Numerical simulation of Black-Scholes model for American options

We consider the penalty method approach, and corresponding numerical schemes, for solving the Black-Scholes model of the American Option. Standard 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 analyze the well-known linearly implicit theta-methods that arise by treating the nonlinear penalty term explicitly thus avoiding iteration. In addition, we have implemented an adaptive time step control strategy to increase computational efficiency.