**D.6 Derivation
of Stock Price Distribution
**

Let
the assumed process for stock prices be described by:

.

The
solution to this stochastic differential equation is given by

which
implies that

Since
*dz* is normally distributed with mean 0 and variance 1, we get

The
solution for *ST*
can be verified by applying Ito's lemma to log(*S*).

If
*f(S,t*) = *log(S),* then

Since *dS = **mSdt + sSdz*,
Ito's lemma yields

so

Thus,
log(S) is also a diffusion process, with drift
(m-s2/2) and volatility
s.

Integrating
from 0 to T yields

This
means that the change in log(S) between time 0 and time T, log(ST)-log(S),
is given by

so

This
completes our characterization of the distributional properties implied from the
geometric Brownian motion model for stock prices.