de Luynes, 75007 Paris, France | Ecole Normale Supérieure, 45 rue d'Ulm, 75005 Paris, France | CEREMADE (Université Paris IX Dauphine) and CREST, Place du Maréchal de Lattre de Tassigny, 75016 Paris, France

Abstract: Consider the second-order differential operators L0=−∂t−a1(y)∂x−a2(y)∂xx and $\tilde{L}$=−b1(y)∂y−b2(y)∂yy−c(y)∂xy and let uε(t,x,y) be the solution of the parabolic problem L0+ε$\tilde{L}$u=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.

DOI: 10.3233/ASY-1997-14404

Journal: Asymptotic Analysis, vol. 14, no. 4, pp. 361-376, 1997