  An American option grants the holder the right to select the time at which to
exercise the option, so pricing an American option entails solving an optimal
stopping problem. Difficulties in applying standard numerical methods to
complex pricing problems have motivated the development of techniques that
combine Monte Carlo simulation with dynamic programming. One class of methods
approximates the option value at each time using a linear combination of basis
functions, and combines Monte Carlo with backward induction to estimate optimal
coefficients in each approximation. We analyze the convergence of such a method
as both the number of basis functions and the number of simulated paths
increase. We get explicit results when the basis functions are polynomials and
the underlying process is either Brownian motion or geometric Brownian motion.
We show that the number of paths required for worst-case convergence grows
exponentially in the degree of the approximating polynomials in the case of
Brownian motion and faster in the case of geometric Brownian motion.
