A high-order finite difference method for option valuation

In this paper, we propose the use of an efficient high-order finite difference algorithm to price options under several pricing models including the Black-Scholes model, the Merton's jump-diffusion model, the Heston's stochastic volatility model and the nonlinear transaction costs or illiquidity models. We apply a local mesh refinement strategy at the points of singularity usually found in the payoff of most financial derivatives to improve the accuracy and restore the rate of convergence of a non-uniform high-order five-point stencil finite difference scheme. For linear models, the time-stepping is dealt with by using an exponential time integration scheme with Carathéodory-Fejér approximations to efficiently evaluate the product of a matrix exponential with a vector of option prices. Nonlinear Black-Scholes equations are solved using an efficient iterative scheme coupled with a Richardson extrapolation. Our numerical experiments clearly demonstrate the high-order accuracy of the proposed finite difference method, making the latter a method of choice for solving both linear and nonlinear partial differential equations in financial engineering problems.