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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




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




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