# Definition of Financial Futures: The price to pay for the underlying asset on delivery date

According to the opening paragraph of the Wikipedia article for "Futures contract", the parties to a futures contract

initially agree to buy and sell an asset for a price agreed upon today (the forward price) with delivery and payment occurring at a future point, the delivery date.

However, later in the same article, a formal definition of a futures contract is given, attributed to Björk, which states that, if $F(t, T)$ designates the quoted market price at time $t$ of a futures contract for delivery of $J$ at time $T$, then, in particular,

At time $T$, the holder pays $F(T,T)$ and is entitled to receive $J$. Note that $F(T,T)$ should be the spot price of $J$ at time $T$.

So, I'm confused: is the amount to be payed on delivery date in exchange of the underlying asset determined on the day that the futures contract is entered, as the opening paragraph indicates, or is it determined by the market quote on the day of delivery, as Björk's definition indicates?

• This is confusing indeed. Physical delivery: upon expiration of the contract, you pay the (forward) price $F$ you have agreed on at inception in exchange for which you receive $J$, which is now worth $S$. Cash settlement : upon expiration you win/lose the cash difference between the spot price $S$and $F$. The problem comes from the notations. $F (t,T)$ was determined in the past. $F (T,T)$ is the price you would agree on at $T$ to receive the underlying at $T$ so indeed it is the spot price of the underlying. – Quantuple Mar 21 '16 at 8:20
• Note that one important aspect of future contracts vs forward contracts (usually traded otc) is the daily margining. – Quantuple Mar 21 '16 at 8:24
• @Quantuple: Thanks. So, if I understand you correctly, the Björk definition does not capture the notion of a futures well, however it can be corrected as follows. Instead of saying "At time $T$, the holder pays $F(T,T)$ and is entitled to receive $J$." it should say "At time $T$, the holder pays $F(0,T)$ and is entitled to receive $J$.", as $F(0,T)$ is the price agreed upon at the time the futures contract was traded, whereas $F(T,T)$ is the price of the asset at the time of delivery. Am I correct? – Evan Aad Mar 21 '16 at 8:37
• Yes $F(0,T)$ or $F (t,T)$ would make more sense in my opinion, with the first argument denoting the time at which you agreed on a price (but it is still true that at $T$, $F (T,T)=S$, thus could say that you pay $F (t,T) and receive$F (T,T) $). – Quantuple Mar 21 '16 at 8:40 ## 1 Answer I think there's confusion here. If a futures contract whose maturity is T is trading at F(t,T) at time t, then the buyer at time t pays no cash at time t, he just "enters into" the contract. At time t+1 he receives an amount F(t+1,T) - F(t,T) of variation margin, and this occurs every day until time T when he receives an amount F(T,T) - F(T-1,T) of variation margin. By definition the value of F(T,T) is the spot price of some asset J at time T. Thus the total cash received over time is F(T,T)-F(t,T). (this may be negative of course). At expiration T, if it is cash settled nothing else happens. If it is physical settled, the holder pays F(T,T) and receives the asset J, although this settlement does not create any positive or negative value because it is an on-market transaction. • Thanks. This is very illuminating. What happens to the funds left in the margin accounts after delivery? Are they returned to the respective parties? – Evan Aad Mar 21 '16 at 16:08 • There are no funds left upon expiry: because of the daily margining each party has already received/paid what it should. Thanks for clarifying the situation dm63. What I described in the comments of the OP question is indeed a forward contract not a futures contract. You receive/lose the same amount of money at the end, but for the forward it comes as a single cash flow at expiry and for the future you have daily margins. – Quantuple Mar 21 '16 at 16:50 • @Quantuple: Suppose the delivery price remains perfectly constant through the life of the futures contract, so that the margins accounts experience no fluctuation. After expiry date, what happens to these funds? Are they returned to the parties? Are they taken by the clearing house? – Evan Aad Mar 21 '16 at 17:50 • Read dm63's answer carefully: you'll see that if$F (t,T) $is constant$\forall t $, you don't win/lose anything on a daily basis: the margins account is left empty. Note that this situation is however impossible in practice because constant$F (t,T) $would be equivalent to knowing the spot price$S_T $at which the underlying will trade at$T $in advance ($F (t, T) = cst \iff F (T,T) = cst $but$F (T,T) = S_T\$ by no arbitrage). – Quantuple Mar 21 '16 at 17:56
• @Quantuple: According to Wikipedia, "To minimize credit risk to the exchange, traders must post a margin or a performance bond, typically 5%-15% of the contract's value." So if I wish to go long on a futures contract whose delivery price is, say 100 USD, I am required to deposit in my margin account a minimum of 5 USD. Now suppose there is minimal or no price fluctuation till expiry. What happens to my 5 USD after expiry date? Do I get them back? Are they absorbed by the clearing house? – Evan Aad Mar 22 '16 at 5:35