Inferring Volatility in the Heston Model and its Relatives — an Information Theoretic Approach

Modeling asset prices via a stochastic volatility approach is vastly popular as these models capture many stylized facts of volatility dynamics, like long-term memory or the leverage effect. Essentially, stochastic volatility models are two-dimensional diffusion processes: the first process captures the dynamics of the random volatility; the second one couples to the volatility driving the returns of the asset. Even though there is an extensive study on deriving properties or even closed, analytic formulae of the distribution of the returns conditioned on the volatility, nearly nothing is known about the conditional distribution the distribution of the volatility if returns are known. But, precisely this distribution expresses our best estimate of the hidden volatility which needs to be inferred from market data. Here, we quantify the uncertainty left about the volatility if the returns are known by the Shannon entropy. In turn, we investigate how much information, the observed market returns actually provide about volatility.

This motivates a careful numerical and analytical study of information theoretic properties of the Heston model. In addition, we study a general class of discrete time models where the volatility process is generated by a Gaussian process. These types of models are motivated from a machine learning perspective and we fit them to different data sets. Using Bayesian model selection, we investigate the influence of structural priors on the volatility dynamics. As before, we interpret our findings from an information theoretic perspective showing that observed market returns are not very informative about the volatility. From these observations, we conclude that the evidence for some of the major stylized facts on volatility dynamics is actually quite weak and prior knowledge might play a major role in volatility modeling.