Delta of a Commodity Future

Generally the price of a future is

$F(t,T) = S(t)e^{r(T-t)},$

and it's delta is:

$\frac{\partial F}{\partial S} = e^{r(T-t)}.$

(As opposed to the delta of a forward which is always one.)

In some explanations this is proven by taking the derivative of the pricing equation above see RRGs response in this link:

Are Futures exactly Delta One?

However for commodity futures the pricing can incorporate a convenience yield (c) that can give rise to backwardation.

$F(t,T) = S(t)e^{(r+c)(T-t)},$

However what should the delta of a commodity future be?

If we take the derivative of the pricing equation it would be:

$\frac{\partial F}{\partial S} = e^{(r+c)(T-t)}.$

Or....

Since the difference in the delta of the futures and forward prices are purely down to the way that futures are settled each day (and nothing else), and so the delta should be that shown below, the same as any other future.

$\frac{\partial F}{\partial S} = e^{(r)(T-t)}.$

Q1.) Is it correct that the delta of a commodity future is the same as any other future?

Q2.) Is this the delta I should use when hedging? i.e. hedging 1 1m futures contract with 1 12m contract will not be completely delta neutral instead I should apply the discount shown above to get the correct hedge?

> because in order to for example hedge $10 exposure today for 2 periods > you would pay$10*e^(0.02*2) for the future. Assuming 0.02r. so the
• OK but here I a have futures on the same underlying with the same maturity: say 1 month $10exp^{0.02*1}=10.2$ and 12 month $10exp^{0.02*12} = 12.71$ therefore the delta in this case is $\frac{10.2}{12.7}=0.8$ correct?. – Bazman Jan 27 '16 at 18:14