**1. Introduction**

Continuous-time stochastic covariance models are crucial in capturing many stylized facts in financial data, from heteroscedasticity and fat tails to changing correlations and leverage effects. Early work in this field focused on discrete time models in the form of generalized autoregressive conditional heteroskedasticity (GARCH) models (see Engle 2002). The best-known representatives in continuous time, are the stochastic Wishart family (see Da Fonseca et al. 2007; Gourieroux 2006) and the Ornstein–Uhlenbeck (OU) family (see Muhle-Karbe et al. 2012) of models, as well as general linear-quadratic jump-diffusions (see Cheng and Scaillet 2007). These approaches are more realistic than the classical Black–Scholes lognormal model, but they quickly become intractable as dimensions increase in terms of the number of parameters and simulation paths, commonly known as the "curse of dimensionality". Recent papers (see De Col et al. 2013; Escobar 2018) have presented models built from linear combination of tractable one-dimensional counterparts. These models involve fewer parameters than Wishart- or OU-type approaches, owing to a reduction in dimensionality while providing a closed-form solution to financial problems.

In this paper, we introduce a multivariate mean-reverting stochastic volatility factor model that combines 1/2 (Heston-type, Heston 1993) and 3/2 processes (Platen 1997) for the modeling of volatility. Such underlying volatility processes were coined 4/2 by Grasselli (2017). Our paper takes advantage of the factor structure in asset prices and allows for a mean-reverting structure on the assets thereby aiming at capturing either multivariate commodity behavior or multiple volatility indexes (see Gnoatto et al. 2018 for an alternative multivariate non-mean-reverting generalization based on a pairwise-structure applied to the exchange-rate market). In particular, our setting reduces the dimension of the parametric space which is a way of controlling the "curse of dimensionality" making parameters identifiable and popular estimation methods feasible. Secondly, the presence of independent common and intrinsic factors, each with its own stochastic volatility, enables an elegant separable structure for characteristic functions (c.f.s) and captures several stylized facts, such as: stochastic volatility, stochastic correlation among stocks (see Engle 2002), co-movements in the

variances (see Diebold and Nerlove 1989), multiple factors in the volatilities (see Heston et al. 2009) and stock correlations (see Da Fonseca et al. 2007). Thirdly, a factor representation is compatible with economical interpretations, where common factors are exogenous variables explaining financial markets, and intrinsic factors relate to companies' intrinsic risks. Lastly, closed-form expressions are available for joint c.f.s; this is useful for derivative pricing and risk managemen<sup>t</sup> calculations via Fourier transformations, and it makes c.f.-based estimations methods feasible (see Carr and Madan 1999; Caldana and Fusai 2013; Fusai et al. 2018).

The rationale for a 4/2 volatility process rather than a 1/2 or 3/2 model is masterly presented in Grasselli (2017) for a one-dimensional structure. For instance, as observed by the author, the 1/2 process predicts that the implied volatility skew will flatten when the instantaneous volatility increases (crises), while the 3/2 model predicts steepening skews. The empirical violation of the Feller condition in the 1/2 model is also noted, which makes volatility paths stay closer to 0 for a longer period than empirically supported, while the 3/2 model admits extreme paths with spikes in instantaneous volatility. The two processes complement each other as they imply very different dynamics for the evolution of the implied volatility surface. It stands to reason that such a convenient underlying drive for multidimensional structures should be used to improve not only marginal volatility behavior, but also the dependence structure.

We obtain an analytical representation for the c.f. of the vector of asset prices, which is in closed-form for non-mean-reverting nested cases. This type of c.f. is helpful for derivative pricing purposes. We also produce a second conditional c.f. that can be used for exact simulations of the non-mean reverting assets given the terminal volatilities, where the latter can be simulated exactly via chi-squares. We identify a set of conditions that not only produces well-defined changes of measure, but also avoids local martingales; hence, it can be used for risk-neutral pricing purposes.

Our results were applied numerically to parameters inspired by commodity prices1. There is a vast literature on commodity modeling (see, for instance, Chiarella et al. 2013; Schwartz 1997, and more recently Schneider and Tavin 2018). In our numerical study, we investigated the impact of the new parameters (*b*, the weight of 3/2 in the overall instantaneous volatility) on the shape of the implied volatility surface and the values of two risk measures: VaR and expected shortfall.
