Penalty approach to a nonlinear obstacle problem governing American put option valuation under transaction costs

We propose a penalty method for a finite-dimensional nonlinear complementarity problem (NCP) arising from the discretization of the infinite-dimensional free boundary/obstacle problem governing the valuation of American options under transaction costs. In this method, the NCP is approximated by a system of nonlinear equations containing a power penalty term. We show that the mapping involved in the system is continuous and strongly monotone. Thus, the unique solvability of both the NCP and the penalty equation and the exponential convergence of the solution to the penalty equation to that of the NCP are guaranteed by an existing theory. Numerical results will be presented to demonstrate the convergence rates and usefulness of this penalty method.