Discounted optimal stopping problems in first-passage time models with random thresholds

We derive closed-form solutions to some discounted optimal stopping problems related to the perpetual American cancellable dividend-paying put and call option pricing problems in an extension of the Black-Merton-Scholes model. The cancellation times are assumed to occur when the underlying risky asset price process hits some unobservable random thresholds. The optimal stopping times are shown to be the first times at which the asset price reaches stochastic boundaries depending on the current values of its running maximum and minimum processes. The proof is based on the reduction of the original optimal stopping problems to the associated free-boundary problems and the solution of the latter problems by means of the smooth-fit and modified normal-reflection conditions. We show that the optimal stopping boundaries are characterised as the maximal and minimal solutions of certain first-order nonlinear ordinary differential equations.