statistique, théorie et gestion de portefeuille - Docs at ISFA

In the sequel, we will first evaluate the moments, which turns out to be easier, and then using eq (66) we
will be able to calculate the cumulants.
8.2 Symmetric assets
We start with the expression of the distribution of the weighted sum of N assets :
PS(s) =
R N
447
N
dx P (x)δ( wixi − s) , (67)
where δ(·) is the Dirac distribution. Using the change of variable (40), allowing us to go from the asset
returns Xi’s to the transformed returns Yi’s, we get
1
PS(s) =
(2π) N/2
det(V )
R N
i=1
1
−
dy e 2 ytV −1y δ(
Taking its Fourier transform ˆ PS(k) = dsPS(s)eiks , we obtain
ˆPS(k)
1
=
(2π) N/2
det(V )
where ˆ PS is the characteristic function of PS.
R N
N
wisgn(yi)f −1 (y 2 i ) − s) . (68)
i=1
1
−
dy e 2 ytV −1y+ikN i=1 wisgn(yi)f −1 (y2 i ) , (69)
In the particular case of interest here where the marginal distributions of the variables Xi’s are the modified
Weibull pdf,
f −1 (yi) = χi| yi
√2 | qi (70)
with
the equation (69) becomes
ˆPS(k)
1
=
(2π) N/2
det(V )
R N
qi = 2/ci , (71)
1
−
dy e 2 ytV −1y+ikN i=1 wisgn(yi)χi| y √2 i | qi . (72)
The task in front of us is to evaluate this expression through the determination of the moments and/or
cumulants.
8.2.1 Case of independent assets
In this case, the cumulants can be obtained explicitely (Sornette et al. 2000b). Indeed, the expression (72)
can be expressed as a product of integrals of the form
We obtain
+∞
0
C2n =
u2
− 2 du e +ikwiχiu i
√2q
. (73)
N
c(n, qi)(χiwi) 2n , (74)
i=1
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