1.3. In all three cases the Gaussian distribution has to be rejected based on

the results of the two applied tests “ the p-values are less than 10’4 . Unfor-

tunately, the critical test values are not known for the ±-stable law. However,

the Anderson-Darling test values are at least an order of magnitude smaller

and the Kolmogorov test values are a few times smaller than for the Gaussian

distribution suggesting a much better ¬t, especially in the tails.

Of course, nothing we have said demonstrates that any one of the above data

sets really has a stable distribution. Although we have shown in Section 1.3.1

that tail index estimates are not su¬cient to reject stability, the converse is

also true: an estimate of ± signi¬cantly less than 2 by no means rules out

a non-stable distribution with power-law tails with ± > 2. Even though it

is relatively easy to reject normality (as we have shown above), alternative

leptokurtic distributions are very hard to tell apart.

1.4 Financial applications of ±-stable laws 29

Table 1.3: ±-stable and Gaussian ¬ts to the the Deutsche Aktienindex (DAX)

index from the period January 2, 1995 “ December 11, 2002.

Parameters ± σ β µ

±-stable ¬t 1.6975 0.0088 -0.3255 -0.0003

Gaussian ¬t 0.0158 0.0003

Test values Anderson-Darling Kolmogorov

±-stable ¬t 1.9714 1.1625

Gaussian ¬t 16.5865 2.8499

STFstab10.xpl

Yet, the central limit property of stable laws, together with good description

of extreme events may justify their application to diverse problems in ¬nance

including portfolio optimization, option pricing and “ most noticeably “ Value-

at-Risk type calculations, where the estimation of low quantiles in portfolio

return distributions is crucial (Khindarova, Rachev, and Schwartz, 2001).

30 1 Stable distributions in ¬nance

Bibliography

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2 Tail dependence

Rafael Schmidt

Tail dependence describes the amount of dependence in the lower-left-quadrant

tail or upper-right-quadrant tail of a bivariate distribution. Multivariate dis-

tributions possessing tail dependence are able to incorporate dependencies of

extremal events. According to Hauksson et al. (2001), Resnick (2002), and

Embrechts et al. (2001) tail dependence plays an important role in extreme

value theory, ¬nance, and insurance models. In particular in credit-portfolio

modelling tail-dependent distributions are of special practical interest, as de-

pendencies of large credit-default events can be modelled.

Here we introduce the tail-dependence concept and characterizing tail depen-

dence by the so-called tail-dependence coe¬cient and we embed this concept

into the general framework of copulae. In Section ??, the tail-dependence co-

e¬cient is calculated for several Archimedean copulae, elliptically contoured

distributions, and other copulae. Section ?? proposes two non-parametric esti-

mators for the tail-dependence coe¬cient based on the empirical copula. Fur-

ther we provide a suitable estimator for the tail-dependence coe¬cient within

the class of elliptically-contoured distributions. The last section presents em-

pirical results comparing the latter estimators.

2.1 Tail dependence and copulae

Tail-dependence de¬nitions for multivariate random vectors are mostly related

to their bivariate marginal distribution functions. Loosely speaking, tail depen-

dence describes the limiting proportion of exceeding one margin over a certain

threshold given that the other margin has already exceeded that threshold. The

following approach from Joe (1997) represents one of many possible de¬nitions

of tail dependence.

Let X = (X1 , X2 ) be a 2-dimensional random vector. We say that X is

34 2 Tail dependence

’1 ’1

Figure 2.1: »U (v) = P{X1 > F1 (v) | X2 > F2 (v)} for a bivariate normal

distribution with ρ = ’0.8, ’0.6, . . . , 0.6, 0.8. Note that »U = 0

for all ρ ∈ (’1, 1).

(bivariate) upper tail-dependent if

’1 ’1

»U := lim’ »U (v) = lim’ P{X1 > F1 (v) | X2 > F2 (v) > 0}, (2.1)

v’1 v’1

’1 ’1

in case the limit exists. F1 , F2 denote the generalized inverse distribu-