# 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 RendlemanBartter model.
• In mathematical finance, GeometricBrownianMotionProcess is used in BlackScholes 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 .
• 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 all

## Basic Examples(3)

Simulate a geometric Brownian motion process:

Mean and variance functions:

Covariance function:

## Scope(13)

### Basic Uses(8)

Simulate an ensemble of paths:

Simulate with arbitrary precision:

Compare paths for different values of the drift parameter:

Compare paths for different values of the volatility parameter:

Simulate a geometric Brownian motion with different starting points:

Process parameter estimation:

Correlation function:

Absolute correlation function:

### Process Slice Properties(5)

Univariate time slice follows a LogNormalDistribution:

First-order probability density function:

Multi-time slice follows a LogMultinormalDistribution:

Second-order PDF:

Compute the expectation of an expression:

Calculate the probability of an event:

Skewness and kurtosis:

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:

Simulate the process:

Mean and variance functions:

Covariance function:

## Applications(2)

Forecast stock prices:

Strip currency units:

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: