Stable count distribution


In probability theory, the stable count distribution is the conjugate prior of a one-sided stable distribution. This distribution was discovered by Stephen Lihn in his 2017 study of daily distributions of S&P 500 index and the VIX index. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it.
Of the three parameters defining the distribution, the stability parameter is most important. Stable count distributions have. The known analytical case of is related to the VIX distribution. All the moments are finite for the distribution.

Definition

Its standard distribution is defined as
where and
Its location-scale family is defined as
where ,, and
In the above expression, is a one-sided stable distribution, which is defined as following.
Let be a standard stable random variable whose distribution is characterized by, then we have
where.
Consider the Lévy sum where, then has the density where. Set, we arrive at without the normalization constant.
The reason why this distribution is called "stable count" can be understood by the relation. Note that is the "count" of the Lévy sum. Given a fixed, this distribution gives the probability of taking steps to travel one unit of distance.

Integral form

Based on the integral form of and, we have the integral form of as
Based on the double-sine integral above, it leads to the integral form of the standard CDF:
where is the sine integral function.

The Wright representation

In "Series representation", it is shown that the stable count distribution is a special case of the Wright function:
This leads to the Hankel integral:

Alternative derivation – lambda decomposition

Another approach to derive the stable count distribution is to use the Laplace transform of the one-sided stable distribution,
Let, and one can decompose the integral on the left hand side as a product distribution of a standard Laplace distribution and a standard stable count distribution,
where .
This is called the "lambda decomposition" since the LHS was named as "symmetric lambda distribution" in Lihn's former works. However, it has several more popular names such as "exponential power distribution", or the "generalized error/normal distribution", often referred to when .
Lambda decomposition is the foundation of Lihn's framework of asset returns under the stable law. The LHS is the distribution of asset returns. On the RHS, the Laplace distribution represents the lepkurtotic noise, and the stable count distribution represents the volatility.

Asymptotic properties

For stable distribution family, it is essential to understand its asymptotic behaviors. From, for small,
This confirms.
For large,
This shows that the tail of decays exponentially at infinity. The larger is, the stronger the decay.

Moments

The n-th moment of is the -th moment of. All positive moments are finite. This in a way solves the thorny issue of diverging moments in the stable distribution.
The analytic solution of moments is obtained through the Wright function:
where
Thus, the mean of is
The variance is

Moment generating function

The MGF can be expressed by a Fox-Wright function or Fox H-function:
As a verification, at, can be Taylor-expanded to via.

Known analytical case – quartic stable count

When, is the Lévy distribution which is an inverse gamma distribution. Thus is a shifted gamma distribution of shape 3/2 and scale,
where,.
Its mean is and its standard deviation is. This called "quartic stable count distribution". The word "quartic" comes from Lihn's former work on the lambda distribution where. At this setting, many facets of stable count distribution have elegant analytical solutions.
The p-th central moments are. The CDF is where is the lower incomplete gamma function. And the MGF is.

Special case when α → 1

As becomes larger, the peak of the distribution becomes sharper. A special case of is when. The distribution behaves like a Dirac delta function,
where, and.

Series representation

Based on the series representation of the one-sided stable distribution, we have:
This series representation has two interpretations:
The proof is obtained by the reflection formula of the Gamma function:, which admits the mapping: in. The Wright representation leads to analytical solutions for many statistical properties of the stable count distribution and establish another connection to fractional calculus.

Applications

Stable count distribution can represent the daily distribution of VIX quite well. It is hypothesized that VIX is distributed like with and . Thus the stable count distribution is the first-order marginal distribution of a volatility process. In this context, is called the "floor volatility". In practice, VIX rarely drops below 10. This phenomenon justifies the concept of "floor volatility". A sample of the fit is shown below:
One form of mean-reverting SDE for is based on a modified Cox–Ingersoll–Ross model. Assume is the volatility process, we have
where is the so-called "vol of vol". The "vol of vol" for VIX is called VVIX, which has a typical value of about 85.
This SDE is analytically tractable and satisfies , thus would never go below. But there is an subtle issue between theory and practice. There has been about 0.6% probability that VIX did go below. This is called "spillover". To address it, one can replace the square root term with, where provides a small leakage channel for to drift slightly below.
Extremely low VIX reading indicates a very complacent market. Thus the spillover condition,, carries a certain significance - When it occurs, it usually indicates the calm before the storm in the business cycle.

Fractional calculus

Relation to Mittag-Leffler function

From Section 4 of, the inverse Laplace transform of the Mittag-Leffler function is
On the other hand, the following relation was given by Pollard,
Thus by, we obtain the relation between stable count distribution and Mittag-Leffter function:
This relation can be verified quickly at where and. This leads to the well-known quartic stable count result:

Relation to time-fractional Fokker-Planck equation

The ordinary Fokker-Planck equation is , where is the Fokker-Planck space operator, is the diffusion coefficient, is the temperature, and is the external field. The time-fractional FPE introduces the additional fractional derivative such that, where is the fractional diffusion coefficient.
Let in, we obtain the kernel for the time-fractional FPE
from which the fractional density can be calculated from an ordinary solution via
Since via change of variable, the above integral becomes the product distribution with, similar to the "lambda decomposition" concept, and scaling of time :
Here is interpreted as the distribution of impurity, expressed in the unit of, that causes the anomalous diffusion.