6.4. HEDGING WITH FUTURES CONTRACTS 237
futures positions in not justeurcurrency but also ineurandusdinterest rates,
and perhaps evensekinterest rates. However, in order to simplify the exposition,
we first consider the case where only one type of futures contract is being used to
hedge a given position.
6.4.1 The Generic Problem and its Theoretical Solution
The problems of currency mismatch and maturity mismatch mean that, at best,
only an approximate hedge can be constructed when hedging with futures. The
standard rule is to look for a futures position that minimizes the variance of the
hedged cash flow. Initially, we shall assume the following:
- There is one unit of foreign currencye(“exposure”) to be received at timeT 1 —for
instance, one Swedish Krona is to be converted intousd, thehc. - A futures contract is available for a “related” currencyh(”hedge”)—for instance,
theeur—with an expiration dateT 2 (≥T 1 ). - The size of the futures contract is one unit of foreign currencyh(for instance,
oneeur). - Contracts are infinitely divisible; that is, one can buy any fraction of the unit
contract.
Items 1 and 3 are easily corrected. Item 4 means we will ignore the fixed-contract-
size problem. The reason is that nothing can be done about it except finding a
theoretical optimum and then rounding to the nearest integer.
Let us show the currency names as superscripts, parenthesized so as to avoid any
possible confusion with exponents. Denote the number of contracts sold byβ.^5 The
total cash flow generated by the futures contracts between timestandT 1 is then
given by the size of the position, –β, times the change in the futures price between
times t andT 1. (True, this ignores time-value effects, but we can’t be too choosy:
the hedge is approximate anyway.)
Example 6.10
Boston Summae Cornucopiae Ltd will receivesekin May, so they want to take a
position in the Juneeurcontract to hedge this. Since a crown is worth about 10
eurocent, they could hedge each crow by 0.10 euros, meaning thatβis set equal to
0.10.
If theeur/sekrate remains constant regardless of the gyrations of theusd/eur
rate, then each pip change in theusd/sekrate is associated with a one-tenth-pip
(^5) Beta should get a double time subscript, as should the variance and covariance in the solution.
But the notation is already cluttered enough.