# A proof that the final payoff on a futures contract is twice that on a forward contract

Following is an argument demonstrating that the final payoff on a futures contract is twice that on a forward contract, contrary to what I believe is the accepted truth that the two payoffs are the same. I'm sure there's fault in my logic, but I can't see where it is and would appreciate it if someone pointed it out for me.

Assuming zero interest rate, the final payoff of a long position on a forward contract is $S_T - F_0$, where $S_T$ is the price of the underlying on expiry date, and $F_0$ is the forward price.

With futures, on the other hand, the stream of daily payoffs of a long position sums up to $$\sum_{k = 0}^{N - 1} (F_{t_{k + 1}} - F_{t_k}) = F_{t_N} - F_{t_0}$$ where $t_0 = 0$, $F_{t_0}$ is the forward price, $N$ is the number of days till expiry, $t_N = T$ and $F_{t_N} = S_T$. In addition, on expiry day the holder of the long position is committed to pay the forward price $F_{t_0}$ in exchange of the underlying asset, so, in total, the final payoff is $$\left(F_{t_N} - F_{t_0}\right) + \left(F_{t_N} - F_{t_0}\right) = 2\left(S_T - F_0\right)$$

In addition, on expiry day the holder (...)

is wrong.

[Short Story]

Due to the daily variation margins calculated by the clearing house on each market close, you have already received/coughed up what you should upon expiry. If the contract is cash-settled, the story thus ends here. In case of physical delivery however, although there will be no additional cash flows, the underlying needs to change hands (from the seller to the buyer) as legally specified by the contract. Note that only a minority of future trades are held until expiry anyway.

[Long Story]

To better understand, take the situation where you are a producer who wishes to secure a certain price for whatever it is you are producing and planning to deliver at some future time $T > 0$. You find that the value at which the future of expiry $T$ trades on the exchange is honest and decide to lock that price by selling a future today. In other words, you enter a legal agreement with a party $A$ which accepts to pay you a certain amount $F(0,T)$ to receive your goods at $T$. So far so good. But now, what if the counterparty with whom you agreed on initially, decides to sell the future contract to some party $B$ at $t^* < T$? By offsetting his position, $A$ is now free of any obligation. These are transferred to new guy $B$ for whom it is as if he had agreed to buy your goods at $F(t^*,T)$ and not your initial price $F(0,T)$. So are you screwed because the price has changed? Well no, thanks to the margins mechanism this is completely transparent for you. Indeed, since the inception of the trade, the clearing house has regularly credited/debited your margins' account starting from the initial future price of $F(0,T)$. The only thing left to do for you, is to meet your obligation upon expiry $T$ i.e. deliver the underlying in case of physical settlement. All in all upon expiry:

• You will have made $F(0,T) - S_T$ (got rid of your goods for the price you wanted)
• Party $A$ will have made $F(t^*,T)-F(0,T)$ (probably a gain since he decided to sell early)
• Party $B$ will have made $S_T - F(t^*,T)$ (received the goods for the price he wanted).

Thus, everyone is happy and it is a zero sum game, as expected. In conclusion, the payout of buying at $t=0$ and holding a future up to its expiry $t=T$ is indeed $$S_T - F(0,T)$$, which is exactly the same payoff as that of a forward (if we neglect second order effects)

• Thanks you. Your explanation was extremely clear and answered many of my questions. – Evan Aad Mar 24 '16 at 5:23
• You're welcome. – Quantuple Mar 24 '16 at 12:52

Given that you have to true up every day, the payoff of the futures contract is already paid off except for that final day on that last day before expiration. There would be no additional amount equal to the difference in the initial and final price. Do you have a certain text that seems to say someone is responsible for both?

• "Delivery usually is a three-day sequence [...] On the third day, the delivery day, delivery takes place and the long pays the short." Introduction to Derivatives and Risk Management, 9th edition by Don M. Chance and Roberts Brooks (South-Western 2013), p. 278 (Google books) – Evan Aad Mar 23 '16 at 21:07
• Oh... non-cash settled. I've never actually taken delivery of an asset, so I'm not speaking from experience. My understanding, though, is that the future is brought back to fair value each day. This is the sum of payments you have in your question. On the last day, the payment that is made is on the fair value of the forward (or on the last day, the fair value of the asset). Effectively, you would have $(S_T - F_T_n) + (F_T_n - F_t_0)$ which is equivalent to the forward. Note the first piece goes to 0 at time = T. – RandyF Mar 23 '16 at 23:06