Pricing American options time-capped by a drawdown event

This paper presents a derivation of the explicit price for the perpetual American put option in the Black–Scholes model, time-capped by the first drawdown epoch beyond a predefined level. We demonstrate that the optimal exercise strategy involves executing the option when the asset price first falls below a specified threshold. The proof relies on martingale arguments and the fluctuation theory of Lévy processes. To complement the theoretical findings, we provide numerical analysis.