# Question about hedging using futures and cross hedge

Hi friends, I have some confusion about some concepts of hedging using futures, as well as cross hedge.

So for example, simplest case would be if an airline company will purchase jet fuel in the future, to hedge it would enter into a long futures contract to take delivery, or close out the position before contract expiration. If jet fuel contract is not available on the market, it would use cross hedge.

My questions are:

Why it must use something like heating oil futures that has high correlation with the asset being hedged? If it enters at t1 and close out at t2, now instead of locking price at F1, the effective price paid would be S2 + F1 - F2, so it will pay S2 at t2 no matter hedge or not, and the result of hedging then only depends on F1 and F2, but F2 is not known at t1 when it does the hedge anyway, seems any futures contract could be used, so what are the benefits to use heating oil futures to cross hedge jet fuel instead of for example corn futures?

There is a concept in cross hedge called minimum variance hedge ratio. To hedge the purchase of 2 million gallons of jet fuel in the future by longing futures on 2 million gallons jet fuel in very intuitive that it is a perfect hedge! And so the hedge ratio must be 1 to 1. However, I get confused for:

(1) Close out before delivery, because of F2 and S2 don’t converge the hedge cannot be perfect. Suppose S2 and F2 are known at t1, and if I want to make it perfect, is the hedge ratio to let F2 = S2? i.e 2.2h = 2 + 2.2h - 1.9h, so h = 2/1.9 = 1.0526 am I correct about this?

(2) When futures underlying is not jet fuel but heating oil, I don’t quite understand why using regression, seems if F2 = S2 i.e the basis is 0, then it is a perfect hedge, but seems I cannot put the case of perfect hedge into this model, I think the result would be exactly the regression line; I played with the numbers a little bit by setting F2 = S2, but seems it’s not a straight line… Probably being a little vague about my idea here, just feel like don’t quite understand why the minimum variance is method to find the hedge ratio…

(3) A little confused about the direction: is it long if need to increase beta of a stock portfolio and short for decrease?

I find the concepts for hedging somewhat difficult, although can still solve practice problems, but it must more or less impact my comprehension of a deeper level. Any help would be greatly appreciated.

Suppose that you hedge jet fuel prices with corn futures. The spot price on jet fuel is USD4.00/gallon and the spot price on corn is USD4.00/bushel.

Six months later, when the futures contracts expire, jet fuel is USD4.50/gallon and corn is USD2.50/bushel. You just lost USD1.50/bushel on corn and you lost USD0.50 on jet fuel. Not much of a hedge.

Suppose that you hedge jet fuel prices with heating oil futures. The spot price on jet fuel is USD4.00/gallon and the spot price on heating oil is USD3.00/gallon.

Six months later, when the futures contracts expire, jet fuel is USD4.50/gallon and heating oil is USD3.30/gallon. You just gained USD0.30/gallon on heating oil and you lost USD0.50 on jet fuel. Not a bad hedge.