Option Pricing: The American Put

The problem of correctly, or perhaps rationally, setting the price of an option on a stock has been attacked many times.' Recently, Black and Scholes developed a very neat argument which provided a solution for the so-called European put and call options, and thus indirectly for the American call option.2 For it can be shown that it is better to retain an American call option until expiration rather than to convert it before expiration.3 In their paper, Black and Scholes also indicated that no formula for the value of an American put option had been constructed as yet. In Section IV of this paper, we present such a formula for the value of an American put option. In order to justify it, in Section II we review and discuss our assumptions about stock prices and their probability distributions. In Section III, we review and discuss the idea of a free and efficient options market and its consequences. In Section IV, we write down a general formula for American put and call options, and then assume for the first time in the paper that the distribution of stock prices is approximately lognormal, thus obtaining an explicit formula for the American put option. A graph of the predictions for a put option is presented; a table of put values is given which is suitable for most applications; and a limited comparison is made between the derived put formula and empirical data. The agreement is as good as can be expected, given the data available. In Section V, there are a summary and conclusions.