e are calculating the
expectation
where the
are given numbers,
is a deterministic function and
is the standard Brownian motion.
According to the Ito formula
(
Ito_formula
)
We integrate the above equality over the time interval
and
obtain
We introduce the notations
according to the
relationship
and
write
for some standard normal variable
.
We proceed with evaluation of the quantity
:
where
is the number defined by the relationship
or
We introduce the
notation
and
continue
Summary
The
expectation
evaluates
to


(No drift Black Scholes)

