**D.3 Ito**'s** Lemma
**

Here we assume that X is the price of some asset, and that price changes are described by the following general diffusion process:

Let
*f(x,t)* be the value at time *t*
of *any* derivative security defined on *x*
(such as a call option). Here we
show that the derivative price, *f*,
also follows a diffusion process:

This
is an important characterization because it describes precisely how the *common* underlying source of uncertainty, *dz*, affects both asset and derivative prices.
Looking ahead, this enables a riskless hedge to be designed between the
asset and its derivative that *eliminates* *dz*, and the
value of the derivative to be derived from this hedge.

An
intuitive proof of the derivative price dynamics can be obtained by taking a
second-order Taylor's series expansion of *f(x,t)*
around a point *(x**0**, y0).
*

Now,
substitute

and

to
get

As
D*t* approaches
dt, all terms involving (D*t*)2
and (D*t*)3/2
go to zero, as do any higher-order terms. The
critical part of the proof then shows that as D*t* goes to
zero, the term b2e2D*t*
becomes non-stochastic and converges to b2*dt*. To see
that the D*t* term does
not go to zero, we can write the discrete time analogue of D*x*
as

.

Therefore, (D*x*)2
retains D*t,* which approaches *dt
*as D*t* goes to zero.

With
this, the formula simplifies to

which
is known as **Ito**'s** Lemma**. It says that if
*f* is a function of *x* and *t*, then *f*
inherits the stochastic properties of *x*,
and the drift rate and volatility have to be adjusted as given by the lemma.
In particular, *df* is also
normally distributed.

For
the stock price, we have

Substituting
for x, a and b in the *df*
equation yields:

This
is the general expression for the change in a derivative's price given the stock
price process.

In
the next topic, we apply Ito's Lemma to derive the __Black-Scholes
Partial Differential Equation__. This
equation expresses the relationship between the “unknown” option valuation
function and some of its partial derivatives.

(C) Copyright 1999, OS
Financial Trading System