# why futures contract has no value

Can any one tell me, why futures contract has no value?

We know that the value of future(Maybe I confuse the concept of future value and future price): $$\textrm{Fut}(t,T) = \widetilde{E}[S(T)|\mathcal{F}_t]$$ How can it be zero? We only have the discounted value of cash flow is zero.

And why in the forward contract, $F$ is not zero in the $\Pi = V(S,t) - \delta S - \delta_1 Z.$ But just set $V = 0$ to obtain the forward price, it is unreasonable.

• @A. Oreo, please read carefully the answers provided. $F(S,r,t)$ above is the future price. It is given by $F(t,T)=\Bbb{E}^Q[S_T \mid \mathcal{F}_t]$ and is clearly not zero. Yet a futures contract value is indeed zero at the moment at which you enter it $V(S,r,t) = \Bbb{E}^\Bbb{Q} \left[ \sum_{i=1}^N e^{r(t_i-t)} (F(t_i,T)-F(t_{i-1},T)) \right]$. Don't confuse functions $V(...)$ and $F(...)$. A similar reasoning applies to forward contract values vs. prices. – Quantuple Mar 13 '17 at 15:34
• @Quantuple I may get it, but even for the any fixed strike $K$ i.e $V(t,T) = e^{-r(T-t)}E[S_T-K|\mathcal{F}_t],$ the cash flow is still $dF?$ But, It seems $dV \neq dF(S,r,t).$ – A.Oreo Mar 14 '17 at 1:48

The mathematical analysis above is correct, but to understand WHY we say that "a futures has no value" it is helpful to understand how a Futures Exchange works.

When you enter into a position (for example go long 1 crude oil contract at 45.25) you do not have to pay anything, nor does the seller of the contract receive any money from you. So it is correct that the futures contract at inception has no value. As the price of oil fluctuates during the day however, the contract does acquire a positive or negative value.

At the close of business on that day (2:30 pm New York Time) a Settlement Price of say 45.35 is declared by the exchange. This is based on the price the contract was trading for at that time. Your contract is now worth 1000*(45.35-45.25) = 100 dollars since there are 1000 bbl in one contract. During the night, you will be paid this amount in cash, thus resetting the value of the contract to zero. This is called the daily Mark to Market process. The next day the price will change again, but again this gain or loss will be settled in cash during the night.

This is why we say that a futures contract at inception or at the close of business has no value. It is being reset to zero each day, and we are analyzing the situation at the time of the reset.

• A contract that truly had a value identical to zero at all times would probably be the most useless economic contract ever invented, but a Future is not like that, its value is merely reset to zero frequently (via a cash payment) and free to change at other times. – noob2 Mar 13 '17 at 15:11
• The value of the contract is zero when you enter it, but it is indeed market to the market at each market close. In fact it works as if the contract was cash settled and reset each day up to the expiry. Thanks for this pragmatic view. – Quantuple Mar 13 '17 at 15:26

The value of a futures contract is the forward value of the payment, discounted back to today -

$$V(t,T) = e^{-r(T-t)} \mathbb{E} \left[ S(T) - F | \mathcal{F}_t \right]$$

and the price of a futures contract is the price $F$ that gives the contract zero present value -

\begin{align} F & = \mathbb{E}\left[ S(T) | \mathcal{F}_t\right] \\ & = S(t)e^{r(T-t)} \end{align}

Note that as soon as you have entered a futures contract, the price $F$ is now fixed, which means that in general the futures contract no longer has zero present value (the same is true of a forward contract).

Your margin account at the exchange is credited or debited at the end of every day to take account of your profit/loss on the contract. This is the primary difference between futures and forwards - on a forward contract, you accumulate a non-zero present value on the contract over time, which is all realized at maturity. On a futures contract you realize your profit/loss as you go, and this requires financing (you need to fund your losses, and can invest your gains).

• Although I of course agree with the point you are trying to make, I think that the equations you propose are in fact better suited for a "forward" contract than a "future" contract. For a future you would have: $$V(t,T) = \Bbb{E}^{\Bbb{Q}} \left[ \sum_{i=1}^N e^{-r(t_i-t)} F(t_{i-1},T)-F(t_i,T) \mid \mathcal{F}_t \right]$$ for each open business day $i=1,...,N$ between now ($t$) and the expiry ($T$) with $F(t,T)$ figuring the future price where indeed $F(t,T) = \Bbb{E}^{\Bbb{Q}^T}[S_T \mid \mathcal{F}_t]$, such that $F(t,T)$ is a martingale and $V(t,T)=0$ – Quantuple Mar 13 '17 at 11:13
• Pls see the update, why for the forward contract, $F$ is not zero, I think they should be similar. – A.Oreo Mar 13 '17 at 11:17
• $F(t,T)$ is zero neither for a futures nor a forward. However the current values $V(t,T)$ of the contract should you enter them at $t$ are. See also here quant.stackexchange.com/questions/31162/…. – Quantuple Mar 13 '17 at 11:19
• @Quantuple Yes, this is a valid point! It takes the financing into account very explicitly. In practice though, I think that most people use the first formula in my answer, and then have a correction between values of $F$ derived from futures and forward contracts where necessary (which is really only for futures on interest rates). Perhaps some desks now do something more advanced to take the financing into account, but I haven't personally seen it. – Chris Taylor Mar 13 '17 at 11:20
• @ChrisTaylor - Never seen it either I must say. As I said I got the point you were trying to make... I guess I was just being pedantic :) – Quantuple Mar 13 '17 at 11:24