10182

# Early Exercise of American Options

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.

### DETAILS

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.
Reference
[1] P. Wilmott and J. Dewynne, Option Pricing, Mathematical Models and Computation, Oxford: Oxford Financial Press, 1993.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.