2 - Background: Discrete-Time Volatility Modeling

Abstract

The most common approach to volatility modeling builds on the generalized autoregressive conditional heteroskedasticity (GARCH) class in which volatility follows a smooth autoregressive transition. This chapter briefly discusses several common approaches to modeling financial volatility, including GARCH, stochastic volatility, and Markov-switching multifractal (MSM) formulations. The key models that facilitate the development of MSM and provide comparisons are discussed. Because GARCH variance follows a smooth autoregressive transition, standard specifications have difficulty capturing the sudden changes in volatility exhibited by many financial series. For this reason, econometricians have considered extensions, called stochastic volatility models, in which volatility is hit by separate shocks. In contrast to standard stochastic volatility models, however, the model generates a closed-form likelihood, which permits convenient and efficient likelihood-based estimation. Standard GARCH models provide good forecasts of short-run volatility dynamics, but often have difficulties capturing lower-frequency cycles. In contrast to the GARCH volatility models discussed earlier, stochastic regime-switching models permit the conditional mean and variance of financial returns to depend on an unobserved latent “state” that may change unpredictably.