Suppose that the parameters in a GARCH$(1,1)$ model are $\backslash$alpha $=0\mathrm{.}03,\backslash$beta $=0\mathrm{.}95$ and $\backslash$omega $=0\mathrm{.}000002.$

$($a$)$ What is the long$-$run average volatility?

$($b$)$ If the current volatility is $1\mathrm{.}5\%$ per day, what is your estimate of the volatility in $20,40,$ and $60$ days?

$($c$)$ What volatility should be used to price $20-,40-,$ and $60-$day options? Hint: use $\backslash$alpha $=$ ln$(1/0\mathrm{.}98$

$($d$)$Suppose that there is an event that increases the volatility from $1\mathrm{.}5\%$ per day to $2\%$ per day. Estimate the effect on the volatility in $20,40,$ and $60$ days.

$($e$)$Estimate by how much the event increases the volatilities used to price $20-,40-,$ and $60-$day options.