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.