**D.4 Derivation
of the PDE
**

We
start with a more general argument. Let
*f(S,t*) denote the value of any derivative security (e.g., a call
option) at time *t* when the stock price
is *S*. Note that, by assumption,
the derivative price does not depend on the path of *S*, but only on the current value of *S*.

By
Ito's lemma,

and
recall that for the stock price

Now,
consider the riskless hedge portfolio in Chapter 2, topic 2.4.

-k
derivative security

1
share of stock.

The
number of derivative securities, k, is

.

The
value of this portfolio, *V*, is given
by

and
the change in the value is

Substituting
the *df* equation and the *dS* equation into this portfolio and rearranging, we get

The
important fact about this equation is that *dV*
is independent of the stochastic term *z*.
This means that all the risk of the derivative can be hedged by holding

units
of the derivative.

Since
*dV* is instantaneously deterministic, the portfolio return must equal
the instantaneous risk-free return. In
other words,

Substituting
for *dV* and *V*, we get

or

This
is called the Black-Scholes (partial) differential equation.
The solution to this equation, subject to the appropriate boundary
condition for *f*, determines the value
of the derivative security. For a
European call option, the boundary condition is

*f*(*S,T*)*
= max*{*0,S-X*}*.
*

The
solution of the differential equation for the call option is given by the Black-Scholes
formula:

where

We
can also apply Ito’s Lemma to derive the implied distribution of stock prices
from a geometric Brownian motion model of stock prices.
The next topic __Distribution of Stock
Prices__ provides a technical overview of this issue.

(C) Copyright 1999, OS
Financial Trading System