Tail Dependence - ETH - Entrepreneurial Risks - ETH Zürich

can directly access the Excel sheets. In a first step the price data provided in the Excel
sheets was converted into ”historical day-to-day return data” given by equation (3.1)
to achieve a measure of relative volatility
return(t) = log(price(t)) − log(price(t − 1)), (3.1)
where log(·) denotes the natural logarithm, the inverse of the exponential function.
The different methods are presented structured in subsections as follows:
The first subsection respectively gives an insight into the underlying theory of the concepts
or in other words, a summary of the most important findings in the papers. The
second subsection respectively provides detailed description concerning the implementation,
analysis and adaptation of variables for better robustness. Within the third
subsection respectively, tail dependence estimates according to the different methods is
analyzed i.e. by bootstrap sampling with replacement for an estimation of error bars.
3.1 Approaches according to Poon, Rockinger, and Tawn
In the first subsection I will start with some concepts for the estimation of dependence
measures for multivariate extreme introduced by Poon, Rockinger, and Tawn published
2004 in [13] (2004) with some further explanations provided by Heffernan 2000 in [14]
(2000), by Coles, Heffernan and Tawn in [15] (1999), and by Ledford and Tawn in
[12] (1996). Then I will go into some details concerning non-parametric estimation of
theses measures and the fitting of parametric models. In the second subsection I will
provide an overwiev on the implementation of non-parametric estimators for asymptotic
dependence χ and asymptotic independence χ.
3.1.1 Theoretical Background
Within this subsection I will provide an overview on the different concepts and estimators
that describe asymptotic dependence and asymptotic independence for the
bivariate case.
Dependence Measures for Multivariate Extreme
In order to reduce the information contained in the copula C to a single parameter, two
measures of extremal dependence χ and χ were introduced in [15] (1999). Both of them
are needed in order to examine whether two variables are asymptotically dependent or
asymptotically independent. First the bivariate returns X and Y are transformed to
unit Fréchet marginals S and T because of fat-tail distributions for risk asset returns:
S = −1/ log (FX(X)) T = −1/ log (FY (Y )), (3.2)
where FX and FY are marginal distribution functions for X and Y . Variables S and
T have the same dependence structure as X and Y and are now on a common scale,
meaning that events of the form S > s and T > s for large s, correspond to equally
extreme events for each variable. Asymptotic dependence χ was given by:
χ = lim
s→∞ Pr(T > s | S > s) (3.3)
= lim
s→∞
Pr(T > s, S > s)
Pr(S > s)
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