Multiple Stopping and Exercise Boundaries for American Put Options

This article studies a finite-horizon American put option with up to $m$ exercise rights under the risk-neutral Black--Scholes model, where consecutive exercises must be separated by a prescribed refractory period $\delta>0$ and unused rights may expire. For each number of remaining rights, we show that optimal exercise is characterized by a single exercise boundary and that these boundaries are ordered by the number of remaining rights. We prove optimality of the first boundary-hitting time and establish local Lipschitz continuity of the exercise boundaries within each open deadline strip. Away from the deadlines, we derive smooth fit, the associated quasi-variational inequality and free-boundary formulation, an early-exercise-premium representation, recursive Volterra-type boundary equations, and a verification theorem. We also extend the recursive formulation to an independent random termination time, thereby allowing for exogenous termination of the option contract.