Geometric Brownian motion

A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with . It is an important example of stochastic processes satisfying a (SDE); in particular, it is used in mathematical finance to model stock prices in the Black–Scholes model.

A stochastic process S_t is said to follow a GBM if it satisfies the following (SDE):

where ${\displaystyle W_{t}}$ is a Wiener process or Brownian motion, and ${\displaystyle \mu }$ ('the percentage drift') and ${\displaystyle \sigma }$ ('the percentage volatility') are constants.

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