This Demonstration shows the optimal value for the exercise of an American option (call or put) in the Black–Scholes model. Unlike the European option, the American option allows early exercise. One can show that for all put options there is a price of the underlying stock such that when the stock is at (or below) this price, the option should be exercised. For call options on a stock that pays a nonzero continuous dividend, there is a stock price such that the option should be exercised when the stock price is at or above this optimal price. It is never optimal to exercise call options that pay no dividend.
This optimal stock price is shown for both put and call options. The optimal exercise point is shown in red on the graph of the Black–Scholes option value of the option. One can show that if an optimal price exists, the tangent to the graph at the red point has slope for call options and for put options. You can see the tangent by checking the “show tangent” checkbox.
The problem of computing the optimal exercise of an American option is known as a free-boundary problem for the associated Black–Scholes partial differential equations. The function FinancialDerivative in Mathematica 8 can efficiently compute the optimal exercise price of an American option (in the Black–Scholes model) without the need to explicitly set up and solve the corresponding free-boundary problem for the Black–Scholes PDE.
 P. Wilmott and J. Dewynne, Option Pricing, Mathematical Models and Computation, Oxford: Oxford Financial Press, 1993.