Stochastic Volatility: Univariate and Multivariate Extensions

Eric Jacquier
Finance Department, HEC Montreal

Nicholas G. Polson
Graduate School of Business, University of Chicago

Peter Rossi
Graduate School of Business, University of Chicago

Discrete time stochastic volatility models (hereafter SVOL) are noticeably harder to estimate than the successful ARCH family of models. In this paper we demonstrate efficient estimation and prediction for a number of univariate and multivariate SVOL models. Namely, we model fat-tailed and skewed conditional distributions, correlated errors distributions (leverage effect), and two multivariate models, a stochastic factor structure model and a stochastic discount dynamic model. These extensions of the basic model are needed if one wants for example to compare SVOL models with ARCH-style models or to implement option pricing and portfolio selection under stochastic volatility.

We specify the models as a hierarchy of conditional probability distributions: Pr(data | volatilities), Pr(volatilities | parameters) and Pr(parameters). This conceptually simple methodology provides a natural environment for the construction of stochastic volatility models that depart from standard distributional assumptions. Given a model and the data, inference and prediction are based on the joint posterior distribution of the volatilities and the parameters which we simulate via Markov chain Monte Carlo (MCMC) methods. Our approach also provides a sensitivity analysis for parameter inference and an outlier diagnostic. We estimate the model for several financial time series, and find that the extensions considered are indeed needed. For the SVOL model we find strong evidence of non-normal conditional distributions for stock returns and exchange rates. We also find evidence of correlated errors for stock returns.

Some key words: stochastic volatility, forecasting and smoothing, Metropolis algorithm.