Geometric Brownian Motion — Section 13: Stochastic Processes

Geometric Brownian Motion (GBM) is the canonical model for asset prices:

dS_t = \mu S_t\, dt + \sigma S_t\, dW_t

The drift and volatility are proportional to the current price, so prices stay positive and percentage changes are constant in distribution. By Itô's Lemma, $\log S_t$ follows a Brownian motion with drift:

\log S_t = \log S_0 + \left(\mu - \frac{1}{2}\sigma^2\right) t + \sigma W_t

Hence $S_t$ is lognormally distributed:

S_t \sim \mathrm{Lognormal}\!\left(\log S_0 + (\mu - \sigma^2/2)t, \sigma^2 t\right)

GBM is the engine inside Black-Scholes option pricing. It captures positivity (prices can't go negative), proportional volatility (a $1\%$ move on a high-priced stock and a low-priced stock are equally likely), and analytical tractability.

Limits: GBM has constant volatility, Gaussian log-returns, and no jumps. Real markets have time-varying volatility, fat tails, and discrete jumps — driving more elaborate models like Heston (stochastic volatility) and Merton (jump-diffusion).

Geometric Brownian Motion (GBM) is the canonical model for asset prices:

dS_t = \mu S_t\, dt + \sigma S_t\, dW_t

\log S_t = \log S_0 + \left(\mu - \frac{1}{2}\sigma^2\right) t + \sigma W_t

Hence $S_t$ is lognormally distributed:

S_t \sim \mathrm{Lognormal}\!\left(\log S_0 + (\mu - \sigma^2/2)t, \sigma^2 t\right)