Statistical & Financial Consulting by Stanford PhD
GEOMETRIC BROWNIAN MOTION

Geometric Brownian Motion (GBM) is a stochastic process $S(t)$ satisfying the following stochastic differential equation (SDE):

$dS(t) = S(t) [\mu dt + \sigma dW(t)],$

where $\mu$ and $\sigma$ are constants and $W(t)$ is a Brownian motion. It can be shown that geometric Brownian motion is given by
$S(t) = S_0 \exp\{(\mu - \frac{\sigma^2}2) t + \sigma W(t)\}.$

It follows from the formula that $S(t)$ has log-normal distribution. Trajectories of geometric Brownian motion are visualized below.

Geometric Brownian motion is the modeling framework in the groundbreaking Black-Scholes model used in asset pricing.

GEOMETRIC BROWNIAN MOTION REFERENCES

Karatzas, I., & Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus (2nd ed). New York: Springer.

Shreve, S. E. (2004). Stochastic Calculus for Finance II: Continuous-Time Models. New York: Springer.

Oksendal, B. K. (2002). Stochastic Differential Equations: An Introduction with Applications (5th ed). Springer-Verlag Berlin Heidelberg.

Protter, P. E. (2005). Stochastic Integration and Differential Equations (2nd ed). Springer-Verlag Berlin Heidelberg.

Gikhman, I. I., & Skorokhod, A. V. (2007). The Theory of Stochastic Processes III. Springer-Verlag Berlin Heidelberg.

BACK TO THE
STATISTICAL ANALYSES DIRECTORY