GeometricBrownianMotionProcess
GeometricBrownianMotionProcess[μ,σ,x0]
represents a geometric Brownian motion process with drift μ, volatility σ, and initial value x0.
Details
- GeometricBrownianMotionProcess is also known as exponential Brownian motion and Rendleman–Bartter model.
- In mathematical finance, GeometricBrownianMotionProcess is used in Black–Scholes model for stock price modeling.
- GeometricBrownianMotionProcess is a continuous-time and continuous-state random process.
- The state
of a geometric Brownian motion satisfies an Ito differential equation 
, where 
follows a standard WienerProcess[]. - The state
follows LogNormalDistribution[(μ-
) t+Log[x0],σ
]. - The parameter μ can be any real number, and σ and x0 any positive real numbers.
- GeometricBrownianMotionProcess can be used with such functions as Mean, PDF, Probability, and RandomFunction.
Examples
open allclose allBasic Examples (3)
Scope (13)
Basic Uses (8)
Process Slice Properties (5)
Univariate time slice follows a LogNormalDistribution:
First-order probability density function:
Multi-time slice follows a LogMultinormalDistribution:
Compute the expectation of an expression:
Calculate the probability of an event:
CentralMoment has no closed form for symbolic order:
FactorialMoment has no closed form for symbolic order:
Cumulant has no closed form for symbolic order:
Generalizations & Extensions (1)
Define a transformed GeometricBrownianMotionProcess:
Applications (2)
Fit a geometric Brownian process to the values:
Simulate future paths for the next half-year:
Calculate the mean function of the simulations to find predicted future values:
Find the trend for index SP500:
Simulate future paths for the next 100 business days:
Calculate the mean function of the simulations to find predicted future values:
Properties & Relations (6)
GeometricBrownianMotionProcess is not weakly stationary:
Geometric Brownian motion process does not have independent increments:
Compare to the product of expectations:
Conditional cumulative probability distribution:
A geometric Brownian motion process is a special ItoProcess:
As well as StratonovichProcess:
Geometric Brownian motion is a solution to the stochastic differential equation 
:
Compare with the corresponding smooth solution:
Use WienerProcess directly to simulate GeometricBrownianMotionProcess:
Apply a transformation to the random sample:
It agrees with the algorithm for simulating corresponding GeometricBrownianMotionProcess:
Neat Examples (3)
Simulate a geometric Brownian motion process in two dimensions:
Simulate a geometric Brownian motion process in three dimensions:
Simulate paths from a geometric Brownian motion process:
Take a slice at 1 and visualize its distribution:
Plot paths and histogram distribution of the slice distribution at 1:
Text
Wolfram Research (2012), GeometricBrownianMotionProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/GeometricBrownianMotionProcess.html (updated 2017).
CMS
Wolfram Language. 2012. "GeometricBrownianMotionProcess." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/GeometricBrownianMotionProcess.html.
APA
Wolfram Language. (2012). GeometricBrownianMotionProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GeometricBrownianMotionProcess.html