Q$5($cross hedging, basis risk, minimum variance hedge ratio$)$

An airline expects to purchase $2$ million gallons of jet fuel in $1$ month and decides to use heating oil

futures for hedging. We suppose that the Table below gives, for $15$ successive months, historical data

on $S_j,$ the jet fuel price per gallon and $S_h,$ the heating oil price per gallon.

$($a$)$ What are the standard deviations of $S_j$ and $S_h,{}_j$ and ${}_h?$ What is the correlation between $S_j$

and $S_h,?$

$($b$\mathrm{.}1)$ Suppose each one$-$month heating oil futures contract has $2$ million gallons of heating oil as the

underlying asset. If the airline company goes long $1$ such contract, what would be the standard

deviation of its total payoff in one month, based on the estimates of standard deviations and

correlations you get in $($a$)?$

$($b$\mathrm{.}2)$ Similar to $($b$\mathrm{.}1),$ but now the airline company goes long $2$ such contract. What would be the standard deviation of its total payoff in one month?

$($c$)$ What is the variance minimizing hedge ratio $h^{*}?$

$($d$)$ Suppose each heating oil contract is on $42,000$ gallons of heating oil. What is the number of contracts $H^{*}$ that minimizes the variance of the airline's total payoff?