  We undertake a systematic study of historic market volatility spanning
roughly five preceding decades. We focus specifically on the time series of
realized volatility (RV) of the S&P500 index and its distribution function. As
expected, the largest values of RV coincide with the largest economic upheavals
of the period: Savings and Loan Crisis, Tech Bubble, Financial Crisis and Covid
Pandemic. We address the question of whether these values belong to one of the
three categories: Black Swans (BS), that is they lie on scale-free, power-law
tails of the distribution; Dragon Kings (DK), defined as statistically
significant upward deviations from BS; or Negative Dragons Kings (nDK), defined
as statistically significant downward deviations from BS. In analyzing the
tails of the distribution with RV > 40, we observe the appearance of
"potential" DK which eventually terminate in an abrupt plunge to nDK. This
phenomenon becomes more pronounced with the increase of the number of days over
which the average RV is calculated -- here from daily, n=1, to "monthly," n=21.
We fit the entire distribution with a modified Generalized Beta (mGB)
distribution function, which terminates at a finite value of the variable but
exhibits a long power-law stretch prior to that, as well as Generalized Beta
Prime (GB2) distribution function, which has a power-law tail. We also fit the
tails directly with a straight line on a log-log scale. In order to ascertain
BS, DK or nDK behavior, all fits include their confidence intervals and
p-values are evaluated for the data points to check if they can come from the
respective distributions.
