Pricing american options under proportional transaction costs using a penalty approach and a finite difference scheme

In this paper we propose a penalty method combined with a finite difference scheme for the Hamilton-Jacobi-Bellman (HJB) equation arising in pricing American options under proportional transaction costs. In this method, the HJB equation is approximated by a nonlinear partial differential equation with penalty terms. We prove that the viscosity solution to the penalty equa-tion converges to that of the original HJB equation when the penalty parameter tends to positive infinity. We then present an upwind finite difference scheme for solving the penalty equation and show that the approximate solution from the scheme converges to the viscosity solution of the penalty equation. A nu-merical algorithm for solving the discretized nonlinear system is proposed and analyzed. Numerical results are presented to demonstrate the accuracy of the method.