Distributions of Historic Market Data - Stock Returns

TL;DR

This paper compares the Heston and multiplicative models for stock return distributions, showing that the Heston model aligns well with empirical data, especially regarding moments and variance behavior.

Contribution

It demonstrates that the Heston model accurately describes the distribution of historic stock returns, unlike the multiplicative model, and derives the return distribution function from the variance distribution.

Findings

Heston model matches empirical moments of stock returns

Mean realized variance is linear with number of days

Distribution of returns scales with days of return

Abstract

We show that the moments of the distribution of historic stock returns are in excellent agreement with the Heston model and not with the multiplicative model, which predicts power-law tails of volatility and stock returns. We also show that the mean realized variance of returns is a linear function of the number of days over which the returns are calculated. The slope is determined by the mean value of the variance (squared volatility) in the mean-reverting stochastic volatility models, such as Heston and multiplicative, independent of stochasticity. The distribution function of stock returns, which rescales with the increase of the number of days of return, is obtained from the steady-state variance distribution function using the product distribution with the normal distribution.