  The goal of this paper is to investigate how the marginal and dependence
structures of a variety of multivariate L\'evy models affect calibration and
pricing. To this aim, we study the approaches of Luciano and Semeraro (2010)
and Ballotta and Bonfiglioli (2016) to construct multivariate processes. We
explore several calibration methods that can be used to fine-tune the models,
and that deal with the observed trade-off between marginal and correlation fit.
We carry out a thorough empirical analysis to evaluate the ability of the
models to fit market data, price exotic derivatives, and embed a rich
dependence structure. By merging theoretical aspects with the results of the
empirical test, we provide tools to make suitable decisions about the models
and calibration techniques to employ in a real context.
