# Normal-inverse Gaussian distribution

Parameters $\mu$ location (real) $\alpha$ tail heaviness (real) $\beta$ asymmetry parameter (real) $\delta$ scale parameter (real) $\gamma = \sqrt{\alpha^2 - \beta^2}$ $x \in (-\infty; +\infty)\!$ $\frac{\alpha\delta K_1 \left(\alpha\sqrt{\delta^2 + (x - \mu)^2}\right)}{\pi \sqrt{\delta^2 + (x - \mu)^2}} \; e^{\delta \gamma + \beta (x - \mu)}$ $K_j$ denotes a modified Bessel function of the third kind[1] $\mu + \delta \beta / \gamma$ $\delta\alpha^2/\gamma^3$ $3 \beta /(\alpha \sqrt{\delta \gamma})$ $3(1+4 \beta^2/\alpha^2)/(\delta\gamma)$ $e^{\mu z + \delta (\gamma - \sqrt{\alpha^2 -(\beta +z)^2})}$ $e^{i\mu z + \delta (\gamma - \sqrt{\alpha^2 -(\beta +iz)^2})}$

The normal-inverse Gaussian distribution (NIG) is 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]

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 $X_1$ and $X_2$ are independent random variables that are NIG-distributed with the same values of the parameters $\alpha$ and $\beta$, but possibly different values of the location and scale parameters, $\mu_1$, $\delta_1$ and $\mu_2,$ $\delta_2$, respectively, then $X_1 + X_2$ is NIG-distributed with parameters $\alpha,$ $\beta,$$\mu_1+\mu_2$ and $\delta_1 + \delta_2.$

## Related Distributions

The class of NIG distributions is a flexible system of distributions that includes fat-tailed and skewed distributions, and the normal distribution, $N(\mu,\sigma^2),$ arises as a special case by setting $\beta=0, \delta=\sigma^2\alpha,$ and letting $\alpha\rightarrow\infty$.

## 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), $W^{(\gamma)}(t)=W(t)+\gamma t$, we can define the inverse Gaussian process $A_t=\inf\{s>0:W^{(\gamma)}(s)=\delta t\}.$ Then given a second independent drifting Brownian motion, $W^{(\beta)}(t)=\tilde W(t)+\beta t$, the normal-inverse Gaussian process is the time-changed process $X_t=W^{(\beta)}(A_t)$. The process $X(t)$ 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

1. ^ 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
2. ^ 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.
3. ^ O. Barndorff-Nielsen, Hyperbolic Distributions and Distributions on Hyperbolae, Scandinavian Journal of Statistics 1978
4. ^ O. Barndorff-Nielsen, Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling, Scandinavian Journal of Statistics 1997
5. ^ S.T Rachev, Handbook of Heavy Tailed Distributions in Finance, Volume 1: Handbooks in Finance, Book 1, North Holland 2003
6. ^ Erik Bolviken, Fred Espen Beth, Quantification of Risk in Norwegian Stocks via the Normal Inverse Gaussian Distribution, Proceedings of the AFIR 2000 Colloquium
7. ^ Anna Kalemanova, Bernd Schmid, Ralf Werner, The Normal inverse Gaussian distribution for synthetic CDO pricing, Journal of Derivatives 2007
8. ^ Ole E Barndorff-Nielsen, Thomas Mikosch and Sidney I. Resnick, Lévy Processes: Theory and Applications, Birkhäuser 2013