An implicit-explicit preconditioned direct method for pricing options under regime-switching tempered fractional partial
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An implicit-explicit preconditioned direct method for pricing options under regime-switching tempered fractional partial differential models Xu Chen1,2 · Deng Ding2 · Siu-Long Lei2 · Wenfei Wang3,4 Received: 6 November 2019 / Accepted: 30 July 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Recently, fractional partial differential equations have been widely applied in option pricing problems, which better explains many important empirical facts of financial markets, but rare paper considers the multi-state options pricing problem based on fractional diffusion models. Thus, multi-state European option pricing problem under regime-switching tempered fractional partial differential equation is considered in this paper. Due to the expensive computational cost caused by the implicit finite difference scheme, a novel implicit-explicit finite difference scheme has been developed with consistency, stability, and convergence guarantee. Since the resulting coefficient matrix equals to the direct sum of several Toeplitz matrices, a precondi¯ log N + S¯ 2 N) operation cost on tioned direct method has been proposed with O(SN each time level with adaptability analysis, where S¯ is the number of states and N is the number of grid points. Related numerical experiments including an empirical example have been presented to demonstrate the effectiveness and accuracy of the proposed numerical method. Keywords Implicit-explicit finite difference method · Direct method · Precondition · Multi-state European options pricing · Tempered fractional partial differential equation Mathematics Subject Classification (2010) 65F05 · 65F08 · 65M06 · 91G80 · 35R11
Siu-Long Lei
[email protected] 1
School of Economics and Commerce, Guangdong University of Technology, Guangzhou, China
2
Department of Mathematics, University of Macau, Macau, China
3
Equity Investments and Trading Department, Haitong Securities Co., Ltd., Shanghai, China
4
Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai, China
Numerical Algorithms
1 Introduction In the classic Black-Scholes (BS) [3] European option pricing models, the asset process St is assumed to follow a random walk or diffusion with a deterministic drift. However, it has been known for many years that the classic BS model suffers from many shortcomings and is not capable of explaining many important empirical facts of financial markets, like skewed and heavy-tailed return distributions or large, sudden movements in stock prices. Thus, despite the superior analytical tractability of the geometric Brownian motion model [3], many authors proposed the more general class of exponential L´evy processes as the underlying model for prices of financial quantities [13], including finite moment log stable (FMLS) model [6], Carr-Geman-Madan-Yor (CGMY) model [5], KoBoL model [4], etc. To incorporate non-stationary behavior into an otherwise stationary exponential L´evy process (e.g., to capture the sudden state movement from the bull market to bear
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