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9.1. Les différentes mesures de dépendances extrêmes 251
of ρs(v). However, most copulas of interest in finance have no simple closed form, so that it is necessary to
resort to numerical computations.
As an example, let us consider the bivariate Gaussian distribution (or copula) with unconditional correlation
coefficent ρ. It is well-known that its unconditional Spearman’s rho is given by
ρs = 6 ρ
· arcsin . (18)
π 2
The left panel of figure 8 shows the conditional Spearman’s rho ρs(v) defined by (17) obtained from a
numerical integration. We observe the same bias as for the conditional correlation coefficient, namely the
conditional rank correlation changes with v eventhough the unconditional correlation is fixed to a constant
value. Nonetheless, this conditional Spearman’s rho seems more sensitive than the conditional correlation
coefficient since we can observe in the left panel of figure 8 that, as v goes to one, the conditional Spearman’s
rho ρs(v) does not go to zero for all values of ρ (at the precision of our bootstrap estimates), as previously
observed with the conditional correlation coefficient (see equation (3)).
The right panel of figure 8 depicts the conditional Spearman’s rho of the Student’s copula with three degres
of freedom. The results are the same concerning the bias, but this time ρs(v) goes to zero for all value
of ρ when v goes to one. Thus, here again, many different behaviours can be observed depending on the
underlying copula of the random variables. Moreover, these two examples show that the quantification of
extreme dependence is a function of the tools used to quantify this dependence. Here, the conditional Spearman’s
ρ goes to a non-vanishing constant for the Gaussian model, while the conditional (linear) correlation
coefficient goes to zero, contrarily to the Student’s distribution for which the situation is exactly the opposite.
2.3 Empirical evidence
Figures 9, 10 and 11 give the conditionnal Spearman’s rho respectively for the Argentina / Brazilian stock
markets, the Brazilian / Chilean stock markets and the Chilean / Mexican stock markets. As previously, the
plain thick line refers to the estimated correlation, while the dashed lines refer to the Gaussian copula and
its 95% confidence levels and and dotted lines to the Student’s copula with three degrees of freedom and its
95% confidence levels.
We first observe that contrarily to the cases of the conditional (linear) correlation coefficient exhibited in
figures 2, 4 and 6, the empirical conditional Spearman’s ρ does not always comply with the Student’s model
(neither with the Gaussian one), and thus confirm the discrepancies observed in figures 3, 5 and 7. In all
cases, for thresholds v larger than the quantile 0.5 corresponding to the positive returns, the Student’s model
with three degrees of freedom is always sufficient to explain the data. In contrast, for the negative returns
and thus thresholds v lower then the quantile 0.5, only the interaction between the Chilean and the Mexican
markets is well described by the Student copula and does not needed any additional ingredient such as the
contagion mechanism. For all other pairs, none of our models explain the data satisfyingly. Therefore, for
these cases and from the perspective of our models, the contagion hypothesis seems to be needed.
There are however several caveats. First, even though we have considered the most natural financial models,
there may be other models, that we have ignored, with constant dependence structure which can account for
the observed evolutions of the conditional Spearman’s ρ. If this is true, then the contagion hypothesis would
not be needed. Second, the discrepancy between the empirical conditional Spearman’s ρ and the prediction
of the the Student’s model does not occur in the tails the distribution, i.e for large and extreme movements,
but in the bulk. Thus, during periods of turmoil, the Student’s model with three degrees fo freedom seems to
remain a good model of co-movements. Third, the contagion effect is never necessary for upwards moves.
Indeed, we observe the same asymmetry or trend dependence as found by (Longin and Solnik 2001) for five
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