A Preconditioned Iterative Method for a Multi-State Time-Fractional Linear Complementary Problem in Option Pricing

Fractal and Fractional

article

Xu Chen¹  Xinxin Gong¹  Siu-Long Lei³  Youfa Sun¹ 

[1] School of Economics, Guangdong University of Technology;Key Laboratory of Digital Economy and Data Governance, Guangdong University of Technology;Department of Mathematics, University of Macau

关键词: preconditioner;    nonlinear finite difference scheme;    linear complementary problem;    time-fractional derivative;    policy iteration method91G60;    65M06;    65F08;    26A33;    65H10;   

DOI  :  10.3390/fractalfract7040334

学科分类：社会科学、人文和艺术（综合）

来源: mdpi

【 摘 要 】

Fractional derivatives and regime-switching models are widely used in various fields of finance because they can describe the nonlocal properties of the solutions and the changes in the market status, respectively. The regime-switching time-fractional diffusion equations that combine both advantages are also used in European option pricing; however, to our knowledge, American option pricing based on such models and their numerical methods is yet to be studied. Hence, a fast algorithm for solving the multi-state time-fractional linear complementary problem arising from the regime-switching time-fractional American option pricing models is developed in this paper. To construct the solution strategy, the original problem has been converted into a Hamilton–Jacobi–Bellman equation, and a nonlinear finite difference scheme has been proposed to discretize the problem with stability analysis. A policy-Krylov subspace method is developed to solve the nonlinear scheme. Further, to accelerate the convergence rate of the Krylov method, a tri-diagonal preconditioner is proposed with condition number analysis. Numerical experiments are presented to demonstrate the validity of the proposed nonlinear scheme and the efficiency of the proposed preconditioned policy-Krylov subspace method.

【 授权许可】

CC BY