3.12 Stochastic Volatility Model

An alternative approach to describe the volatility evolution of a financial time series is to introduce an innovation to the conditional variance equation of at; see Melino and Turnbull (1990), Taylor (1994), Harvey, Ruiz, and Shephard (1994), and Jacquier, Polson, and Rossi (1994). The resulting model is referred to as a stochastic volatility (SV) model. Similar to EGARCH models, to ensure positiveness of the conditional variance, SV models use instead of . A SV model is defined as

(3.40)

where the ϵt are iid N(0, 1), the vt are iid , {ϵt} and {vt} are independent, α0 is a constant, and all zeros of the polynomial are greater than 1 in modulus. Adding the innovation vt substantially increases the flexibility of the model in describing the evolution of , but it also increases the difficulty in parameter estimation. To estimate an SV model, we need a quasi-likelihood method via Kalman filtering or a Monte Carlo method. Jacquier, Polson, and Rossi (1994) provide some comparison of estimation results between quasi-likelihood and Markov chain Monte Carlo (MCMC) methods. The difficulty in estimating an SV model is understandable because for each shock at the model uses two innovations ϵt and vt. We discuss an MCMC method to estimate SV models in Chapter 12. For more discussions on stochastic volatility models, see Taylor (1994).

The appendixes of Jacquier, Polson, and Rossi (1994) provide some properties of the SV model when m = 1. For instance, with m = 1, we have

and , , and corr() = . Limited experience shows that SV models often provided improvements in model fitting, but their contributions to out-of-sample volatility forecasts received mixed results.