statistique, théorie et gestion de portefeuille - Docs at ISFA

272 9. Mesure de la dépendance extrême entre deux actifs financiers
C Proof of equation (9)
We assume that X and Y are related by the equation
X = αY + ɛ , (C.75)
where α is a non random real coefficient and ɛ an idiosyncratic noise independent of Y , whose distribution
is assumed to admit a moment of second order σ 2 ɛ . Let us also denote by σ 2 y the second moment of the
variable Y .
We have
Cov(X, Y | Y ∈ A) = Cov(αY + ɛ, Y | Y ∈ A), (C.76)
since Y and ɛ are independent. We have also
= αVar(Y | Y ∈ A) + Cov(ɛ, Y | Y ∈ A), (C.77)
= αVar(Y | Y ∈ A), (C.78)
Var(X | Y ∈ A) = = α 2 Var(Y | Y ∈ A) + 2 Cov(ɛ, Y | Y ∈ A) + Var(ɛ | Y ∈ A), (C.79)
= α 2 Var(Y | Y ∈ A) + σ 2 ɛ , (C.80)
where, again, we have used the independence of Y and ɛ. This allows us to write
Since
we finally obtain
which conclude the proof.
ρA =
=
αVar(Y | Y ∈ A)
, (C.81)
Var(Y | Y ∈ A)(α2Var(Y | Y ∈ A) + σ2 ɛ )
sgn(α)
1 + σ2 ɛ
α2 1 · Var(Y | Y ∈A)
ρA =
ρ =
sgn(α)
1 + σ2 ɛ
α 2 · 1
Var(Y )
ρ
ρ2 + (1 − ρ2 )
34
. (C.82)
Var(y)
Var(y | y∈A)
, (C.83)
, (C.84)