International Finance: Putting Theory Into Practice

240 CHAPTER 6. THE MARKET FOR CURRENCY FUTURES

believe that de distribution of ∆Sis constant. So if we usesas shorthand for ∆S/S,
we would use the following equation:

s ̃(te)=α′′+γ ̃s(th)+ ̃′′t. (6.7)

The assumption now is thatγis constant, notβ. If you run a regression between
percentages, you need to transform the slopeγfrom an elasticity into a partial
derivative:^6

if ̃s(te)=α′′+γ ̃s(th)+ ̃′′t thenβ = γ

S(te)

St(h)

(6.8)

= γ St(cross), (6.9)

whereS(cross)is the cross rate. In our example,

St(e)

S(th)

has dimension

usd/sek

usd/eur

=eur/sek, (6.10)

so the cross rate is the value of onesekineur, which is euros per crown or, generally,
h/e.

Ifγis unity, we get the relative-value rule that we started out with in the first
example, where we hedged each crown with 0.10 euros because the initial cross rate
iseur/sek0.10. This is a rule that practitioners often use. They do not actually
run this regression: instead, they just guess that the gamma equals unity. For
instance, Boston SC’s expects that every percentage in theeur(against theusd)
on average leads to a similar change in theusdvalue of thesek. [This is slightly
more general than our earlier story of a fixed cross rate: now each percentage change
in the euro’s value is assumed to lead to the same percentage change in the crown’s
valueon average; sometimes it may be more, sometimes less, but on average the
change is the same.] Then the hedge ratio would simply be set equal to the cross
rate:
Rule of thumb for cross hedge:γ= 1 soβ=St(h/e). (6.11)

Example 6.12
Again assume spot rates of 1.40 for theeurand 0.14 for thesek. The quick-and-
dirty hedge ratio would be set equal to the cross rate, the value of onsekineur,
which equals 0.14/1.40 = 0.10. The reason is that you think that percentage changes
of the two currencies will be similar (γ= 1), but since theeuris worth about ten
Kronar now, oneeurwould change by as much as would ten Kronar. Therefore,

(^6) In terms of a regression ofyonx, the exposure is written as∆y
∆x. An elasticity equalse=
∆y
∆x×
x
y,
so∆∆xy=e×yx.