Quadratic Finite Element and Preconditioning for Options Pricing in the SVCJ Model

Journal of Computational Finance, Forthcoming

20 Pages Posted: 11 Jan 2012

See all articles by Ying-Ying Zhang

Ying-Ying Zhang

Chongqing University

Hong-Kui Pang

Xuzhou Normal University

Liming Feng

University of Illinois at Urbana-Champaign - Department of Industrial and Enterprise Systems Engineering

Xiao-Qing Jin

University of Macau

Date Written: December 31, 2011

Abstract

We consider option pricing problems in the stochastic volatility jump diffusion model with correlated and contemporaneous jumps in both the return and the variance processes (SVCJ). The option value function solves a partial integro-differential equation (PIDE). We discretize this PIDE in space by the quadratic FE method and integrate the resulting ordinary differential equation in time by an implicit-explicit Euler based extrapolation scheme. The coefficient matrix of the resulting linear systems is block penta-diagonal with penta-diagonal blocks. The preconditioned bi-conjugate gradient stabilized (PBiCGSTAB) method is used to solve the linear systems. According to the structure of the coefficient matrix, several preconditioners are implemented and compared. The performance of preconditioning techniques for solving block-tridiagonal systems resulting from the linear FE discretization of the PIDE is also investigated. The combination of the quadratic FE for spatial discretization, the extrapolation scheme for time discretization, and the PBiCGSTAB method with an appropriate preconditioner is found to be very efficient for solving the option pricing problems in the SVCJ model. Compared to the standard second order linear finite element method combined with the popular successive over-relaxation (SOR) linear system solver, the proposed method reduces computational time by about twenty times at the accuracy level of 1 cent and more than fifty times at the accuracy level of 0.1 cent for the barrier option example tested in the paper.

Keywords: stochastic volatility jump diffusion, barrier option, partial integro-differential equation, quadratic finite element, preconditioning, BiCGSTAB, modified incomplete LU preconditioner, block circulant preconditioner

Suggested Citation

Zhang, Ying-Ying and Pang, Hong-Kui and Feng, Liming and Jin, Xiao-Qing, Quadratic Finite Element and Preconditioning for Options Pricing in the SVCJ Model (December 31, 2011). Journal of Computational Finance, Forthcoming, Available at SSRN: https://ssrn.com/abstract=1983106

Ying-Ying Zhang

Chongqing University ( email )

Shazheng Str 174, Shapingba District
Shazheng street, Shapingba district
Chongqing 400044, Chongqing 400030
China

HOME PAGE: http://user.qzone.qq.com/93347989/blog/1308306747

Hong-Kui Pang

Xuzhou Normal University ( email )

Xuzhou, Jiangsu
China

Liming Feng (Contact Author)

University of Illinois at Urbana-Champaign - Department of Industrial and Enterprise Systems Engineering ( email )

104 S. Mathews Avenue
Urbana, IL 61801
United States

Xiao-Qing Jin

University of Macau ( email )

P.O. Box 3001
Macau
Macau

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