A Spectral Element Method for Option Pricing Under Regime-Switching with Jumps

In this paper, we propose the spectral element method to price European, digital, butterfly, American, discrete and continuous barrier options in a Markovian jump-diffusion regime-switching economy. The spectral element method discretisation is considered for the approximation of the spatial derivatives in a system of partial integro-differential equations and is chosen because it possesses spectral accuracy such that highly accurate option prices can be obtained using a small number of grid discretisation nodes. Essentially, the spectral element method consists of splitting the computational domain into as many elements as needed and approximating the basis functions by high-order orthogonal polynomials within each element. In order to sustain the high-order convergence in time, we also use an exponential time integration scheme to solve the semi-discrete system. Our numerical examples support our error analysis and indicate that the spectral element method converges exponentially for the values and the hedging parameters of the regime-dependent options. Therefore, the proposed scheme provides a viable alternative to the finite difference or finite element methods which usually converge only algebraically.