Perpetual American cancellable standard options in models with last passage times

We derive explicit solutions to the perpetual American cancellable standard put and call options in an extension of the Black-Merton-Scholes model. It is assumed that the contracts are cancelled at the last hitting times for the underlying asset price process of some constant upper or lower levels which are not stopping times with respect to the observable filtration. We show that the optimal exercise times are the first times at which the asset price reaches some lower or upper constant levels. 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 conditions.