e already established in (
Change
of measure recipe section
) and
(
Change of numeraire recipe
section
) that the change of numeraire is all about the modification of SDE
drift term. However, the lengthy calculation of the
(
Two asset change of
numeraire section
) creates uneasy feeling. We would like to have a formula
for direct transformation of the drift that is independent of the choice of
the driving Brownian motion.
We are changing from numeraire
to numeraire
and look at price of some traded asset
:
The notation
is chosen according to the rule "drift of Z in X-based probability". We
perform the change (
Change of Brownian
motion
)
then the SDE for
changes as
follows
Hence,
We represent the result in symmetrical
form:
|
|
(Change of drift recipe)
|
where
is the drift of
under the
-based
probability measure,
is the drift of
under
-based
probability measure. We obtain another useful representation as
follows
Therefore,
|
|
(Change of drift recipe 1)
|
for any traded asset
and numeraires
and
.
|
