  Rough volatility is a well-established statistical stylised fact of financial
assets. This property has lead to the design and analysis of various new rough
stochastic volatility models. However, most of these developments have been
carried out in the mono-asset case. In this work, we show that some specific
multivariate rough volatility models arise naturally from microstructural
properties of the joint dynamics of asset prices. To do so, we use Hawkes
processes to build microscopic models that reproduce accurately high frequency
cross-asset interactions and investigate their long term scaling limits. We
emphasize the relevance of our approach by providing insights on the role of
microscopic features such as momentum and mean-reversion on the
multidimensional price formation process. We in particular recover classical
properties of high-dimensional stock correlation matrices.
