Log Heston Model for Monthly Average VIX

TL;DR

This paper introduces a log-Heston model for monthly VIX and stock returns, demonstrating that normalized returns approximate Gaussianity and capturing heavy tails with a mean-reverting stochastic volatility framework.

Contribution

It proposes a novel log-Heston model for monthly VIX and stock returns, highlighting the normalization of returns to Gaussianity and modeling non-Gaussian innovations.

Findings

Normalized returns are close to Gaussian

Model captures Pareto-like tails

Works for various indices and return types

Abstract

We model time series of VIX (monthly average) and monthly stock index returns. We use log-Heston model: logarithm of VIX is modeled as an autoregression of order 1. Our main insight is that normalizing monthly stock index returns (dividing them by VIX) makes them much closer to independent identically distributed Gaussian. The resulting model is mean-reverting, and the innovations are non-Gaussian. The combined stochastic volatility model fits well, and captures Pareto-like tails of real-world stock market returns. This works for small and large stock indices, for both price and total returns.