Implied and Realized Volatility: A Study of Distributions and the Distribution of Difference

Abstract: We study distributions of realized variance (squared realized volatility) and squared implied volatility, as represented by VIX and VXO indices. We find that Generalized Beta distribution provide the best fits. These fits are much more accurate for realized variance than for squared VIX and VXO -- possibly another indicator that the latter have deficiencies in predicting the former. We also show that there are noticeable differences between the distributions of the 1970-2017 realized variance and its 1990-2017 portion, for which VIX and VXO became available. This may be indicative of a feedback effect that implied volatility has on realized volatility. We also discuss the distribution of the difference between squared implied volatility and realized variance and show that, at the basic level, it is consistent with Pearson's correlations obtained from linear regression.