Geometric Brownian Motion

Plain Brownian Motion is a bad model for a stock price: it can drift below zero, and it moves in dollars rather than percentages, so a $10 wiggle would mean the same thing for a $20 stock as for a $2,000 one. Geometric Brownian motion (GBM) fixes both problems with one idea — apply the random walk to the logarithm of the price. Because you are compounding percentage returns rather than adding dollar amounts, the price can drift but never turns negative, and a 1% move means 1% whether the stock is cheap or expensive. This is the model underneath Black-Scholes and most of the options world.

The equation and its solution

GBM says the price $S_t$ changes by a steady drift plus a random shock, both proportional to the current price:

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

In words: over a tiny slice of time $dt$, the price grows by $\mu S_t\,dt$ on average (a percentage drift $\mu$) and gets jostled by $\sigma S_t\,dW_t$, a random kick whose size is set by the volatility $\sigma$ and a Brownian increment $dW_t$. Solving it (with Itô's lemma) gives an explicit formula for the price at any future time:

$$S_t = S_0 \exp\!\Big(\big(\mu - \tfrac12\sigma^2\big)t + \sigma W_t\Big).$$

Here $S_0$ is today's price and $W_t$ is a random draw that is normal with mean 0 and variance $t$. Because the exponent is normal, the price itself is lognormal — a right-skewed bell that can't go negative. That single fact drives everything else.

The volatility drag

Look closely at the exponent: the average growth rate is not $\mu$ but $\mu - \tfrac12\sigma^2$. The random shocks quietly subtract $\tfrac12\sigma^2$ from your compound growth. This is the volatility drag (or variance drain): swings hurt compounding because a loss needs a bigger gain to recover from — down 50% then up 50% leaves you at 75%, not back to even. So there are two different growth numbers:

• the average price grows at rate $\mu$: $\;E[S_t] = S_0\,e^{\mu t}$,
• the typical (median) price grows at the slower rate $\mu - \tfrac12\sigma^2$.

Worked example: mean versus median

A stock trades at $100 today ($S_0 = 100$), with expected return $\mu = 8\%$ and volatility $\sigma = 20\%$ per year. Where do we expect it in one year?

The average future price:

$$E[S_1] = 100\,e^{0.08} \approx 100 \times 1.0833 = 108.33.$$

The median future price uses the drag-adjusted rate $\mu - \tfrac12\sigma^2 = 0.08 - \tfrac12(0.20)^2 = 0.08 - 0.02 = 0.06$:

$$\text{median} = 100\,e^{0.06} \approx 100 \times 1.0618 = 106.18.$$

So while the average outcome is $108.33, the middle outcome is only $106.18. Half the time the stock ends below $106.18 — the higher mean is propped up by a thin tail of large winners. Confuse the two and you will systematically overstate how the typical path compounds.

Where it misleads

• Volatility is not constant. GBM assumes a single fixed $\sigma$, but real volatility clusters and jumps around — the very thing the Black-Scholes flaw known as the volatility smile exposes.
• Prices gap; GBM does not. The paths are continuous, so a crash of $-20\%$ overnight has essentially zero probability under GBM, yet markets deliver those. Real returns have fatter tails than lognormal allows.
• Drift is nearly unknowable. The formula treats $\mu$ as known, but expected returns are estimated with enormous error — which is exactly why sizing rules like the Kelly criterion lean on the $\tfrac12\sigma^2$ term and treat $\mu$ with suspicion.

GBM is the workhorse price model: it turns the abstract Wiener process into something that behaves like a tradable asset, and its lognormal payoff is the foundation the whole option-pricing machinery is built on.