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Construction of a Third-Order K-Scheme and Its Application to Financial Models

Research paper by Yuji Shinozaki

Indexed on: 22 Dec '17Published on: 21 Nov '17Published in: SIAM Journal on Financial Mathematics



Abstract

SIAM Journal on Financial Mathematics, Volume 8, Issue 1, Page 901-932, January 2017. The author proposes a new discretization method of stochastic differential equations (SDEs) that lies in the framework of K-scheme (Kusuoka approximation, Kusuoka--Lyons--Ninomiya--Victoir method). K-scheme is a higher-order discretization framework for performing weak approximations of SDEs. The Ninomiya--Victoir and Ninomiya--Ninomiya methods are practically feasible discretization methods that belong to the K-scheme class. These are second-order weak discretization methods, and some extrapolations of them have been proposed. The new method proposed herein is a third-order weak discretization method that involves no extrapolation. Polynomials of Gaussian random variables that approximate iterated Wiener integrals play a key role in the new method. In addition, the author discusses the applications of the proposed method to the pricing of derivatives in practically important financial models, achieving the desired theoretical order and computational efficiency.