Pricing of American options in discrete time using least squares estimates with complexity penalties

Pricing of American options in discrete time is considered, where the option is allowed to be based on several underlying stocks. It is assumed that the price processes of the underlying stocks are given by Markov processes. We use the Monte Carlo approach to generate artificial sample paths of these price processes, and then we use nonparametric regression estimates to estimate from this data so-called continuation values, which are defined as mean values of the American option for given values of the underlying stocks at time t subject to the constraint that the option is not exercised at time t. As nonparametric regression estimates we use least squares estimates with complexity penalties, which include as special cases least squares spline estimates, least squares neural networks, smoothing splines and orthogonal series estimates. General results concerning rate of convergence are presented and applied to derive results for the special cases mentioned above. Furthermore the pricing of American options is illustrated by simulated data.