A Polynomial Scheme of Asymptotic Expansion for Backward SDEs and Option pricing (Revised in December 2014; Forthcoming in “Quantitative Finance”)
A new asymptotic expansion scheme for backward SDEs (BSDEs) is proposed. The perturbation parameter “ε” is introduced just to scale the forward stochastic variables within a BSDE. In contrast to the standard small-diffusion asymptotic expansion method, the dynamics of variables given by the forward SDEs is treated exactly. Although it requires a special form of the quadratic covariation terms, it allows rather generic drift as well as jump components to exist. The resultant approximation is given by a polynomial function in terms of the unperturbed forward variables whose coefficients are uniquely specified by the solution of the recursive system of linear ODEs. Applications to a jumpextended Heston and λ-SABR models for European contingent claims, as well as the utility-optimization problem in the presence of a terminal liability are discussed.