Multivariate Stochastic Volatility via Wishart Processes

Financial models for asset and derivatives pricing, risk management, portfolio optimization, and asset allocation rely on volatility forecasts. Time-varying volatility models, such as generalized autoregressive conditional heteroscedasticity and stochastic volatility (SVOL), have been successful in improving forecasts over constant volatility models. We develop a new multivariate SVOL framework for modeling financial data that assumes covariance matrices stochastically varying through a Wishart process. In our formulation, scalar variances naturally extend to covariance matrices rather than to vectors of variances as in traditional SVOL models. Model fitting is performed using Markov chain Monte Carlo simulation from the posterior distribution. Because of the model's complexity, an efficiently designed Gibbs sampler is described that produces inferences with a manageable amount of computation. Our approach is illustrated on a multivariate time series of monthly industry portfolio returns. A test of the economic value of our model found that minimum-variance portfolios based on our SVOL covariance forecasts outperformed out-of-sample portfolios based on alternative covariance models, such as dynamic conditional correlations and factor-based covariances.