Geometric Brownian Motion: The Useful Baseline Behind Black–Scholes
For educational purposes only; not investment advice.
Direct answer
Section titled “Direct answer”Geometric Brownian motion (GBM) is a continuous-time model in which an asset’s percentage change combines a constant drift and a random Brownian shock:
dS(t) / S(t) = μ dt + σ dW(t).
Its solution is S(T) = S0 exp[(μ − 0.5σ²)T + σW(T)]. Log returns are normal and prices are lognormal, so modeled prices remain positive. GBM is central to the classic Black–Scholes framework because it is analytically tractable—not because real stocks follow it exactly.
Drift, Ito correction, and pricing measure
Section titled “Drift, Ito correction, and pricing measure”Over horizon T, ln[S(T)/S0] has mean (μ − 0.5σ²)T and variance σ²T. The −0.5σ² Ito correction offsets exponential convexity so that under the real-world specification E[S(T)] = S0e^(μT). Expected price growth μ is therefore not the same as mean log-return growth.
For risk-neutral option pricing, μ is replaced by r − q, the risk-free rate less continuous dividend yield. This does not predict that the stock’s real return equals r − q; it imposes the no-arbitrage martingale condition on the discounted tradable asset. Confusing real-world and risk-neutral measures mixes a forecast into a pricing calculation.
GBM assumes continuous paths, independent stationary Gaussian increments, and constant volatility. It omits earnings gaps, heavy tails, volatility clustering, changing skew, trading halts, default mechanics, liquidity, and transaction costs. Smaller time steps cannot repair those structural omissions.
One-year distribution from $100
Section titled “One-year distribution from $100”Let S0 = $100, real-world μ = 8%, σ = 20%, and T = 1. Mean log return is 8% − 20%²/2 = 6%, while its standard deviation is 20%.
- Median terminal price:
$100e^0.06 = $106.18. - Expected terminal price:
$100e^0.08 = $108.33. - With standard-normal shock
Z = −1:$100e^(0.06−0.20) = $86.94. - With
Z = +1:$100e^(0.06+0.20) = $129.69.
The mean exceeds the median because the lognormal price distribution is right-skewed. The Z = ±1 values are scenarios, not bounds. A real market can also jump to prices that a continuous GBM path would never traverse instantaneously.
For exact discrete simulation over Δt, use S(t+Δt)=S(t)exp[(μ−0.5σ²)Δt+σsqrt(Δt)Z]. With daily Δt=1/252, the random one-standard-deviation scale is 20%/sqrt(252)=1.26%. Fix the random seed and report Monte Carlo standard error; more paths reduce sampling error, not model error.
Modeling checklist
Section titled “Modeling checklist”- State whether the task is real-world forecasting, risk-neutral pricing, stress testing, or teaching.
- Record
S0, drift, volatility, dividend, rates, horizon, day count, time step, and data timestamp. - Keep annualized inputs and
Δtin consistent units; do not mix trading-day and calendar-year conventions silently. - Use the exact exponential step for GBM and an auditable random seed; verify simulated log-return mean and variance.
- Report path count, standard error, convergence, and confidence interval for Monte Carlo outputs.
- Compare GBM with jump, volatility-regime, heavy-tail, and liquidity stress scenarios.
- For barriers or path dependence, test observation frequency and Brownian-bridge corrections.
- Do not estimate short-horizon direction from
μ; drift is small relative to return noise and hard to estimate. - Treat a good numerical fit as conditional on the model, not proof that its assumptions describe future markets.
Common misconceptions
Section titled “Common misconceptions”- “GBM predicts stock prices.” It specifies a conditional distribution under chosen parameters.
- “Log-return drift is
μ.” It isμ − 0.5σ²under this notation. - “Risk-neutral probabilities are real-world forecasts.” They serve no-arbitrage valuation and include risk pricing.
- “More simulation steps make the market model realistic.” They reduce discretization error but do not add jumps or stochastic volatility.
- “Lognormal means downside is limited.” Price stays above zero in finite model time but can approach it arbitrarily closely.
- “Constant volatility is harmless.” It conflicts with observed term structure, skew, clustering, and crisis behavior.