Approximating the Optimal Exercise Boundary for American Options via Least-Squares Monte Carlo

The least-squares Monte Carlo method of Longstaff-Schwartz is utilized to construct the optimal exercise boundary (OXB) of an American put option when the underlying follows a geometric Brownian motion (GBM). The optimal exercise price at each time step is obtained by solving numerically the equation of the exercising boundary condition. The set of such exercise prices, along with their ldquo;standard deviations,rdquo; is then fitted to a smooth, monotonic model of a sum of three exponential functions to approximate the OXB, which turns out to be very close to the exact solution of the boundary. The approach can be efficiently implemented and readily computed in practice, and should be applicable to cases when the underlying price process is not GBM