Estimation of stochastic volatility models through adaptive Kalman filtering methods

Volatility is a central concept when dealing with financial applications. It is usually equated with the risk and plays a central role in the pricing of derivative securities. It is also widely acknowledged nowadays that volatility is both time-varying and predictable, and stochastic volatility models are commonplace. The approach based on autoregressive conditional heteroscedasticity (ARCH) introduced by Engle, and later generalized to GARCH by Bollerslev, was the first attempt to take into account the changes in volatility over time. The class of stochastic volatility (SV) models is now recognized as a powerful alternative to the traditional and widely used ARCH/GARCH approach. We focus on the maximum likelihood estimation of the class of stochastic volatility models. The main technique is based on the Kalman filter (KF), which is known to be numerically unstable. Using the advanced array square-root form of the KF, we construct a new square-root algorithm for the log-likelihood gradient (score) evaluation. This avoids the use of the conventional KF with its inherent numerical instabilities and improves the robustness of computations against roundoff errors. The proposed square-root adaptive KF scheme is ideal for simultaneous parameter estimation and extraction of the latent volatility series.