statistique, théorie et gestion de portefeuille - Docs at ISFA

414 13. Gestion de portefeuille sous contraintes de capital économique
Thus
f(x1, · · · , xN) = f(x ∗ 1, · · · , x ∗ N) + ∂f
=
Sc ( 1
+
wiσi) c−1 2
i
ij
∂xi
hi + 1
2
ij
∂2f hihj + · · · (158)
∂xi∂xj
∂2f hihj + · · · , (159)
∂xi∂xj
where, as in the previous section, hi = xi − x ∗ i and the derivatives of f are expressed at x∗ 1 , ..., x∗ N .
It is easy to check that the nth-order derivative of f with respect to the xi’s evaluated at {x∗ i } is proportional
to Sc−n . In the sequel, we will use the following notation :
∂n
f
∂xi1 · · · ∂xin
We can write :
f(x1, · · · , xN) =
up to the fourth order. This leads to
P (S) ∝ e −
Sc ( Sc−2
+
wiσi) c−1 2
( w
S c
iσi ) c−1
{x ∗ i }
ij
= M (n)
i1···in Sc−n . (160)
M (2)
ij hihj + Sc−3
6
ijk
Sc−2ij
(2)
− M
dh1 · · · dhNe 2
ij hihj
δ wihi
×
⎡
⎣1 + Sc−3
6
Using the relation δ ( wihi) = dk
2π e−ikj wjhj , we obtain :
P (S) ∝ e −
or in vectorial notation :
( w
S c
iσi ) c−1
P (S) ∝ e −
( w
dk
2π
S c
iσi ) c−1
×
M (3)
ijk
M (3)
ijk hihjhk + · · · (161)
ijk hihjhk + · · ·
×
⎤
Sc−2ij
(2)
− M
dh1 · · · dhNe 2
ij hihj−ikj wjhj ×
×
dk
2π
⎡
⎡
⎣1 + Sc−3
6
⎣1 + Sc−3
6
Let us perform the following standard change of variables :
M (3)
ijk
ijk hihjhk + · · ·
Sc−2
−
dh e 2 htM (2) h−ikwth ×
⎤
M (3)
(M (2)−1 exists since f is assumed convex and thus M (2) positive) :
S c−2
ijk
ijk hihjhk + · · ·
⎦ . (162)
⎤
⎦ , (163)
⎦ . (164)
h = h ′ − ik
S c−2 M(2)−1 w . (165)
2 ht M (2) h + ikw t h = Sc−2
2 h′t M (2) h ′ + k2
2S c−2 wt M (2)−1 w . (166)
26