  We introduce a multivariate stochastic volatility model for asset returns
that imposes no restrictions to the structure of the volatility matrix and
treats all its elements as functions of latent stochastic processes. When the
number of assets is prohibitively large, we propose a factor multivariate
stochastic volatility model in which the variances and correlations of the
factors evolve stochastically over time. Inference is achieved via a carefully
designed feasible and scalable Markov chain Monte Carlo algorithm that combines
two computationally important ingredients: it utilizes invariant to the prior
Metropolis proposal densities for simultaneously updating all latent paths and
has quadratic, rather than cubic, computational complexity when evaluating the
multivariate normal densities required. We apply our modelling and
computational methodology to $571$ stock daily returns of Euro STOXX index for
data over a period of $10$ years. MATLAB software for this paper is available
at http://www.aueb.gr/users/mtitsias/code/msv.zip.
