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2.1.2 Forward contract and futures markets

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A Forward contract is the simplest form of a contingent claim that can be derived from an asset, since it does not contain any element of choice. Two parties agree, on a future delivery date $ T$ , to exchange an underlying asset for a predetermined amount of cash called the delivery price $ K$ . The underlying can be any kind of asset (e.g. commodities, shares, currencies) that has a fluctuating spot price $ S(t)$ ; on the delivery date $ T$ , the terminal payoff $ \Lambda(S)$ is simply calculated from the difference between the spot and the delivery price

$\displaystyle \Lambda_\mathrm{long}=S-K, \qquad\qquad\qquad\qquad \Lambda_\mathrm{short}=K-S.$ (2.1.2#eq.1)

The value (2.1.2#eq.1, left) plotted in (2.1.2#fig.1, left) shows that a long forward position (where the holder has the right and the obligation to buy the underlying for a price $ K$ ) increases in value and becomes profitable when the underlying exceeds the delivery price; the maximum losses in a long position occur if the underlying loses all of its market value $ S=0$ and the contract obliges the holder to buy for the delivery price $ \Lambda=-K$ .

Figure 2.1.2#fig.1: Terminal payoff diagrams $ \Lambda (S)=V(S,T)$ of forward contracts struck for a delivery at a price $ K$ on a date $ T$ ; the value of a long (left) and a short position (right) is plotted as a function of possible realizations of the underlying spot price $ S$ .
\includegraphics[width=6cm]{figs/payFutLong.eps}        \includegraphics[width=6cm]{figs/payFutShort.eps}

The opposite is true for the party who enters a short forward position (right): the holder has both the right and the obligation to sell the underlying with a maximum profit of $ \Lambda=+K$ and potential losses that are unlimited if the underlying becomes arbitrarily expensive $ S\gg K$ . To avoid the unnecessary exchange of cash on the day $ t_0<T$ when the contract is written, the delivery price is sometimes chosen equal to the forward price $ F(t_0,T)$ , which, by definition, makes the initial value of the contract worthless $ K=F(t_0,T)=S(t_0)$ .

A futures contract is a special type of forward contract with standardized delivery dates and sizes that allow trading on an exchange: (2.1.2#tab.1) shows an example of a commodity future that enables the owner of a contract to buy one tone of wheat some time in the future. A system of margin requirements is designed to protect both parties against default: instead of realizing the profit or the loss at the expiry date, futures are evaluated every day and margin payments are made across gradually over the lifetime of the contract.

Table 2.1.2#tab.1: Futures of wheat (GBP/tone) quoted on Oct 22, 2002 in the press
Delivery Settlement Volume Open interest
date price High Low
Nov 59.90 60.75 59.90 160 950
Jan 62.25 62.25 62.25 50 1550
Mar 64.10 64.75 64.10 20 900
May 65.75 66.55 65.75 140 3410


Despite these differences, futures prices can be shown to be equal to the forward prices if both parties can be trusted and the interest rate is fixed.

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