Fourth Order Accurate Implicit Finite Difference Method for Evaluating American Options

We present a numerical method for valuing vanilla American options on a single asset that is fourth order accurate in the log of the asset price, and second order accurate in time. The method overcomes the standard difficulty encountered in developing high order accurate finite difference schemes for valuing American options, that is the lack of smoothness in the option price at the critical boundary. To do this we make special corrections to the right hand side of the difference equations near the boundary so they retain their level of accuracy. These corrections are easily evaluated using estimates of the boundary location and jump in the gamma that occurs there, such as those developed by Carr. The method can also be used for evaluating American options depending on more than one asset whenever estimates of the location of the critical boundary are available. Furthermore, the provable error estimates we obtain also allow development of extrapolation techniques. Results of numerical experiments comparing our method with more standard finite difference methods are provided.