194 E. Luciano and P. Semeraro
In the financial literature, different univariate Levy processes – able to capture ́
non-normality – have been applied in order to model stock returns (as a reference
for Levy processes, see for example Sato [10]). Their multivariate extensions are ́
still under investigation and represent an open field of research. One of the most
popular Levy processes in Finance is the Variance Gamma intr ́ oduced by Madan
and Seneta [8]. A multivariate extension has been introduced by Madan and Seneta
themselves. A generalisation of this multivariate process, namedα-VG, has been
introduced by Semeraro [11]. The generalisation is able to capture independence and
to span a wide range of dependence. For fixed margins it also allows various levels
of dependence to be modelled. This was impossible under the previous VG model.
A thorough application to credit analysis is in Fiorani et al. [5]. Theα-VG process
depends on three parameters for each margin(μj,σj,αj)and an additional common
parametera. The linear correlation coefficient is known in closed formula and its
expression is independent of time. It can be proved [7] that the process also has
non-linear dependence.
How can we study dynamic dependence of theα-VG process? Powerful tools
to study non-linear dependence between random variables are copulas. In a seminal
paper, Embrechts et al. [4] invoked their use to represent both linear and non-linear
dependence. Copulas, which had been introduced in the late 1950s in statistics and
had been used mostly by actuaries, do answerstaticdependence representation needs.
However, they hardly cover all thedynamicrepresentation issues in finance. For Levy ́
processes or the distributions they are generated from, the reason is that, for given
infinitely divisible margins, the conditions that a copula has to satisfy in order to
provide an infinitely divisible joint distribution are not known [3].
In contrast, if one starts from a multivariate stochastic process as a primitive entity,
the corresponding copula seldom exists inclosed format every point in time. Indeed,
copula knowledge at a single point in time does not help in representing dependence
at later maturities. Apart from specific cases, such as the traditional Black Scholes
process, the copula of the process is time dependent. And reconstructing it from the
evolution equation of the underlying process is not an easy task. In order to describe
the evolution of dependence over time we need a family of copulas{Ct,t≥ 0 }.Most
of the time, as in the VG case, it is neither possible to deriveCtfrom the expression
ofC 1 nor to getC 1 in closed form. However, via Sklar’s Theorem [12], a numerical
version of the copula at any timetcan be obtained. The latter argument, together with
the fact that for theα-VG process the linear correlation is constant in time, leads us to
compare theα-VG empirical copula for different tenurestwith the Gaussian closed
form one. We study the evolution over time of the distance between the empirical
and the Gaussian copula as a measure of the corresponding evolution of non-linear
dependence.
The paper is organised as follows: Section 2 reviews the VG model and its depen-
dence; it illustrates how we reconstruct the empirical copula. Section 3 compares the
approximating (analytical) and actual (numerical) copula, while Section 4 concludes.