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.