A fast preconditioned policy iteration method for solving the tempered fractional HJB equation governing American options valuation

A fast preconditioned policy iteration method is proposed for the Hamilton–Jacobi–Bellman (HJB) equation involving tempered fractional order partial derivatives, governing the valuation of American options whose underlying asset follows exponential Lévy processes. An unconditionally stable upwind fi...

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Bibliographic Details
Published inComputers & mathematics with applications (1987) Vol. 73; no. 9; pp. 1932 - 1944
Main Authors Chen, Xu, Wang, Wenfei, Ding, Deng, Lei, Siu-Long
Format Journal Article
LanguageEnglish
Published Oxford Elsevier Ltd 01.05.2017
Elsevier BV
Subjects
American options
Convergence
Fast Fourier transformations
Finite difference method
Finite element analysis
Fourier transforms
Fractions
Hamilton–Jacobi–Bellman equation
Iterative methods
Mathematical analysis
Mathematical models
Matrix methods
Numerical analysis
Preconditioner
Studies
Tempered fractional derivative
Unconditional stability
Unconditional stability
Hamilton–Jacobi–Bellman equation
American options
Preconditioner
Tempered fractional derivative
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Summary:A fast preconditioned policy iteration method is proposed for the Hamilton–Jacobi–Bellman (HJB) equation involving tempered fractional order partial derivatives, governing the valuation of American options whose underlying asset follows exponential Lévy processes. An unconditionally stable upwind finite difference scheme with shifted Grünwald approximation is first developed to discretize the established HJB equation under the tempered fractional diffusion models. Next, the policy iteration method as an outer iterative method is utilized to solve the discretized HJB equation and proven to be convergent in finite steps to its numerical solution. Given the Toeplitz-like structure of the coefficient matrix in each policy iteration, the resulting linear system can be fast solved by the Krylov subspace method as an inner iterative method via fast Fourier transform (FFT). Furthermore, a novel preconditioner is proposed to speed up the convergence rate of the inner Krylov subspace iteration with theoretical analysis to ensure the linear system can be solved in O(NlogN) operations under some mild conditions, where N is the number of spatial node points. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed fast preconditioned policy method.
ISSN:0898-1221
1873-7668
DOI:10.1016/j.camwa.2017.02.040