## Black-Scholes and American-style options

##### Published: Feb 2005

The Black-Scholes model values options as the difference between the present value of the stock on expiration day and the present value of exercising the option on expiration day. There is more detail on the derivation of the model in Treasury Today October 2004, December 2004 and January 2005.

In Treasury Today November 2004, we identified the main assumptions of the model. One of the assumptions is that the option is European-style. This means that the option can only be exercised on expiration day.

The problem with this assumption is that it undervalues American-style options, which can be exercised at any time up to expiration date. Likewise, it undervalues exchange-traded options which can be resold at any point up to expiration.

By comparing two options, one European and one American, otherwise with the same terms, it is clear that the American style option is of higher value to the holder. This is because it offers the holder more opportunity to exercise the option.

Although there is no strict formula to value an American-style option, it is possible to identify an approximate value by employing one of the following techniques:

#### A basic approach

The simplest method is to value the American-style option as the higher of the option’s European value and its intrinsic value.

• The European value is calculated using the model we explained in recent issues and will give the option’s value were it to be exercised on expiration day.
• The intrinsic value (the stock price minus the option strike price) will be positive as long as the option is ‘in-the-money’. The intrinsic value equals the value of the option if it is exercised today.

This approach computes the value of an option which can be exercised on two dates – today or on expiration day. Where there is still some time until expiration, it does not give a value to the holder (of an American-style option) of having the opportunity to exercise the option between today and expiration day.

An alternative method is to use a quadratic approximation method developed by Barone-Adesi and Whaley. This defines the value of an American call option as follows:

Where:

• Ca is the value of the American-style call option.
• C is the value of a European-style call option with the same terms.
• Q is a quadratic equation providing an approximation of the value of the early exercise option.

The quadratic equation is solved using an iterative process. It requires significant computational power and is beyond the scope of this article.

More information can be found from a number of locations. One site providing more detail on the equations and various methods of calculation is www.global-derivatives.com/options/american-options.php