A finite difference method for pricing European and American options under a geometric Lévy process

In this paper we develop a numerical approach to a fractional-order differential Linear Complementarity Problem (LCP) arising in pricing European and American options under a geometric Lévy process. The LCP is first approximated by a nonlinear penalty fractional Black-Scholes (fBS) equation. We then propose a finite difference scheme for the penalty fBS equation. We show that both the continuous and the discretized fBS equations are uniquely solvable and establish the convergence of the numerical solution to the viscosity solution of the penalty fBS equation by proving the consistency, stability and monotonicity of the numerical scheme. We also show that the discretization has the 2nd-order truncation error in both the spatial and time mesh sizes. Numerical results are presented to demonstrate the accuracy and usefulness of the numerical method for pricing both European and American options under the geometric Lévy process

Super-diffusive Transport Processes in Porous Media

The basic assumption of models for the transport of contaminants through soil is that the movements of solute particles are characterized by the Brownian motion. However, the complexity of pore space in natural porous media makes the hypothesis of Brownian motion far too restrictive in some situations. Therefore, alternative models have been proposed. One of the models, many times encountered in hydrology, is based in fractional differential equations, which is a one-dimensional fractional advection diffusion equation where the usual second-order derivative gives place to a fractional derivative of order α, with 1 < α ≤ 2. When a fractional derivative replaces the second-order derivative in a diffusion or dispersion model, it leads to anomalous diffusion, also called super-diffusion. We derive analytical solutions for the fractional advection diffusion equation with different initial and boundary conditions. Additionally, we analyze how the fractional parameter α affects the behavior of the solutions

A spectral collocation method for nonlinear fractional boundary value problems with a Caputo derivative

In this paper, we consider the nonlinear boundary value problems involving the Caputo fractional derivatives of order α∈ (1 , 2) on the interval (0, T). We present a Legendre spectral collocation method for the Caputo fractional boundary value problems. We derive the error bounds of the Legendre collocation method under the L2- and L∞-norms. Numerical experiments are included to illustrate the theoretical results.MOE (Min. of Education, S’pore