Enhanced policy iteration for American options via scenario selection

Kolodko and Schoenmakers (2006 Kolodko, A and Schoenmakers, J. 2006. Iterative construction of the optimal Bermudan stopping time. Finan. Stochast., 10: 27–49. [Crossref], [Web of Science ®] , [Google Scholar]) and Bender and Schoenmakers (2006 Bender, C, Kolodko, A and Schoenmakers, J. 2006. Iterating cancelable snowballs and related exotics in a many-factor Libor model. Risk, September: 126–130.  [Google Scholar]) introduced a policy iteration that allows the achievement of a tight lower approximations of the price for early exercise options via a nested Monte Carlo simulation in a Markovian setting. In this paper we enhance the algorithm by a scenario selection method. It is demonstrated by numerical examples that the scenario selection can significantly reduce the number of inner simulations actually performed, and thus can greatly speed up the method (by up to a factor of 15 in some examples). Moreover, it is shown that the modified algorithm retains the desirable properties of the original, such as the monotone improvement property, termination after a finite number of iteration steps, and numerical stability.