Option pricing formulas based on a non-Gaussian stock price model

Options are financial instruments that depend on the underlying stock. We explain their non-Gaussian fluctuations using the nonextensive thermodynamics parameter q. A generalized form of the Black-Scholes (BS) partial differential equation and some closed-form solutions are obtained. The standard BS 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.