238 CHAPTER 6. THE MARKET FOR CURRENCY FUTURES
change in theusd/eurrate. To hedge this one-tenth-pip change in theusd/eur
rate, one tenth of a euro then suffices. For instance, if the spot rate changes from
usd/sek0.14 to 0.15 and theeur/sekrate remains at 0.10, then the euro goes up
fromusd/eur1.4 to 1.5; so holding 0.10 euro would be enough to hedge the crown
position.
The above example shows you one simple rule for chosing the hedge ratio: set it
equal to the relative value, theeur/sekcross rate in our case. The example also
makes it clear that this rule assumes a fixed cross rate, which cannot be literally
true. So our issue is under what assumptions the above simple rule still works and
how we can do better if the rule of thumb fails. In general, the hedged cash flow
equals
Cash flow at timeT 1 =S ̃(Te 1 )−β×(f ̃T(h 1 ),T 2 −ft,T(h) 2 ). (6.2)
Example 6.11
If Boston SC has set beta equal to 0.10, and thesekthen appreciates from 0.14 to
0.15 while theeurappreciates from 1.40 to 1.48, Boston SC’s cash flow is
0. 15 − 0. 10 ×(1. 48 − 1 .40) = 0. 15 −.08 = 0. 142 ,which is 0.02 above the initial rate (instead of 0.10, if unhedged).
In the example, setting beta equal to 0.10 clearly lowers the risk. The standard
approach is to chooseβso as to make the variance of the hedged cash flow as small
as possible. But we already know the solution. If we had written the problem as
one of minimizing var( ̃) where ̃:= ̃y−β ̃x, you would immediately have recognized
this to be a “regression” problem, with the usual regression beta as the solution:
β = the slope coefficient fromS ̃T(e 1 )=α+βf ̃T(h 1 ),T 2 + ̃,=
cov(f ̃T(h 1 ,T) 2 ,S ̃(Te 1 ))
var(f ̃T(h 1 ,T) 2 ). (6.3)
DoItYourself problem 6.2
Formally derive this result. First write out the variance of the hedged cash flow for
a givenβ, using the fact that the (known) current futures price does not add to the
variance. Then find the value forβthat minimizes the variance of the remaining
risk.
We now look at a number of special cases.6.4.2 Case 1: The Perfect Match
There is a perfect match if the futures contract expires atT 1 (that is,T 2 =T 1 ) and
e=h. For example, assume there is asekcontract with exactly the same date