Finite Element and Discontinuous Galerkin Methods with Perfect Matched Layers for American Options

摘要

This paper is devoted to the American option pricing problem governed by the Black-Scholes equation.The existence of an optimal exercise policy makes the problem a free boundary value problem of a parabolic equation on an unbounded domain.The optimal exercise boundary satisfies a nonlinear Volterra integral equation and is solved by a high-order collocation method based on graded meshes.This free boundary is then deformed to a fixed boundary by the front-fixing transformation.The boundary condition at infinity (due to the fact that the underlying asset's price could be arbitrarily large in theory),is treated by the perfectly matched layer technique.Finally,the resulting initial-boundary value problems for the option price and some of the Greeks on a bounded rectangular space-time domain are solved by a finite element method.In particular,for Delta,one of the Greeks,we propose a discontinuous Galerkin method to treat the discontinuity in its initial condition.Convergence results for these two methods are analyzed and several numerical simulations are provided to verify these theoretical results.