Distributions of Historic Market Data -- Implied and Realized Volatility

TL;DR

This paper systematically compares implied volatility measures VIX and VXO with realized volatility, analyzing their distributional properties and the ratio's heavy-tailed nature, while also evaluating stochastic volatility models against historical data.

Contribution

It introduces a comprehensive statistical comparison of implied and realized volatility distributions, including the ratio's heavy-tailed characteristics and model-based variance analysis.

Findings

The ratio of implied to realized variance is best fitted by heavy-tailed distributions.

No substantial difference in accuracy between VIX and VXO.

Stochastic volatility models' variance estimates are compared with historical data.

Abstract

We undertake a systematic comparison between implied volatility, as represented by VIX (new methodology) and VXO (old methodology), and realized volatility. We compare visually and statistically distributions of realized and implied variance (volatility squared) and study the distribution of their ratio. We find that the ratio is best fitted by heavy-tailed -- lognormal and fat-tailed (power-law) -- distributions, depending on whether preceding or concurrent month of realized variance is used. We do not find substantial difference in accuracy between VIX and VXO. Additionally, we study the variance of theoretical realized variance for Heston and multiplicative models of stochastic volatility and compare those with realized variance obtained from historic market data.