An Asymptotic Expansion for Forward-Backward SDEs: A Malliavin Calculus Approach (Revised in September 2013)

Abstract

This paper proposes a new closed-form approximation scheme for the representation of the forward-backward stochastic differential equations (FBSDEs) of Ma and Zhang (2002). In particular, we obtain an error estimate for the scheme applying Malliavin calculus method of Kunitomo and Takahashi (2001, 2003), Kusuoka (2003), Takahashi and Yamada (2012) for the forward SDEs combined with the Picard iteration scheme for the BSDEs. We also show numerical examples for pricing options with counterparty risk under the local and stochastic volatility models, where the credit value adjustment (CVA) is taken into account.