Stochastic Volatility with Feedback

An important empirical feature in many financial time series is the effect in which the future volatility increases when the price of an asset drops. The negative dependence between a price and volatility is typically called the leverage effect. In signal processing, this condition is known as feedback. The stochastic volatility model is a nonlinear and non-Gaussian state space model for which the volatility is a latent process and the observations are the returns (percent change) of an asset. Including feedback can often result in a better fit and can yield improvements in option pricing applications. The asymmetry in the relationship between volatility and the value of an asset is evaluated in various ways, but in the econometrics literature, it is often captured by allowing a negative correlation between the state noise and observation noise processes. In fact, the term leverage now seems to be synonymous with correlated errors rather than directly including feedback in the model. Here, a simple and computationally fast approach to fitting a model taking into account leverage as feedback is presented. Because of the model complexity, the likelihood cannot be evaluated analytically and various approaches to dealing with intractable likelihoods have been suggested. A fairly simple approach is offered here that involves using mixtures of Gaussian distributions. Including direct feedback and mixtures in the model makes it unnecessary to include correlated errors.