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9.1. Les différentes mesures de dépendances extrêmes 245
which goes to one as v goes to infinity as 1 − ρ s v ∼v→∞ 1−ρ2
ρ 2
These two simple examples clearly show that, in the case of two Gaussian random variables, the two conditional
correlation coefficients ρ + v and ρ s v exhibit opposite behavior since the conditional correlation coefficient
ρ + v is a decreasing function of v which goes to zero as v → +∞ while the conditional correlation
coefficient ρ s v is an increasing function of v and goes to one as v → ∞. These opposite behaviors are very
general and do not depend on the particular choice of the joint distribution of X and Y , namely the Gaussian
distribution studied until now, as it will be seen in the sequel.
This result underlines the importance of the choice of the conditioning set with the following two caveats.
First, as already stressed by many authors, the conditional correlation ρ + v ou ρ s v change with the value of the
threshod v even if the unconditional correlation ρ remains unchanged. Thus, the observation of a change
in the conditional correlation does not provide a reliable signature of a change in the true (unconditional)
correlation. Second, the conditional correlations can exhibit really opposite behaviors depending on the
conditioning sets. Specifically, we have seen that accounting for a signed trend or only for its amplitude
may yield a decrease or an increase of the conditional correlation with respect to the unconditional one, so
that these changes cannot be interpreted as a strengthening or a weakening of the correlations.
v −2 .
1.3 Influence of the underlying distribution for a given conditioning set
For a fixed conditioning set defining a specific conditional correlation coefficient like ρ + v or ρ s v, the behavior
of these coefficients can be dramatically different from a pair of random variables to another one, depending
on their underlying joint distribution. As an example, let the variables X and Y have a multivariate Student’s
distribution with ν degrees of freedom and an (unconditional) correlation coefficient ρ. According to the
proposition stated in appendix B.1, we have the exact formula
ρA =
ρ
ρ2 + E[E(X2 | Y )−ρ2Y 2 . (5)
| Y ∈A]
Var(Y | Y ∈A)
Appendix B.1 gives the explicit formulas of E[E(X 2 | Y ) − ρ 2 Y 2 | Y ∈ A] and Var(Y | Y ∈ A).
Expression (5) is the analog for a Student bivariate distribution to (2) derived above for the Gaussian bivariate
distribution. Again, ρ and ρA share the following properties: they have the same sign, ρA equals zero if and
only if ρ equals zero and ρA can be either greater or smaller than ρ. We now apply this general formula (5)
to the calculus of ρ + v and ρ s v and we find (see appendices B.3 and B.4) that, conditioning on large returns,
ρ + v −→
while when conditionning on large volatility,
ρ s v −→
ρ 2 + (ν − 1)
ρ 2 + 1
(ν−1)
ρ
ν−2
ν (1 − ρ2 )
ρ
ν−2
ν (1 − ρ 2 )
, (6)
. (7)
In both cases, ρ + v and ρ s v goes, at infinity, to non vanishing constant (exepted for ρ = 0). Moreover, for ν
larger than νc 2.839, this constant is smaller than the unconditional correlation coefficient ρ, for all value
of ρ, in the case of ρ + v , while for ρ s v it is always larger than ρ, whatever ν (larger than two) may be.
These results show that, conditioned on large returns, ρ + v is a decreasing function of the threshold v (at
least when ν ≥ 2.839), while, conditioned on large volatilities, ρ s v is an increasing function of v. Thus, for
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