A Theory of Non-Gaussian Option Pricing

A generalized option-pricing formula is found, based on a nonGaussian stock price model. The dynamics of the underlying stock are assumed to follow a stochastic process with anomalous nonlinear diffusion, phenomenologically modelled as a statistical feedback process within the framework of the generalized thermostatistics proposed by Tsallis. A generalized form of the Black-Scholes differential equation is derived. Solutions are obtained through risk-free asset valuation techniques. To this end, we formulate equivalent martingale measures for this class of stochastic processes. We obtain a closed form solution for the price of a European call option, characterized by the so-called entropic index q which arises within the Tsallis framework. The standard Black-Scholes pricing equations are recovered as a special case (q = 1). The distribution of stock returns is well-modelled with q circa 1.5. Using that value of q in the option pricing model we capture features found in real option prices such as the volatility smile. Acknowledgements Many insightful discussions with Roberto Osorio and Jeremy Evnine are gratefully acknowledged.