An independent refiner who is exposed to the risk of increasing crude oil costs and falling refined product prices runs the risk that his refining margin will be less than anticipated. He decides to lock$-$in the current favorable cracking margins, using the $3$:$2$:$1$ crack spread strategy, which closely matches the cracking margin at the refinery. It means that $3$ contracts of crude oil make $2$ contracts of gasoline and $1$ contract of ULSD.

On September $15,$ the refiner decides to use the $3$:$2$:$1$ crack spread to hedge. December gasoline futures are traded at $$1\mathrm{.}60$ per gallon and one December ULSD futures are traded at $$1\mathrm{.}70$ per gallon, and November crude oil futures are traded at $$50\mathrm{.}00$ per barrel.

$1)$ What is the Crack spread in the futures market? How should the refiner hedge in September?

One month later, on October $15,$ the refiner purchases the crude oil at $$60\mathrm{.}00$ per barrel in the spot market for refining into products. At the same time, he also sells gasoline from his existing stock in the cash market for $$1\mathrm{.}70$ per gallon and diesel $($ULSD$)$ fuel for $$1\mathrm{.}80$ per gallon. Since the futures market closely reflects the cash market, November crude oil futures are also selling at $$60\mathrm{.}00$ per barrel. December gasoline futures are also trading higher at $$1\mathrm{.}70$ per gallon and December ULSD futures are trading at $$1\mathrm{.}80$ per gallon.

$2)$ What is the crack spread in the futures market now? How should the refiner close out the futures position?

$3)$ What is the profit$/$loss of the hedged position? What is the profit$/$loss if the refiner did not hedge?