A polynomial scheme of asymptotic expansion for backward SDEs and option pricing

Masaaki Fujii

Quantitative Finance, 2016, vol. 16, issue 3, 427-445

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.

Date: 2016
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DOI: 10.1080/14697688.2015.1036770

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