R. Geske and H. E. Johnson (1984) develop an equation for the American put price and obtain accurate prices using a method requiring quadrivariate normal integrals evaluated over an interval containing four equally spaced exercise points. The authors show that a modification of their method, which uses optimal placement of exercise points, yields, in most cases, accurate values using nothing more than bivariate normals. In the more difficult (deep-in-the-money) cases, trivariate normals suffice.

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.

Geske and Johnson (1984) develop an equation for the American put price and obtain accurate prices using a method requiring quadrivariate normal integrals evaluated over an interval containing four equally spaced exercise points. We show that a modification of their method which uses optimal placement of exercise points yields in most cases accurate values using nothing more than bivariate normals. In the more difficult (deep‐in‐the‐money) cases, trivariate normals suffice.

Geske and Johnson (1984) develop an equation for the American put price and obtain accurate prices using a method requiring quadrivariate normal integrals evaluated over an interval containing four equally spaced exercise points. We show that a modification of their method which uses optimal placement of exercise points yields in most cases accurate values using nothing more than bivariate normals. In the more difficult (deep-in-the-money) cases, trivariate normals suffice.