A long-range memory stochastic model of the return in financial markets

TL;DR

This paper introduces a nonlinear stochastic differential equation model that captures the long-range memory and power law statistics of financial market returns, effectively reproducing empirical NYSE data.

Contribution

It proposes a novel SDE model based on nonextensive statistical mechanics that accurately mimics return distributions and spectral properties in financial markets.

Findings

Model reproduces empirical return PDFs and power spectra

Captures long-range memory effects in market volatility

Aligns with observed data for NYSE one-minute returns

Abstract

We present a nonlinear stochastic differential equation (SDE) which mimics the probability density function (PDF) of the return and the power spectrum of the absolute return in financial markets. Absolute return as a measure of market volatility is considered in the proposed model as a long-range memory stochastic variable. The SDE is obtained from the analogy with earlier proposed model of trading activity in the financial markets and generalized within the nonextensive statistical mechanics framework. The proposed stochastic model generates time series of the return with two power law statistics, i.e., the PDF and the power spectral density, reproducing the empirical data for the one minute trading return in the NYSE.