---
tag: D48DR46B97
kind: concept
name: Geometric Brownian motion
msc: [60J65, 91G30]
specializes: [9HDQ094ADQ]
sources:
  - url: https://en.wikipedia.org/wiki/Geometric_Brownian_motion
same_as:
  wikidata: Q1503307
  wikipedia: Geometric Brownian motion
---

## Summary

Geometric Brownian motion is a positive process whose logarithm is a Brownian motion with drift.
It solves the stochastic differential equation $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$.
It is the model of stock prices in the Black–Scholes theory of option pricing.

## Explanations

### L1 human

In geometric Brownian motion, the random changes are proportional to the current value, so it grows or shrinks by percentages rather than by fixed amounts.
This is why it never becomes negative, which suits prices of shares.
Over a time $t$, the value is multiplied by a [[log-normal|4W40NHWPQT]] random factor.
For example, with drift $\mu = 0.05$ and volatility $\sigma$ the expected value grows like $e^{0.05 t}$, while the median value grows only like $e^{(0.05 - \sigma^2/2) t}$.