6.4. HEDGING WITH FUTURES CONTRACTS 241
one Euro shorted would hedge about tensek. In other words, 0.10 Euros persek
will do.
Suppose, alternatively, that you prefer to run a regression between monthly per-
centage changes onsekandeur, and the slope is 0.96 with anR^2 of 0.864. Then
regression-based hedge ratio = 0. 96 × 0 .10 = 0. 096.That is, you’d lower your hedge ratio.
The rule of thumb is almost surely biased, which is bad, but has one big ad-
vantage: it has zero sampling error. Let us explain each statement. First, the
assumption of unit gamma’s across the board does not make sense, statistically. For
example, if itwere true, then the reverse regression, betweeneurandsekrather
than the inverse, would also produce a unit gamma, but this is mathematically pos-
sible only when there is no noise. It is easy to verify that the product of the two
gamma’s—the one fromyonxand the one fromxony— is theR^2 , which is surely
a number below unity; so one expects at least one of the two gamma’s to be below
unity, and normally both will be below unity. I elaborate on this in Teknote 6.1.
But while the drawback of the rule of thumb is a bias, it has the advantage of
no sampling error. If you actually run regressions, then the estimated sample will
randomly deviate, depending on sampling coincidences, even if nothing structural
has changed. Now from the point of view of the user, sampling error is as bad
as bias. For instance, if the true gamma (known to the Great Statistician in the
Sky only) is 0.95, then the error introduced by an estimated gamma of 0.90 is as
bad as the bias introduced by the rule-of-thumb value, unity. Likewise, hedging
with a unit gamma would be as bad as hedging with an estimated gamma that
equals, with equal probability, 1.00 or 0.90. So it all depends on squared bias versus
estimation variance. Experiments (Sercu and Wu, 2000) show that the rule of thumb
does better than the regression-based hedge if the relation betweeniandjis close,
which is the case for theusd/sekandusd/eurrates. When the link between the
two variables becomes lower, sampling-error variance increases but so does the bias,
and in fact bias tends to become the worse of the two evils.
6.4.4 Case 3: The Delta hedge
Suppose now that there is asekcontract, but for the wrong date instead of for the
wrong currency. Our money comes in Feb 15, while the contract expires March 20,
for example. So our futures contract will still have a 35-day remaining life when it
is liquidated. In principle we’d have to regress possible spot values for thesekon
the corresponding 35-day futures price of thesek. One problem is that we do not
have time-series data on 35-day futures: the real-world data have a daily-changing
maturity.
There are two ways out, both connected toIRP. Since futures are almost indis-