American Options with Stopping Time Constraints to Executive Stock Options

This paper concerns the pricing of American options with stochastic stopping time constraints expressed in terms of the states of a Markov process. Following the ideas of Menaldi, Robin, and Sun (1996) we transform the constrained into an unconstrained optimal stopping problem. The transformation replaces the original payoff by the value of a generalized barrier option. We suggest a new Monte Carlo method to numerically calculate the option value also for multidimensional Markov processes. Because of presence of stopping time constraints the classical Longstaff-Schwartz least-square Monte Carlo algorithm or its extension introduced in Egloff (2005) cannot be directly applied. We adapt the Longstaff-Schwartz algorithm to solve the stochastic Cauchy-Dirichlet problem related to the valuation problem of the barrier option along a set of simulated trajectories of the underlying Markov process.