Multivariate Variance Gamma and Gaussian dependence: a study with copulas 201Fig. 1.Level curves of the Gaussian (Gaus) and empirical (emp) copula of theα-VG calibrated
price processes, S & P - Nasdaq, after one year
Ta b le 4 .Simulated values ofρ ̃s(t)andτ( ̃t)for the numerical (Cˆ) and Gaussian copula (Gauss)
over different horizons. S & P/Nasdaq pair
Pair Cˆ 0. 1 Cˆ 1 Cˆ 10 Cˆ 100 Gauss
S&P/Nasdaqρ ̃s 0.74 0.78 0.79 0.79 0.79
τ ̃ 0.54 0.59 0.58 0.59 0.594 Conclusions and further research
This paper measures the non-linear dependence of theα-VG process, calibrated to
a set of stock market data, by means of a distance between its empirical copula at
timetand the corresponding Gaussian one, which is characterised by the (constant)
correlation coefficient of the process.
Our empirical analysis suggests that non-linear dependence is “decreasing” in
time, since the approximation given by the Gaussian copula improves in time. As
expected, non-linear dependence coefficients confirm the result. The tentative con-
clusion is that, similarly to marginal non-Gaussianity, which is usually stronger on
short-horizon than on long-horizon returns, joint non-linear dependence and non-
Gaussianity fade over time.
This represents an important point of departure for practical, large-scale imple-
mentations of theα-VG model and of its subcase, the traditional VG model. Any
multivariate derivative price or portfolio risk measure indeed is based on the joint
distribution of returns. If we use a time-varying empirical copula in order to re-assess
prices and risk measures over time, and we want the results to be reliable, exten-
sive and time-consuming simulations are needed. If, on the contrary, we can safely