10.5 Forecasting Conditional Volatility from ARCH Models

An important task of modeling conditional volatility is to generate accurate forecasts for both the future value of a financial time series as well as its conditional volatility. Volatility forecasts are used for risk management, option pricing, portfolio allocation, trading strategies and model evaluation. ARCH and GARCH models can generate accurate forecasts of future daily return volatility, especially over short horizons, and these forecasts will eventually converge to the unconditional volatility of daily returns. This section illustrates how to forecast volatility using the GARCH(1,1) model.

10.5.1 Forecasting daily return volatility from the GARCH(1,1) model

For simplicity, consider the basic GARCH$(1,1)$ model $$\begin{eqnarray*} R_{t} & = & \mu+\epsilon_{t}\\ \epsilon_{t} & = & \sigma_{t}z_{t}\\ \sigma_{t}^{2} & = & \omega+\alpha_{1}\epsilon_{t-1}^{2}+\beta_{1}\sigma_{t-1}^{2}\\ z_{t} & \sim & iid\,N(0,1) \end{eqnarray*}$$ Assume the sample period is $t=1,2,\cdots,T$ and forecasts for volatility are desired for the out-of-sample periods $T+1,T+2,\ldots,T+k$. For ease of notation, assume the GARCH(1,1) parameters are known.⁵⁸ The optimal, in terms of mean-squared error, forecast of $\sigma_{T+k}^{2}$ given information at time $T$ is $E[\sigma_{T+k}^{2}|I_{T}]$ and can be computed using a simple recursion known as the chain-rule of forecasting. For $k=1,$ $$\begin{align} E[\sigma_{T+1}^{2}|I_{T}] & =\omega+\alpha_{1}E[\varepsilon_{T}^{2}|I_{T}]+\beta_{1}E[\sigma_{T}^{2}|I_{T}]\tag{10.30}\\ \quad & =\omega+\alpha_{1}\varepsilon_{T}^{2}+\beta_{1}\sigma_{T}^{2},\nonumber \end{align}$$ where it is assumed that $\varepsilon_{T}^{2}$ and $\sigma_{T}^{2}$ are known at time $T$. Similarly, for $k=2$ $$\begin{align} E[\sigma_{T+2}^{2}|I_{T}] & =\omega+\alpha_{1}E[\varepsilon_{T+1}^{2}|I_{T}]+\beta_{1}E[\sigma_{T+1}^{2}|I_{T}].\tag{10.31} \end{align}$$ Now, in the GARCH(1,1) model, $E[\varepsilon_{T+1}^{2}|I_{T}]=E[z_{T+1}^{2}\sigma_{T+1}^{2}|I_{T}]=E[\sigma_{T+1}^{2}|I_{T}]$ so that (10.30) becomes $$\begin{eqnarray*} E[\sigma_{T+2}^{2}|I_{T}] & = & \omega+(\alpha_{1}+\beta_{1})E[\sigma_{T+1}^{2}|I_{T}]\\ & = & \omega+(\alpha_{1}+\beta_{1})\left(\alpha_{1}\varepsilon_{T}^{2}+\beta_{1}\sigma_{T}^{2}\right) \end{eqnarray*}$$ In general, for $k\geq2$ we have $$\begin{align} E[\sigma_{T+k}^{2}|I_{T}] & =\omega+(\alpha_{1}+\beta_{1})E[\sigma_{T+k-1}^{2}|I_{T}]\tag{10.32} \end{align}$$ which, by recursive substitution, reduces to $$E[\sigma_{T+k}^{2}|I_{T}]=\omega\sum_{i=0}^{k-1}(\alpha_{1}+\beta_{1})^{i}+(\alpha_{1}+\beta_{1})^{k-1}(\alpha_{1}\varepsilon_{T}^{2}+\beta_{1}\sigma_{T}^{2}).$$ Now, since $0<\alpha_{1}+\beta_{1}<1$, as $k\rightarrow\infty$ $$\begin{eqnarray*} \sum_{i=0}^{k-1}(\alpha_{1}+\beta_{1})^{i} & \rightarrow & \frac{1}{1-(\alpha_{1}+\beta_{1})},\\ (\alpha_{1}+\beta_{1})^{k-1} & \rightarrow & 0, \end{eqnarray*}$$ and so $$E[\sigma_{T+k}^{2}|I_{T}]\rightarrow\frac{\omega}{1-(\alpha_{1}+\beta_{1})}=\bar{\sigma}^{2}=\mathrm{var}(R_{t}).$$ Notice that the speed at which $E[\sigma_{T+k}^{2}|I_{T}]$ approaches $\bar{\sigma}^{2}$ is captured by approaches $\bar{\sigma}^{2}$ is captured by the GARCH(1,1) persistence $\alpha_{1}+\beta_{1}.$

An alternative representation of the forecasting equation (10.32) starts with the mean-adjusted form $$\sigma_{T+1}^{2}-\bar{\sigma}^{2}=\alpha_{1}(\varepsilon_{T}^{2}-\bar{\sigma}^{2})+\beta_{1}(\sigma_{T}^{2}-\bar{\sigma}^{2}),$$ where $\bar{\sigma}^{2}=\omega/(1-\alpha_{1}-\beta_{1})$ is the unconditional variance. Then by recursive substitution $$E[\sigma_{T+k}^{2}|I_{T}]-\bar{\sigma}^{2}=(\alpha_{1}+\beta_{1})^{k-1}(E[\sigma_{T+1}^{2}|I_{T}]-\bar{\sigma}^{2}).$$

The forecasting algorithm (10.32) produces unbiased forecasts for the conditional variance $\sigma_{T+k}^{2}.$ The forecast for the conditional volatility, $\sigma_{T+k},$ is computed as the square root of the forecast for $\sigma_{T+k}^{2}$ which is not unbiased because $$E[\sigma_{T+k}|I_{T}]\neq\sqrt{E[\sigma_{T+k}^{2}|I_{T}]}.$$

Example 2.11 ( Forecasting daily volatility from fitted GARCH(1,1) models using rugarch )

To be completed

$\blacksquare$

10.5.2 Forecasting multi-day return volatility using a GARCH(1,1) model

To be completed

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58. In practice, the GARCH forecasting algorithm will use the GARCH parameters estimated over the sample period.↩︎