The optimal stopping problem is a classical one within stochastic calculus theory. Formally, given a gain process, the optimal stopping problem is about finding the stopping time that maximizes the expected gain. In discrete time, this problem is solved using dynamic programming techniques and setting up a backward recursion. This delivers both the optimal stopping rule and the optimal value process. This optimal value process coincides with the Snell Envelope of the gain process, namely, the smallest supermartingale dominating it. 

Optimal stopping theory closely relates to the American derivatives valuation problem. American derivatives are financial contracts characterized by a payoff process that depends on an underlying stochastic process (usually the price of a traded asset). The holder of an American derivative chooses when to cash in the payoff, trying to do so optimizing the gain. Therefore, the fair price of this derivate depends on its optimal stopping time. 

During my talk, which will be accessible also to non-specialists, I will first describe and then show how to solve the optimal stopping problem in a discrete time setting. Secondly, I will show how to apply these techniques for the valuation of American derivatives in the most standard setting. Finally, I will present a result on how to price these derivatives in a more sophisticated market model.