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244 9. Mesure de la dépendance extrême entre deux actifs financiers
then present an empirical illustration of the evolution of the correlation between several stock indexes of
Latin American markets.
1.1 Definition
We study the correlation coefficient ρA of two real random variables X and Y conditioned on Y ∈ A, where
A is a subset of R such that Pr{Y ∈ A} > 0.
By definition, the conditional correlation coefficient ρA is given by
ρA =
Cov(X, Y | Y ∈ A)
Var(X | Y ∈ A) · Var(Y | Y ∈ A) . (1)
Applying this general expression of the conditional correlation coefficient, we will give closed formula for
several standard distributions and models. This will allow us to investigate the influence of the conditionning
set and the underlying model on the behavior of ρA.
1.2 Influence of the conditioning set
Let the variables X and Y have a multivariate Gaussian distribution with (unconditional) correlation coefficient
ρ. The following result have been proved by (Boyer et al. 1997) :
ρ
. (2)
ρA =
ρ2 + (1 − ρ2 Var(Y )
) Var(Y | Y ∈A)
We can note that ρ and ρA have the same sign, that ρA = 0 if and only if ρ = 0 and that ρA does not depend
directly on Var(X). Note also that ρA can be either greater or smaller than ρ since Var(Y | Y ∈ A) can be
either greater or smaller than Var(Y ). We will illustrate this property in the two following examples, where
we consider a conditioning on large positive (or negative) returns and a conditioning on large volatility.
The difference comes from the fact that in the first case, we account for the trend while we neglect this
information in the second case.
Example 1: conditioning on large (positive) returns. Let us first consider the conditioning set A =
[v, +∞), with v ∈ R+. Thus ρA is the correlation coefficient conditioned on the returns Y larger than a
given positive threshold v. It will be denoted by ρ + v in the sequel. Assuming for simplicity, but without loss
of generality, that Var(Y ) = 1, we can easily show (see appendix A.1.1 for an exact calculation) that for
large v
ρ + v ∼v→∞
ρ 1
· , (3)
1 − ρ2 v
which slowly goes to zero as v goes to infinity. Obviously, by symmetry, the conditional correlation coefficient
ρ − v , conditioned on Y smaller than v, obeys the same formula.
Example 2: conditioning on large volatilities. Let now the conditioning set be A = (−∞, −v] ∪
[v, +∞), with v ∈ R+. Thus ρA is the correlation coefficient conditioned on |Y | larger than v, i.e., it
is conditioned on large volatility of Y . Still assuming Var(Y ) = 1, we denote this correlation coefficient by
ρ s v and, as shown in appendix A.1.2, we can conclude that, for large v,
ρ s v ∼v→∞
ρ
ρ2 + 1−ρ2
2+v2 , (4)
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