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. In the next year Barndorff-Nielsen published the NIG in another paper. It was introduced in the mathematical finance literature in 1997.
The parameters of the normal-inverse Gaussian distribution are often used to construct a heaviness and skewness plot called the NIG-triangle.
Properties
Moments
The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available.56
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 ∼ N I G ( α , β , δ , μ ) and y = a x + b , {\displaystyle x\sim {\mathcal {NIG}}(\alpha ,\beta ,\delta ,\mu ){\text{ and }}y=ax+b,}then7
y ∼ N I G ( α | a | , β a , | a | δ , a μ + b ) . {\displaystyle y\sim {\mathcal {NIG}}{\bigl (}{\frac {\alpha }{\left|a\right|}},{\frac {\beta }{a}},\left|a\right|\delta ,a\mu +b{\bigr )}.}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 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} are independent random variables that are NIG-distributed with the same values of the parameters α {\displaystyle \alpha } and β {\displaystyle \beta } , but possibly different values of the location and scale parameters, μ 1 {\displaystyle \mu _{1}} , δ 1 {\displaystyle \delta _{1}} and μ 2 , {\displaystyle \mu _{2},} δ 2 {\displaystyle \delta _{2}} , respectively, then X 1 + X 2 {\displaystyle X_{1}+X_{2}} is NIG-distributed with parameters α , {\displaystyle \alpha ,} β , {\displaystyle \beta ,} μ 1 + μ 2 {\displaystyle \mu _{1}+\mu _{2}} and δ 1 + δ 2 . {\displaystyle \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 ( μ , σ 2 ) , {\displaystyle N(\mu ,\sigma ^{2}),} arises as a special case by setting β = 0 , δ = σ 2 α , {\displaystyle \beta =0,\delta =\sigma ^{2}\alpha ,} and letting α → ∞ {\displaystyle \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 ( γ ) ( t ) = W ( t ) + γ t {\displaystyle W^{(\gamma )}(t)=W(t)+\gamma t} , we can define the inverse Gaussian process A t = inf { s > 0 : W ( γ ) ( s ) = δ t } . {\displaystyle A_{t}=\inf\{s>0:W^{(\gamma )}(s)=\delta t\}.} Then given a second independent drifting Brownian motion, W ( β ) ( t ) = W ~ ( t ) + β t {\displaystyle W^{(\beta )}(t)={\tilde {W}}(t)+\beta t} , the normal-inverse Gaussian process is the time-changed process X t = W ( β ) ( A t ) {\displaystyle X_{t}=W^{(\beta )}(A_{t})} . The process X ( t ) {\displaystyle X(t)} at time t = 1 {\displaystyle t=1} 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 I G {\displaystyle {\mathcal {IG}}} denote the inverse Gaussian distribution and N {\displaystyle {\mathcal {N}}} denote the normal distribution. Let z ∼ I G ( δ , γ ) {\displaystyle z\sim {\mathcal {IG}}(\delta ,\gamma )} , where γ = α 2 − β 2 {\displaystyle \gamma ={\sqrt {\alpha ^{2}-\beta ^{2}}}} ; and let x ∼ N ( μ + β z , z ) {\displaystyle x\sim {\mathcal {N}}(\mu +\beta z,z)} , then x {\displaystyle x} follows the NIG distribution, with parameters, α , β , δ , μ {\displaystyle \alpha ,\beta ,\delta ,\mu } . 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
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. doi:10.1098/rspa.1977.0041. JSTOR 79167. /wiki/Doi_(identifier) ↩
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 ↩
Paolella, Marc S (2007). Intermediate Probability: A computational Approach. John Wiley & Sons. ↩
Ole E Barndorff-Nielsen, Thomas Mikosch and Sidney I. Resnick, Lévy Processes: Theory and Applications, Birkhäuser 2013 ↩
Karlis, Dimitris (2002). "An EM Type Algorithm for ML estimation for the Normal–Inverse Gaussian Distribution". Statistics and Probability Letters. 57: 43–52. ↩