$8.$ Suppose that the $3-$year and $3\mathrm{.}5-$year zero rates with continuous compounding are $2\mathrm{.}5\%$ and $2\mathrm{.}55\%,$ respectively.

$($a$)$ What is the forward rate for the six$-$month period beginning in $3$ years $(3$R$35)($from Year $3$ to Year $3\mathrm{.}5)$ with continuous compounding?

$($b$)$ What is the forward rate for the six$-$month period beginning in $3$ years $(3$R$3\mathrm{.}5)($from Year $3$ to Year $3\mathrm{.}5)$ with semiannual compounding?

$($c$)$ What is the $($Year $0)$ value of an FRA that promises to pay the lender $2\mathrm{.}75\%($compounded semiannually$)$ on a principal of $$3$ million for the six$-$month period starting in $3$ years $($from Year $3)$

$9.$ At the end of one day $($t$=1),$ a clearinghouse member is long $90$ futures contracts, and the settlement price is $$650$ per contract. On the following day $($t$=2),$ the member becomes responsible for clearing additional new $25$ short futures contracts $($day trades$),$ entered into at a price of $$685$ per contract. The settlement price at the end of this day is $$665.$ The initial margin is $$2,200$ per contract. How much does the member still have to add to his margin account with the exchange clearinghouse $($on t$=2)$ to meet the initial margin requirement for the $25$ new futures contracts, after considering the gain$/$loss of the $90$ old futures contracts and the $25$ new futures contracts?

$10.$ The standard deviation of monthly changes in the spot prices $($in dollars per barrel$)$ of crude oil is $0\mathrm{.}096.$ The standard deviation of monthly changes in the futures prices of crude oil for the closest contract is $0\mathrm{.}087.$ The correlation between the futures price changes and the spot price changes is $0\mathrm{.}92.$ It is now April $15.$ A oil producer is committed to purchasing $65,000$ barrels of crude oil on December $15.$ The producer wants to use the December crude oil futures contracts to hedge its risk. Each contract is for the delivery of $1,000$ barrels of crude oil.

What strategy should the oil producer follow? How many December crude oil futures contracts should the oil producer sell or buy now?