Suppose that you are presented with $2$ models: $($i$)$ an EWMA model with parameter

$=0\mathrm{.}91$; and $($ii$)$ a GARCH$(1,1)$ model with parameters $=0\mathrm{.}09,=0\mathrm{.}90,$ and $=$

$0\mathrm{.}000008.$

a$)$ What is the long$-$run average volatility in both models?

$[4$ marks$]$

b$)$ Assume that the most recent return un$-1$ is estimated at $2\%$ and the most recent

volatility $$n$-1$ is estimated at $1\%.$ Update the volatility estimate in both models.

$[4$ marks$]$

For all the remaining items, consider only the GARCH$(1,1)$ model:

c$)$ If the current volatility is $0\mathrm{.}5\%$ per day, what is your estimate of the volatility in $5,$

$100,$ and $500$ days.

$[4$ marks$]$

d$)$ What volatility should be used to price $5-,100-,$ and $500-$day options?

$[4$ marks$]$

e$)$ Suppose that there is an event that increases the volatility from $0\mathrm{.}5\%$ per day to

$1\mathrm{.}5\%$ per day. Estimate the effect on the volatility in $5,100,$ and $500$ days.

$[4$ marks$]$

f$)$

Estimate by how much the event increases the volatilities used to price $5-,100-,$

and $500-$day options.