Rethinking Generalized Beta Family of Distributions

TL;DR

This paper introduces a stochastic differential equation approach to the Generalized Beta family of distributions, providing new insights into their properties, limits, and applications to financial market volatility analysis.

Contribution

It offers a novel SDE-based framework for understanding GB distributions, clarifies their limits, and derives an alternative PDF form for better tail analysis.

Findings

The SDE approach explains the limits of GB distributions.

Derived an alternative form of the GB PDF emphasizing finite-range power-law behavior.

Applied the model to S&P 500 volatility data, identifying negative Dragon Kings.

Abstract

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.