Closed Form Option Pricing Formulas based on a non-Gaussian Stock Price Model

An option pricing formula is obtained, based on a stochastic model with statistical feedback. The fluctuations evolve according to a Tsallis distribution which fits empirical data for stock returns. A generalized form of the Black-Scholes partial differential equation is derived, parametrized by the Tsallis entropic index q. A martingale representation is found, which allows us to use concepts of risk-free asset pricing theory. We explicity solve the case of European options and the exact closed-form solutions which are found capture features of real option prices, such as the volatility smile. Although empirical stock price returns clearly do not follow the lognormal distribution, many of the most famous results of mathematical finance are based on that distribution. For example, Black and Scholes [1] were able to derive the prices of options and other derivatives of the underlying stock based on such a model. An option is the right to buy or sell the underlying stock at some set price (called the strike) at some time in the future. While of great importance and widely used, such theoretical option prices do not quite match the observed ones. In particular, the Black-Scholes model underestimates the prices of options in situations when the stock price at the time of exercise is different from the strike. In order to match the observed