Options, Futures, and Other Derivatives, 10th Edition

23.6 USING GARCH(1,1) TO FORECAST FUTURE VOLATILITY

The variance rate estimated at the end of day $n-1$ for day n, when GARCH(1,1) is used, is

$${\sigma }_n^2=\left(1-\alpha -\beta \right)V_L+\alpha u_{n-1}^2+\beta {\sigma }_{n-1}^2$$

so that

$${\sigma }_n^2-V_L=\alpha \left(u_{n-1}^2-V_L\right)+\beta \left({\sigma }_{n-1}^2-V_L\right)$$

On day $n+t$ in the future,

$${\sigma }_{n+t}^2-V_L=\alpha \left(u_{n+t-1}^2-V_L\right)+\beta \left({\sigma }_{n+t-1}^2-V_L\right)$$

The expected value of $u_{n+t-1}^2$ is ${\sigma }_{n+t-1}^2$. Hence,

$$E\left[{\sigma }_{n+t}^2-V_L\right]=\left(\alpha +\beta \right)E\left({\sigma }_{n+t-1}^2-V_L\right)$$

where E denotes expected value. Using this equation repeatedly yields

$$E\left[{\sigma }_{n+t}^2-V_L\right]=(\alpha +\beta {)}^t\left({\sigma }_n^2-V_L\right)$$

or

$$E[{\sigma }_{n+t}^2]-V_L+{(\alpha +\beta )}^t\left({\sigma }_n^2-V_L\right)$$(12.13)

This equation forecasts the volatility on day $n+t$ using the information available at the end of day $n-1$. In the EWMA model, $\alpha +\beta =1$ and equation (23.13) shows that the expected future variance ...