# 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?

## 1 Answer

Since all futures are linear instruments you can achieve a perfect hedge by going short or long into the same future depending on your position.

If however there are no available futures you can use cross-hedging as explained by Hull (2007)

i get an error bellow I'm not sure why so I'll put it in code format:

> To answer your question the delta of a future is not perfectly 1
> 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
> delta is 10/10.41

• 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
• well you would need to roll over your hedge month after month if you hedge for 12 months using futures of 1 month maturity. since you started with t+1 there what you calculated is the delta of the future at t+1 with that of t+12 so that's 11 months. – Alex Bădoi Jan 27 '16 at 18:19
• Thanks again,Ok lets forget about the rolling issue assume the position will be unwound prior to the roll date. Does the delta you calculated above apply to futures on the same underlying but with different maturities or is it for one future v spot? – Bazman Jan 27 '16 at 18:27