Stationarity of the detrended price return in stock markets

TL;DR

This paper introduces a new stochastic model for stock index returns that incorporates non-stationary effects and demonstrates the stationarity of detrended returns using empirical data analysis.

Contribution

It develops a linear Fokker-Planck and associated SDE model with q-Gaussian noise for stock returns, accounting for non-stationarity and validating it with real market data.

Findings

Detrended stock returns are stationary.

The model captures non-stationary effects in stock markets.

Empirical validation confirms the model's effectiveness.

Abstract

This paper proposes a governing equation for stock market indexes that accounts for non-stationary effects. This is a linear Fokker-Planck equation (FPE) that describes the time evolution of the probability distribution function (PDF) of the price return. By applying Ito's lemma, this FPE is associated with a stochastic differential equation (SDE) that models the time evolution of the price return in a fashion different from the classical Black-Scholes equation. Both FPE and SDE equations account for a deterministic part or trend, and a stationary, stochastic part as a q-Gaussian noise. The model is validated using the S\&P500 index's data. After removing the trend from the index, we show that the detrended part is stationary by evaluating the Hurst exponent of the multifractal time series, its power spectrum, and its autocorrelation.