# 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.

• 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 '14 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 '14 at 1:44
• So what is the price of the futures at expiry in your formula? – RRG May 13 '14 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 May 13 '14 at 7:24
• An easy way to see that a forward's value is delta one is that a forward can be replicated with a long call and short put. – user217285 Dec 21 '17 at 23:18

## 5 Answers

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.

• 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 '14 at 1:25
• The delta for a forward is not 1. It's exp(r(T-t)) like a futures. – RRG May 13 '14 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 May 13 '14 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 '14 at 5:30
• I edited my answer to make it more precise when practitioners refer to a forward delta as 1 and when they define it to be exp(r(T-t)). Generally though forward delta of 1 is considered because most traders concern themselves with changes in valuations and with setting up precise hedges and not how forward prices change in the future (the difference between price and value of a forward contract is important). – Matt May 13 '14 at 6:59

I think there is confusion around the forward price and the value of a forward contract. A forward contract obligates an exchange of an asset at some future time $T$. By convention, this forward contract has initial value zero (at time $0$). The forward contract, being an exchange of an asset for a set dollar amount in the future, has at some $t \in [0, T]$ a value of $f(t, T)=S_t-Ke^{-r(T-t)}$. This contract clearly has delta equal to one.

Now consider the problem of the "correct" price $K$ at time zero. By convention, $f(0, T)=0$. Using the equation $S_t-Ke^{-r(T-t)}$ and solving for K at $t=0$ yields $K=S_0e^{rT}$.

$K$ is not time dependent: it is fixed at time zero. However, at time $t$ another forward contract may be initiated with maturity $T$. The same argument as above yields the price of $K$ at time $t$ of $S_t e^{r(T-t)}$. To explicitly show this dependence of $K$ on $t$ I will now let $F(t, T)$ denote the value of $K$ for a forward contract with expiration $T$ initiated at time $t$. Since $F(t, T)=S_t e^{r(T-t)}$ the "delta" of $F(t, T)$ is $e^{r(T-t)}$.

It is important to note that $F(t, T)$ is not an asset: after all, the discounted value of $F(t, T)$ is clearly not a martingale under the risk-neutral measure. It is more natural to take the delta of the forward contract, which is an asset.

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 $t<T$.

• 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 '14 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 '14 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 '14 at 2:13
• But RPG thanks for your effort! – Swab.Jat May 13 '14 at 3:26
• The price of a futures contract originated at $t$ is $S_t - F(t, T)e^{-r(T-t)}$. The "future price" is $F(t, T)=S_t e^{r(T-t)}$ so that the contract at origination has zero value. The delta of a futures contract is thus 1. – user9403 Feb 17 '15 at 19:02

For Forward contract, I agree with @Matt that its delta is exactly one.

This can be seen by the usual no-arbitrage argument, where long 1 Forward contract, short 1 underlying, and invest the shortsell proceeding in cash account at time 0. Then at Forward maturity T, everything will be settled with zero P&L. (i.e. use cash account at T to payoff forward price payment F, get underlying, and use it to close shortsell position.)

As during the entire life of this self-financing hedging portfolio, I only shortsell 1 underlying, therefore the hedge is exactly delta one at any time.

For Futures contract however, the hedge is not exactly delta one, but exp{r(T-t)}

For a long position in Futures contract, the interim cash flows from marked-to-market will go into the cash account. This part will grow by risk free interest rate (assuming it is not random). Hence, there is no hedge to be considered for these cash flows as it is not a Stochastic term. (although it does impact the Futures price as @Matt pointed out due to correlation between interest rate and underlying, but it is another question.)

The only Stochastic term in long Futures position, is the change of Futures price (one can show that dF=sigmaFdB). It is well known that F=S*exp{r(T-t)}. For every 1 unit change of S, Futures price will change by exp{r(T-t)}, and that contributes to the change in value of Futures position.

Thus, the delta of the Futures contract, is exp{r(T-t)}

Because the delta is time-dependent, the hedge will be dynamic and require frequent adjustment to hedge position, as compared to a static hedge of Forward position (always delta one).

I have another proof from my professor, but I think I can only share that privately. :)

Looking at the post - it seems it is the definition of delta itself, not the details of the formulae, that is different

I thought the delta was the ratio of change in value of the derivative to the change in the same (unit) amount of underlier

The post appears to be saying that the delta is the ratio of change of the derivative to the change in the equivalent amount of the underlier

• The confusion because @RPG incorrectly confused forward price and contract. Forward price is not a derivative, but forward contract is. – SmallChess Jan 28 '16 at 0:37