Steven R. DunbarDepartment of Mathematics203 Avery HallUniversity of Nebraska-LincolnLincoln, NE 68588-0130http://www.math.unl.edu

Voice: 402-472-3731Fax: 402-472-8466

Math 489/Math 889

Stochastic Processes and

Advanced Mathematical FinanceDunbar, Fall 2009

Properties of Geometric Brownian Motion

Rating

Mathematically Mature: may contain mathematics beyond calculus withproofs.

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Section Starter Question

For the ordinary differential equation

dx

dt= rx x(0) = x0

what is the rate of growth of the solution?

Key Concepts

1. Geometric Brownian Motion is the continuous time stochastic processz0 exp(t+ W (t)) where W (t) is standard Brownian Motion.

2. The mean of Geometric Brownian Motion is

z0 exp(t+ (1/2)2t).

3. The variance of Geometric Brownian Motion is

z20 exp(2t+ 2t)(exp(2t) 1).

Vocabulary

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

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2. A random variable X is said to have the lognormal distribution (withparameters and ) if log(X) is normally distributed (log(X) N(, 2)).The p.d.f. for X is

fX(x) =1

2pixexp((1/2)[(ln(x) )/]2).

Mathematical Ideas

Geometric Brownian Motion

Geometric Brownian Motion is the continuous time stochastic processX(t) = z0 exp(t+ W (t)) where W (t) is standard Brownian Motion. Mosteconomists prefer Geometric Brownian Motion as a model for market pricesbecause it is everywhere positive (with probability 1), in contrast to BrownianMotion, even Brownian Motion with drift. Furthermore, as we have seenfrom the stochastic differential equation for Geometric Brownian Motion,the relative change is a combination of a deterministic proportional growthterm similar to inflation or interest rate growth plus a normally distributedrandom change

dX

X= r dt+ dW .

See Itos Formula and Stochastic Calculus. On a short time scale this is asensible economic model.

Theorem 1. At fixed time t, Geometric Brownian Motion z0 exp(t+W (t))has a lognormal distribution with parameters (ln(z0) + t) and

t.

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Figure 1: The p.d.f. for a lognormal random variable

Proof.

FX(x) = P [X x]= P [z0 exp(t+ W (t)) x]= P [t+ W (t) ln(x/z0)]= P [W (t) (ln(x/z0) t)/]= P

[W (t)/

t (ln(x/z0) t)/(

t)]

=

(ln(x/z0)t)/(t)

12pi

exp(y2/2) dy

Now differentiating with respect to x, we obtain that

fX(x) =1

2pixt

exp((1/2)[(ln(x) ln(z0) t)/(t)]2).

Calculation of the Mean

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

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Theorem 2. E [z0 exp(t+ W (t))] = z0 exp(t+ (1/2)2t)

Proof.

E [X(t)] = E [z0 exp(t+ W (t))]= z0 exp(t)E [exp(W (t))]= z0 exp(t)E [exp(W (t)u)] |u=1= z0 exp(t) exp(

2tu2/2)|u=1= z0 exp(t+ (1/2)

2t)

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

Calculation of the Variance

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

Var [X] = E[(X E [X])2] = E [X2] (E [X])2

and the previously obtained formula for E [X].Theorem 3. Var [z0 exp(t+ W (t))] = z

20 exp(2t+

2t)[exp(2t) 1]Proof. First compute:

E[X(t)2

]= E

[z20 exp(t+ W (t))

2]

= z20E [exp(2t+ 2W (t))]= z20 exp(2t)E [exp(2W (t))]= z20 exp(2t)E [exp(2W (t)u)] |u=1= z20 exp(2t) exp(4

2tu2/2)|u=1= z20 exp(2t+ 2

2t)

Therefore,

Var [z0 exp(t+ W (t))] = z20 exp(2t+ 2

2t) z20 exp(2t+ 2t)= z20 exp(2t+

2t)[exp(2t) 1].

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Note that this has the consequence that the variance starts at 0 and thenincreases. The variation of Geometric Brownian Motion starts small, andthen increases, so that the motion generally makes larger and larger swingsas time increases.

Parameter Summary

If a Geometric Brownian Motion is defined by the stochastic differential equa-tion

dX = rX dt+X dW X(0) = z0

then the Geometric Brownian Motion is

X(t) = z0 exp((r (1/2)2)t+ W (t)).

At each time the Geometric Brownian Motion has lognormal distributionwith parameters (ln(z0)+rt(1/2)2t) and

t. The mean of the Geometric

Brownian Motion is E [X(t)] = z0 exp(rt). The variance of the GeometricBrownian Motion is

Var [X(t)] = z20 exp(2rt)[exp(2t) 1]

If the primary object is the Geometric Brownian Motion

X(t) = z0 exp(t+ W (t)).

then by Itos formula the SDE satisfied by this stochastic process is

dX = (+ (1/2)2)X(t) dt+X(t) dW X(0) = z0.

At each time the Geometric Brownian Motion has lognormal distributionwith parameters (ln(z0)+t) and

t. The mean of the Geometric Brownian

Motion is E [X(t)] = z0 exp(t + (1/2)2t). The variance of the GeometricBrownian Motion is

z20 exp(2t+ 2t)[exp(2t) 1].

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Ruin and Victory Probabilities for Geometric BrownianMotion

Because of the exponential-logarithmic connection between Geometric Brow-nian Motion and Brownian Motion, many results for Brownian Motion canbe immediately translated into results for Geometric Brownian Motion. Hereis a result on the probability of victory, now interpreted as the condition ofreaching a certain multiple of the initial value. For A < 1 < B define theduration to ruin or victory, or the hitting time as

TA,B = min{t 0 : z0 exp(t+ W (t))z0

= A,z0 exp(t+ W (t))

z0= B}

Theorem 4. For a Geometric Brownian Motion with parameters and ,and A < 1 < B,

P[z0 exp(tA,B + W (TA,B))

z0= B

]=

1 A1(22)/2B1(22)/2 A1(22)/2

Quadratic Variation of Geometric Brownian Motion

The quadratic variation of Geometric Brownian Motion may be deduced fromItos formula:

dX = ( 2/2)X dt+X dWso that

(dX)2 = ( 2/2)2X2dt2 + ( 2/2)X2 dt dW +2X2(dW )2.Operating on the principle that terms of order (dt)2 and dt dW are small

and may be ignored, and that (dW )2 = dt, we obtain:

(dX)2 = 2X2 dt .

Sources

This section is adapted from: A First Course in Stochastic Processes, SecondEdition, by S. Karlin and H. Taylor, Academic Press, 1975, page 357; AnIntroduction to Stochastic Modeling 3rd Edition, by H. Taylor, and S. Karlin,

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Academic Press, 1998, pages 514-516; and Introduction to Probability Models9th Edition, S. Ross, Academic Press, 2006.

Problems to Work for Understanding

1. Differentiate (ln(x/z0)t)/(t)

12pi

exp(y2/2) dy

to obtain the p.d.f. of Geometric Brownian Motion.

2. What is the probability that Geometric Brownian Motion with param-eters = 2/2 and (so that the mean is constant) ever rises tomore than twice its original value? In economic terms, if you buy astock or index fund whose fluctuations are described by this GeometricBrownian Motion, what are your chances to double your money?

3. What is the probability that Geometric Brownian Motion with param-eters = 0 and ever rises to more than twice its original value? Ineconomic terms, if you buy a stock or index fund whose fluctuations aredescribed by this Geometric Brownian Motion, what are your chancesto double your money?

4. Derive the probability of ruin (the probability of Geometric BrownianMotion hitting A < 1 before hitting B > 1).

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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:

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Information on this website is subject to change without notice.Steve Dunbars Home Page, http://www.math.unl.edu/~sdunbar1Email to Steve Dunbar, sdunbar1 at unl dot eduLast modified: Processed from LATEX source on June 29, 2011

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