# Normal-inverse Gaussian distribution

The **normal-inverse Gaussian distribution (NIG)** is a continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the inverse Gaussian distribution. The NIG distribution was noted by Blaesild in 1977 as a subclass of the generalised hyperbolic distribution discovered by Ole Barndorff-Nielsen.[2] In the next year Barndorff-Nielsen published the NIG in another paper.[3] It was introduced in the mathematical finance literature in 1997.[4]

Parameters |
location (real) tail heaviness (real) asymmetry parameter (real) scale parameter (real) | ||
---|---|---|---|

Support | |||

denotes a modified Bessel function of the third kind[1] | |||

Mean | |||

Variance | |||

Skewness | |||

Ex. kurtosis | |||

MGF | |||

CF |

The parameters of the normal-inverse Gaussian distribution are often used to construct a heaviness and skewness plot called the NIG-triangle.[5]

## Properties

### Moments

The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available.[6][7]

### Linear transformation

This class is closed under affine transformations, since it is a particular case of the Generalized hyperbolic distribution, which has the same property.

### Summation

This class is infinitely divisible, since it is a particular case of the Generalized hyperbolic distribution, which has the same property.

### Convolution

The class of normal-inverse Gaussian distributions is closed under convolution in the following sense:[8] if and are independent random variables that are NIG-distributed with the same values of the parameters and , but possibly different values of the location and scale parameters, , and , respectively, then is NIG-distributed with parameters and

## Related distributions

The class of NIG distributions is a flexible system of distributions that includes fat-tailed and skewed distributions, and the normal distribution, arises as a special case by setting and letting .

## Stochastic process

The normal-inverse Gaussian distribution can also be seen as the marginal distribution of the normal-inverse Gaussian process which provides an alternative way of explicitly constructing it. Starting with a drifting Brownian motion (Wiener process), , we can define the inverse Gaussian process Then given a second independent drifting Brownian motion, , the normal-inverse Gaussian process is the time-changed process . The process at time 1 has the normal-inverse Gaussian distribution described above. The NIG process is a particular instance of the more general class of Lévy processes.

## References

- Ole E Barndorff-Nielsen, Thomas Mikosch and Sidney I. Resnick, Lévy Processes: Theory and Applications, Birkhäuser 2013
*Note: in the literature this function is also referred to as Modified Bessel function of the third kind* - 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*. The Royal Society.**353**(1674): 401–409. doi:10.1098/rspa.1977.0041. JSTOR 79167. - O. Barndorff-Nielsen, Hyperbolic Distributions and Distributions on Hyperbolae, Scandinavian Journal of Statistics 1978
- O. Barndorff-Nielsen, Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling, Scandinavian Journal of Statistics 1997
- S.T Rachev, Handbook of Heavy Tailed Distributions in Finance, Volume 1: Handbooks in Finance, Book 1, North Holland 2003
- Erik Bolviken, Fred Espen Beth, Quantification of Risk in Norwegian Stocks via the Normal Inverse Gaussian Distribution, Proceedings of the AFIR 2000 Colloquium
- Anna Kalemanova, Bernd Schmid, Ralf Werner, The Normal inverse Gaussian distribution for synthetic CDO pricing, Journal of Derivatives 2007
- Ole E Barndorff-Nielsen, Thomas Mikosch and Sidney I. Resnick, Lévy Processes: Theory and Applications, Birkhäuser 2013