statistique, théorie et gestion de portefeuille - Docs at ISFA

264 9. Mesure de la dépendance extrême entre deux actifs financiers
Expression (A.8) is the same as (A.4) as it should. This gives the following conditional variance:
Var(Y | |Y | > v) = 1 +
√
2v
1
v2
, (A.9)
and finally yields, for large v,
A.1.3 Intuitive meaning
ρ s v ∼v→∞
ρ
ρ 2 + 1−ρ2
2+v 2
√ v
πe 2 = v
v√2
2 erfc
2 + 2 + O
∼v→∞ 1 − 1
2
1 − ρ 2
ρ 2
1
. (A.10)
v2 Let us provide an intuitive explanation (see also (Longin and Solnik 2001)). As seen from (A.1), ρ + v is
controlled by the dependence Var(Y | Y > v) ∝ 1/v 2 derived in Appendix A.1.1. In contrast, as seen from
(A.6), ρ s v is controlled by Var(Y | |Y | > v) ∝ v 2 given in Appendix A.1.2. The difference between ρ + v and
ρ s v can thus be traced back to that between Var(Y | Y > v) ∝ 1/v 2 and Var(Y | |Y | > v) ∝ v 2 for large v.
This results from the following effect. For Y > v, one can picture the possible realizations of Y as those of
a random particle on the line, which is strongly attracted to the origin by a spring (the Gaussian distribution
that prevents Y from performing significant fluctuations beyond a few standard deviations) while being
forced to be on the right to a wall at Y = v. It is clear that the fluctuations of the position of this particle
are very small as it is strongly glued to the unpenetrable wall by the restoring spring, hence the result
Var(Y | Y > v) ∝ 1/v 2 . In constrast, for the condition |Y | > v, by the same argument, the fluctuations
of the particle are hindered to be very close to |Y | = v, i.e., very close to Y = +v or Y = −v. Thus, the
fluctuations of Y typically flip from −v to +v and vice-versa. It is thus not surprising to find Var(Y | |Y | >
v) ∝ v 2 .
This argument makes intuitive the results Var(Y | Y > v) ∝ 1/v2 and Var(Y | |Y | > v) ∝ v2 for large
v and thus the results for ρ + v and for ρs v if we use (A.1) and (A.6). We now attempt to justify ρ + v ∼v→∞ 1
v
and 1 − ρs v ∼v→∞ 1/v2 directly by the following intuitive argument. Using the picture of particles, X
and Y can be visualized as the positions of two particles which fluctuate randomly. Their joint bivariate
Gaussian distribution with non-zero unconditional correlation amounts to the existence of a spring that ties
them together. Their Gaussian marginals also exert a spring-like force attaching them to the origin. When
Y > v, the X-particle is teared off between two extremes, between 0 and v. When the unconditional
correlation ρ is less than 1, the spring attracting to the origin is stronger than the spring attracting to the
wall at v. The particle X thus undergoes tiny fluctuations around the origin that are relatively less and less
attracted by the Y -particle, hence the result ρ + v ∼v→∞ 1
v → 0. In constrast, for |Y | > v, notwithstanding
the still strong attraction of the X-particle to the origin, it can follow the sign of the Y -particle without
paying too much cost in matching its amplitude |v|. Relatively tiny fluctuation of the X-particle but of the
same sign as Y ≈ ±v will result in a strong ρs v, thus justifying that ρs v → 1 for v → +∞.
A.2 Conditioning on both X and Y larger than u
By definition, the conditional correlation coefficient ρu, conditioned on both X and Y larger than u, is
ρu =
=
Cov[X, Y | X > u, Y > u]
Var[X | X > u, Y > u] Var[Y | X > u, Y > u] , (A.11)
m11 − m10 · m01
√
m20 − m10 2√ , (A.12)
m02 − m01
2
26