arXiv:cond-mat/0204331v7 [cond-mat.stat-mech] 18 Sep 2002 Option Pricing Formulas based on a non-Gaussian Stock Price Model Lisa Borland Iris Financial Engineering and Systems 456 Montgomery Street, Suite 800 San Francisco, CA 94104, USA February 1, 2008 Abstract Options are financial instruments that depend on the underlying stock. We explain their non-Gaussian fluctuations using the nonexten- sive thermodynamics parameter q. A generalized form of the Black- Scholes (B-S) partial differential equation, and some closed-form so- lutions are obtained. The standard B-S equation (q = 1) which is used by economists to calculate option prices requires multiple values of the stock volatility (known as the volatility smile). Using q =1.5 which well models the empirical distribution of returns, we get a good description of option prices using a single volatility. Although empirical stock price returns clearly do not follow the log- normal distribution, many of the most famous results of mathematical fi- nance are based on that distribution. For example, Black and Scholes (B-S) [1] were able to derive the prices of options and other derivatives of the un- derlying 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 B-S 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 1