We want a rule to "differentiate" expression of the form
,
where
is a differentiable function. If
were also differentiable, then the ordinary chain rule would
give

which
could be written in differentiable notation
as

However,
B(t) is not differentiable, and in particular has nonzero quadratic variation,
so the correct formula has an extra term,
namely,

This
is Ito's formula in differential form. Integrating this, we obtain Ito's
formula in integral
form:

Consider
so that

Let
x
be numbers. Taylor's formula
implies

Fix
and
be a partition of
Using Taylor's formula, we
write

We
let
to
obtain

This
is Ito's formula in integral form for the special
case

Homework 6. Page 251, #1, #3. Due 4/4.

Geometric Brownian motion
is

where
and
are constant. Define

so

Then

According
to Ito's
formula,

Thus,
Geometric Brownian motion in differential form
is

and
Geometric,Brownian motion in integral form
is

In the integral form of Geometric Brownian
motion,

the
Riemann
integral

is
differentiable with
This term has zero quadratic variation. The Ito integral

is
not differentiable. It has quadratic
variation

Thus
the quadratic variation of S is given by the quadratic variation of G. In
differentiable notation, we
write

An investor begins with nonrandom initial wealth
X
and at each time t, holds
shares of stock. Stock is modelled by a geometric Brownian
motion:

can be random, but must be adapted. The investor finances his investing by
borrowing or lending at interest rate r. Let X(t) denote the wealth of the
investor at time
.
Then

Consider an European option which pays
at time T. Let
denote the value of this option at time t if the stock price is
.
In other words, the value of the option at each time
is

The
differential of this value
is

A
hedging portfolio starts with some initial wealth
X
and invests so that the wealth
at each time tracks
We saw above
that

To
ensure that
for all t, we equate coefficients in their differentials. Equating the dB
coefficients, we obtain the
-hedging
rule:

Equating
the dt coefficients, we
obtain

But
we have set
and we are seeking to cause
to agree with
.
Making these substitutions, we
obtain

which
simplies
to

In
conclusion, we should let
be the solution of the Black-Scholes partial differential
equation

satisfying
the terminal
condition

If
an investor starts with
and uses the
hedge

then
he will
have
for all t, and in particular,