Stochastic volatility

We give an overview of a broad class of models designed to capture stochastic volatility in financial markets, with illustrations of the scope of application of these models to practical finance problems. In a broad sense, this model class includes GARCH, but we focus on a narrower set of specifications in which volatility follows its own random process and is therefore a latent factor. These stochastic volatility specifications fit naturally in the continuous-time finance paradigm, and therefore serve as a prominent tool for a wide range of pricing and hedging applications. Moreover, the continuous-time paradigm of financial economics is naturally linked with the theory of volatility modeling and forecasting, and in particular with the practice of constructing ex-post volatility measures from high-frequency intraday data (realized volatility). One drawback is that in this setting volatility is not measurable with respect to observable information, and this feature complicates estimation and inference. Further, the presence of an additional state variable—volatility—renders the model less tractable from an analytic perspective. New estimation methods, combined with model restrictions that allow for closed-form solutions, make it possible to address these challenges while keeping the model consistent with the main properties of the data.