Hyperbolic Distribution

The hyperbolic distribution, introduced by Barndorff-Nielsen in 1977, is a continuous probability distribution characterized by its semi-heavy tails and potential asymmetry. It belongs to the family of normal variance-mean mixtures and provides a versatile model for financial returns, particle sizes, and wind speeds.

Definition

The hyperbolic distribution is characterized by four key parameters that define its shape and location:

$α > 0$ : Controls tail heaviness and decay rate
$| β | < α$ : Determines asymmetry
$δ > 0$ : Sets the scale
$μ$ : Specifies the location

The probability density function (PDF) is defined as:

f (x) = \frac{γ}{2 α δ K_{1} (δ γ)} e^{- α \sqrt{δ^{2} + (x - μ)^{2}} + β (x - μ)}

where $γ = \sqrt{α^{2} - β^{2}}$ and $K_{1}$ is the modified Bessel function of the second kind of order 1.

While the cumulative distribution function (CDF) can be expressed using special functions, it lacks a closed-form solution. The distribution is defined over all real numbers, making it suitable for modeling various continuous phenomena.

Applications

The hyperbolic distribution finds extensive use in financial modeling, particularly for modeling asset returns where traditional normal distributions fail to capture the heavy tails and skewness observed in real market data.

In environmental science, it serves as an effective model for wind speed distributions under turbulent atmospheric conditions, providing better fits than simpler distributions.

Risk managers and portfolio analysts employ the hyperbolic distribution for more accurate risk assessment and portfolio optimization, as it better accounts for extreme market events.

Environmental scientists use this distribution to analyze various environmental data sets where asymmetric patterns and heavy tails are common features.

Properties

Statistical Properties

Property	Value
Mean	$μ + \frac{δ β K_{2} (δ γ)}{γ K_{1} (δ γ)}$
Variance	$\frac{δ}{γ} \frac{K_{2} (δ γ)}{K_{1} (δ γ)} + {(\frac{β δ}{γ})}^{2} \frac{K_{1} (δ γ) K_{3} (δ γ) - K_{2} (δ γ)^{2}}{K_{1} (δ γ)^{2}}$

Notable characteristics of the hyperbolic distribution include:

The distribution exhibits semiheavy tails that are heavier than the normal distribution but lighter than power law distributions.
The logarithm of the probability density function forms a hyperbola in the plane.
The distribution remains stable under affine transformations, meaning it preserves its form under linear transformations.

Relationships to Other Distributions

The hyperbolic distribution approaches the normal distribution as $α$ approaches infinity with $β / α$ held constant.
The hyperbolic distribution is a special case of the generalized hyperbolic distribution.
The hyperbolic distribution is closely related to normal variance-mean mixtures in probability theory.

The HyperbolicDistribution class

The hyperbolic distribution is implemented by the HyperbolicDistribution class. It has one constructor that takes the 4 parameters in the following order: location, tail heaviness, asymmetry, and scale. The following constructs a hyperbolic distribution with location 0, scale 1, $α = 1.5$ and $β = 0.8$ :

var hyperbolic = new HyperbolicDistribution(0, 1.5, 0.8, 1);

Visual Basic

Dim hyperbolic = New HyperbolicDistribution(0, 1.5, 0.8, 1)

Visual Basic

No code example is currently available or this language may not be supported.

let hyperbolic = new HyperbolicDistribution(0, 1.5, 0.8, 1)

Note that evaluation of the Cumulative Distribution Function (CDF) of a hyperbolic distribution is very expensive because no closed form exists and it has to be evaluated using numerical integration of the Probability Density Function. Evaluating the inverse CDF is also expensive.

The HyperbolicDistribution class has four specific properties that correspond to the parameters of the distribution. The TailHeavinessParameter and AsymmetryParameter properties return the shape parameters, $α$ and $β$ . The LocationParameter and ScaleParameter properties return the location and scale parameters, respectively.

References

For more information on the hyperbolic distribution, see the following sources:

Barndorff-Nielsen, Ole (1977). "Exponentially decreasing distributions for the logarithm of particle size". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 353 (1674). The Royal Society: 401–409.