Optimal stopping and American options with discrete dividends and exogenous risk

In this paper we analyze some problems arising in the evaluation of American options when the underlying security pays discrete dividends. To this aim, we study the problem of maximizing the expected gain process over stopping times taking values in the union of disjoint, real compact sets. The results we obtain can be applied to evaluate options with restrictions on exercise periods, but are also useful for the evaluation of American options on assets that pay discrete dividends. In particular, we generalize the evaluation formula for American call options due to Whaley [Journal of Financial Economics 9 (1981) 207], allowing for a stochastic jump of the underlying security at the ex-dividend date and discuss the existence of the optimal stopping time. In the same framework, we analyze American put options, justifying the procedure used in Meyer [Journal of Computational Finance 5 (2) (2002)] to account for the presence of discrete dividends in the free boundary formulation from the perspective of optimal stopping theory.