A Second-Order Difference Scheme for the Penalized Black–Scholes Equation Governing American Put Option Pricing

Abstract

In this paper we present a stable finite difference scheme on a piecewise uniform mesh along with a power penalty method for solving the American put option problem. By adding a power penalty term the linear complementarity problem arising from pricing American put options is transformed into a nonlinear parabolic partial differential equation. Then a finite difference scheme is proposed to solve the penalized nonlinear PDE, which combines a central difference scheme on a piecewise uniform mesh with respect to the spatial variable with an implicit time stepping technique. It is proved that the scheme is stable for arbitrary volatility and arbitrary interest rate without any extra conditions and is second-order convergent with respect to the spatial variable. Furthermore, our method can efficiently treats the singularities of the non-smooth payoff function. Numerical results support the theoretical results.