Mathematical and Statistical Methods for Actuarial Sciences and Finance

Multivariate Variance Gamma and Gaussian dependence: a study with copulas 197

Since the marginal and joint distributions in (5) and (6) cannot be written in
closed form, the copula of theα-VG process and the ensuing non-linear dependence
measures, such as Spearman’s rho and Kendall’s tau, cannot be obtained analytically.
The only measure of dependence one can find in closed form is the linear corre-
lation coefficient:
ρX(t)=

μ 1 μ 2 α 1 α 2 a

√

(σ 12 +μ^21 α 1 )(σ 22 +μ^22 αj)

. (10)

This coefficient is independent of time, but depends on both the marginal and the
common parametera. For given marginal parameters the correlation is increasing in
the parameteraifμ 1 μ 2 >0, as is the case in most financial applications. Sincea

has to satisfy the following bounds: 0≤a≤min

(

1

α 1 ,

1

α 2

)

the maximal correlation

allowed by the model corresponds toa=min

(

1

α 1 ,

1

α 2

)

.

However, it can be proved that linear dependence is not exhaustive, since even
whenρ=0 the components of the process can be dependent [7]. In order to study
the whole dependence we should evaluate empirical versions of the copula obtained
from (9) using the integral expression of the marginal and joint distributions in (5)
and (6). A possibility which is open to the researcher, in order to find numerically the
copula of the process at timet, is then the following:

• fix a grid(ui,vi),i= 1 ,...,Non the square [0,1]^2 ;


• for eachi= 1 ,...,Ncompute(Ft^1 )−^1 (ui)and(Ft^2 )−^1 (vi)by numerical ap-
  proximation of the integral expression: let(F ̃t^1 )−^1 (ui)and(F ̃t^2 )−^1 (vi)be the
  numerical results;


• find a numerical approximation for the integral expression (6), let it beFˆt(xi,yi);


• find the approximated value ofCt(ui,vi):

Cˆt(ui,vi)=Fˆ((F ̃t^1 )−^1 (ui),(F ̃t^1 )−^1 (ui)), i=^1 ,...,N.

We name the copulaCˆtnumerical,empiricaloractual copulaof theα-VG dis-
tribution at timet.
In order to discuss the behaviour of non-linear dependence we compare the em-
pirical copula and the Gaussian one with the same linear correlation coefficient, for
different tenorst. We use the classicalL^1 distance:

dt(Ct,Ct′)=

∫ 1

0

|Ct(u,v)−C′t(u,v)|dudv. (11)

It is easy to demonstrate that the distancedis consistent with concordance order,
i.e.,Ct≺Ct′≺C′′t impliesd(Ct,C′t)≤d(Ct,C′′t)[9]. It follows that the nearer the
copulas are in terms of concordance, the nearer they are in terms ofdt. Observe that
the maximal distance between two copulas is^16 , i.e., the distance between the upper
and lower Fr ́echet bounds.
Therefore for eachtwe:

• fix the marginal parameters and a linear correlation coefficient;