  We introduce a multivariate diffusion model that is able to price derivative
securities featuring multiple underlying assets. Each asset volatility smile is
modeled according to a density-mixture dynamical model while the same property
holds for the multivariate process of all assets, whose density is a mixture of
multivariate basic densities. This allows to reconcile single name and
index/basket volatility smiles in a consistent framework. Our approach could be
dubbed a multidimensional local volatility approach with vector-state dependent
diffusion matrix. The model is quite tractable, leading to a complete market
and not requiring Fourier techniques for calibration and dependence measures,
contrary to multivariate stochastic volatility models such as Wishart. We prove
existence and uniqueness of solutions for the model stochastic differential
equations, provide formulas for a number of basket options, and analyze the
dependence structure of the model in detail by deriving a number of results on
covariances, its copula function and rank correlation measures and
volatilities-assets correlations. A comparison with sampling simply-correlated
suitably discretized one-dimensional mixture dynamical paths is made, both in
terms of option pricing and of dependence, and first order expansion
relationships between the two models' local covariances are derived. We also
show existence of a multivariate uncertain volatility model of which our
multivariate local volatilities model is a Markovian projection, highlighting
that the projected model is smoother and avoids a number of drawbacks of the
uncertain volatility version. We also show a consistency result where the
Markovian projection of a geometric basket in the multivariate model is a
univariate mixture dynamics model. A few numerical examples on basket and
spread options pricing conclude the paper.
