Stable Count Distribution for the Volatility Indices and Space-Time Generalized Stable Characteristic Function

Abstract

This paper collects statistical and mathematical properties of the stable count distribution, with dispersed treatises on economic regime related to volatility. It is hypothesized that the stable count distribution is the marginal distribution of the volatility indices, such as CBOE VIX and VXTYN. Mathematically, its PDF is a special case of the Wright function. Analytical solutions for its moments and moment-generating function are carried out in terms of the Fox-Wright function. We show that it can be the kernel of a product distribution in fractional dynamics and quantitative finance. In the second half of the paper, the stable distribution family is generalized. A novel mathematical approach is taken by defining a space-time generalized stable characteristic function via the Mittag-Leffler function. The fractional PDF it produces is shown to be consistent with the fractional calculus of Riesz-Feller space derivative and Caputo-Mainardi time derivative. Small deviations in both time-fractional and space-fractional scenarios are studied to explain the ultra-high volatility tail observed in the VIX data during bear markets. The solutions match VIX data up to the last one percentile in the right tail, which corresponds to the largest market crashes in financial history.

Keywords: stable count distribution, volatility index, fractional calculus, generalized stable characteristic function, the Wright function, the Fox-Wright function, the Mittag-Leffler function

JEL Classification: C02, C58, G01