A welcome to the jungle of continuous-time multivariate non-Gaussian models based on Lévy processes applied to finance

In this paper we review the large and growing literature on continuous-time multivariate non-Gaussian models based on Levy processes applied to finance and proposed in the literature in the last years. We explain the empirical motivation and the idea behind each approach. Then, we study the models focusing on the parsimony of the number of parameters, the properties of the dependence structure, and the computational tractability. For each parametric class we analyze the main features, we provide the characteristic function, the marginal moments up to order four, the covariances and the correlations. Furthermore, we survey the methods proposed in literature to calibrate these models on the time-series of log-returns, with a view toward practical applications and possible numerical issues. Finally, to empirically assess the differences between models, we conduct an analysis on a five-dimensional series of stock index log-returns.