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

H |

n this topic, we apply the theory developed in Topic 6.7 to an actual hedging example. Recall that initially, the arbitrage-free present value of the futures price is zero, but as subsequent shifts occur in interest rates, this is no longer the case. Here we consider the case of shifts in interest rates that occur *subsequently to the time of settlement of the futures contract*.

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.