  We present a new approximation scheme for the price and exercise policy of
American options. The scheme is based on Hermite polynomial expansions of the
transition density of the underlying asset dynamics and the early exercise
premium representation of the American option price. The advantages of the
proposed approach are threefold. First, our approach does not require the
transition density and characteristic functions of the underlying asset
dynamics to be attainable in closed form. Second, our approach is fast and
accurate, while the prices and exercise policy can be jointly produced. Third,
our approach has a wide range of applications. We show that the proposed
approximations of the price and optimal exercise boundary converge to the true
ones. We also provide a numerical method based on a step function to implement
our proposed approach. Applications to nonlinear mean-reverting models, double
mean-reverting models, Merton's and Kou's jump-diffusion models are presented
and discussed.
