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GEOMETRIC BROWNIAN MOTION
 = S(t) [\mu dt + \sigma dW(t)], )
where
and
are constants and
)
is a Brownian motion. It can be shown that geometric Brownian motion is given by
 = S_0 \exp\{(\mu - \frac{\sigma^2}2) t + \sigma W(t)\}. )
It follows from the formula that)
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.
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Geometric Brownian Motion (GBM) is a stochastic process
satisfying the following stochastic differential equation (SDE):
where
It follows from the formula that
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
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