The Pricing of Stock Options

We describe how a stock price model based on nonextensive statistics can be used to derive a generalized theory for pricing stock options. A review of theoretical and empirical results is presented…. In 1973, Black and Scholes [1] and Merton [12] published their seminal papers which developed a theory of the fair price of options. Scholes and Merton were later to receive the 1997 Nobel prize for this famous work (Fisher Black had unfortunately passed away two years earlier). Options are important financial instruments which are traded in a huge volume all around the world on a variety of exchanges. There are options on underlying assets ranging from orange juice to gold, stocks to currency. In principle, an option is simply the right—but not the obligation—to execute some previously agreed upon action, for example, the right to buy or sell the underlying asset at some predetermined price, called the strike. It is not difficult to understand that the existence of such instruments could be extremely useful—for example, the right to buy an asset at a certain price protects against unforeseen events which could lead to huge price rises and thereby losses to someone who knows that they will need the asset at some time in the future. Similarly, the right to sell the asset at a certain price will protect against unforeseen drops in its value. These examples illustrate the use of options to hedge oneself against possible future events. Another use is more speculative: If a trader believes that the price of a stock will rise above a certain price at some date in the future, then it is in his interest to secure an option to buy the stock at some fixed lower price. Then, if the price of the stock does rise above that price, the trader can execute his option, just to turn around and resell the stock again at the higher market price.