This article summarizes several methods for pricing American options, and provides free spreadsheets for each.
European options are commonly traded in the commodity markets. They have closed-form pricing equations, derived from the traditional Black-Scholes analysis. The equations are easily implemented in spreadsheets or programming languages.
Most exchange-traded options are, however, American options. American options can be exercised at or before expiry; this greater flexibility for the option holder results in greater risk for the option writer. This means American options are more expensive than European options
American options do not have closed-form pricing equations. Accordingly, many numerical techniques and approximations for pricing American options have been developed. Several of the most popular methods are summarized below.
Barone-Adesi & Whaley
This method separates the value of American options into two parts. The first is the value of an European option, and the second is the value of early exercise. The latter is given by a partial differential equations, which Barone-Adesi & Whaley approximate with a quadratic equation (hence the alternative name of the method, the Quadratic Method)
This approximation model can be used for options with a continuous dividend, a constant dividend yield, and discrete dividends.
Detailed technical information is given in this reference: “Efficient analytic approximation of american option values”, Journal of Finance 42, 301-320. C code for the Barone-Adesi & Whaley approximation is given here (the Excel spreadsheet is partly based on this code).
Bjerksund & Stensland
The Bjerksund & Stensland approximation was developed in 1993. The method is fast and computationally efficient. For long-dated options, the Bjerksund & Stensland model is more accurate than the Barone-Adesi & Whaley method.
If the value of the underlying asset is greater than or equal to a trigger price, the option should be exercised. However, if the risk-free rate is less than or equal to the cost of carry, the the value of the option is calculated by the Black-Scholes Model
Ju & Zhong
This method, first published in 1999, is more accurate than the quadratic approximation for options with small or large maturity times. The spreadsheet code is is derived from the original Ju & Zhong paper.
Binomial and Trinomial Trees
Binomial and trinomial option pricing methods give the price of an underlying stock over a period of time. This makes them particularly suitable for pricing American options, which can be exercised at any time before expiry.
Both methods involves three general steps
- A tree for stock prices is constructed. At each time step, the price can either go up or down (for binomial trees). Additionally, trinomial trees allow the stock price to remain the same at each time step
- The value of the option at maturity is calculated
- The value of the option at any time befory expiry is calculated through backwards induction
- are easily understood and do not require complex mathematics,
- can be quickly implemented in spreadsheets,
- and can be modified to include dividends.
Excel Spreadsheets for Pricing American Options
These Excel spreadsheets implement the pricing approximations described above. Any of these Excel spreadsheets can be easily adapted to calculated the implied volatility of an American option by using Excel’s Goal Seek functionality.