ATLASGeometric Brownian motion (GBM)
Investing glossary · Models and probabilities
The classic model of stock prices in which returns are random, with a steady drift and constant volatility, so prices can never go negative.
Definition
Geometric Brownian motion (GBM) is the textbook model for how a stock price evolves: each instant brings a small random shock whose size is proportional to the current price, plus a steady drift. It is the model behind the Black-Scholes option formula and the starting point of many Monte Carlo simulations.
Under GBM the logarithm of the price follows a normal distribution, so prices are log-normal: they cannot fall below zero and their distribution is skewed to the upside. A consequence is that the median outcome is lower than the mean, and the difference grows with volatility. Real prices have fatter tails and changing volatility, so GBM is a useful simplification, not a literal truth.
Formula
dS = μ·S·dt + σ·S·dW → S_t = S₀ · exp( (μ − σ²/2)·t + σ·W_t )
S = price, μ = drift (expected annual return), σ = volatility, W = a Wiener process (random walk), t = time in years. The term σ²/2 corrects for the effect of compounding volatility.
Worked example
Take Tesla (TSLA) at $407.23, a drift of 7% a year and an annual volatility of 41%. After a year the mean outcome under GBM is $436.76, but the median is only $401.09, because high volatility drags down the typical path.
The more volatile the stock, the further the median falls below the mean: a few big winners pull the average up while most paths end lower.
How ATLAS uses it
ATLAS uses the logic of GBM, a drift and a volatility, as the base of its simulations, then corrects it with fat-tailed distributions fitted to past forecasts.
Related in ATLAS: Scenario simulation →Browse all ATLAS tools →See it on a real stock
The example above uses these companies. Open their pages for the full numbers:
Frequently asked questions
Why is the median below the mean in GBM?
Because returns compound. A loss of 50% needs a gain of 100% to recover, so volatility reduces the typical growth of a path below the average growth.
Is GBM realistic?
Only roughly. Real returns have fat tails and volatility that changes with time, which GBM ignores, so models built on it are usually adjusted.
Related terms
Put the idea to work on your own stocks
Open any company in ATLAS to see its fair value, its odds of rising and its risk numbers side by side.
Try ATLAS →Not investment advice. ATLAS estimates are produced by statistical models from public data; they can be wrong and past performance does not predict future results. Consider your own situation before investing. Disclaimer