Hedging with futures

If an company knows that it has to sell a particular asset at a particular time in the future, it can hedge by taking a short position, therefore locking in the price of delivery. This is called a short hedge. Similarly, a company that knows that it will need an asset in the future can take a long hedge, thus locking in the price of purchase. It is very important to note that hedging does not necessarily improve the financial outcome, it just reduces the uncertainty. In practice, hedging is not perfect, the basis risk arises due to a number of reasons, some of which were discussed in the introduction:

• The asset being hedged might be different that the one underlying the futures contract, i.e. using a 30y T-bill to hedge a 10y T-note;

• The hedger might be uncertain about the exact time that the delivery has to take place, i.e. a new oil ring that is expected to start extracting next summer, without knowing exactly when; and

• The futures contract matures after the delivery date that the hedger has in mind, i.e. the hedger needs to buy steel in January but steel futures expire on March.

The basis $b$ is defined as

$$b\left( t\right) =S\left( t\right) -F\left( t,T\right)$$   ,

where $S\left( t\right)$ is the spot price of the underlying asset and $F\left( t,T\right)$ is the price of the futures contract that has been utilized. If the asset to be hedged is the same as the one underlying the futures, then the basis on expiration is equal to zero. If the delivery date is not the same as the one that the futures matures, then the basis will signify the ``losses'' or ``gains'' of the hedge than are not known when the hedge is constructed.

Example 4 (Basis risk: different maturities)   Today, the gold price is $S\left( t\right)$. Say that one has to deliver gold at time $\tau$, and in order to hedge takes a short position on a futures contract that matures at time $T>\tau$. The price of this contract is equal to $F\left( t,T\right)$, and the basis today is $b\left( t\right) =S\left( t\right) -F\left( t,T\right) <0$. The basis on the delivery date will be $b\left( \tau \right) =S\left( \tau \right) -F\left( \tau ,T\right)$, will indicate the gains [or losses] due to the hedge. See figure 2.2 for a visual representation of the basis risk.

Figure 2.2: Basis risk

At time $\tau$ the company will close the futures contract by taking a long position and at the same time sell the gold at the current price. The marking-the-market procedure will leave the company with a loss of $F\left( \tau ,T\right) -F\left( t,T\right)$ since the futures was sold at time $t$ and bought at time $\tau$; while by selling the asset the income is $%% S\left( \tau \right)$. The total income is therefore is

$$S\left( \tau \right) +F\left( t,T\right) -F\left( \tau ,T\right) =F\left( t,T\right) +b\left( \tau \right)$$   .

The value $F\left( t,T\right)$ is known at time $t$, while the quantity $%% b\left( \tau \right)$ represents the basis risk. Obviously, if $\tau =T$, then $b\left( \tau \right) =b\left( T\right) =0$, and there is no basis risk.

If one takes into account the time value of money -something that we did not do in the previous example, then for most investment assets the basis risk is very small, due to arbitrage arguments that we will explore later. Generally speaking, the basis risk arises from uncertainty about the future interest rates and uncertainty about the future yields of the underlying asset. For investments that are difficult or costly to store, the basis risk might increase substantially. The delivery month that is as close as possible -but not earlier than, the date when the hedge matures.

As we noted before, the asset used for hedging might be different then the one being hedged. Say that there were no futures contracts for gold, and one had to use futures on silver.

Example 5 (Basis risk: different assets)   Denote the price of gold by $S$ and the price of silver by $\tilde{S}$. Denote by $F$ and $\tilde{F}$ the corresponding futures prices. As before, say that one has to deliver gold at time $\tau$, but such futures do not exist. Futures contracts on silver are going to be used instead. As before, the total income of this hedge will be

$$S\left( \tau \right) +\tilde{F}\left( t,T\right) -\tilde{F}\left( \tau ,T\right)$$   ,

which can be rewritten as

$$\tilde{F}\left( t,T\right) +\left[ S\left( \tau \right) -\tilde{S... ...t] -\left[ \tilde{F}\left( \tau ,T\right) -\tilde{S}\left( \tau \right) \right]$$   .

The quantity $\tilde{F}\left( \tau ,T\right) -\tilde{S}\left( \tau \right)$ is the basis of silver $\tilde{b}\left( \tau \right)$; it would exist if the asset to be actually delivered was indeed silver. The quantity $S\left( \tau \right) -\tilde{S}\left( \tau \right)$ is the basis arising due to the differences of the assets.

In the example above observe that if $\tau =T$, that is to say if the maturity of the silver futures was the same as the maturity of the hedge, then $\tilde{F}\left( \tau ,T\right) =\tilde{S}\left( \tau \right)$ and the basis of the hedge would be only due to the differences between the two assets. On the other hand, if the two assets were perfectly correlated, then $S\left( \tau \right) =\tilde{S}\left( \tau \right)$, and there would only be basis due to the maturity differences. This makes clear that when there is no futures contract on the asset being hedged, one has to choose the futures that has the highest correlation with the underlying asset.