A refinery needs $120,000$ barrels of oil in $6$ months. The $6-$month futures price is $$80$ per barrel $($contract size: $1,000$ barrels$),$ and the spot price is $$78.$ Spot volatility is $0\mathrm{.}22,$ futures volatility is $0\mathrm{.}25,$ and correlation is $0\mathrm{.}9.$ Calculate the optimal number of futures contracts for a minimum$-$variance hedge and specify the position. b$)(10$ pts$)$ At maturity, the spot price is $$82,$ and the futures price converges to $$82.$ Compute the initial and final basis, and determine the net cost per barrel after the hedge, assessing basis risk. c$)(8$ pts$)$ With SOFR at $4\%$ per annum $($continuously compounded$)$ and no storage costs or convenience yield, calculate the theoretical $6-$month futures price. If the actual price is $$80,$ explain the impact on the hedging decision. d$)(14$ pts$)$ The refinery now hedges oil purchases of $60,000$ barrels every $3$ months for $9$ months using $3-$month futures. Today, the spot price is $$78,$ the $3-$month futures price is $$79,$ SOFR is $4\%,$ spot volatility is $0\mathrm{.}22,$ futures volatility is $0\mathrm{.}25,$ and correlation is $0\mathrm{.}9.$ At $3$ months, the spot price is $$80,$ and the new $3-$month futures price is $$81$; at $6$ months, the spot price is $$82,$ and the new $3-$month futures price is $$83.$ Calculate the optimal number of contracts for the initial hedge, then compute the net cost per barrel for each $3-$month period, accounting for futures gains$/$losses and basis risk. Analyze how rolling impacts total cost compared to a single $9-$month hedge at $$81($assume theoretical price aligns with market$).$