Likelihood-based estimation and specification analysis of one- and two-factor SV models with leverage effects
Techniques for simulated maximum likelihood (SML) estimation, filtering, and assessing the fit of stochastic volatility models are examined. Both one- and two-factor models (with leverage effects) are considered. The techniques are computationally efficient, robust, straightforward to implement, and easy to adapt to new models. Using these techniques, it is possible to estimate single-factor models over data sets of several thousand observations in several seconds. The computational efficiency of the techniques means that Monte Carlo studies assessing both the small sample statistical properties as well as the numerical properties of the estimators are easy to do. Such studies are important for all simulation estimators, including simulation-based Bayesian and method of moments estimators. The application looks at S\&P 500 index returns. Even the simple single-factor models adequately capture the dynamics of volatility; the problem is to get the shape of the returns distribution right. Although including a second volatility factor improves the fit over the basic single-factor models, a new formulation of the SV-t model (a single factor model, but with $t$ rather than normal errors in the observation equation) provides the best fit. However, all the models considered fail in a similar manner: they are unable to capture the left tail of the distribution. Fitting this part of the distribution is important for option-pricing and risk management. Although it may be possible to come up with ad hoc parametric models that fit particular data series and sample periods, a promising alternative might be to look at single-factor models with flexible forms for the error distributions
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