242 CHAPTER 6. THE MARKET FOR CURRENCY FUTURES
tinguishible from forwards, we know that
f ̃T(e)
1 ,T 2 ≈
S ̃T(e)
11 + ̃rT 1 ,T 2
1 + ̃rT(e 1 ),T 2, (6.12)
where the risk-free ratesrnow get tildes because we do not yet know what they will
be, on Feb 15. So one way to solve the ever-changing-maturity problem in the data
is to construct forward rates from spot and interest data, probably using 30-day
p.a.rates to approximate the 35p.a.data.^7 The other way out is to use a rule of
thumb. Inverting Equation [6.12], we get
S ̃T(e)
1 =1 + ̃r(Te 1 ),T 2
1 + ̃rT 1 ,T 2
f ̃T(e)
1 ,T 2. (6.13)The rule of thumb then follows under a not very harmful assumption, namely that
there is no uncertainty about the interest rates. For instance, suppose you knew
that the ratio (1 +r(e))/(1 +r) 35 days would be 1.005 on Feb 15. Then Equation
[6.12] would specialize into
S ̃T(e)
1 = 1.^005
f ̃T(e)
1 ,T 2 , (6.14)which tells us immediately that the forward-looking regression coefficient ofS(e)on
f(e)is 1.005. So the rule of thumb for the delta hedge is to set the hedge ratio
equal to the forecasted ratio (1 +r(e))/(1 +r) 35 days for Feb 15. Experiments show
that it hardly matters how you implement this: take the current 35-day rates, or
forecasts implicit on forward interest rates (if available). Also, since the regression
(if you would run it) has a very highR^2 , the bias is tiny and the rule of thumb does
quite well.
6.4.5 Case 4: The Cross-and-Delta hedge
Now combine the problems: we use aeurcontract expiring March 20 to hedgesek
that come forth on Feb 15. In principle we have to regress possiblesekspot rates
on 35-dayeurfutures.
The rule of thumb is a combination of the two preceding ones: set the hedge ratio
equal to the current cross rate times the forecast ratio (1 +r(e))/(1 +r). Again, the
rule of thumb does quite well when the currenciesiandj are closely related and
theR^2 , therefore, is high.
(^7) This would also solve synchronization problems in data: spot and interest rates are observed
at the same time, while the futures prices may be from a different data base and observed at a
different time of the day.