Steven R.

Dunbar
Department of Mathematics
203 Avery Hall
University of Nebraska-Lincoln
Lincoln, NE 68588-0130
http://www.math.unl.edu
Voice: 402-472-3731
Fax: 402-472-8466

Stochastic Processes and

Advanced Mathematical Finance

Properties of Geometric Brownian Motion

Rating
Mathematically Mature: may contain mathematics beyond calculus with
proofs.

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Section Starter Question
What is the relative rate of change of a function?
For the function defined by the ordinary differential equation
dx
= rx x(0) = x0
dt
what is the relative rate of growth? What is the function?

Key Concepts
1. Geometric Brownian Motion is the continuous time stochastic process
z0 exp(µt + σW (t)) where W (t) is standard Brownian Motion.

2. The mean of Geometric Brownian Motion is

z0 exp(µt + (1/2)σ 2 t).

3. The variance of Geometric Brownian Motion is

z02 exp(2µt + σ 2 t)(exp(σ 2 t) − 1).

Vocabulary
1. Geometric Brownian Motion is the continuous time stochastic pro-
cess z0 exp(µt + σW (t)) where W (t) is standard Brownian Motion.

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 2. A random variable X is said to have the lognormal distribution (with
parameters µ and σ) if log(X) is normally distributed (log(X) ∼ N (µ, σ 2 )).
The p.d.f. for X is
1
fX (x) = √ exp((−1/2)[(ln(x) − µ)/σ]2 ).
2πσx

Mathematical Ideas
Geometric Brownian Motion
Geometric Brownian Motion is the continuous time stochastic process
X(t) = z0 exp(µt + σW (t)) where W (t) is standard Brownian Motion. Most
economists prefer Geometric Brownian Motion as a simple model for market
prices because it is everywhere positive (with probability 1), in contrast to
Brownian Motion, even Brownian Motion with drift. Furthermore, as we
have seen from the stochastic differential equation for Geometric Brownian
Motion, the relative change is a combination of a deterministic proportional
growth term similar to inflation or interest rate growth plus a normally dis-
tributed random change
dX
= r dt +σ dW .
X
(See Itô’s Formula and Stochastic Calculus.) On a short time scale this is a
sensible economic model.
A random variable X is said to have the lognormal distribution (with
parameters µ and σ) if log(X) is normally distributed (log(X) ∼ N (µ, σ 2 )).
The p.d.f. for X is
1
fX (x) = √ exp((−1/2)[(ln(x) − µ)/σ]2 ).
2πσx
Theorem 1. At fixed time t, Geometric Brownian Motion z0 exp(µt+σW
√ (t))
has a lognormal distribution with parameters (ln(z0 ) + µt) and σ t.

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Proof.

FX (x) = P [X ≤ x]
= P [z0 exp(µt + σW (t)) ≤ x]
= P [µt + σW (t) ≤ ln(x/z0 )]
= P [W (t) ≤ (ln(x/z0 ) − µt)/σ]
h √ √ i
= P W (t)/ t ≤ (ln(x/z0 ) − µt)/(σ t)
Z (ln(x/z0 )−µt)/(σ√t)
1
= √ exp(−y 2 /2) dy
−∞ 2π
Now differentiating with respect to x, we obtain that
1 √
fX (x) = √ √ exp((−1/2)[(ln(x) − ln(z0 ) − µt)/(σ t)]2 ).
2πσx t

Calculation of the Mean

We can calculate the mean of Geometric Brownian Motion by using the m.g.f.
for the normal distribution.

Theorem 2. E [z0 exp(µt + σW (t))] = z0 exp(µt + (1/2)σ 2 t)

Proof.

E [X(t)] = E [z0 exp(µt + σW (t))]

= z0 exp(µt)E [exp(σW (t))]
= z0 exp(µt)E [exp(σW (t)u)] |u=1
= z0 exp(µt) exp(σ 2 tu2 /2)|u=1
= z0 exp(µt + (1/2)σ 2 t)

since σW (t) ∼ N (0, σ 2 t) and E [exp(Y u)] = exp(σ 2 tu2 /2) when Y ∼ N (0, σ 2 t).
See Moment Generating Functions, Theorem 4.

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 1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
0 1 2 3 4
x

Figure 1: The p.d.f. and c.d.f. for a lognormal random variable with m = 1,
s = 1.5.

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Calculation of the Variance
We can calculate the variance of Geometric Brownian Motion by using the
m.g.f. for the normal distribution, together with the common formula

Var [X] = E (X − E [X])2 = E X 2 − (E [X])2

   

and the previously obtained formula for E [X].

Theorem 3. Var [z0 exp(µt + σW (t))] = z02 exp(2µt + σ 2 t)[exp(σ 2 t) − 1]

Proof. First compute:

E X(t)2 = E z02 exp(µt + σW (t))2

   

= z02 E [exp(2µt + 2σW (t))]

= z02 exp(2µt)E [exp(2σW (t))]
= z02 exp(2µt)E [exp(2σW (t)u)] |u=1
= z02 exp(2µt) exp(4σ 2 tu2 /2)|u=1
= z02 exp(2µt + 2σ 2 t)

Therefore,

Var [z0 exp(µt + σW (t))] = z02 exp(2µt + 2σ 2 t) − z02 exp(2µt + σ 2 t)

= z02 exp(2µt + σ 2 t)[exp(σ 2 t) − 1].

Note that this has the consequence that the variance starts at 0 and then
grows exponentially. The variation of Geometric Brownian Motion starts
small, and then increases, so that the motion generally makes larger and
larger swings as time increases.

Stochastic Differential Equation and Parameter Sum-

mary
If a Geometric Brownian Motion is defined by the stochastic differential equa-
tion
dX = rX dt +σX dW X(0) = z0

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then the Geometric Brownian Motion is
X(t) = z0 exp((r − (1/2)σ 2 )t + σW (t)).
At each time the Geometric Brownian Motion √ has lognormal distribution
with parameters (ln(z0 )+rt−(1/2)σ 2 t) and σ t. The mean of the Geometric
Brownian Motion is E [X(t)] = z0 exp(rt). The variance of the Geometric
Brownian Motion is
Var [X(t)] = z02 exp(2rt)[exp(σ 2 t) − 1]
If the primary object is the Geometric Brownian Motion
X(t) = z0 exp(µt + σW (t)).
then by Itô’s formula the SDE satisfied by this stochastic process is
dX = (µ + (1/2)σ 2 )X(t) dt +σX(t) dW X(0) = z0 .
At each time the Geometric Brownian
√ Motion has lognormal distribution
with parameters (ln(z0 )+µt) and σ t. The mean of the Geometric Brownian
Motion is E [X(t)] = z0 exp(µt + (1/2)σ 2 t). The variance of the Geometric
Brownian Motion is
z02 exp(2µt + σ 2 t)[exp(σ 2 t) − 1].

Ruin and Victory Probabilities for Geometric Brownian

Motion
Because of the exponential-logarithmic connection between Geometric Brow-
nian Motion and Brownian Motion, many results for Brownian Motion can
be immediately translated into results for Geometric Brownian Motion. Here
is a result on the probability of victory, now interpreted as the condition of
reaching a certain multiple of the initial value. For A < 1 < B define the
“duration to ruin or victory”, or the “hitting time” as
z0 exp(µt + σW (t)) z0 exp(µt + σW (t))
TA,B = min{t ≥ 0 : = A, = B}
z0 z0
Theorem 4. For a Geometric Brownian Motion with parameters µ and σ,
and A < 1 < B,
2 2
1 − A1−(2µ−σ )/σ
 
z0 exp(µTA,B + σW (TA,B ))
P = B = 1−(2µ−σ2 )/σ2
z0 B − A1−(2µ−σ2 )/σ2

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Quadratic Variation of Geometric Brownian Motion
The quadratic variation of Geometric Brownian Motion may be deduced from
Itô’s formula:
dX = (µ − σ 2 /2)X dt +σX dW
so that
( dX)2 = (µ − σ 2 /2)2 X 2 dt2 + (µ − σ 2 /2)σX 2 dt dW +σ 2 X 2 ( dW )2 .
Guided by the heuristic principle that terms of order ( dt)2 and dt · dW ≈
( dt)3/2 are small and may be ignored, and that ( dW )2 = dt, we obtain:
( dX)2 = σ 2 X 2 dt .

Continuing heuristically, the expected quadratic variation is

Z T  Z T 
2 2 2
E ( dX) = E σ X dt
0 0
Z T
2
  2 
=σ E X dt
0
Z T 
2 2 2
=σ z0 exp(2µt + 2σ t) dt
0
2 2
σ z0
exp((2µ + 2σ 2 )T ) − 1 .


=
2µ + 2σ 2
Note the assumption that the order of the integration and the expectation
can be interchanged.

Sources
This section is adapted from: A First Course in Stochastic Processes, Second
Edition, by S. Karlin and H. Taylor, Academic Press, 1975, page 357; An
Introduction to Stochastic Modeling 3rd Edition, by H. Taylor, and S. Karlin,
Academic Press, 1998, pages 514–516; and Introduction to Probability Models
9th Edition, S. Ross, Academic Press, 2006.

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Algorithms, Scripts, Simulations
Algorithm
Given values for µ, σ and an interval [0, T ], the script creates trials sam-
ple paths of Geometric Brownian Motion, sampled at N equally-spaced values
on [0, T ]. The scripts do this by creating trials Brownian Motion sam-
ple paths sampled at N equally-spaced values on [0, T ] using the definition
of Brownian Motion having normally distributed increments. Adding the
drift term and then exponentiating the sample paths creates trials Ge-
ometric Brownian Motion sample paths sampled at N equally-spaced values
on [0, T ]. Then the scripts use semi-logarithmic least-squares statistical fit-
ting to calculate the relative growth rate of the mean of the sample paths.
The scripts also compute the predicted relative growth rate to compare it
to the calculated relative growth rate. The problems at the end of the sec-
tion explore plotting the sample paths, comparing the sample paths to the
predicted mean with standard deviation bounds, and comparing the mean
quadratic variation of the sample paths to the theoretical quadratic variation
of Geometric Brownian Motion.

Scripts
Geogebra GeoGebra applet for Geometric Brownian Motion

R R script for Geomtric Brownian Motion.

1 mu <- 1
2 sigma <- 0.5
3 T <- 1
4 # length of the interval [0 , T ] in time units
5
6 trials <- 200
7 N <- 200
8 # number of end - points of the grid including T
9 Delta <- T / N
10 # time increment
11
12 t <- t ( seq (0 ,T , length = N +1) * t ( matrix (1 , trials , N +1) )
)
13 # Note use of the R matrix recycling rules , by columns ,
so transposes

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 14 W <- cbind (0 , t ( apply ( sqrt ( Delta ) * matrix ( rnorm ( trials
* N ) , trials , N ) , 1 , cumsum ) ) )
15 # Wiener process , Note the transpose after the apply , (
side effect of
16 # apply is the result matches the length of individual
calls to FUN ,
17 # then the MARGIN dimension / s come next . So it ’ s not so
much
18 # " transposed " as that being a consequence of apply in 2 D
.) Note
19 # use of recycling with cbind to start at 0
20
21 GBM <- exp ( mu * t + sigma * W )
22
23 meanGBM <- colMeans ( GBM )
24
25 meanGBM _ rate <- lm ( log ( meanGBM ) ~ seq (0 ,T , length = N +1) )
26 predicted _ mean _ rate = mu + (1 / 2) * sigma ^2
27

28 cat ( sprintf ( " Observed meanGBM relative rate : % f \ n " ,

coefficients ( meanGBM _ rate ) [2] ) )
29 cat ( sprintf ( " Predicted mean relative rate : % f \ n " ,
predicted _ mean _ rate ) )
30

Octave Octave script for Geometric Brownian Motion

1 mu = 1;
2 sigma = 0.5;
3 T = 1;
4 # length of the interval [0 , T ] in time units
5
6 trials = 200;
7 N = 100;
8 # number of end - points of the grid including T
9 Delta = T / N ;
10 # time increment
11
12 W = zeros ( trials , N +1) ;
13 # initialization of the vector W approximating
14 # Wiener process
15 t = ones ( trials , N +1) .* linspace (0 , T , N +1) ;
16 # Note the use of broadcasting ( Octave name for R
recylcing )

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 17 W (: , 2: N +1) = cumsum ( sqrt ( Delta ) * stdnormal_rnd ( trials
, N ) , 2) ;
18
19 GBM = exp ( mu * t + sigma * W ) ;
20
21 meanGBM = mean ( GBM ) ;
22

23 A = [ transpose ((0: Delta : T ) ) ones ( N +1 , 1) ];

24 meanGBM_rate = A \ transpose ( log ( meanGBM ) )
25 p r ed i c te d _ me a n _r a t e = mu + (1/2) * sigma ^2
26

Perl Perl PDL script for Geometric Brownian Motion

1 $mu = 1;
2 $sigma = 0.5;
3 $T = 1.;
4 # length of the interval [0 , T ] in time units
5
6 $trials = 200;
7 $N = 100;
8 # number of end - points of the grid including T
9 $Delta = $T / $N ;
10 # time increment
11
12 $W = zeros ( $N + 1 , $trials ) ;
13 # initialization of the vector W approximating
14 # Wiener process
15 $t = ones ( $N +1 , $trials ) * zeros ( $N + 1 ) -> xlinvals (
0 , $T ) ;
16 # Note the use of PDL dim 1 threading rule ( PDL name for
R recycling )
17 $W ( 1 : $N , : ) .= cumusumover ( sqrt ( $Delta ) * grandom (
$N , $trials ) ) ;
18
19 $GBM = exp ( $mu * $t + $sigma * $W ) ;
20
21 $meanGBM = sumover ( $GBM - > xchg (0 ,1) ) / $trials ;
22
23 use PDL :: Fit :: Linfit ;
24
25 $fitFuncs = cat ones ( $N + 1) , zeros ( $N + 1 ) -> xlinvals (
0 , $T ) ;

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 26 ( $linfitfunc , $coeffs ) = linfit1d log ( $meanGBM ) ,
$fitFuncs ;
27 print " Observed Mean GBM Rate : " , $coeffs (1) , " \ n " ;
28 print " Predicted Mean GBM Rate : " , $mu + (1./2.) * $sigma
**2 , " \ n " ;
29

SciPy Scientific Python script for Geometric Brownian Motion

1 import scipy
2
3 mu = 1.;
4 sigma = 0.5;
5 T = 1.
6 # length of the interval [0 , T ] in time units
7
8 trials = 200
9 N = 100
10 # number of end - points of the grid including T
11 Delta = T / N ;
12 # time increment
13
14 W = scipy . zeros (( trials , N +1) , dtype = float )
15 # initialization of the vector W approximating
16 # Wiener process
17 t = scipy . ones ( ( trials , N +1) , dtype = float ) * scipy .
linspace (0 , T , N +1)
18 # Note the use of recycling
19 W [: , 1: N +1] = scipy . cumsum ( scipy . sqrt ( Delta ) * scipy . random
. standard_normal ( ( trials , N ) ) , axis = 1 ,)
20

21 GBM = scipy . exp ( mu * t + sigma * W )

22 meanGBM = scipy . mean ( GBM , axis =0)
23 p r e d i c t e d _ m e a n _ G B M _ r a t e = mu + (1./2.) * sigma **2
24
25 meanGBM_rate = scipy . polyfit ( scipy . linspace (0 , T , N +1) ,
scipy . log ( meanGBM ) , 1)
26
27 print " Observed Mean GBM Relative Rate : " , meanGBM_rate
[0];
28 print " Predicted Mean GBM Relative Rate : " ,
predicted_mean_GBM_rate ;

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Problems to Work for Understanding
1. Differentiate
√
Z (ln(x/z0 )−µt)/(σ t)
1
√ exp(−y 2 /2) dy
−∞ 2π
to obtain the p.d.f. of Geometric Brownian Motion.

2. What is the probability that Geometric Brownian Motion with param-

eters µ = −σ 2 /2 and σ (so that the mean is constant) ever rises to
more than twice its original value? In economic terms, if you buy a
stock or index fund whose fluctuations are described by this Geometric
Brownian Motion, what are your chances to double your money?

3. What is the probability that Geometric Brownian Motion with param-

eters µ = 0 and σ ever rises to more than twice its original value? In
economic terms, if you buy a stock or index fund whose fluctuations are
described by this Geometric Brownian Motion, what are your chances
to double your money?

4. Derive the probability of ruin (the probability of Geometric Brownian

Motion hitting A < 1 before hitting B > 1).

5. Modify the scripts to plot several sample paths of Geometric Brownian

Motion all on the same set of axes.

6. Modify the scripts to plot several sample paths of Geometric Brownian

Motion and the mean function of Geometric Brownian Motion and the
mean function plus and minus one standard deviation function, all on
the same set of axes.

7. Modify the scripts to measure the quadratic variation of each of many

sample paths of Geometric Brownian Motion, find the mean quadratic

13
 variation and compare to the theoretical quadratic variation of Geo-
metric Brownian Motion.

Reading Suggestion:
References
[1] S. Karlin and H. Taylor. A Second Course in Stochastic Processes. Aca-
demic Press, 1981.

[2] Sheldon M. Ross. Introduction to Probability Models. Academic Press,

9th edition, 2006.

[3] H. M. Taylor and Samuel Karlin. An Introduction to Stochastic Modeling.

Academic Press, third edition, 1998.

Outside Readings and Links:

1. Graphical Representations of Brownian Motion and Geometric Brow-
nian Motion

2. Wikipedia Geometric Brownian Motion

I check all the information on each page for correctness and typographical
errors. Nevertheless, some errors may occur and I would be grateful if you would

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alert me to such errors. I make every reasonable effort to present current and
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terests and opinions of its author. They do not explicitly represent official positions
or policies of my employer.
Information on this website is subject to change without notice.
Steve Dunbar’s Home Page, http://www.math.unl.edu/~sdunbar1
Email to Steve Dunbar, sdunbar1 at unl dot edu
Last modified: Processed from LATEX source on August 4, 2016

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