A trader owns $55,000$ units of a particular asset and decides to hedge the value of

her position with futures contracts on another related asset. Each futures contract

is on $5,000$ units. The standard deviation of the change in the spot price over the

life of the hedge is estimated to be $0\mathrm{.}43.$ The standard deviation of the change in the

futures price over the life of the hedge is $0\mathrm{.}40.$ The coefficient of correlation

between the spot price change and futures price change is $0\mathrm{.}95.$

$($a$)$ What is the minimum variance hedge ratio?

$($b$)$ Should the hedger take a long or short futures position?

$($c$)$ What is the optimal number of futures contracts that the trader needs to use?