Geometric Brownian motion

by Nikolai V. Shokhirev

Up: Stochastic Labs
-  Pseudorandom numbers
-  Probability transformation
-  Diffusion, Wiener process
-  Geometric Brownian motion
-  Ornstein-Uhlenbeck process

Geometric Brownian motion

This process is very popular in financial engineering and quantitative analysis.

Stochastic Differential Equation (SDE)

The above verbal definition can be presented in the following compact form

dy = μ y dt + σ y dW_t

By substituting z = ln y it can be reduced to diffusion with modified drift:

dz = d ln y = (μ - ½ σ²) dt + σ dW_t

A formal solution is:

y = y₀ exp[(μ - ½ σ²) t + σ W_t]

Fokker-Planck Equation

The corresponding PDE obviously has the solution:

Performing the transformation to the original variable we have

Using this density function it can be shown that the average value of y exponentially depends on time:

<y> = y₀ exp(μ t)

This result can be easy obtained directly from SDE.

Stochastic simulation program

Info	Geometric Brownian motion

Feel free to download and play (at your own risk). I am adding several new features, please check later.

Downloads

Stochastic simulation program.

Some references

• Geometric Brownian motion by Alan Cairns http://www-maths.mcs.st-andrews.ac.uk/~rac/MT4551/FM6.pdf
• Geometric Brownian motion (From Wikipedia, the free encyclopedia), http://en.wikipedia.org/wiki/Geometric_Brownian_motion .
• Stochastic Processes with Focus in Petroleum Applications http://www.puc-rio.br/marco.ind/stochast.html#log-Brownian
• Monte Carlo Simulation of Stochastic Processes http://www.puc-rio.br/marco.ind/sim_stoc_proc.html#mc-gbm
   

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