A linearly implicit predictor–corrector scheme for pricing American options using a penalty method approach

Abstract

Pricing of an American option is complicated since at each time we have to determine not only the option value but also whether or not it should be exercised (early exercise constraint). This makes the valuation of an American option a free boundary problem. Typically at each time there is a particular value of the asset, which marks the boundary between two regions: to one side one should hold the option and to other side one should exercise it. Assuming that investors act optimally, the value of an American option cannot fall below the value that would be obtained if it were exercised early. Effectively, this means that the American option early exercise feature transforms the original linear pricing partial differential equation into a nonlinear one. We consider a penalty method approach in which the free and moving boundary is removed by adding a small and continuous penalty term to the Black–Scholes equation; consequently,the problem can be solved on a fixed domain. Analytical solutions of the Black–Scholes model of American option problems are seldom available and hence such derivatives must be priced by stable and efficient numerical techniques. Standard numerical methods involve the need to solve a system of nonlinear equations, evolving from the finite difference discretization of the nonlinear Black–Scholes model, at each time step by a Newton-type iterative procedure. We implement a novel linearly implicit scheme by treating the nonlinear penalty term explicitly, while maintaining superior accuracy and stability properties compared to the well-known θ-methods.