In order to reduce the oscillations of the numerical solution of fractional exotic options pricing model, a class of numerical schemes are developed and well studied in this paper which are based on the 4th-order Padé approximation and 2nd-order weighted and shifted Grünwald difference scheme. Since the spatial discretization matrix is positive definite and has lower Hessenberg Toeplitz structure, we prove the convergence of the proposed scheme. Numerical experiments on fractional digital option and fractional barrier options show that the (0, 4)-Padé scheme is fast, and significantly reduces the oscillations of the solution and smooths the Delta value.

Fractional regime-switching option models have recently attracted much attention because they can capture the sudden state movement of the market, and deal with the non-stationary behavior. A second-order numerical scheme is proposed to solve the regime-switching option pricing models with fractional derivatives in space. The sufficient conditions of the stability and convergence of the proposed scheme are studied in details. An alternating direction implicit (ADI) method is implemented to accelerate the computation in every time layer. Numerical experiments are presented to verify the convergence and efficiency of the proposed method, compared with classical Krylov subspace solvers.