Statistical & Financial Consulting by Stanford PhD
Home Page
GEOMETRIC BROWNIAN MOTION

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



where µ and σ are constants and W(t) is a Brownian motion. It can be shown that geometric Brownian motion is given by


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


IMPORTANT LINKS ON THIS SITE