Option pricing under deformed Gaussian distributions

•The paper presents and studies a class of deformed geometric Brownian motions.•Deformed Gaussian distributions are obtained generalizing Tsallis distribution.•Such deformed processes show an ability in describing fat tails.•These models describe a complete market: no price of risk is required.•Implied volatilities for European options are evaluated.

In financial literature many have been the attempts to overcome the option pricing drawbacks that affect the Black and Scholes model. Starting from the Tsallis deformation of the usual exponential function, this paper presents, in a complete market setup, a class of deformed geometric Brownian motions flexible enough to reproduce fat tails and to capture the volatility behavior observed in models that consider both stochastic volatility and jumps.