A Theory of Non_Gaussian Option Pricing

TL;DR

This paper develops a non-Gaussian option pricing model based on a nonlinear Fokker-Planck equation and Tsallis entropy, providing a generalized Black-Scholes formula that captures market phenomena like the volatility smile.

Contribution

It introduces a novel non-Gaussian framework for option pricing using Tsallis entropy, extending the Black-Scholes model to better fit empirical market data.

Findings

The model accurately reproduces the volatility smile.

It closely matches market prices for Japanese Yen options.

The generalized formula reduces to Black-Scholes when q=1.

Abstract

Option pricing formulas are derived from a non-Gaussian model of stock returns. Fluctuations are assumed to evolve according to a nonlinear Fokker-Planck equation which maximizes the Tsallis nonextensive entropy of index $q$. A generalized form of the Black-Scholes differential equation is found, and we derive a martingale measure which leads to closed form solutions for European call options. 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 reproduce the volatility smile. The partial derivatives (or Greeks) of the model are also calculated. Empirical results are demonstrated for options on Japanese Yen futures. Using just one value of $\sigma$ across strikes we closely reproduce market prices, for expiration times ranging from weeks to several months.