Comparison between the probability distribution of returns in the Heston model and empirical data for stock indexes

TL;DR

This paper compares the probability distribution of stock index returns with the Heston model's analytical formula, finding good agreement from 1982-1999 but deviations in 2000-2002, indicating a market regime change.

Contribution

It provides empirical validation of the Heston model's distribution against real market data over different periods.

Findings

Good fit between model and data from 1982-1999

Deviations observed in 2000-2002 data

Evidence of market regime change from positive to negative growth

Abstract

We compare the probability distribution of returns for the three major stock-market indexes (Nasdaq, S&P500, and Dow-Jones) with an analytical formula recently derived by Dragulescu and Yakovenko for the Heston model with stochastic variance. For the period of 1982-1999, we find a very good agreement between the theory and the data for a wide range of time lags from 1 to 250 days. On the other hand, deviations start to appear when the data for 2000-2002 are included. We interpret this as a statistical evidence of the major change in the market from a positive growth rate in 1980s and 1990s to a negative rate in 2000s.