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Concept# Geometric Brownian motion

Summary

A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. It is an important example of stochastic processes satisfying a stochastic differential equation (SDE); in particular, it is used in mathematical finance to model stock prices in the Black–Scholes model.
A stochastic process St is said to follow a GBM if it satisfies the following stochastic differential equation (SDE):
where is a Wiener process or Brownian motion, and ('the percentage drift') and ('the percentage volatility') are constants.
The former parameter is used to model deterministic trends, while the latter parameter models unpredictable events occurring during the motion.
For an arbitrary initial value S0 the above SDE has the analytic solution (under Itô's interpretation):
The derivation requires the use of Itô calculus. Applying Itô's formula leads to
where is the quadratic variation of the SDE.
When , converges to 0 faster than ,
since . So the above infinitesimal can be simplified by
Plugging the value of in the above equation and simplifying we obtain
Taking the exponential and multiplying both sides by gives the solution claimed above.
The process for , satisfying the SDE
or more generally the process solving the SDE
where and are real constants and for an initial condition , is called an Arithmetic Brownian Motion (ABM). This was the model postulated by Louis Bachelier in 1900 for stock prices, in the first published attempt to model Brownian motion, known today as Bachelier model. As was shown above, the ABM SDE can be obtained through the logarithm of a GBM via Itô's formula. Similarly, a GBM can be obtained by exponentiation of an ABM through Itô's formula.

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