Normal-inverse Gaussian distribution explained
The normal-inverse Gaussian distribution (NIG, also known as the normal-Wald distribution) 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.[1] In the next year Barndorff-Nielsen published the NIG in another paper.[2] It was introduced in the mathematical finance literature in 1997.[3]
The parameters of the normal-inverse Gaussian distribution are often used to construct a heaviness and skewness plot called the NIG-triangle.[4]
Properties
Moments
The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available.[5] [6]
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. If
x\siml{NIG}(\alpha,\beta,\delta,\mu)andy=ax+b,
then
[7] y\siml{NIG}l( | \alpha | \left|a\right| | , | \beta |
a |
,\left|a\right|\delta,a\mu+br).
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
\beta=0,\delta=\sigma2\alpha,
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),
W(\gamma)(t)=W(t)+\gammat
, we can define the inverse Gaussian process
Then given a second independent drifting Brownian motion,
W(\beta)(t)=\tildeW(t)+\betat
, the normal-inverse Gaussian process is the time-changed process
. The process
at time
has the normal-inverse Gaussian distribution described above. The NIG process is a particular instance of the more general class of
Lévy processes.
As a variance-mean mixture
Let
denote the
inverse Gaussian distribution and
denote the
normal distribution. Let
z\siml{IG}(\delta,\gamma)
, where
\gamma=\sqrt{\alpha2-\beta2}
; and let
, then
follows the NIG distribution, with parameters,
. This can be used to generate NIG variates by ancestral sampling. It can also be used to derive an
EM algorithm for
maximum-likelihood estimation of the NIG parameters.
[9] References
- 10.1098/rspa.1977.0041. 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. 401–409. 79167. The Royal Society.
- 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
- Book: Paolella . Marc S . Intermediate Probability: A computational Approach . 2007 . John Wiley & Sons.
- Ole E Barndorff-Nielsen, Thomas Mikosch and Sidney I. Resnick, Lévy Processes: Theory and Applications, Birkhäuser 2013
- Karlis . Dimitris . An EM Type Algorithm for ML estimation for the Normal–Inverse Gaussian Distribution . Statistics and Probability Letters . 2002 . 57 . 43-52.