geometric Brownian motion and the Black-Scholes model
Plain Brownian motion makes a poor stock model: it can go negative (prices cannot), and a one-dollar wiggle should mean more to a one-dollar stock than to a thousand-dollar stock. The fix is to let the RETURNS, not the prices, be Brownian. Geometric Brownian motion (GBM) is what you get when percentage changes follow Brownian motion: the price grows by a steady percentage drift while shaking by a percentage-sized random jitter. The result stays positive, scales proportionally, and is the standard textbook model of a stock price — and the foundation of the Black-Scholes option-pricing formula that won a Nobel Prize.
GBM is the solution of the SDE dS = mu S dt + sigma S dB, where mu is the expected growth rate and sigma is the volatility. Note both terms are proportional to S itself — that is what makes it 'geometric'. Solving it is a beautiful one-line use of Ito's lemma: apply Ito to log(S). The naive chain rule would give d(log S) = (dS)/S = mu dt + sigma dB, but Ito's extra second-order term subtracts (1/2) sigma^2 dt, giving d(log S) = (mu - sigma^2/2) dt + sigma dB. So log(S) is ordinary Brownian motion with drift, which means S(t) = S(0) exp((mu - sigma^2/2) t + sigma B(t)). The price is the exponential of a Normal, so S(t) is lognormally distributed and always positive. That little -sigma^2/2 'volatility drag', a direct gift of Ito's lemma, is why the median price grows slower than the mean.
Black-Scholes assumes the stock follows GBM and then asks: what is the fair price of an option (a contract to buy the stock at a fixed strike later)? The genius is the hedging argument: you can continuously trade the stock to perfectly cancel an option's risk, building a portfolio that grows at the risk-free rate. That logic, combined with Ito's lemma, produces the Black-Scholes partial differential equation and its famous closed-form price. The honest caveats are large and important: real markets have fat tails, jumps, and volatility that itself moves randomly, none of which GBM captures; volatility sigma is not constant (the 'volatility smile' in real option prices is the market admitting the model is wrong); and the perfect hedge assumes frictionless, continuous trading that no one can actually do. GBM is a brilliant first model, not the truth.
A stock starts at 100 with drift mu = 8 percent and volatility sigma = 20 percent per year. After one year its log-price is Normal with mean log(100) + (0.08 - 0.02) = log(100) + 0.06 and standard deviation 0.20. The expected price is 100 exp(0.08) = 108.3, but the median is only 100 exp(0.06) = 106.2 — the gap is the volatility drag sigma^2/2 = 0.02.
Prices are the exponential of a Brownian motion, so they stay positive and are lognormally distributed.
GBM assumes constant volatility, no jumps, and Normal returns — all false in real markets (which have fat tails and a 'volatility smile'). It is a useful baseline, not reality; the median price grows slower than the mean by the sigma^2/2 drag.