Geometric Brownian Motion for Dummies

What to remember

• Gains and losses come from different bases: $(1 + r)(1 - r) = 1 - r^2$, so an up-and-down round trip of size $r$ loses $r^2$.
• The typical (median) path grows at $\mu - \sigma^2/2$; the average grows at $\mu$.
• GBM: $S_T = S_0 \exp\big((\mu - \sigma^2/2)T + \sigma W_T\big)$, so $S_T$ is lognormal with mean $S_0 e^{\mu T}$ and median $S_0 e^{(\mu - \sigma^2/2)T}$.
• $P(S_T > E[S_T]) = 1 - \Phi(\sigma\sqrt T / 2)$, which goes to $0$ as $T$ grows: a few extreme paths carry the average.
• The same $-\sigma^2/2$ appears in Black-Scholes $d_2$, leveraged funds and Kelly betting.
• Each daily tick contributes $\ln(1 + x) \approx x - \tfrac12 x^2$, and the squared ticks add up to $\sigma^2 t$, so the naive solution $e^{\mu t + \sigma W_t}$ runs ahead of the true path by $\tfrac12\sigma^2 t$ in the exponent.