Monte Carlo methods for estimating, smoothing, and filtering one- and two-factor stochastic volatility models

One- and two-factor stochastic volatility models are assessed over three sets of stock returns data: S&P 500, DJIA, and Nasdaq. Estimation is done by simulated maximum likelihood using techniques that are computationally efficient, robust, straightforward to implement, and easy to adapt to different models. The models are evaluated using standard, easily interpretable time-series tools. The results are broadly similar across the three data sets. The tests provide no evidence that even the simple single-factor models are unable to capture the dynamics of volatility adequately; the problem is to get the shape of the conditional returns distribution right. None of the models come close to matching the tails of this distribution. Including a second factor provides only a relatively small improvement over the single-factor models. Fitting this aspect of the data is important for option pricing and risk management.