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Recently, regime-switching option pricing based on fractional diffusion models has been used, which explains many significant empirical facts about financial markets better. There are many methods to solve the problem, but to the best of our knowledge, effective preconditioners for the second-order schemes have not been proposed. Thus, in this article, an implicit numerical scheme is developed for a regime-switching European option pricing problem under a multi-state tempered fractional model. The scheme is proven to be unconditionally stable and converges quadratically in space and linearly in time. Besides, the resulting linear system is solved using an iterative method, and a preconditioner is proposed to accelerate the rate of convergence. The preconditioner is constructed through circulant approximations to the Toeplitz blocks due to the coefficient matrix, which is is a block matrix with Toeplitz blocks. The spectral analysis of the preconditioned matrix is given, which demonstrates that the spectrum of the preconditioned matrix is clustered around 1. Numerical examples show the efficiency of the proposed method, and an empirical study is also provided.

In this paper, finite difference schemes for differential equations with both temporal and spatial fractional derivatives are studied. When the order of the time fractional derivative is in, an alternating direction implicit (ADI) scheme with second-order accuracy in both space and time is constructed. For equations with time fractional derivatives of order lying in, a scheme is derived and solved by the generalized minimal residual method. We also propose a preconditioner to improve the efficiency for the implementation of the scheme in this situation. [ABSTRACT FROM PUBLISHER]