The American Put Option and Its Critical Stock Price

We derive an expression for the critical stock price for the American put. We start by expressing the put price as an integral involving first-passage probabilities. This approach yields intuition for Merton's result for the perpetual put. We then consider the finite-lived case. Using (1) the fact that the put value ceases to depend on time when the critical stock price is reached and (2) the result that an American put equals a European put plus an early-exercise premium, we derive the critical stock price. We approximate the critical-stock-price function to compute accurate put prices.