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

TL;DR

This paper introduces a polynomial asymptotic expansion scheme for backward SDEs that accurately handles complex dynamics, including jumps and drifts, providing explicit solutions useful for option pricing and utility optimization.

Contribution

It presents a novel polynomial expansion method for BSDEs that treats forward SDE dynamics exactly, accommodating jumps and complex drifts, with explicit recursive ODE solutions.

Findings

Effective approximation for jump-extended Heston and lambda-SABR models

Applicable to European options and utility optimization problems

Provides explicit polynomial solutions with recursive ODE coefficients

Abstract

A new asymptotic expansion scheme for backward SDEs (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 the recursive system of linear ODEs. Applications to a jump-extended Heston and lambda-SABR models for European contingent claims, as well as the utility-optimization problem in the presence of a terminal liability are discussed.