  We approach the Generalized Beta (GB) family of distributions using a
mean-reverting stochastic differential equation (SDE) for a power of the
variable, whose steady-state (stationary) probability density function (PDF) is
a modified GB (mGB) distribution. The SDE approach allows for a lucid
explanation of Generalized Beta Prime (GB2) and Generalized Beta (GB1) limits
of GB distribution and, further down, of Generalized Inverse Gamma (GIGa) and
Generalized Gamma (GGa) limits, as well as describe the transition between the
latter two. We provide an alternative form to the "traditional" GB PDF to
underscore that a great deal of usefulness of GB distribution lies in its
allowing a long-range power-law behavior to be ultimately terminated at a
finite value. We derive the cumulative distribution function (CDF) of the
"traditional" GB, which belongs to the family generated by the regularized beta
function and is crucial for analysis of the tails of the distribution. We
analyze fifty years of historical data on realized market volatility,
specifically for S\&P500, as a case study of the use of GB/mGB distributions
and show that its behavior is consistent with that of negative Dragon Kings.

Statistical Finance;Econometrics;Mathematical Finance