The Log-Normal Distribution of Geometric Brownian Motion

Students often struggle with the

-\frac{1}{2}\sigma^2

term. It arises from Ito’s Lemma because

dS_t

is not a standard derivative; the second-order term

(dW_t)^2 \approx dt

cannot be ignored, pulling the distribution toward zero compared to a naive exponentiation of the drift.

Institutional Reference: Advanced Stochastic Processes