Black-Scholes-Like Derivative Pricing With Tsallis Non-extensive Statistics

TL;DR

This paper extends the Black-Scholes derivative pricing model by incorporating Tsallis non-extensive statistics and nonlinear Fokker-Planck dynamics, providing a more accurate framework for markets with heavy tails and anomalous diffusion.

Contribution

It introduces a modified Black-Scholes formula based on non-extensive statistics, linking microscopic stochastic equations to derivative pricing.

Findings

The model accounts for heavy-tailed distributions in asset returns.

It demonstrates how non-extensive statistics alter option pricing.

The approach applies to markets exhibiting anomalous diffusion.

Abstract

We recently showed that the S&P500 stock market index is well described by Tsallis non-extensive statistics and nonlinear Fokker-Planck time evolution. We argued that these results should be applicable to a broad range of markets and exchanges where anomalous diffusion and `heavy' tails of the distribution are present. In the present work we examine how the Black-Scholes derivative pricing formula is modified when the underlying security obeys non-extensive statistics and Fokker-Planck time evolution. We answer this by recourse to the underlying microscopic Ito-Langevin stochastic differential equation of the non-extensive process.