Pricing American Options Time-Capped by a Drawdown Event 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 time beyond a predefined level. The underlying asset price follows a geometric Lévy process with downward exponentially distributed jumps. We show that the optimal exercise strategy is to stop the first time the asset price falls below a certain threshold. The proof combines martingale techniques with fluctuation theory for Lévy processes. A numerical study illustrating the main results is also provided.