Title: Distributions of Historic Market Data -- Implied and Realized Volatility,
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.