statistique, théorie et gestion de portefeuille - Docs at ISFA

266 9. Mesure de la dépendance extrême entre deux actifs financiers
A.2.2 Asymptotic behavior of the first moment m10
The first moment m10 = E[X | X > u, Y > u] is given by (A.13). For large u,
1 − ρ
1 − Φ
1 + ρ u
= 1
2 erfc
1 − ρ
2(1 + ρ) u
=
1 + ρ
1 − ρ
1 + ρ
−15
1 − ρ
1−ρ
−
e 2(1+ρ) u2
√
2π u
3
(A.24)
2 1 + ρ 1 1 + ρ
1 − · + 3 ·
1 − ρ u2 1 − ρ
1
u4 · 1
1
+ O
u6 u8 , (A.25)
so that multiplying by (1 + ρ) φ(u), we obtain
u2
−
(1 + ρ)2 e 1+ρ
2
1 + ρ 1 1 + ρ
m10 L(u, u; ρ) = 1 − · + 3 ·
1 − ρ2 2π u 1 − ρ u2 1 − ρ
1
3 1 + ρ
− 15 ·
u4 1 − ρ
1
1
+ O
u6 u8 .
(A.26)
Using the result given by equation (A.22), we can conclude that
m10 = u + (1 + ρ) · 1
u − (1 + ρ)2 (2 − ρ)
·
(1 − ρ)
1
u3 + (10 − 8ρ + 3ρ2 )(1 + ρ) 3
(1 − ρ) 2
In the sequel, we will also need the behavior of m10 2 :
m10 2 = u 2 + 2 (1 + ρ) − (1 + ρ)2 (3 − ρ)
(1 − ρ)
A.2.3 Asymptotic behavior of the second moment m20
· 1
u2 + 2(8 − 5ρ + 2ρ2 )(1 + ρ) 3
(1 − ρ) 2
· 1
1
+ O
u5 u7
. (A.27)
· 1
1
+ O
u4 u6
. (A.28)
The second moment m20 = E[X2 | X > u, Y > u] is given by expression (A.14). The first term in the
right hand side of (A.14) yields
(1 + ρ 2
1 − ρ
) u φ(u) 1 − Φ
1 + ρ u
= (1 + ρ 2 u2
−
1 + ρ e 1+ρ
2
1 + ρ 1 1 + ρ
)
1 − · + 3 ·
1 − ρ 2π 1 − ρ u2 1 − ρ
1
u4 3 1 + ρ
−15 ·
1 − ρ
1
1
+ O
u6 u8 (A.29) ,
while the second term gives
ρ 1 − ρ 2
√ 2π
2
φ
1 + ρ u
= ρ 1 − ρ
2 e− u2
1+ρ
2π
. (A.30)
Putting these two expressions together and factorizing the term (1 + ρ)/(1 + ρ 2 ) allows us to obtain
m20 L(u, u; ρ) =
(1 + ρ)2
1 − ρ 2
u2
−
e 1+ρ
2π
1 + ρ2 1
1 − ·
1 − ρ
−15 (1 + ρ2 )(1 + ρ) 2
(1 − ρ) 3
28
u2 + 3(1 + ρ2 )(1 + ρ)
(1 − ρ) 2
· 1
+ O
u6 1
u 8
· 1
u 4
+ L(u, u; ρ) , (A.31)