On This Page
- What is geometric Brownian motion (GBM)?
- The GBM solution and the Ito correction
- GBM example: mean vs median price after one year
- GBM percentile bands over time
- How to simulate geometric Brownian motion
- GBM drift: risk-neutral pricing vs estimation from data
- Limitations of geometric Brownian motion
- Bottom line
- FAQ
Key Takeaways
- Geometric Brownian motion writes the instantaneous change in price as dS = mu S dt + sigma S dW, so both the expected drift and the random shock scale with the current price level.
- Applying Ito’s lemma to the logarithm produces the closed-form solution S_t = S_0 exp((mu - sigma^2 / 2) t + sigma W_t), and the sigma^2 / 2 term is the Ito correction.
- Because log price is normally distributed, price itself is lognormal, strictly positive, and right-skewed, so the mean S_0 e^(mu t) sits above the median S_0 e^((mu - sigma^2 / 2) t).
- Uncertainty about log price grows with the square root of elapsed time, which is the source of the square-root-of-time scaling in expected moves and option pricing.
- Black-Scholes runs on GBM with the drift set to the risk-free rate, less any continuous dividend yield, which is why its terminal price is lognormal.
- Use the model’s drift, volatility and time inputs to calculate price percentiles, simulate paths and compare option-payoff scenarios.
What is geometric Brownian motion (GBM)?
Geometric Brownian motion (GBM) is a continuous-time model in which an asset price changes in proportion to its current level, with constant drift and volatility. It is written dS = mu × S × dt + sigma × S × dW, so both the drift and the random-shock terms scale with price. Under GBM, log returns are normal and prices are lognormal, and it is the price process behind the Black-Scholes model.
mu is the proportional price drift. sigma is log-return volatility per square root of time. W is standard Brownian motion, whose nonoverlapping increments are independent normal variables.
The GBM solution and the Ito correction
The exact solution is St = S0 × exp((mu − sigma² / 2) × t + sigma × Wt).
Log return is normal with mean (mu − sigma² / 2) × t and variance sigma² × t. Starting from a positive price, terminal price is lognormal and stays positive at finite times.
Mean price is S0 × exp(mu × t). Median price is S0 × exp((mu − sigma² / 2) × t). The mean counts the big winners at full weight; the median is the outcome half the paths beat.
Formula
dS = μ S dt + σ S dW; solution S_t = S_0 exp((μ − σ² / 2) t + σ W_t); mean E[S_t] = S_0 e^(μ t); median = S_0 e^((μ − σ² / 2) t)- S is the asset price and dS is its instantaneous change over the time increment dt, with t measured in years.
- μ is the drift, the expected instantaneous rate of return per year, expressed as a decimal.
- σ is annualized log-return volatility. Over an interval of length dt, the log-return standard deviation is σ × √(dt).
- W is a standard Brownian motion, so dW is a normal increment with mean zero and variance dt, and W_t is normal with mean zero and variance t.
- The σ² / 2 term is the Ito correction. It appears because the logarithm is a concave function of a random variable, and it is what separates the median path from the mean path.
GBM example: mean vs median price after one year
A trader models a $100 stock over one year as geometric Brownian motion with a 12 percent annual drift and 40 percent annualized volatility, then compares the average outcome with the typical one.
- Median terminal price: 100 × exp((0.12 − 0.40² / 2) × 1) = 100 × exp(0.04) = $104.08
- Mean terminal price: 100 × exp(0.12 × 1) = $112.75
- One standard deviation band on log price: 100 × exp(0.04 − 0.40) = $69.77 up to 100 × exp(0.04 + 0.40) = $155.27
- Share of paths finishing below the mean: N((0.12 − 0.04) / 0.40) = N(0.20) = 57.9 percent
At one year, mean price is $112.75 and median price is $104.08, a difference of $8.67. The 12% proportional drift gives 12.75% expected price growth, while the 4% mean log-return rate gives 4.08% growth for the typical path. Of the terminal outcomes, 57.9% finish below the mean, so a minority of large winners pulls the average up.
Start at $100 with mu = 12%, sigma = 40% and t = 1 year. Half the annual variance is 8%, so expected log return is 4%.
Median price is $104.08, a 4.08% gain. Mean price is $112.75, a 12.75% expected gain. Their dollar gap is $8.67.
The 8-percentage-point correction sits between the two log rates, 12% and 4%, and the exponentials turn it into the $8.67 dollar gap. Of the terminal outcomes, 57.9% finish below the mean price, so the mean overstates what a typical holder ends with.
GBM percentile bands over time
At one year, mean log return plus or minus one log standard deviation maps to $69.77 through $155.27. That terminal interval holds 68.27% of the distribution, and it sits lopsided around the $100 start, with more room above than below.
The chart repeats the calculation at 25 times across one year. Each vertical slice is a pointwise interval: 68.27% of outcomes sit inside it on that date, and fewer paths stay inside the shaded band for the entire year.
Log-return standard deviation grows as sigma × √t. The dollar endpoints follow exponentials of it, so the dollar band stretches further above the median than below it.
How to simulate geometric Brownian motion
For a time step dt, draw an independent standard normal Z and calculate Snext = S × exp((mu − sigma² / 2) × dt + sigma × sqrt(dt) × Z).
One draw produces a terminal value for a terminal-only payoff, and repeated steps produce a sampled path. Each step’s log return is an independent draw, so the dollar size of a move scales with the price at that step.
GBM drift: risk-neutral pricing vs estimation from data
Risk-neutral Black-Scholes stock pricing uses mu = r − q for constant interest rate r and continuous dividend yield q. A return scenario uses your own mu.
From equally spaced historical log returns, estimate sigma using their standard deviation and interval length. Add sigma² / 2 to the annualized mean log return to recover the proportional drift estimate.
Limitations of geometric Brownian motion
- Price the skew separately. The model carries one constant volatility, while option prices imply different volatilities across strikes and expirations, so check each strike’s implied volatility before trusting a GBM price.
- Size gap risk outside the model. GBM moves continuously with normal log returns, so its tails are thinner than real returns, and earnings gaps and crashes need their own scenario.
- Stress regime behavior by hand. Independent increments leave momentum, mean reversion and volatility clustering out of the simulated paths.
- Estimate drift from a long history. With constant parameters and known volatility, drift precision grows only with the total time observed, so a longer record beats finer sampling.
- Model default separately. GBM prices stay strictly positive, so a bankruptcy case needs jump-to-zero dynamics or a different process.
Bottom line
Geometric Brownian motion is a standard baseline for prices that compound: proportional moves, lognormal outcomes and log-price uncertainty that grows with the square root of time. Use it for percentiles, simulations and Black-Scholes intuition, and treat constant volatility, continuous paths and independent returns as assumptions to check against the market.
Clear answers
Frequently Asked Questions
Sources and Further Reading
- Stojkoski, Sandev, Basnarkov, Kocarev and Metzler: Generalised Geometric Brownian Motion, Theory and Applications to Option Pricing (Entropy 22(12), 2020)
- Choi, Kwak, Tee and Wang: A Black-Scholes User's Guide to the Bachelier Model (Journal of Futures Markets, 2022)
- Bollerslev and Zhou: Expected Stock Returns and Variance Risk Premia (Federal Reserve FEDS 2007-11)
How entries are sourced, calculated and updated: editorial methodology.