Testing stationarity of the detrended price return in stock markets

This paper proposes a generalized porous media equation with drift as the governing equation for stock market indexes. The proposed governing equation can be expressed as a Fokker–Planck equation (FPE) with a non-constant diffusion coefficient. The governing equation accounts for non-stationary effects and describes the time evolution of the probability distribution function (PDF) of the price return. By applying Ito's Lemma, the 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 stochastic part or q-Gaussian noise. The q-Gaussian noise can be decomposed into a Gaussian noise affected by a standard deviation or volatility. The presented model is validated using the S&P500 index's data from the past 25 years per minute. We show that the price return becomes Gaussian, consequently stationary by normalizing the detrended data set. The normalization of the data is calculated by subtracting the trend and then dividing by the standard deviation of the detrended price return. The stationarity test consists of representing the power spectrum in terms of the time series's autocorrelation. Additionally, this paper presents the multifractal analysis for the detrended and normalized price return to describe the Hurst exponent dynamics over the dataset.