UM
A fast preconditioned policy iteration method for solving the tempered fractional HJB equation governing American options valuation
Xu Chen1; Wenfei Wang2,3; Deng Ding1; Siu-Long Lei1
2017-05-01
Source PublicationComputers and Mathematics with Applications
ISSN0898-1221
Volume73Issue:9Pages:1932-1944
Abstract

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.

KeywordAmerican Options Hamilton–jacobi–bellman Equation Preconditioner Tempered Fractional Derivative Unconditional Stability
DOI10.1016/j.camwa.2017.02.040
URLView the original
Indexed BySCI
Language英语
WOS Research AreaMathematics
WOS SubjectMathematics, Applied
WOS IDWOS:000400878700005
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Cited Times [WOS]:4   [WOS Record]     [Related Records in WOS]
Document TypeJournal article
CollectionUniversity of Macau
Personal research not belonging to the institution
Corresponding AuthorSiu-Long Lei
Affiliation1.Department of Mathematics, University of Macau, Macau, China
2.Shenwan Hongyuan Securities Postdoctoral Research Station, Shanghai, China
3.Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai, China
First Author AffilicationUniversity of Macau
Corresponding Author AffilicationUniversity of Macau
Recommended Citation
GB/T 7714
Xu Chen,Wenfei Wang,Deng Ding,et al. A fast preconditioned policy iteration method for solving the tempered fractional HJB equation governing American options valuation[J]. Computers and Mathematics with Applications,2017,73(9):1932-1944.
APA Xu Chen,Wenfei Wang,Deng Ding,&Siu-Long Lei.(2017).A fast preconditioned policy iteration method for solving the tempered fractional HJB equation governing American options valuation.Computers and Mathematics with Applications,73(9),1932-1944.
MLA Xu Chen,et al."A fast preconditioned policy iteration method for solving the tempered fractional HJB equation governing American options valuation".Computers and Mathematics with Applications 73.9(2017):1932-1944.
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