e have description of market under some numeraire
in
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 also introduce the reciprocal quantity
.
We proceed to calculate the drift of
.
Suppose we have one pound at time
.
We may invest into pound bonds
and convert to dollars at maturity. We may also convert to dollars right away
and invest into 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
The above is to be compared with the formula (
Bond
SDE
) under the condition
.
Hence,
or
where the
is standard Brownian motion with respect to 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 (otherwise there would be arbitrage).
By similar
argument
We also
have
Hence,


(X to Y connection)

We now collect results for general case. We want to change numeraire from
to
,
where
is a price of a traded asset denominated in
.
We execute the
program
Hence,
or
