reviously we obtained (
Change of drift
recipe 1
) that the drift of an SDE changes according
to
We now combine the
program
into a similar expression. Note, that the general formula does not work for
the second transformation between the risk neutral measures. We use the
explicit result for that case. Let
be a
-price
of a traded asset.
Then
We put the recipes together:
Therefore,
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(Currency change of numeraire recipe)
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The above expression may be understood by considering units of measure. The
is an absolute quantity but
and
are measured in units of currency: pound for
and dollar for
.
Hence, there must be a multiplication by
.
Note that the transition
does not involve multiplication of
by exchange rate. The path of
remains the same. This is different from the
change
In such situation SDE transformation is slightly
different:
thus
We use the formula (
X to Y connection
) to
convert everything to
quantities:
and
conclude
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