Building multivariate Sato models with linear dependence

Abstract

The increased trading in multi-name financial products has required the development of stateof-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.

Multivariate Lévy Models: Calibration and Pricing

“…Moreover, it would be meaningful to compare versions of LS and BB models that embed Sato margins. Such processes, studied by Marena et al (2018a) and Boen and Guillaume (2019), allow for non-stationary increments and are thus better suited to value path-dependent contracts. However, as these processes keep the same dependence structure of the original versions, a further direction for research could be to empirically assess models that allow for time-dependent correlations (see e.g., Semeraro (2022)), that are more consistent with market data.…”