  The Multi Variate Mixture Dynamics model is a tractable, dynamical,
arbitrage-free multivariate model characterized by transparency on the
dependence structure, since closed form formulae for terminal correlations,
average correlations and copula function are available. It also allows for
complete decorrelation between assets and instantaneous variances. Each single
asset is modelled according to a lognormal mixture dynamics model, and this
univariate version is widely used in the industry due to its flexibility and
accuracy. The same property holds for the multivariate process of all assets,
whose density is a mixture of multivariate basic densities. This allows for
consistency of single asset and index/portfolio smile. In this paper, we
generalize the MVMD model by introducing shifted dynamics and we propose a
definition of implied correlation under this model. We investigate whether the
model is able to consistently reproduce the implied volatility of FX cross
rates once the single components are calibrated to univariate shifted lognormal
mixture dynamics models. We consider in particular the case of the Chinese
renminbi FX rate, showing that the shifted MVMD model correctly recovers the
CNY/EUR smile given the EUR/USD smile and the USD/CNY smile, thus highlighting
that the model can also work as an arbitrage free volatility smile
extrapolation tool for cross currencies that may not be liquid or fully
observable. We compare the performance of the shifted MVMD model in terms of
implied correlation with those of the shifted Simply Correlated Mixture
Dynamics model where the dynamics of the single assets are connected naively by
introducing correlation among their Brownian motions. Finally, we introduce a
model with uncertain volatilities and correlation. The Markovian projection of
this model is a generalization of the shifted MVMD model.
