# Are Futures exactly Delta One?

Delta of Future is exactly one I thought. This post here, says otherwise.

However, quoting John Hull again:

$$f = \text{Value of Future contract} = S_{t=0} - K \exp(-rT)$$

where $S$ it the spot price, $S_{t=0}$ is the spot price today, $r$ is the risk-free rate and $T$ is the time to maturity.

$$\Delta = \frac{df}{dS} = \frac{dS}{dS} - \frac{d[K \exp(-rT)]}{dS} = 1 - 0 = 1.0$$

As $K$ is constant, $T$ is constant, and the risk-free rate is not dependent on $S$. So I don't see why Delta of future contracts isn't exactly 1.0 (in contrary to argument from Riskprep.com article).

Futures are traded on Delta One desks after all.

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Your formula for for the price of a futures contracts is not correct. For example consider the price at expiry with T=0. Your formula states f_{T=0}=S-K which can't be true. –  RRG May 13 at 1:37
T is not time. It's time-to-maturity. You don't substitute zero into it. The second term discount K to present value. value of contract is diff between spot, and pv(strike) –  Swab.Jat May 13 at 1:44
So what is the price of the futures at expiry in your formula? –  RRG May 13 at 1:50
For the sake of clarity, some confusion arose because of the difference between forward price and forward value. @Swap.Jat, can you please specify what exactly you try to determine? –  Matt Wolf May 13 at 7:24

Forward delta is 1 (defined as change in the value of the forward with respect to an instantaneous change in the price of the underlying, holding everything else constant).

However for a meaningful discussion of the differences in forward and futures pricing, the forward price delta of forwards should be considered and it is exp(r(T-t)).Though the delta of the two are identical the value of a portfolio holding a forward vs futures contract will change over time and here is why: The difference arises from the fact that interest rates are not constant but random and forwards are OTC products that are settled at maturity while futures are settled daily. This subtle difference leads to different cash flows because money that is deposited into your account or that you need to cough up because of daily margin settlements can be invested/must be borrowed at prevailing interest rates.

For example, if the underlying discount rate process and underlying asset price process are positively correlated then if asset prices rise conversely interest rates will be lower and surpluses that are deposited into your account on a daily basis must be invested at lower rates. The opposite when asset prices fall, you need to deposit variation margin and need to borrow at higher rates. Hence, the futures contract must be priced lower than the forward in this example to make the futures contract equally attractive.

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Thanks Matt. But, if we forget daily margining for future for the moment?... Can we derive how delta not exactly = 1 from formula: f = value of Future contract = S(t=0) - K exp(-rT)? I take derivative of f, r comes from yield curve is a number/float for a given t (Sure over time it's not a constant but we do read off a number from yield curve). I can't see why 1st derivative of second term with respect to S isn't zero exactly. –  Swab.Jat May 13 at 1:25
The delta for a forward is not 1. It's exp(r(T-t)) like a futures. –  RRG May 13 at 1:41
I disagree. Can you please walk me through your derivation of forward delta? You need to discount the change in value back hence exp(r(T-t)) cancels out. –  Matt Wolf May 13 at 5:08
@Matt Wolf. Since you agree that the forward price is the discounted spot price it should be clear that the delta cannot be 1. The financing cost to buy the spot changes with the discounted spot price. The delta is therefore the discount factor. –  RRG May 13 at 5:30
I never agreed with your statement, in fact I did not even discuss the "discounted spot price". But let me be more specific that I was referring to the delta of a forward of a non-dividend paying stock. My answer stands, the delta of such contract is 1. –  Matt Wolf May 13 at 5:32

At time $t$ the price of a futures contract with maturity at time $T$ is

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

where $S(t)$ is the spot price at time $t$ and $r$ is the interest rate. The delta of the futures contract is hence

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

For $r>0$ we therefore have $\partial F/\partial S>1$ for $t<T$.

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F(t,T)=S(t)er(T−t) is how you calculate "fair" future/foward price. But once you enter into a contract, future/forward price becomes constant K. Both K and r are not function of S. If you take first derivative of f = [Value of Future contract] = diff between Spot and PV(K) = S(t=0) - K exp(-rT) ... first term = 1.0 exactly, and the second term should go to zero (As K/r/T all constant with respect to S) –  Swab.Jat May 13 at 2:04
I don't know what you mean with "the price becomes constant". Obviously the price of the futures contract that you own is the current fair price of the futures contract (in an efficient market). –  RRG May 13 at 2:09
Thanks RPG, but I didn't say "Price becomes constant". I said K (forward/future price) of any particular future contract you took position is a constant number. Once you enter into a contract, you can't change K. –  Swab.Jat May 13 at 2:13
But RPG thanks for your effort! –  Swab.Jat May 13 at 3:26