e have description of the market under some numeraire
in the
 -denomination
and we would like to change to some
 -denominated
numeraire
(The
is measured in
and the
is measured in
 .
We introduce
- pound price of a dollar. A
 -amount
should be multiplied by
to obtain a
 -amount.
We introduce the reciprocal quantity
 .
is what is regularly quoted: 2.00 dollars per pound or around that.
We proceed to calculate the drift of
 .
Suppose we have one pound at time
 .
We may invest into the pound bonds
and convert to dollars at maturity. We may also convert to dollars right away
and invest into the dollar bonds
 .
We get a dollar outcome in both situations and the dollar risk neutral
expectation of both strategies should be the same. We express such conclusion
below:
We move the time
 -known
quantities out of the expectation sign and
obtain
Let
 .
We
get
and
consequently
Note that the expectation
is the drift that we are calculating and the bonds have
expansions
Hence,
or
where the
is the standard Brownian motion with respect to the risk neutral probability
measure on dollar market. The result agrees with the intuition that when the
dollar MMA rate is higher than the pound MMA rate then the exchange rate
should drift against dollar.
By a similar
argument
We also
have
Hence,
We now collect the results for the general case. We want to change from
to
 .
We execute the
program
Hence,
or
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