A Polynomial Scheme of Asymptotic Expansion for Backward SDEs and Option pricing

TL;DR: In this article, a new asymptotic expansion scheme for backward stochastic differential equations (BSDEs) is proposed, which allows rather generic drift as well as jump components to exist, and gives a polynomial function in terms of the unperturbed forward variables whose coefficients are uniquely specified by the solution of a recursive system of linear ordinary differential equations.

Abstract: A new asymptotic expansion scheme for backward stochastic differential equations (BSDEs) is proposed. The perturbation parameter ‘’ 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 of the continuous part, 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 a recursive system of linear ordinary differential equations. Applications to the jump-extended Heston and -SABR models for European contingent claims, as well as the utility-optimization problem in the presence of a terminal liability, are discussed.