Stochastic Volatility

Let $y_t$ be a stock series returns or the difference of logged exchange rates. Such a series will normally be approximately white noise. However it may not be independent because of serial dependence in the variance. This can be modelled by \begin{equation}\label{sv1} y_t = \sigma_t \epsilon_t = \sigma \epsilon_t \text{exp}(h_t/2), \quad \epsilon_t\sim \text{IID}(0, 1), \quad t = 1,\ldots,T \end{equation} where \begin{equation}\label{sv2} h_{t+1} = \phi h_t + \eta_t, \quad \eta_t \sim \text{NID}(0,\sigma^2_{\eta}), \quad |\phi| \leq 1. \end{equation} The term $\sigma^2$ is a scale factor, $\phi$ is a persistence parameter, and $\eta_t$ is a disturbance term which in the simplest model is uncorrelated with $\epsilon_t$. Literature reviews are given by Shephard (2005) and Ghysels, Harvey, and Renault (1996). This stochastic volatility (SV) model has two main attractions. The first is that it is the natural (Euler) discrete time analogue of the continuous time model used in papers on option pricing, such as Hull and White (1987). The second is that its statistical properties are easy to determine. The disadvantage with respect to the conditional variance models of the GARCH class is that likelihood based estimation can only be carried out by a computer intensive technique such as that described in Kim, Shephard, and Chib (1998) and Sandmann and Koopman (1998). However, a quasi-maximum likelihood (QML) method is relatively easy to apply and is often reasonably efficient. This method is based on transforming the observations to give: \begin{equation}\label{sv3} \text{log}\, y^2_t = \kappa + h_t + \xi_t, \quad t = 1,\ldots,T \end{equation} where \begin{equation} \xi_t = \text{log}\, \epsilon^2_t - E(\text{log}\, \epsilon^2_t) \end{equation} and \begin{equation} \kappa = \text{log}\, \sigma^2 + E(\text{log}\, \epsilon^2_t) \end{equation} As shown in Harvey, Ruiz, and Shephard (1994), the state space form given by equations \eqref{sv1} and \eqref{sv2} provides the basis for QML estimation via the Kalman filter and also enables smoothed estimates of the variance component, $h_t$, to be constructed and predictions made. One of the attractions of the QML approach is that it can be applied without the assumption of a particular distribution for $\epsilon_t$. In Harvey, Ruiz, and Shephard (1994), the volatility in the daily exchange rate of the US dollar against four currencies is examined; see also Mahieu and Schotman (1998).

Daily exchange rates

The data are in the file EXCH.csv which is part of any TSL installer file with a version number higher than v1.00 and can be found in the data folder located in the install folder of TSL. The file EXCH.csv consists of daily exchange rates for the US dollar. The four series are the daily exchange rates of the dollar against the pound, deutschmark, yen and Swiss franc. The data were recorded at the end of each weekday from 1/10/81 until 28/6/85. Thus the sample size is 946.

Step 1 is to load the time series data and transform it so that we can model the data as in equation \eqref{sv3}.

Database ► Load database ► Load the EXCH.csv file.

In the figure below, the EXCH dataset is loaded into TSL and the Pound / Dollar exchange rate is highlighted.

Subsequently go to the Build your own model page and select an ARMA(1,0) and a Fixed Level. Go to
Model setup page ► Select a fixed Level (no slope) ► select an ARMA process of order p=1 and q=0.
Go to Estimate page ► Estimate.
We should now see the following figure. The smoothed estimate of the volatility process, $h_t$, is displayed in the usual way by selecting the ARMA(p,q) process on the Graphics and Diagnostics page.