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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 preconditioned direct method has been proposed with O (S ̄ N log N + S ̄ 2 N) operation cost on 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. [ABSTRACT FROM AUTHOR]

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 ( N log N ) 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. [ABSTRACT FROM AUTHOR]

In recent years, considerable literature has proposed the more general class of exponential Lévy processes as the underlying model for prices of financial quantities, which thus better explain many important empirical facts of financial markets. Finite moment log stable, Carr–Geman–Madan–Yor and KoBoL models are chosen from those above-mentioned models as the dynamics of underlying equity prices in this paper. With such models pricing barrier options, one kind of financial derivatives is transformed to solve specific fractional partial differential equations (FPDEs). This study focuses on numerically solving these FPDEs via the fully implicit scheme, with the shifted Grünwald approximation. The circulant preconditioned generalized minimal residual method which converges very fast with theoretical proof is incorporated for solving resultant linear systems. Numerical examples are given to demonstrate the effectiveness of the proposed preconditioner and show the accuracy of our method compared with that done by the Fourier cosine expansion method as a benchmark. [ABSTRACT FROM PUBLISHER]

In recent years, fractional partial differential equation (FPDE) has been widely applied in options pricing problems, which better explains many important empirical facts of financial markets. However, the vast majority of the literatures focus on pricing the single asset option under the FPDE framework. In this paper, a two-dimensional FPDE governing the valuation of rainbow options is established, where two underlying assets are assumed to follow independent exponential Lévy processes, and its boundary conditions are determined by solving one-dimensional FPDEs. A second-order accurate finite difference scheme is proposed to discretize the two-dimensional FPDE. Given the block Toeplitz with Toeplitz block structure of the coefficient matrix, a fast preconditioned Krylov subspace method is developed for solving the resulting linear system with O (N log N) computational complexity per iteration, where N is the matrix size. The proposed preconditioner accelerates the convergence of the iterative method with theoretical analysis. Numerical examples are given to demonstrate the accuracy and efficiency of our proposed numerical methods. [ABSTRACT FROM AUTHOR]