Generalized Normal Mean Variance Mixture and Subordinated Brownian Motion

Normal mean variance mixtures are extensively applied in finance. Under conditions for infinite divisibility they generate subordinated Brownian motions, used to represent stocks returns. The standard generalization to the multivariate setting of normal mean variance mixture does not allow for independence and can incorporate only limited dependence. In this paper we propose a multivariate definition of normal mean variance mixture, named generalized normal mean variance mixture, which includes both independence and high dependence. We give conditions for infinite divisibility and prove that the multivariate Lévy process defined from it is a subordinated Brownian motion. We analyze both the distribution and the related process. In the second part of the paper we use the construction to introduce a multivariate generalized hyperbolic distribution (and process) with generalized hyperbolic margins. We conclude with a numerical example to show the case of calibration and the flexibility of the model in describing dependence.