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21-11-2017 | Issue 3/2018

# A Fast Preconditioned Penalty Method for American Options Pricing Under Regime-Switching Tempered Fractional Diffusion Models

Journal:
Journal of Scientific Computing > Issue 3/2018
Authors:
Siu-Long Lei, Wenfei Wang, Xu Chen, Deng Ding
Important notes
This work was supported by the research Grants MYRG2016-00202-FST and MYRG068(Y4-L2)-FST13-DD from University of Macau, 081/2016/A2 and 048/2017/A from Macao Science and Technology Development Fund (FDCT).

## Abstract

A fast preconditioned penalty method is developed for a system of parabolic linear complementarity problems (LCPs) involving tempered fractional order partial derivatives governing the price of American options whose underlying asset follows a geometry Lévy process with multi-state regime switching. By means of the penalty method, the system of LCPs is approximated with a penalty term by a system of nonlinear tempered fractional partial differential equations (TFPDEs) coupled by a finite-state Markov chain. The system of nonlinear TFPDEs is discretized with the shifted Grünwald approximation by an upwind finite difference scheme which is shown to be unconditionally stable. Semi-smooth Newton’s method is utilized to solve the finite difference scheme as an outer iterative method in which the Jacobi matrix is found to possess Toeplitz-plus-diagonal structure. Consequently, 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. With the above-mentioned preconditioning technique via FFT, under some mild conditions, the operation cost in each Newton’s step can be expected to be $$\mathcal{O}(N\mathrm{log}N)$$, where N is the size of the coefficient matrix. Numerical examples are given to demonstrate the accuracy and efficiency of our proposed fast preconditioned penalty method.

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