We start with an exposition of the univariate Generalized Hyperbolic (GH) Distribution in- troduced in the literature by Ole Barndorff-Neilsen in 1977 while modeling particle size from a diamond mine (see, e.g., Barndorff-Neilsen, 1977) and the subclasses which are relevant for application in this Thesis. The distribution is well applied in economics particularly in the fields of modeling financial markets and risk management due to its semi-heavy tails.
5.4.1 The Generalized Hyperbolic Distribution
A random variable X is said to follow a Generalized Hyperbolic (GH) distribution if its prob- ability density function is given by
fGH(x;α, β, δ, λ, µ) = (γ/δ)γ
√2πKλ(δγ) Kλ−1
2(αp
δ2+ (x−µ)2) (p
δ2+ (x−µ)2)/α)12−λβ(x−µ) (5.4.1) where γ = p
α2−β2, µ, λ, α, β, δ ∈ R and µ, βand δ are location, asymmetry and scale parameters respectively while Kλ is the modified Bessel function of the third kind with index λ. δ ≥ 0 and 0 ≤ |β| < α . The mean and variance of this distribution are respectively given by
E[X] =µ+ δβKλ+1(δγ)
γKλ(δγ) (5.4.2)
and
V[X] = δKλ+1(δγ) γKγ(δγ) +
βδ γ
2
Kλ+2(δγ)
Kλ(δγ) − k2λ+1(δγ) Kλ2(δγ)
(5.4.3) Special cases of the generalized hyperbolic distribution (see, e.g., Jørgensen (1982), Barndorff- Neilsen and Stelzer (2004)) are
(i) Whenλ=−12, the GH specializes to the Normal Inverse Gaussian (NIG) and (ii) Whenλ= 1, the GH becomes the Hyperbolic distribution.
Definition 5.4.1 (Modified Bessel Function of the Third Kind with Index λ). 1 The integral representation of the modified Bessel function of the third kind with index λ can be found in
1In addition we have an explicit from of the Bessel fuction,K1
2(x) =pπ
2xexp(−x)
Barndorff-Neilsen et al (1982) and Abramowitz and Stegun (1972):
Kλ(x) = 1 2
Z ∞ 0
yλ−1exp n
−x
2 y−y−1o
dy, x >0. (5.4.4) The substitution y=xp
χ/ψ can be used to obtain the following relation which allows one to bring the GH (5.4.1) into a closed-form expression
Z ∞ 0
yλ−1exp
−1 2
χ y +yψ
dy= 2 χ
ψ λ/2
Kλp ψχ
. (5.4.5)
Asymptotic relations for small arguments x can be used for calculating the densities of special cases of the GH density as follows
Kλ(x)∼Γ(λ)2λ−1x−λ asx↓0 and λ >0 (5.4.6) and
Kλ(x)∼Γ(λ)2λ−1x−λ asx↓0 and λ <0 (5.4.7) The asymptotic relation for large arguments x is given in footnote 3.
5.4.2 The Normal Inverse Gaussian
A random variable X follows a Normal Inverse Gaussian (NIG) distribution with parameter vector (α, β, µ, δ) if its probability density function is
fN IG(x:α, β, µ, δ) = αδexp [p(x)]
πq(x) K1[αq(x)] (5.4.8)
wherep(x) =δp
(α2−β2) +β(x−µ), q(x) = q
(x−µ)2+δ2 andK1 is the modified Bessel function2 of the third kind with order one (see e.g., Abramowitz and Stegun 1972). Hereµ∈R is a location density, β ∈ R is the skewness parameter and, if β < 0, the NIG is negatively skewed; α ≥ |β| measures the heaviness of the tails (shape of the distribution) and finally δ >0 is the scale parameter. The NIG is a very flexible member of the family of distributions enjoying the convolution property as shown in Kalemanova and Werner (2006):
2SpecificallyK1(x) =x 4
R∞ 0 exp
t+x2
4t
t−2dt, x∈R.
Property 5.4.1
The NIG is a mixture of normal and inverse Gaussian distributions. Let X|Y =
y∼N(µ+βy, y)
Y ∼IG(δγ, γ2) with γ :=p
α2−β2
(5.4.9)
then X ∼N IG(α, β, µ, δ) is what is denoted by the density function fN IG(α, β, µ, δ) =
Z ∞ 0
fN(x;µ+βy, y).fIG y;δγ, γ2
dy. (5.4.10)
Property 5.4.2
The NIG distribution is closed under convolution. In fact it is the only member of the family of general hyperbolic distributions to have the property that for independent random variables, X ∼ N IG(α, β, µX, δX) and Y ∼ N IG(α, β, µY, δY), their sum is NIG distributed, that is,
X+Y ∼N IG(α, β, µX, δX)∗N IG(α, β, µY, δY) =N IG(α, β, µX +µY, δX +δY) (5.4.11) The mean, variance, skewness and kurtosis of this random variable X are, respectively,
E[X] =µ+ δβ
pα2−β2, (5.4.12)
V[X] = δα2
(α2−β2)3/2, (5.4.13)
S[X] = 3 (β/α)
δp
α2−β2
1/2 (5.4.14)
and
K[X] = 3 1 + 4(β/α)2 δp
α2−β2 . (5.4.15)
However, moment estimators as starting values of the NIG distribution may be used. If ¯mi, i= 1,2,3,4, are the sample mean, variance, skewness and kurtosis respectively, then define
ˆ
γ = 3
¯ m3p
3 ¯m4−5 ¯m23.
The moment estimators are then given by ˆ
µ= ¯m1−βˆδˆ. ˆ γ, βˆ= m¯3m¯2ˆγ2
3, δˆ= m¯22ˆγ3.
βˆ2+ ˆγ2 and
ˆ α=
βˆ2+ ˆγ21/2
.
These initial values can also be estimated by the method of moments (Bolviken and Benth 2000) from a given samplex1, x2, ..., xn for X ∼N IG(α, β, µ, δ) through the ratio
(S[X])2
K[X] ,K[X]>0.
5.4.3 The Hyperbolic Distribution
The random variableXis said to have a Hyperbolic (HYP) distribution if its probability density function is given by
fHY P(x;α, β, δ, µ) =
pα2−β2 2αδK1
δp
α2−β2exp{−α(u(x)) +β(x−µ)}, (5.4.16) whereu(x) =
q
δ2+ (x−µ)2
and−∞ ≤x≤ ∞.The domain of variation of the parameters is µ ∈ R, δ > 0, and 0 ≤ |β| < α. The first application of the hyperbolic distribution to finance is in Eberlein and Keller (1995). The alternative set of distributions for modeling skew and heavy-tailed data is the skew extension to the Student’s t−distribution. Hansen (1994) was the first to propose a skew extension to the Student’st−distribution for modeling financial returns. There are several versions of this distribution, for details, see for example Fernandez and Steel (1998), Branco and Dey (2001), Jones and Faddy (2003) and Azzalini and Capitanio (2003). However, all these skew-type distributions have both tails behaving like polynomials which mean that they fit fat-tailed data well but deficient in handling substantial skewness.
The probability density function derived by Aas and Haff (2006) as a limiting case of the GH distribution λ=−ν2 and α→ |β|
in (5.4.1) which they referred to as GH skew Student’s
t−distribution. The main attraction of this distribution is that unlike any other member of the GH family, it has one tail determined by a polynomial and the other by exponential behaviour.
In addition, it is almost as analytically tractable as the NIG distribution. Therefore, the skew Student’s t−distribution has one heavy and one semi-heavy tail.
5.4.4 The Skewed Student’s t−distribution
A random variableX is said to follow a GH skew Student’st−distribution (SSt) if its (Aas and Haff, 2006) probability density function is given by
fSSt(x;ν, µ, β, δ) =
2(1−ν)/2δν|β|(ν−1)/2K(ν+1)/2(|β|u(x)) Γν
2 √
π(u(x))(ν+1)/2
exp{β(x−µ)} for β 6= 0, Γ
ν+ 1 2
δΓν 2
√ π
"
1 + (x−µ)2 δ2
#−(ν+1)/2
for β = 0,
(5.4.17) where u(x) =
q
δ2 + (x−µ)2
. It can be recognized that the density in (5.4.17) is that of a non-central (scaled) Student’st−distribution withν degrees of freedom whenβ = 0.The mean and variance of a SSt distributed random variable X are respectively
E[X] =µ+ βδ2
ν−2 (5.4.18)
and
V[X] = 2β2δ4
(ν−2)2(ν−4) + δ2
ν−2. (5.4.19)
Another subclass of the GH distributions family is the Variance-Gamma distribution that we consider in the next Subsection. It is the normal variance-mean mixture where the mixing density is the gamma distribution. The tails of the distribution decrease more slowly than the normal distribution. It is therefore suitable to model phenomena where numerically large values are more probable than is the case for the normal distribution. The distribution was introduced in the finance literature by Madan and Seneta (1990) and has been successful applied in diverse fields such as modeling returns from financial assets and turbulent wind speeds.
5.4.5 The Variance-Gamma distribution
Let X be a continuous random variable. X is said to be distributed as the Variance-Gamma (VG) distribution if its probability density function is of the form
fV G(x;α, µ, λ, β) = (α2−β2)λ|x−µ|λ−1/2Kλ−1/2(α|x−µ|)
√πΓ (λ) (2α)λ−1/2 exp (β(x−µ)), (5.4.20) where−∞< x <∞,µ(location parameter), α,β(asymmetry parameter) are real andλ >0.
Here, Γ (.) denotes the Gamma function, and Kλ, the Bessel function of the third kind. The mean and variance of X are
E[X] =µ+ 2βλ
α2−β2 (5.4.21)
and
V[X] = 2λ (α2 −β2)
1 + 2β α2−β2
. (5.4.22)
The class of Variance-Gamma distributions is closed under convolution in the following sense that if X1 and X2 are independent random variables that are variance-gamma distributed with the same values of the parameters α and β, but possibly different values of the other parameters, λ1, µ1 and λ2, µ2 respectively, then X1+X2 is variance-gamma distributed with parameters α, β, λ1+λ2 and µ1+µ2.