A Multivariate Time-Changed Lévy Model for Financial Applications

The purpose of this paper is to introduce a bivariate Lévy process constructed by subordination of a Brownian motion with independent components by a bivariate subordinator. A class of subordinated processes extensively studied in finance is the Variance Gamma one. They have been first introduced in literature by Madan and Seneta [5], [2], as models for stocks return. Let B̃ be a Brownian motion and G a Gamma process (A process {G(t), t ≥ 0} is a Gamma process with parameters (a, b) if it is a Lévy process so that the defining distribution of X(1) is Gamma with parameters (a, b) (shortly L(X(1)) = Γ(a, b))) and let them be independent. The Variance Gamma process X = {X(t) : t ≥ 0} is defined by subordination as: X(t) = B̃(G(t)), ∀t ≥ 0. The model we present here is inspired by the bivariate generalization of the Variance Gamma made by Luciano and Schoutens [4]. They considered two independent Brownian motion subordinated by a common Gamma process. The main contribute here is to introduce a bivariate subordinator G with correlated Gamma margins (subordination by a multivariate subordinators is studied in a more general setting in Barndorff-Nielsen, O.E. et al. [1]).