Building multivariate Sato models with linear dependence

The increased trading in multi-name financial products has required the development of state-of-the-art multivariate models. These models should be computationally tractable and, at the same time, flexible enough to explain the stylized facts of asset log-returns and of their dependence structure. The popular class of multivariate Lévy models provides a variety of tractable models, but suffers from one major shortcoming: Lévy models can replicate single-name derivative prices for a given time-to-maturity, but not for the whole range of quoted strikes and maturities, especially during periods of market turmoil. Moreover, there is a significant discrepancy between the moment term structure of Lévy models and the one observed in the market. Sato processes on the other hand exhibit a moment term structure that is more in line with empirical evidence and allow for a better replication of single-name option price surfaces. In this paper, we propose a general framework for multivariate models characterized by independent and time-inhomogeneous increments, where the asset log-return processes at unit time are modeled as linear combinations of independent self-decomposable random variables, where at least one self-decomposable random variable is shared by all the assets. As examples, we consider two general subclasses within this new framework, where we assume a normal variance-mean mixture with a one-sided tempered stable mixing density or a difference of one-sided tempered stable laws for the distribution of the risk factors. Particular attention is given to the models' ability to explain the asset dependence structure. A numerical study reveals the advantages of these new types of models.