e take the view of a dollar-based observer. We are valuing a contract
dependent on the pound price of a traded asset
.
The state variable is given by
.
We assume that the rates are constants to limit the size of the calculation.
We compose the dollar valued portfolio
where
refers to the pound denominated MMA. The calculation is similar to the section
( Transformation
of SDE based on delta hedging section). We proceed to calculate the
differential
where the bracket
refers to the sum of the second derivatives in
and
,
see ( XY_bracket). If we
set
then we
obtain
Finally,
Assuming that the only cashflow that the contract pays is the final cashflow
we may represent
as an
expectation
This result agrees with the result of the previous section because the price
is given by the
SDE
in the risk neutral
-measure.
We change the numeraire to the risk-neutral
$-measure
and
obtain
