Normal Inverse Gaussian Distribution

The normal-inverse Gaussian distribution (NIG distribution) is a continuous probability distribution that is a normal variance-mean mixture with an inverse Gaussian distribution as the mixing density. It is also known as the NIG distribution.

The normal-inverse Gaussian distribution has a location parameter $\mu$, a scale parameter $\delta$, and two shape parameters $\alpha$ and $\beta$. Its probability density function (PDF) is:

where $K_1$ is the modified Bessel function of the second kind. The parameters must satisfy the following conditions:

• $\alpha >0$

• $|\beta |<\alpha$

• $\delta >0$

• Financial mathematics uses the distribution to model asset returns with heavy tails and skewness.

• Risk management employs it for modeling financial risk factors.

• Statistical modeling uses it for handling data with asymmetry and heavy tails.

• Turbulence modeling applies it to describe velocity increments in fluid dynamics.

Notable properties include:

• The characteristic function is $\varphi (t)=\exp (i\mu t+\delta (\sqrt{{\alpha }^2-{\beta }^2}-\sqrt{{\alpha }^2-(\beta +it{)}^2}))$.

• The distribution is infinitely divisible.

• The tails are heavier than the normal distribution but lighter than the Cauchy distribution.

• The normal-inverse Gaussian distribution is a special case of the generalized hyperbolic distribution.

• When $\alpha \to \mathrm{\infty }$ with $\beta /\alpha$ fixed, it approaches a normal distribution.

• It arises as a normal variance-mean mixture with the inverse Gaussian as the mixing distribution.

The normal inverse Gaussian distribution is implemented by the NormalInverseGaussianDistribution 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, $\alpha =1.5$ and $\beta =0.8$:

var nig = new NormalInverseGaussianDistribution(6.8, 4.1, 1.3, 3.2);

Dim nig = new NormalInverseGaussianDistribution(6.8, 4.1, 1.3, 3.2)

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

let nig = new NormalInverseGaussianDistribution(6.8, 4.1, 1.3, 3.2)

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 NormalInverseGaussianDistribution class has four specific properties that correspond to the parameters of the distribution. The LocationParameter and ScaleParameter properties return the location and scale parameters, respectively. The Alpha and Beta properties return the shape parameters, $\alpha$ and $\beta$.

• Barndorff-Nielsen, O. E. (1997). Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling. Scandinavian Journal of Statistics, 24(1), 1-13.

• Bibby, B. M., & Sørensen, M. (2003). Hyperbolic Processes in Finance. In Handbook of Heavy Tailed Distributions in Finance (pp. 211-248). Elsevier.

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