MetaA firm will sell $1$ million gallons of jet fuel in $3$ months.

$($a$)$ What position should the firm enter, short or long and why? $[2$ mark$]$

Suppose that the firm chooses to hedge its position with NYMEX heating oil futures.

One NYMEX heating oil contract is for $42,000$ gallons. Assume that the variability of price changes of oil is $0\mathrm{.}16,$

the riskiness of futures price of oil is $=0\mathrm{.}14,$ and the coefficient of linear correlation between the two changes is $0\mathrm{.}5.$

$($b$)$ Calculate the optimal hedge ratio $[2$ marks$]$

$($c$)$ What does it mean? $[2$ marks$]$

$($d$)$ Calculate how many contracts should the firm go long or short to minimize its risk? $[2$ marks$]$

$($e$)$ How many futures contracts would the company buy or sell under the na$$ve hedge ratio?

Why is there a difference between answer $($d$)$ and answer $($e$)?$