# Q-Gaussian diffusion in stock markets

## TL;DR

This paper investigates the statistical properties of S&P 500 stock returns over 22 years, revealing two regimes characterized by q-Gaussian distributions and deriving their governing equations and diffusion coefficients.

## Contribution

It introduces a dual-regime analysis of stock return distributions using q-Gaussian models and derives the governing equations from the porous media framework.

## Key findings

- Identification of two distinct super-diffusive regimes
- Both regimes are accurately modeled by q-Gaussian distributions
- Explicit derivation of the Black-Scholes diffusion coefficient

## Abstract

We analyze the Standard & Poor's 500 stock market index from the last 22 years. The probability density function of price returns exhibits two well-distinguished regimes with self-similar structure: the first one displays strong super-diffusion together with short-time correlations, and the second one corresponds to weak super-diffusion with weak time correlations. Both regimes are well-described by q-Gaussian distributions. The porous media equation is used to derive the governing equation for these regimes, and the Black-Scholes diffusion coefficient is explicitly obtained from the governing equation.