statistique, théorie et gestion de portefeuille - Docs at ISFA

242 9. Mesure de la dépendance extrême entre deux actifs financiers
Our contribution to the literature is both methological and empirical. On the methodological front, firstof-all,
we review the existing tools available for probing the dependence between large or extreme events
for several models of interest for financial time series; second, we provide explicit analytical expressions
for these measures of dependence between two variables; third, this allows us to quantify the misleading
intrepretations of certain conditional coefficients commomly used for exploring the evolution of the dependence
associated with a change in the market conditions (an increase of the volatility, for instance). On the
empirical front, we apply our theoretical results to the controversial problem of the occurrence or not of a
contagion phenomenon across Latin American markets during the turmoil period associated with the Mexican
crisis in 1994 or with the recent Argentina crisis. In this purpose, we use the novel insight derived from
our analysis on several measures of dependence and apply them to the question of a possible evolution of
the dependence between the Argentina, Brazilian, Chilean and Mexican markets with respect to the market
conditions.
The dependence measures we study are the conditional correlation coefficients ρ + v , ρ s v, ρu, the conditional
Spearman’s rho ρs(v) and the tail dependence coefficients λ and ¯ λ, whose properties are investigated for
several models among which are the bivariate Gaussian distribution, the bivariate Student’s distribution,
and the one factor model for various distributions of the factor. Initially, we hoped to show the existence
of logical links between some of these measures, such as a vanishing tail-dependence parameter λ implies
vanishing asymptotic conditional correlation coefficients. In fact, we will show that this turns out to be
wrong and one can construct simple examples for which all possible combinations occur. Therefore, each
of these measures probe a different quality of the dependence between two variables for large or extreme
events. In addition, even if the conditional correlation coefficients are asymptotically zero, they decay in
general extremely slowly, as inverse powers of the value of the threshold, and may thus remain significant
for most pratical applications. These results will allow us to assert that, somewhat similarly to risks whose
adequate characterization requires an extension beyond the restricted one-dimensional measure in terms of
the variance (volatility) to include all higher order cumulants or more generally the knowledge of the full
distribution (Sornette et al. 2000a, Sornette et al. 2000b, Andersen and Sornette 2001), our results suggest
that tail-dependence has also a multidimensional character.
The paper is organized as follows.
Section 1 describes three conditional correlation coefficients, namely the correlation ρ + v conditioned on
signed exceedance of one variable, or on both variables (ρu) and the correlation ρ s v conditioned on absolute
value exceedance (or large volatility) of one variable. (Boyer et al. 1997) have already provided the general
expression of ρ + v and ρ s v for the Gaussian bivariate model, which are used to derive their v dependence for
large v, and to show that, for a given distribution, the conditional correlation coefficient changes even if the
unconditional correlation is let unchanged and the nature of this change depends on the conditioning set.
We then provide the general expression of ρ + v and ρ s v for the Student’s bivariate model with ν degrees of
freedom and for the factor model X = αY + ɛ, for which we give a general expression of the conditional
correlation coefficient whatever the distributions of Y and ɛ may be. This leads us to conclude by comparision
with the Gaussian model that, for a fixed conditioning set, the behavior of the conditional correlation
change dramatically from a distribution to another one. Conditioning now on both variables, we are able to
provide the asymptotic dependence of ρu only for the bivariate Gaussian model and show that it essentially
behaves like ρ + v . We then apply these results to show that we cannot entirely explain the behavior of the conditional
correlation coefficient of the Latin American stock indexes by the conditioning effect, suggesting
the existence of a possible contagion.
In section 2, to account for several deficiencies of the correlation coefficient, we propose an alternative
measure of dependence, the conditional Spearman’s rho, which is related to the probability of concordance
and discordance of several events drawn from the same probability distribution. This measure provides
an important improvement with respect to the correlation coefficient since it only takes into account the
4