13. Time Series with Stochastic Volatility

In the previous chapters we have already indicated that volatility plays an important role in modelling financial systems and time series. Unlike the term structure, volatility is unobservable and thus must be estimated from the data.

Reliable estimation and forecast of volatility are important for large credit institutes where volatility is directly used to measure risk. The risk premium for example is often specified as a function of volatility. It is interesting to find an appropriate model for volatility. In the literature, the capability of macroeconomic factors to forecast volatility has been examined. Although macroeconomic factors have some forecasting capabilities, the most
important factor seems to be the lagged endogenous return. As a result recent studies are mainly concentrated on time series models.

Stock, exchange rate, interest rate and other financial time series have stylized facts that are different from other time series. A good candidate for
modelling financial time series should represent the properties of the stochastic processes. Neither the classical linear AR or ARMA processes nor the nonlinear generalizations can fulfil the task. In this chapter we will describe the most popular volatility class of models: the ARCH (autoregressive conditional heteroscedasticity) model that can replicate these stylized facts appropriately.

Stylized fact 1: Time series of share prices $ X_t$ and other basic financial instruments are not stationary time series and possess at least a local trend.

Similar to the ARIMA model in Chapter 11, we transform the original data by taking first differences to get a stationary time series. Here we consider the log return (see Definition 10.15) instead of the original share prices. We simply call it return in this chapter. One could consider the simple return $ R_t$ as well (see Definition 10.14).

Stylized fact 2: Returns $ r_t$ have a leptokurtic distribution. The empirically estimated kurtosis is mostly greater than 3.

We have discussed the properties of the return's distribution in Section 3.3 and Section 10.2. The leptokurtosis can be illustrated in a comparison of the density of a normal distribution and a kernel estimator of the adjusted data (see Figure 14.1). We can see in Theorem 12.3 that an ARCH process has a kurtosis greater than 3 even if the innovation of the process itself is normally distributed.

Stylized fact 3: The return process is white noise (Definition 10.8) since the sample autocorrelation $ \hat{\rho}_{\tau,n}, k \not= 0$ (11.23) is not significantly different from 0. Furthermore the white noise is not independent since the sample autocorrelations of squared and absolute returns are clearly greater than 0.

ARCH models possess the characteristic (Theorem 12.1) that we have already described in Section 10.2. A stronger condition than pairwise uncorrelation of returns is that returns are unpredictable, which is connected to the no arbitrage condition. As in Section 10.3 $ {\cal F}_t$ denotes the information set at time t. The best prediction of return $ r_{t+1}$ at day $ t$ for day $ t+1$ is the conditional expectation $ r_{t+1\vert t} = \mathop{\text{\rm\sf E}}[r_{t+1} \vert{\cal F}_t]$ (Theorem 10.1) based on the information set $ {\cal F}_t$. The time series of the return is called unpredictable if

$\displaystyle r_{t+1\vert t} = \mathop{\text{\rm\sf E}}[r_{t+1} \vert{\cal F}_t] = \mathop{\text{\rm\sf E}}[r_{t+1}],$

i.e. the best prediction of the next return is simply its unconditional mean. The information set $ {\cal F}_t$ gives no hints for predicting future prices. ARCH processes are automatically unpredictable (Definition 12.1).

An unpredictable time series is always white noise because the autocorrelation is equal to 0. It is even possible that a linear prediction is better than the expectation estimated only by the unpredictable time series (see the proof of Theorem 12.1). The condition of unpredictability is actually stronger than pairwise uncorrelation. A predictable white noise is for example $ \varepsilon_t = \eta _t + \gamma \eta_{t-1} \eta_{t-2}$, where $ \eta_t$ is independent white noise with expectation of 0. This bilinear process has vanishing autocorrelations but $ \mathop{\text{\rm\sf E}}[ \varepsilon_{t+1} \vert {\cal F}_t] = \gamma \eta_t \eta_{t-1} \not= 0 = \mathop{\text{\rm\sf E}}[\varepsilon_{t+1}].$

If the returns were predictable we could develop a trading strategy based on the resulting predictions of price, which would give us a positive profit. The existence of a stochastic arbitrage probability obviously contradicts the assumption of a perfect financial market (Section 2.1).

Stylized fact 4: Volatility tends to form cluster: After a large (small) price change (positive or negative) a large (small) price change tends to occur. This effect is called volatility clustering.

We will consider the properties of financial time series in more detail in the following section. According to the stylized fact 4, the squared returns are positively correlated. Thus returns are conditionally heteroscedastic i.e.

$\displaystyle \mathop{\text{\rm Var}}[r_{t+1} \vert {\cal F}_t] \neq \mathop{\text{\rm Var}}[r_{t+1}].$

The returns $ r_t$ are not independent but their variability depends on recent changes of price.