Pricing American options time-capped by a drawdown even in a Lévy market

This paper presents a derivation of the explicit price for the perpetual American put option time-capped by the first drawdown epoch beyond a predefined level. We consider the market in which an asset price is described by geometric L\'evy process with downward exponential jumps. We show that the optimal stopping rule is the first time when the asset price gets below a special value. The proof relies on martingale arguments and the fluctuation theory of L\'evy processes. We also provide a numerical analysis.