Mathematical and Statistical Methods for Actuarial Sciences and Finance

Multivariate Variance Gamma and Gaussian

dependence: a study with copulas∗

Elisa Luciano and Patrizia Semeraro

Abstract.This paper explores the dynamic dependence properties of a L ́evy process, the

Variance Gamma, which has non-Gaussian marginal features and non-Gaussian dependence.

By computing the distance between the Gaussian copula and the actual one, we show that even

a non-Gaussian process, such as the Variance Gamma, can “converge” to linear dependence

over time. Empirical versions of different dependence measures confirm the result over major

stock indices data.

Key words:multivariate variance Gamma, L ́evy process, copulas, non-linear dependence

1 Introduction

Risk measures and the current evolution of financial markets have spurred the interest

of the financial community towards models of asset prices which present both non-

Gaussian marginal behaviour and non-Gaussian, or non-linear, dependence. When

choosing from the available menu of these processes, one looks for parsimoniousness

of parameters, good fit of market data and, possibly, ability to capture their dependence

and the evolution of the latter over time. It is difficult to encapsulate all of these

features – dynamic dependence, in particular – in a single model. The present paper

studies an extension of the popular Variance Gamma (VG) model, namedα-VG,

which has non-Gaussian features both at the marginal and joint level, while succeeding

in being both parsimonious and accurate in data fitting. We show that dependence

“converges” towards linear dependence over time. This represents good news for

empirical applications, since over long horizons one can rely on standard dependence

measures, such as the linear correlation coefficient, as well as on a standard analytical

copula or dependence function, namely the Gaussian one, even starting from data

which do not present the standard Gaussian features of the Black Scholes or log-

normal model. Let us put the model in the appropriate context first and then outline

the difficulties in copula towards dynamic dependence description then.

∗©c2008 by Elisa Luciano and Patrizia Semeraro. Any opinions expressed here are those of

the authors and not those of Collegio Carlo Alberto.

M. Corazza et al. (eds.), Mathematical and Statistical Methodsfor Actuarial Sciencesand Finance
© Springer-Verlag Italia 2010