e are evaluating the
expectation
where the
are numbers,
are deterministic functions and
is a standard Brownian motion. We use the notation and results of
(
No drift Black Scholes
). Note
that
hence,
We introduce the
notation
and
assume
at a particular time moment
.
Then
and
We introduce the
notation
It is conventional to introduce the
quantities
Summary
The
expectation
evaluates
to
The expressions in the above summary are not very convenient for calculations.
Hence, we perform another transformation. With the
notation
the expressions for
take the
form


(Black Scholes formula)

One notable property of the formula (
Black
Scholes formula
) is revealed by the following
calculation
In
addition,
Notation
We will use the following
notation


(BlackScholesUndiscountedCall)

