  We introduce a simple stochastic volatility model, whose novelty consists in
taking into account hitting times of the asset price, and study the optimal
stopping problem corresponding to a put option whose time horizon (after the
asset price hits a certain level) is exponentially distributed. We obtain
explicit optimal stopping rules in various cases one of which is interestingly
complex because of an unexpected disconnected continuation region. Finally, we
discuss in detail how these stopping rules could be used for trading an
American put when the trader expects a market drop in the near future.
