Rolling the hedge

We saw before what the optimal instrument is in two cases where the hedge is not a perfect one. Namely we deduced that

1. If the asset being hedged is different that the one underlying the futures contract, then one has to choose the futures contract that has the highest possible correlation with the asset to be hedged; and

2. If there is no futures contract on the asset to be hedged that expires on the maturity date of the hedge, then one has to choose the futures that expires as close as possible -but not earlier than, the date when the hedge matures.

Of course there is another possibility: suppose that the hedge matures in a very distant point in time and all futures contracts available at the moment expire before that time. The hedger must then roll the hedge forward. This is done in the following fashion:

$$\begin{tabular}{ll} Time & Action \hline $t$ $\left[ =\tau _{... ...ontract $n$ \\ $\tau $ & Close futures contract $n$ \hline \end{tabular}$$

In the above strategy there are $n$ sources of uncertainty, resulting in $n$ different basis risks. $n-1$ of them are referred to as the rollover basis: they represent the uncertainty about the difference of the futures price of the contract that is closed and the new contract that is opened when the hedge is rolled forward. The last basis risk is the classical one: it is the uncertainty about the price of the $n$th contract and the spot price of the underlying asset at time $\tau$. The next example illustrates the rollover basis.

Figure 2.3: The basis when rolling the hedge

Example 6 (Rollover basis)   Figure 2.3 describes the mechanics of the rollover basis. Say that one wants to deliver some asset at time $\tau$, where the futures contracts mature at times $\tau_1$, $\tau_2$ and $\tau_3$, as shown in figure 2.3:

• The hedger initially at time $t$ shorts the available futures $F_1$.

• At time $\tau_1$ she closes her position. The futures price will now be $S(\tau_1)$ due to the futures-spot convergence. Therefore, through marking to market from the first futures she realizes an income $F(t,\tau_1)-S(\tau_1)$.

• Simultaneously, the hedger short a new futures contract which matures at $\tau_2$ and has price $F(\tau_1,\tau_2)$.

• At time $\tau_2$ she repeats the previous two steps. The income is $F(\tau_1,\tau_2)-S(\tau_2)$, and she has shorted a futures with price $F(\tau_2,\tau_3)$.

• Finally, at time $\tau$ she closes the final position. The futures price will be $F(\tau,\tau_3)$ and the underlying's price will be $S(\tau)$. The income will be $F(\tau_2,\tau_3)-F(\tau,\tau_3)$.

• She also delivers the asset and receives the market price $S(\tau)$.

The total income of this strategy is

$$F(t,\tau_1)-S(\tau_1)+F(\tau_1,\tau_2)-S(\tau_2)+F(\tau_2,\tau_3)-F(\tau,\tau_3)+S(\tau)$$

which can be rewritten in terms of the bases as

$$F(t,\tau_1)-b_2(\tau_1)-b_3(\tau_2)+b_3(\tau)$$