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

On Trading American Put Options with Interactive Volatility