Mathematical and Statistical Methods for Actuarial Sciences and Finance

Multivariate Variance Gamma and Gaussian dependence: a study with copulas 199

Ta b le 1 .Calibrated parameters for theα-VG price processes for the stock indices in the sample

Asseti μi σi αi

S&P − 0. 65 0.22 0.10

Nasdaq − 0. 67 0.11 0.13

CAC 40 − 0. 46 0.10 0.11

FTSE − 0. 59 0.045 0.031

Nikkei − 0. 34 0.16 0.10

DAX − 0. 27 0.13 0.14

Hang Seng − 1. 68 0.8 0.03

Ta b le 2 .Maximal correlation anda-parameter (in parentheses) for the calibratedα-VG models,
all stock indices

S&PNasdaqCAC 40 FTSE Nikkei Dax

Nasdaq 0.803

(7.590)

CAC 40 0.795 0.701

(9.020)(7.590)

FTSE 0.505 0.410 0.406

(9.791)(7.590) (9.020)

Nikkei 0.556 0.461 0.457 0.284

(9.593)(7.590) (9.020) (9.593)

Dax 0.512 0.536 0.447 0.261 0.294

(7.092)(7.092) (7.092) (7.092)(7.092)

Hang Seng 0.500 0.406 0.403 0.834 0.282 0.259

(9.791)(7.590) (9.020)(31.976)(9.593)(7.092)

3.3 Copula results

We computed the empirical copulaCˆtfor the following tenors:t= 0. 1 , 1 , 10 ,100.
We report in Table 3 the distancesdtcorresponding toeach pair of stocks and each
timet.
In order to give a qualitative idea of the distances obtained we also provide a
graphical representation of the copula level curves for the pair Nasdaq andS&Pat
timet=1.
We observe that the distance in Table 3 is very low and decreasing in time. The plot
(and similar, unreported ones, for other couples and tenors) reinforces the conclusion.
Therefore the Gaussian copula seems to be a good approximation of the true copula,
at least for long horizons.