statistique, théorie et gestion de portefeuille - Docs at ISFA

442 14. Gestion de Portefeuilles multimoments et équilibre de marché
6.3 Determination of the joint distribution : maximum entropy and Gaussian copula
Let us now consider N random variables Xi with marginal distributions pi(xi). Using the transformation
(38), we define N standard normal variables Yi. If these variables were independent, their joint distribution
would simply be the product of the marginal distributions. In many situations, the variables are not
independent and it is necessary to study their dependence.
The simplest approach is to construct their covariance matrix. Applied to the variables Yi, we are certain that
the covariance matrix exists and is well-defined since their marginal distributions are Gaussian. In contrast,
this is not ensured for the variables Xi. Indeed, in many situations in nature, in economy, finance and in
social sciences, pdf’s are found to have power law tails ∼ A
x1+µ for large |x|. If µ ≤ 2, the variance and the
covariances can not be defined. If 2 < µ ≤ 4, the variance and the covariances exit in principle but their
sample estimators converge poorly.
We thus define the covariance matrix:
V = E[yy t ] , (42)
where y is the vector of variables Yi and the operator E[·] represents the mathematical expectation. A
classical result of information theory (Rao 1973) tells us that, given the covariance matrix V , the best joint
distribution (in the sense of entropy maximization) of the N variables Yi is the multivariate Gaussian:
1
P (y) =
(2π) N/2det(V ) exp
− 1
2 ytV −1
y . (43)
Indeed, this distribution implies the minimum additional information or assumption, given the covariance
matrix.
Using the joint distribution of the variables Yi, we obtain the joint distribution of the variables Xi:
∂yi
P (x) = P (y)
∂xj
, (44)
where ∂yi
is the Jacobian of the transformation. Since
we get
∂xj
This finally yields
P (x) =
∂yi
∂xj
= √ 2πpj(xj)e 1
2 y2 i δij , (45)
∂yi
N
∂xj
= (2π)N/2
i=1
pi(xi)e 1
2 y2 i . (46)
1
exp −
det(V ) 1
2 yt (x) (V −1
N
− I)y (x) pi(xi) . (47)
i=1
As expected, if the variables are independent, V = I, and P (x) becomes the product of the marginal
distributions of the variables Xi.
Let F (x) denote the cumulative distribution function of the vector x and Fi(xi), i = 1, ..., N the N corresponding
marginal distributions. The copula C is then such that
F (x1, · · · , xn) = C(F1(x1), · · · , Fn(xn)) . (48)
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