A. Concepts of dependence
2. The generalized correlation model (CRE model)
Note that α* is increasing in ρ1 and ρ2 and that
( )
=( )
= −p=lim→ , lim→ * 1, 2 1
2 1 1
* 1
*
2 1
max α ρ ρ α ρ ρ
α ρ ρ .
Figure 36 gives a visual example of Theorem 3 and its implications.
Loss distributions of homogenous portfolios in the normal correlation model
(λ = 1, p = 0.5%)
0,001 0,01 0,1 1
0,0001 0,001
0,01 0,1
1
Confidence level α
Loss relative to portfolio exposure
ρ = 0 ρ = 0.1 ρ = 0.2 ρ = 0.3 ρ = 0.4 ρ = 0.5
ρ = 0.6 ρ = 0.7 ρ = 0.8 ρ = 0.9 ρ = 0.99 ρ = 0.999
0 0.9 0.99 0.999 0.9999
α = 99.5%
Figure 36: Loss distributions of homogenous portfolios in the normal correlation model 2
2. The generalized correlation model (CRE model)
While being economically intuitive, a major drawback of the normal correlation model is the somewhat arbitrary choice of the multivariate normal distribution to describe the joint move-ments of clients’ individual risk indices. Historical reasons certainly were dominant in this selection because normal distributions appear as finite dimensional marginal distributions of the log-returns of the geometric Brownian motion, the standard model of continuous stochas-tic processes. This is used for example in the classic Black-Scholes-Merton model and is also by far the best understood continuous multivariate distribution.
In the present discussion, however, a typical criticism of the normal distribution is that it is not well adapted to the specific features of much financial data. This assessment refers espe-cially to the phenomenon that many empirical distributions have long tails, i.e. that large
de-viations from the mean of a distribution are observed much more frequently than one would expect if the underlying distribution were normal.
1 10 100 1000
-0.04 -0.02 0 0.02 0.04
Häufigkeit
Logarithmische Returns DAX
Frequency
A long tail distribution in finance:
daily DAX-returns
DAX-returns (logarithmic)
blue: normal distribution red: normal inverse gauss distribution
1 10 100 1000
-0.04 -0.02 0 0.02 0.04
Häufigkeit
Logarithmische Returns DAX
Frequency
A long tail distribution in finance:
daily DAX-returns
DAX-returns (logarithmic)
blue: normal distribution red: normal inverse gauss distribution
Figure 37: An empirical long tail distribution in finance: DAX-returns
Figure 37 gives an example of the normal and a long tail distribution fitted to the same finan-cial data, daily DAX-returns from January 1, 1991 to November 30, 2000. It is apparent that the long tail distribution216 is much better adapted to the frequency of extreme returns than the normal distribution while both distributions are quite similar in the region of small deviations from the mean.
In the normal correlation model, two things were fundamental: the marginal distributions that were needed to calculate clients’ default and transition thresholds and the correlation matrix of clients’ risks indices. In order to extend the model, note that a multivariate distribution is in general not uniquely determined by its marginal distributions and its correlation structure.
However, exceptions in that respect are spherical and elliptical distributions.
A distribution D is called spherical217 if it is invariant under orthogonal transformations, i.e. if for a random vector X∈Rn with X ~D and any orthogonal map U∈Rn×n the equation
( ) ( )
X =L UX Lholds218. If D has a density d than this definition is equivalent to saying that d is constant on spheres.219
216 We chose the normal inverse gauss distribution in the example, a family of distributions where the length of the tails can be continuously adapted due to a further parameter besides mean and covariance matrix. See below for details.
217 Or ‘spherically symmetric’.
114 Let S be the family of all spherical distributions. A distribution D is called elliptical220 if it is
an affine linear transformation of a spherical distribution221, i.e. if for a random vector X∈Rn with X ~D and a random vector Y∈Rn with L
( )
Y ∈S there exists µ∈Rn andn
A∈Rn× such that
Y A X =µ+ ⋅ .
The best known example of spherical or elliptical distributions, respectively, is the family of multivariate normal distributions so prominent in the normal correlation model. Figure 38 shows plots of densities and their contours of the bivariate normal distribution for correlations
=0
ρ (the spherical case) and for ρ=75% (an elliptical case).
The bivariate normal distribution Contours
Densities
ρρρρ= 0 ρρρρ= 75%
Spherical and elliptical distributions The bivariate normal distribution
Contours
Densities
ρρρρ= 0 ρρρρ= 75%
Spherical and elliptical distributions
Figure 38: Spherical and elliptical distributions
218 L
( )
X denotes the law of X. The expression denotes that the distributions of X and of UX are equal.219 Fang et al. 1989, definition 2.1., p. 29.
220 Or ‘elliptically symmetric’.
221 Fang et al. 1989, definition 2.2., p. 31.
115 Elliptical distributions are an interesting generalization of the normal distribution in the
corre-lation model because a multivariate elliptical distribution is uniquely determined by its uni-variate marginals, its mean and its covariance matrix since the type of all marginals is the same222.
Not all symmetric univariate distributions are possible as marginal distributions of an ellipti-cal distribution in R for any n n∈N. It can be shown, however, that a univariate distribution D is the marginal distribution of a spherical distribution in R for any n n∈N if and only if it is a variance mixture of centered normals223. Hence224, D can be defined by its density func-tion
( )
ss dW x s
x
f = ∞
∫
− 0
2
exp 2 1 2 ) 1
( π
where the weight or mixing distribution W only takes values on (0,∞), i.e. a variance mixture of normals is a normal distribution with random variance. This definition immediately implies that a random variable X ~D can be written as
Y w X = ⋅
where Y is standard normally distributed, w ~W, and Y and w are stochastically independent.
Example 1
A well known example of a normal variance mixture is the Student-t distribution with n de-grees of freedom. Here the mixing distribution w is given as
ϑ w= n
where ϑ~χn2. Example 2
A more flexible family of mixture distributions is the generalized hyperbolic distribution. The one-dimensional centered and symmetric version of the generalized hyperbolic distribution has three free parameters λ, α, δ and is defined by its Lebesgue-density
(
x; , ,) (
a , ,) (
x2)
( 1/2)/2 1/2(
2 x2)
gh λα δ = λα δ ⋅ δ − − λ− α δ +
λ K
222 Cf. Embrechts et al. 1999, p. 11.
223 Fang et. al, 1989, theorem 2.21, p. 48. Note that there exist univariate distributions that are no variance mixtures of normals that can be marginals of spherical distributions for some, but not all n∈N.
224 Note that any mixture of normals has a density with respect to Lebesgue measure.
116 Alterna-tively, the generalized hyperbolic distribution can be defined by its mixing distribution. This is the generalized inverse Gauss distribution with density
( ) ( )
The generalized hyperbolic distribution is continuous in its parameters and has the normal and the t-distribution as limiting cases:
For α,δ →∞, σ2
αδ → and any given λ, the generalized hyperbolic distribution converges towards N
( )
0,σ2 .225On the other hand, for α =0, δ = ν and λ=−ν/2 it is equal to the t-distribution with ν degrees of freedom.226
Example 3
An interesting special case of the generalized hyperbolic distribution is the normal inverse Gauss distribution (NIG). It is obtained for λ=−1/2 and has the inverse Gauss distribution
( )
+ as mixing distribution.
The NIG is particularly interesting as an alternative for the normal distribution in the correla-tion model because it is not only infinitely divisible227, but also closed under convolution.
Hence, similar to the normal distribution, it generates a Lévy-motion whose finite dimen-sional marginals are all NIG-distributed228. Therefore, the intuitive interpretation in the
225 Cf. Prause 1999, p. 3.
226 Cf. Prause 1999, p. 5.
227 As is any generalized hyperbolic distribution. Cf. Barndorff-Nielsen and Halgreen 1977.
228 Cf. Eberlein et al. 1998, p. 6f., who use the Lévy-motion generated by the NIG instead of the classical geometric Brownian motion to model financial price processes.