An Analytic Approximation for the American Put Price

Black and Scholes 1 derived the pricing equation for a European put when the stock price follows geometric Brownian motion. For this same case, Merton 5 derived the pricing equation for an American put with infinite time to maturity. Brennan and Schwartz 2, Rubinstein and Cox 7, and Parkinson 6 have developed numerical solutions for the price of an American put. Numerical solutions are expensive and do not provide much intuition. Naturally, an analytic solution would be much preferred; unfortunately, pricing the American put requires solving a formidable and presumably intractable boundary value problem.