Minimum Variance Hedge

I'm wondering about how valid the statement in the ST5 notes about the minimum variance hedge ratio is:

The notes state that if you are hedging an index or share with future contracts on the identical underlying (same term as well), then the hedge ratio is 1.
Surely this cannot be the case. Future contracts are margined daily which leads to the profits acrueing daily. This being the case, a future contract on the same underlying share is far more sensitive to movements in the share. The ideal hedge ratio surely would be the present value of 1, not 1.

The textbook Options, Futures, and Other Derivatives (6th edition, J. C. Hull), page 348 gives an explanation and a formula. The formula is that if H units of the underlying asset are required to hedge a position, then H*e^(-rT) future contracts would be required to hedge the same position. Hence in the case in ST5, if the underlying was a single share, then e^(-rT) futures would be the ideal minimum hedge ratio, not 1?

Do the notes mean to say forward contracts instead of future contracts?

 

I'm replying to myself as it seems I am not going to get a reply.

My guess is that the notes simply discuss the differences between futures and forwards as background information, and then assume no differences for simplicity.

 

The statement in ST5 notes is the same as in Hull page 57.

In the first case we have the minimum variance hedge.

In the second case we have the delta neutral hedge.

These are different types of hedges. Another type of hedge is the beta neutral hedge and (for bonds) the duration neutral hedge.

 

Surely the minimum variance hedge would be a delta-neutral position (if it is possible to achieve such a delta-neutral hedge)?

Delta-neutrality, if possible, would eradicate the randomness - I fail to see how it is possible to achieve a lower variance hedge if a delta-neutral hedge is possible.

 

"Surely the minimum variance hedge would be a delta-neutral position (if it is possible to achieve such a delta-neutral hedge)?"



Your misunderstanding of this is explained by your bracketed sentence. Often there is no perfectly correlated hedge

 

I think I can propose a solution (to original query) with the following example:

If you're hedging a long position in a share with a future (say 1 year horizon), written on that exact share, the delta-neutral strategy, and hence minimum variance strategy would be to:

Go short e^(-r*364/365) futures at t0. This is because the first margining takes places the next day and the interest on the margining would be there for a further 364 days.
1 day later, the short future holding would need to be increased to e^(-r*363/365) futures.
This would continue until the day before the termination, where the short future holding would need to be increased to e^(-r*0/365) = 1.

Hedging using futures requires continual rebalancing.

To derive the above, you need to try to replicate the forward contract with the future contract(s). The change in value in the forward is only realised at termination, and the early arrival of the changes in value of the future contract should be discounted by the amount of time by which it is going to grow.

Hence, the minimum variance hedge ratio can only be a constant of 1 with forwards, not futures.
The above idea is basically the same proof that forward and future prices are the same with deterministic internet rates: since you can replicate the forward contract with future contracts (of the same price) by the above procedure, any mismatch with prices can be arbitraged.