Suppose that the monthly log returns, in percentages, of a stock follow the following Markov switching model:

\[ \begin{aligned} r_{t} & =1.25+a_{t}, \quad a_{t}=\sigma_{t} \epsilon_{t}, \\ \sigma_{t}^{2} & = \begin{cases}0.10 a_{t-1}^{2}+0.93 \sigma_{t-1}^{2} & \text { if } s_{t}=1 \\ 4.24+0.10 a_{t-1}^{2}+0.78 \sigma_{t-1}^{2} & \text { if } s_{t}=2\end{cases} \end{aligned} \]

where the transition probabilities are

\[ P\left(s_{t}=2 \mid s_{t-1}=1\right)=0.15, \quad P\left(s_{t}=1 \mid s_{t-1}=2\right)=0.05 \]

Suppose that \(a_{100}=6.0, \sigma_{100}^{2}=50.0\), and \(s_{100}=2\) with probability 1.0. What is the 1 -step-ahead volatility forecast at the forecast origin \(t=100\) ? Also, if the probability of \(s_{100}=2\) is reduced to 0.8 , what is the 1 -step-ahead volatility forecast at the forecast origin \(t=100\) ?