In this section we present thegeneralized hyperbolic distributionsand describe their most important properties. We will also discuss thegeneralized inverse Gaussian distributions which play an important role in the theory of generalized hyperbolic distributions and processes. As mentioned earlier, this class of distributions is also of independent interest as a model of positive quantities in finance. We will present a few examples of how well these distributions fit financial data.
1.1. The generalized hyperbolic distribution
The generalized hyperbolic distributions were introduced by Barndorff-Nielsen (1977) and include, among others, thehyperbolic distributions, the normal-inverse Gaussian(NIG) distributions, thescaled t-distributionsand the variance-gamma distributions. We shall discuss these sub-classes in more detail later. First we present the generalized hyperbolic distributions and their properties.
A generalized hyperbolic distribution has five parameters. If X follows a generalized hyperbolic distribution we write
X∼H (λ, α, β, δ, à).
The probability density function of a generalized hyperbolic distribution is given by (γ /δ)λ
√2πKλ(δγ )ãKλ−1/2(α
δ2+(x−à)2) (
δ2+(x−à)2/α)1/2−λ ãeβ(x−à), x∈R, (1) whereγ2=α2−β2, andKλis the modified Bessel function of the third kind with indexλ.
Definitions and results concerning Bessel functions are collected in an appendix.
The parameter domain for the class of generalized hyperbolic distributions is given by δ0, α >0, α2> β2, ifλ >0,
δ >0, α >0, α2> β2, ifλ=0, δ >0, α0, α2β2, ifλ <0.
In all cases à∈R. Ifδ=0 orα2=β2 the generalized hyperbolic density in (1) is de- fined as the limit expression obtained by using (A.5). Note that ifβ is equal to zero, the distribution is symmetric.
The class of generalized hyperbolic distributions is closed under affine transformation.
That is, ifX∼H (λ, α, β, δ, à)andY is defined asY=aX+b, for some positivea, then we have that
Y ∼H
λ,α a,β
a, aδ, aà+b
. (2)
214 B.M. Bibby and M. Sứrensen
From (2) we also see that the parameterλ is invariant under affine transformations of a generalized hyperbolic random variable.
From (A.3) it follows that the mode points for the generalized hyperbolic distribution are solutions to the equation
x−à
δ2+(x−à)2ãKλ−3/2(α
δ2+(x−à)2) Kλ−1/2(α
δ2+(x−à)2)=β
α. (3)
Ifβ=0, it follows immediately that the distribution is unimodal with mode pointà. If λ32, the ratio of the modified Bessel functions in (3) increases monotonically from 0 to 1, and therefore the distribution is unimodal. See Blổsild (1978) for further discussion of features of the generalized hyperbolic density function.
The Laplace transform of the generalized hyperbolic distribution is given by L(z)=eàzãγλãKλ(δγz)
γzλãKλ(δγ ), |β+z|< α, (4)
whereγz2=α2−(β+z)2. From (A.3) we get that EX=à+δβKλ+1(δγ )
γ Kλ(δγ ) , (5)
and
VarX=δKλ+1(δγ ) γ Kλ(δγ ) +β2δ2
γ2
Kλ+2(δγ )
Kλ(δγ ) −Kλ2+1(δγ ) Kλ2(δγ )
. (6)
Expressions for the skewness and kurtosis involve modified Bessel functions in a rather complicated way and can be found in Barndorff-Nielsen and Blổsild (1980).
Sometimes it is useful to reparametrize the generalized hyperbolic density in terms of the parametersλ,τ,ζ,δ, andà, whereτ=β/γ andζ =δγ. Using this parametrization, the generalized hyperbolic density has the form,
√ζ
√2π δKλ(ζ )ãKλ−1/2(ζ√
1+τ2
1+((x−à)/δ)2) (
1+((x−à)/δ)2/√
1+τ2)1/2−λ ãeτ ζ (x−à)/δ, x∈R. (7) The parametersτ,ζ, andλare invariant under affine transformations of a random variable following the generalized hyperbolic distribution. More precisely, the result equivalent to (2) is thatY ∼H (λ, τ, ζ, aδ, aà+b). From this result we see thatδis a scaling parameter andàis a location parameter. In Figure 1 generalized hyperbolic densities are drawn for different values ofλ,τ, andζ. In all cases the mean value is 0 and the variance is 1. The tail behaviour of the distributions is more easily seen in Figure 2, where the logarithm of the same densities are plotted.
Ch. 6: Hyperbolic Processes in Finance 215
Fig. 1. Generalized hyperbolic densities with mean 0 and variance 1 for different values of the parametersλ,τ, andζ.
Fig. 2. The logarithm of generalized hyperbolic densities with mean 0 and variance 1 for different values of the parametersλ,τ, andζ.
216 B.M. Bibby and M. Sứrensen
We shall now consider the important special cases of the generalized hyperbolic distri- bution mentioned earlier. Thehyperbolic distributionsis the subclass obtained whenλis equal to 1. Withλequal to 1 in (1), we get the following expression for the density of a hyperbolic distribution,
γ
2αδK1(δγ )exp
−α
δ2+(x−à)2+β(x−à)
, x∈R. (8)
From (8) we see that the logarithm of the density of a hyperbolic distribution is a hyperbola, which should be compared to the parabolic log-density of the normal distribution. The name of the hyperbolic distribution stems from this observation. In fact, the definition of the hyperbolic distributions was inspired by the empirical finding by the founding father of the physics of wind blown sand, Brigadier R.A. Bagnold, that the log-density of the distribution of the logarithm of the grain size of natural sand deposits looks more like a hyperbola than like a parabola, as had previously been assumed by geomorphologists, see Bagnold (1941).
For the hyperbolic distributions Equation (3), which determines the mode points of the generalized hyperbolic distribution, simplifies to
x−à
δ2+(x−à)2=β α,
implying that the distribution is unimodal with mode point x=à+δβ
γ .
Lettingδtend to zero and using (A.5), we get the asymmetric Laplace distribution as a special case of the hyperbolic distribution, that is,
α2−β2
2α eβ(x−à)−α|x−à|, x∈R.
The normal distribution can also be obtained as a limit case of the hyperbolic distribution.
Lettingα, δ→ ∞in such a way thatδ/α→σ2, we get, using (A.6), the normal density:
√ 1
2π σ2e−
1 2σ2(x−à)2
, x∈R.
According to Barndorff-Nielsen et al. (1985) we have that the skewness (γ1) and the kurtosis (γ2) for large values ofζ and small values ofβ/αsatisfy that
(γ1, γ2)∼
3χ ,3ξ2 ,
Ch. 6: Hyperbolic Processes in Finance 217
where
χ= β/α
√1+ζ and ξ= 1
√1+ζ.
Based on this observation Barndorff-Nielsen et al. (1985) suggested that the parameters χ andξ are natural measures of asymmetry and “kurtosis” for the hyperbolic distribution.
Note that they are invariant under location-scale transformations. The parametersχ andξ vary in the so-calledshape triangledefined by
(χ , ξ )∈R2|0|χ|< ξ <1
. (9)
Note that the normal and the (possibly skew) Laplace distributions are obtained as limit distributions when ξ →1 and ξ →0, respectively. In Figure 3 hyperbolic log density functions are plotted for different values ofχandξ in the shape triangle.
In Figure 4 a histogram based on 2666 observations of the daily returns of IBM-stocks (returns are increments on a logarithmic scale of the stock prices) in the period from 1 Jan- uary 1990 to 20 March 2000 is given. Each point indicates the mid-point of the top of a column in the histogram. The best generalized hyperbolic, hyperbolic, and normal densities are superimposed on the histogram. The parameter values corresponding to the generalized
Fig. 3. Hyperbolic log densities with mean 0 and variance 1 for different values of the parameters χ andξ (−0.8,−0.6, . . . ,0.8 forχand 0.0,0.25, . . . ,1.0 forξ). The log densities are placed at the corresponding values
ofχandξ.
218 B.M. Bibby and M. Sứrensen
Fig. 4. A histogram of 2666 daily IBM-stock returns. Superimposed are the best fitting generalized hyperbolic, hyperbolic, and normal densities. The parameter values corresponding to the generalized hyperbolic density are α=5.174,β=0.0048,δ=0.0262,à=0.0002, andλ= −1.933. The parameter values corresponding to the
hyperbolic density areα=82.26,β=3.725,δ=0.0060, andà= −0.0007.
hyperbolic density areα=5.174,β=0.0048,δ=0.0262,à=0.0002, andλ= −1.933.
For the hyperbolic density the parameter values areα=82.26, β=3.725,δ=0.0060, andà= −0.0007. In Figure 5 the logarithms of the same histogram points and the same densities are plotted.
Log-histograms and log-densities are very useful when the interest is focussed on tail behaviour. From Figures 4 and 5 it is evident that a heavy-tailed distribution such as a gen- eralized hyperbolic or hyperbolic distribution provides a good fit to the data, and certainly a much better fit than the normal distribution, in particular in the tails. A plot like Figure 5, which emphasizes differences in tail behaviour, reveals that the extreme tails of the his- togram are a bit heavier than those of the fitted generalized hyperbolic distribution. There is no reason to be overly concerned about this minor discrepancy, because, first, it should be remembered that it is measured on a logarithmic scale, and secondly, the two log-histogram points in the extreme left tail are based on only 1 and 2 observations, respectively, while each of the two points in the extreme right tail represents 2 observations.
Thenormal-inverse Gaussian(NIG)distributionsis the subclass obtained forλequal to−12. The density of the normal-inverse Gaussian distribution is given by
αδ
π eδγ ãK1(α
δ2+(x−à)2)
δ2+(x−à)2 ãeβ(x−à), x∈R. (10)
Ch. 6: Hyperbolic Processes in Finance 219
Fig. 5. The logarithm of the histogram in Figure 4 of 2666 daily IBM-stock returns. Superimposed are the loga- rithms of the best fitting generalized hyperbolic, hyperbolic, and normal densities. The parameter values are as in
Figure 4.
If the distribution ofXhas density function (10), we write X∼NIG(α, β, δ, à).
If we letαtend to zero, it follows from (A.5) that theNIG-distribution converges to the Cauchy distribution with location parameteràand scale parameterδ.
The Laplace transform of aNIG-distribution is especially simple:
L(z)=eàz+δ(γ−γz), |β+z|< α, (11)
whereγz2=α2−(β+z)2. Expressions for the mean and variance are also simple in the case of aNIG-distribution:
EX=à+δβ
γ , VarX=δα2 γ3.
The skewness is 3δα2βγ−5and the kurtosis is 3δα2(α2+4β2)γ−7. Although these ex- pressions are quite simple, it is also for theNIG-distributions informative to use the shape triangle, which can be defined in complete analogy with that for the hyperbolic distribu- tions, see, e.g., Rydberg (1997). In Figure 6NIGlog-density functions are drawn for dif- ferent values ofχandξin the shape triangle defined in the same way as for the hyperbolic distribution.
220 B.M. Bibby and M. Sứrensen
Fig. 6. Normal-inverse Gaussian log densities with mean 0 and variance 1 for different values of the parametersχ andξ(−0.8,−0.6, . . . ,0.8 forχand 0.0,0.25, . . . ,1.0 forξ). The log-densities are placed in the shape triangle
at the corresponding values ofχandξ.
Finally, but not least, the class of normal-inverse Gaussian distributions is closed under convolution when the parametersαandβ are fixed, that is ifX1andX2are independent so thatXi ∼NIG(α, β, δi, ài),i=1,2, then we have that
X1+X2∼NIG(α, β, δ1+δ2, à1+à2). (12) Only two subclasses of the generalized hyperbolic distributions are closed under convo- lution. The other class with this important property is the class ofvariance-gamma(VG) distributions, which is obtained whenδis equal to 0. This is only possible whenλ >0 and α >|β|. The variance-gamma distributions (withβ=0) were introduced in the financial literature by Madan and Seneta (1990). Another and perhaps more natural name for the full class is thenormal-gamma(NG) distributions. The density function is given by
γ2λ
√π (λ)(2α)λ−1/2|x−à|λ−1/2Kλ−1/2
α|x−à|
eβ(x−à), x∈R, (13) where denotes the gamma-function. If X follows a variance-gamma distribution, we write
X∼VG(λ, α, β, à).
Ch. 6: Hyperbolic Processes in Finance 221
The reader is reminded that the parameter domain isλ >0,α >|β|0 andà∈R. The Laplace transform of aVG-distribution is simple:
L(z)=eàz γ
γz 2λ
, |β+z|< α, (14)
where againγz2=α2−(β+z)2. From (14) (or from (5) and (6)) it easily follows that
EX=à+2βλ
γ2 , VarX=2λ γ2
1+2
β γ
2 .
The class of variance-gamma distributions is closed under convolution whenαandβ are fixed. IfX1 andX2 are independent random variables such thatXi ∼VG(λi, α, β, ài), i=1,2, then we have that
X1+X2∼VG(λ1+λ2, α, β, à1+à2). (15)
This convolution property follows from (14).
By (A.6), the tails of a VG-distribution decrease as|x|λ−1e−α|x|+βx whenx→ ±∞.
The logarithm of the densities of variance-gamma distributions are plotted for different values ofλin Figure 7. In all casesβ=0, the mean is zero, and the variance is one. From this figure appears a disadvantage of the class ofVG-distributions. The probability density is very peaked at the centre forλ <1, while forλ1 the tail-behaviour does not fit the tails found in typical financial data like those in Figure 5 as well as other generalized hyperbolic distributions like for instance theNIG-distribution.
We will finally consider the subclass of the generalized hyperbolic distributions that is obtained whenα= |β|, or equivalentlyγ=0. This is only possible whenλ <0 and δ >0. It is convenient to introduce the reparametrizationν= −2λ. Forγ=0 we obtain the density function
δν
√π2(ν−1)/2(ν/2)ãK(ν+1)/2(|β|
δ2+(x−à)2) (
δ2+(x−à)2/|β|)(ν+1)/2 ãeβ(x−à), x∈R, (16) whereν >0,δ >0,β∈Randà∈R. A natural name for this distribution is theasymmetric scaled t-distribution,as will soon be clear. From (A.6) it follows that whenβis positive, the left-hand tail decreases as|x|−(ν/2+1)e2βx, while the right-hand tail decreases asx−(ν/2+1). Whenβ is negative, the behaviour of the two tails is interchanged. The expectation exists provided ν >2, and the variance exists whenν >4. More generally, the n-th moment exists whenν >2n. The Laplace transform of the distribution given by (16) is
eàz(−δz(z+2β))ν/2Kν/2(−δz(z+2β))
(ν/2)2ν/2−1 (17)
222 B.M. Bibby and M. Sứrensen
Fig. 7. The logarithm of the densities of variance-gamma distributions withβ=0, mean 0, and variance 1 for different values of the parameterλ.
with domain−2β < z0 whenβ >0 and 0z <−2β whenβ <0. Whenβ=0, the domain is the set{0}, and we obtain the density function
((ν+1)/2) δ√
π (ν /2)(1+((x−à)/δ)2)(ν+1)/2, x∈R,
which is the well-known density of the scaledt-distribution withνdegrees of freedom.
1.2. The generalized inverse Gaussian distribution
The second class of distributions, that we consider in this section, is the class ofgener- alized inverse Gaussian(GIG) distributions. TheGIG-distributions are described by three
Ch. 6: Hyperbolic Processes in Finance 223
parameters and defined on the positive half axis. The generalized inverse Gaussian density is of the form
(γ /δ)λ
2Kλ(δγ )ãxλ−1ãexp −1 2
δ2x−1+γ2x
, x >0. (18)
The parameter domain is given by δ >0, γ0, ifλ <0, δ >0, γ >0, ifλ=0, δ0, γ >0, ifλ >0.
The class of generalized inverse Gaussian distributions was first proposed in 1946 by Éti- enne Halphen, who used it to model the distribution of the monthly flow of water in hy- droelectric stations, see Seshardi (1997). The class was rediscovered by Sichel (1973) who used it to construct mixtures of Poisson distributions and by Barndorff-Nielsen (1977) who used it to construct the class of generalized hyperbolic distributions, but also realized its broad usefulness and initiated an in depth study of the class. We shall return to the relation to the generalized hyperbolic distributions later. The generalized inverse Gaussian distribu- tions were briefly mentioned by Goog (1953) as an intermediate between Pearson’s curves of Type III and V. The class of generalized inverse Gaussian distributions was investigated extensively in Jứrgensen (1982).
Using (A.5) we see that forλ >0 andγ >0 the gamma distribution emerges as limit distribution whenδtends to zero, that is we get the following density for positiveλandγ,
(γ2/2)λ
(λ) ãxλ−1ãeγ2x/2, x >0.
Similarly, the inverse gamma distribution with density given by (2/δ2)λ
(−λ) ãxλ−1ãe(δ2/2)/x, x >0,
is obtained whenγ tends to zero forλ <0 andδ >0. This distribution has a tail of the Pareto type. Finally, for λ= −12 we get the inverse Gaussian distribution with density function given by
√ δ
2π x3ãe−γ (x−δ/γ )2/(2x), x >0.
224 B.M. Bibby and M. Sứrensen
The generalized inverse Gaussian distributions are unimodal with mode point given by
λ−1+
(λ−1)2+δ2γ2
γ2 ifγ >0, δ2
2(1−λ) ifγ=0.
IfX has a generalized inverse Gaussian distribution, we writeX∼GIG(λ, δ, γ ). In Fig- ure 8 generalized inverse Gaussian densities are plotted for different values of λ and ω=δγ. In all cases the variance is 1.
The Laplace transform of theGIG(λ, δ, γ )-distribution is L(z)= Kλ(ω
1−2z/γ2)
Kλ(ω)(1−2z/γ2)λ/2 (19)
forδ >0 andγ >0. The domain ofLisz < γ2/2 whenλ0 andzγ2/2 whenλ <0.
In the casesδ=0 or γ =0, the Laplace transform is obtained from (19) by (A.5). For δ=0,
L(z)=
1− 2z γ2
−λ
, z <γ2 2 ,
Fig. 8. Generalized inverse Gaussian densities with variance 1 for different values of the parametersλand ω=δγ.
Ch. 6: Hyperbolic Processes in Finance 225
which is the well-known Laplace transform of the gamma-distribution. Forγ =0 we ob- tain
L(z)= 2Kλ(√
−2δ2z)
(−λ)(−δ2z/2)λ/2, z0.
For positive values ofδandγ the moments ofXare given by EXj=
δ γ
jKλ+j(ω)
Kλ(ω) , j=1,2, . . . . (20)
When eitherδorγ is zero, the moments ofXare also known and are obtained as limits of (20). The variance ofXis given by
VarX= δ
γ 2
Kλ+2(ω)
Kλ(ω) −Kλ2+1(ω) Kλ2(ω)
. (21)
In Figure 9 a histogram of 307 monthly observations of interest rates in the period from June 1964 to December 1989 is given along with a fitted generalized inverse Gaussian density corresponding to the parameter valuesδ=0.2693,γ=11.23, andλ= −7.0707.
More precisely, the data are annualized monthly yields of U.S. one-month Treasury bills.
The same data set was studied in Chan et al. (1992).
There is the following important relationship between the generalized hyperbolic distri- bution and the generalized inverse Gaussian distribution, which was, in fact, how the gen-
Fig. 9. A histogram of 307 monthly interest rates. The generalized inverse Gaussian density with parameters δ=0.2693,γ=11.23, andλ= −7.0707 is superimposed.
226 B.M. Bibby and M. Sứrensen
eralized hyperbolic distribution was originally derived in Barndorff-Nielsen (1977). The generalized hyperbolic distribution is a normal variance–mean mixture where the mixing distribution is generalized inverse Gaussian. What is meant by this is that if
X|W=w∼N (à+βw, w),
andW∼GIG(λ, δ, γ ), then the marginal distribution ofXwill be generalized hyperbolic, X∼H (λ, α, β, δ, à), whereα2=β2+γ2. This property provides a possible interpretation of non-Gaussian stochastic variation described by a generalized hyperbolic distribution.
As special cases we have that the normal-inverse Gaussian distribution appears when the mixing distribution is an inverse Gaussian distribution, and the variance-gamma dis- tribution emerges as a normal variance–mean mixture where the mixing distribution is a gamma distribution. This explains the names of the distributions. The asymmetric scaled t-distribution is a normal variance–mean mixture with an inverse gamma mixing distrib- ution. As a special case we get the well-known result that thet-distribution is a normal variance mixture (β=0) with an inverse gamma mixing distribution.
The mixing result implies that there is the following simple relationship between the Laplace transform,LX, of the generalized hyperbolic distributionH (λ, α, β, δ, à)and that of theGIG(λ, δ,
α2−β2)-distribution,LW: LX(z)=eàzãLW
βz+1
2z2
.
Barndorff-Nielsen and Halgreen (1977) showed that generalized inverse Gaussian dis- tributions areinfinitely divisible. Using that the generalized hyperbolic distributions are normal variance-mean mixtures with generalized inverse Gaussian mixing distributions, they also proved that generalized hyperbolic distributions are infinitely divisible. Halgreen (1979) showed that generalized hyperbolic distributions and generalized inverse Gaussian distribution are evenself-decomposable. In the following section, the properties of infinite divisibility and self-decomposability will turn out to be important because they allow the construction of certain hyperbolic stochastic process models.
1.3. Statistical inference
Inference for the parameters when dealing with independent and identically generalized hyperbolic or generalized inverse Gaussian distributed observations should be based on the likelihood function. The C-program HYP described in Blổsild and Sứrensen (1992) can be used for maximum likelihood estimation in the situation where independent and identi- cally (possibly multi-dimensional) hyperbolic distributed observations are considered. The program HYP also has the facility of basing the inference on the multinomial likelihood function obtained by only observing the number of observations in given intervals. More precisely, ifI1, . . . , Ik are disjoint intervals with union the entire real line andyj denotes
Ch. 6: Hyperbolic Processes in Finance 227
the number of observations inIj,j=1, . . . , k, then the multinomial log-likelihood func- tion is given by
*(α, β, δ, à)= k j=1
yjlogpj, (22)
wherepj is the probability that a hyperbolic distributed random variable takes a value inIj, that is,
pj=
Ij
γ
2αδK1(δγ )exp
−α
δ2+(x−à)2+β(x−à)
dx, j=1, . . . , k.
(23) Inference based on grouped observations from other distributions can of course be car- ried out in a similar way using (22) and the equivalent of (23). Küchler et al. (1999) note that if the observations are not independent then inference based on the multinomial like- lihood function for grouped observations will be more robust to effects of the dependence than inference based on the original likelihood function for independent observations.