In contrast with European options which have fixed maturities, the holder of an American option is allowed to exercise it at any given (random) time. This transforms the valuation problem into an optimization problem in which one has to find the optimal time to exercise in order to maximize the payoff of the option. As will be seen in the first section below, not all random times can be considered in this process, and we restrict ourselves to stopping times whose value at time t be can decided based on the historical data available.