Simulating Theta and Gamma of American Options

Abstract

This article derives explicit expressions to simulate theta and gamma for American options using the pathwise derivative method. Although the pathwise derivative formulas for delta, rho, and vega of American options have been studied in the literature, no correct explicit results for theta and gamma exist. In addition, we propose a simulation-based least-square method to compute the optimal stopping boundary for American options. The optimal stopping boundary is needed to evaluate our pathwise derivative expression for gamma and can be used in the integral method to calculate the price and Greeks of American options. Our proposed least-square approach to compute the optimal stopping boundary provides an alternative to the traditional recursive method of solving a system of equations. We also incorporate a Brownian bridge in the computation of the Greeks and extend the application of our results to American basket options.