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GeometricBrownianMotionProcess
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GeometricBrownianMotionProcess

✖

represents a geometric Brownian motion process with drift μ, volatility σ, and initial value x0.

Details

Examples

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Basic Examples  (3)Summary of the most common use cases

Simulate a geometric Brownian motion process:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-d1w4w3
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-2cv7lb
Out[2]=2

Mean and variance functions:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-h7sccn
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-p4sqxp
Out[2]=2

Covariance function:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-ckcrvm
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-wgmtgx
Out[2]=2

Scope  (13)Survey of the scope of standard use cases

Basic Uses  (8)

Simulate an ensemble of paths:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-fpxoi1
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-oqxhn9
Out[2]=2

Simulate with arbitrary precision:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-big92j
Out[1]=1

Compare paths for different values of the drift parameter:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-2vtftp
In[2]:=2
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-1hrt14
Out[2]=2

Compare paths for different values of the volatility parameter:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-go5d2j
In[2]:=2
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-1t61wr
Out[2]=2

Simulate a geometric Brownian motion with different starting points:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-0pbe4h
In[2]:=2
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-37l3wb
In[3]:=3
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-sl7ej3
Out[3]=3

Process parameter estimation:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-b4q52
In[2]:=2
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-e93vf3
Out[2]=2

Correlation function:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-gs88g1
Out[1]=1

Absolute correlation function:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-1y2afr
Out[1]=1

Process Slice Properties  (5)

Univariate time slice follows a LogNormalDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-iuh4hp
Out[1]=1

First-order probability density function:

In[2]:=2
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-1ou3ti
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-19dhvn
Out[3]=3

Multi-time slice follows a LogMultinormalDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-m0rwoh
Out[1]=1

Second-order PDF:

In[2]:=2
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-qdgkbp
Out[2]=2

Compute the expectation of an expression:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-edmw6r
Out[1]=1

Calculate the probability of an event:

In[2]:=2
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-miucta
Out[2]=2

Skewness and kurtosis:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-xgr42o
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-bmnu2m
Out[2]=2

Moment:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-84cmmy
Out[1]=1

CentralMoment has no closed form for symbolic order:

In[2]:=2
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-eawm2x
Out[2]=2

FactorialMoment has no closed form for symbolic order:

In[3]:=3
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-o10xs2
Out[3]=3

Cumulant has no closed form for symbolic order:

In[4]:=4
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-fmfg53
Out[4]=4

Generalizations & Extensions  (1)Generalized and extended use cases

Define a transformed GeometricBrownianMotionProcess:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-dpyj6l

Simulate the process:

In[2]:=2
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-bc670
In[3]:=3
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-wjjis
Out[3]=3

Mean and variance functions:

In[4]:=4
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-itzp8c
Out[4]=4

Covariance function:

In[5]:=5
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-fqg18c
Out[5]=5

Applications  (2)Sample problems that can be solved with this function

Forecast stock prices:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-5lz7tf
Out[1]=1

Strip currency units:

In[2]:=2
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-esnm56
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-0yex7h
Out[3]=3

Fit a geometric Brownian process to the values:

In[4]:=4
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-ot0o4y
Out[4]=4

Simulate future paths for the next half-year:

In[5]:=5
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-yidrji
In[6]:=6
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-1uy8ty
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-7ymhs
Out[7]=7

Calculate the mean function of the simulations to find predicted future values:

In[8]:=8
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-s9vd1z
In[9]:=9
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-tufahb
Out[9]=9

Find the trend for index SP500:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-8x63b7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-y4glca
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-x37dze
Out[3]=3

Simulate future paths for the next 100 business days:

In[4]:=4
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-enmj5r
In[5]:=5
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-ko7hkh
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-0kprru
Out[6]=6

Calculate the mean function of the simulations to find predicted future values:

In[7]:=7
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-otoyrq
In[8]:=8
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-mpekny
Out[8]=8

Properties & Relations  (6)Properties of the function, and connections to other functions

GeometricBrownianMotionProcess is not weakly stationary:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-krsxqw
Out[1]=1

Geometric Brownian motion process does not have independent increments:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-4qm7tz
In[2]:=2
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-c8sa4l
Out[2]=2

Compare to the product of expectations:

In[3]:=3
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-oz53le
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-gyuvmt
Out[4]=4

Conditional cumulative probability distribution:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-ih0g5c
Out[1]=1

A geometric Brownian motion process is a special ItoProcess:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-ewmdo
Out[1]=1

As well as StratonovichProcess:

In[2]:=2
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-fx97pi
Out[2]=2

Geometric Brownian motion is a solution to the stochastic differential equation :

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-ek8mx8
In[2]:=2
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-wv0y4
In[3]:=3
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-h8873w
Out[3]=3

Compare with the corresponding smooth solution:

In[4]:=4
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-bw6r6p
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-hym4j8
Out[5]=5

Use WienerProcess directly to simulate GeometricBrownianMotionProcess:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-uis1qj
In[2]:=2
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-4dew5q

Apply a transformation to the random sample:

In[3]:=3
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-8kr7zd

It agrees with the algorithm for simulating corresponding GeometricBrownianMotionProcess:

In[4]:=4
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-mklly2
In[5]:=5
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-mwwpbw
Out[5]=5

Neat Examples  (3)Surprising or curious use cases

Simulate a geometric Brownian motion process in two dimensions:

In[23]:=23
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-pwk6zq
Out[26]=26

Simulate a geometric Brownian motion process in three dimensions:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-ceyk91
In[2]:=2
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-kojc8
Out[2]=2

Simulate paths from a geometric Brownian motion process:

In[1]:=1
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-q7hoj9

Take a slice at 1 and visualize its distribution:

In[2]:=2
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-8s9gfo
In[3]:=3
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-nbi1ug

Plot paths and histogram distribution of the slice distribution at 1:

In[4]:=4
✖
https://wolfram.com/xid/0vvexylhokp9zu0u4i6-kwifgz
Out[4]=4
Wolfram Research (2012), GeometricBrownianMotionProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/GeometricBrownianMotionProcess.html (updated 2017).
✖
Wolfram Research (2012), GeometricBrownianMotionProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/GeometricBrownianMotionProcess.html (updated 2017).

Text

Wolfram Research (2012), GeometricBrownianMotionProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/GeometricBrownianMotionProcess.html (updated 2017).

✖
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.

✖
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

✖
Wolfram Language. (2012). GeometricBrownianMotionProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GeometricBrownianMotionProcess.html

BibTeX

@misc{reference.wolfram_2025_geometricbrownianmotionprocess, author="Wolfram Research", title="{GeometricBrownianMotionProcess}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/GeometricBrownianMotionProcess.html}", note=[Accessed: 26-March-2025 ]}

✖
@misc{reference.wolfram_2025_geometricbrownianmotionprocess, author="Wolfram Research", title="{GeometricBrownianMotionProcess}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/GeometricBrownianMotionProcess.html}", note=[Accessed: 26-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_geometricbrownianmotionprocess, organization={Wolfram Research}, title={GeometricBrownianMotionProcess}, year={2017}, url={https://reference.wolfram.com/language/ref/GeometricBrownianMotionProcess.html}, note=[Accessed: 26-March-2025 ]}

✖
@online{reference.wolfram_2025_geometricbrownianmotionprocess, organization={Wolfram Research}, title={GeometricBrownianMotionProcess}, year={2017}, url={https://reference.wolfram.com/language/ref/GeometricBrownianMotionProcess.html}, note=[Accessed: 26-March-2025 ]}