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

Math 489/Math 889
Stochastic Processes and
Advanced Mathematical Finance
Dunbar, Fall 2009

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Properties of Geometric Brownian Motion

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Rating

Mathematically Mature: may contain mathematics beyond calculus with proofs.

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Section Starter Question

For the ordinary differential equation

\frac{dx}{dt} = r x x (0) = x_{0}

what is the rate of growth of the solution?

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Key Concepts

Geometric Brownian Motion is the continuous time stochastic process $z_{0} exp (μ t + σ W (t))$ where $W (t)$ is standard Brownian Motion.
The mean of Geometric Brownian Motion is $z_{0} exp (μ t + (1 ∕ 2) σ^{2} t) .$
The variance of Geometric Brownian Motion is $z_{0}^{2} exp (2 μ t + σ^{2} t) (exp (σ^{2} t) - 1) .$

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Vocabulary

Geometric Brownian Motion is the continuous time stochastic process $z_{0} exp (μ t + σ W (t))$ where $W (t)$ is standard Brownian Motion.
A random variable $X$ is said to have the lognormal distribution (with parameters $μ$ and $σ$ ) if $log (X)$ is normally distributed ( $log (X) \sim N (μ, σ^{2})$ ). The p.d.f. for $X$ is $f_{X} (x) = \frac{1}{\sqrt{2 π} σ x} exp ((- 1 ∕ 2) {[(ln (x) - μ) ∕ σ]}^{2}) .$

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Mathematical Ideas

Geometric Brownian Motion

Geometric Brownian Motion is the continuous time stochastic process $X (t) = z_{0} exp (μ t + σ W (t))$ where $W (t)$ is standard Brownian Motion. Most economists prefer Geometric Brownian Motion as a 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 distributed random change

\frac{dX}{X} = r dt + σ dW .

See Itô’s Formula and Stochastic Calculus.. On a short time scale this is a sensible economic model.

Theorem 1. At fixed time $t$ , Geometric Brownian Motion $z_{0} exp (μ t + σ W (t))$ has a lognormal distribution with parameters $(ln (z_{0}) + μ t)$ and $σ \sqrt{t}$ .

Proof.

\begin{array}{l} F_{X} (x) & = ℙ [X \leq x] \\ = ℙ [z_{0} exp (μ t + σ W (t)) \leq x] \\ = ℙ [μ t + σ W (t) \leq ln (x ∕ z_{0})] \\ = ℙ [W (t) \leq (ln (x ∕ z_{0}) - μ t) ∕ σ] \\ = ℙ [W (t) ∕ \sqrt{t} \leq (ln (x ∕ z_{0}) - μ t) ∕ (σ \sqrt{t})] \\ = \int_{- \infty}^{(ln (x ∕ z_{0}) - μ t) ∕ (σ \sqrt{t})} \frac{1}{\sqrt{2 π}} exp (- y^{2} ∕ 2) d y \end{array}

Now differentiating with respect to $x$ , we obtain that

f_{X} (x) = \frac{1}{\sqrt{2 π} σ x \sqrt{t}} exp ((- 1 ∕ 2) {[(ln (x) - ln (z_{0}) - μ t) ∕ (σ \sqrt{t})]}^{2}) .

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Figure 1:

The p.d.f. for a lognormal random variable

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. $E [z_{0} exp (μ t + σ W (t))] = z_{0} exp (μ t + (1 ∕ 2) σ^{2} t)$

Proof.

\begin{array}{l} E [X (t)] & = E [z_{0} exp (μ t + σ W (t))] \\ = z_{0} exp (μ t) E [exp (σ W (t))] \\ = z_{0} exp (μ t) E [exp (σ W (t) u)] |_{u = 1} \\ = z_{0} exp (μ t) exp (σ^{2} t u^{2} ∕ 2) |_{u = 1} \\ = z_{0} exp (μ t + (1 ∕ 2) σ^{2} t) \end{array}

since $σ W (t) \sim N (0, σ^{2} t)$ and $E [exp (Y u)] = exp (σ^{2} t u^{2} ∕ 2)$ when $Y \sim N (0, σ^{2} t)$ . See Moment Generating Functions, Theorem 4.. □

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. $Var [z_{0} exp (μ t + σ W (t))] = z_{0}^{2} exp (2 μ t + σ^{2} t) [exp (σ^{2} t) - 1]$

Proof. First compute:

\begin{array}{l} E [X {(t)}^{2}] & = E [z_{0}^{2} exp {(μ t + σ W (t))}^{2}] \\ = z_{0}^{2} E [exp (2 μ t + 2 σ W (t))] \\ = z_{0}^{2} exp (2 μ t) E [exp (2 σ W (t))] \\ = z_{0}^{2} exp (2 μ t) E [exp (2 σ W (t) u)] |_{u = 1} \\ = z_{0}^{2} exp (2 μ t) exp (4 σ^{2} t u^{2} ∕ 2) |_{u = 1} \\ = z_{0}^{2} exp (2 μ t + 2 σ^{2} t) \end{array}

Therefore,

\begin{array}{l} Var [z_{0} exp (μ t + σ W (t))] & = z_{0}^{2} exp (2 μ t + 2 σ^{2} t) - z_{0}^{2} exp (2 μ t + σ^{2} t) \\ = z_{0}^{2} exp (2 μ t + σ^{2} t) [exp (σ^{2} t) - 1] . \end{array}

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

Parameter Summary

If a Geometric Brownian Motion is defined by the stochastic differential equation

dX = r X dt + σ X dW X (0) = z_{0}

then the Geometric Brownian Motion is

X (t) = z_{0} exp ((r - (1 ∕ 2) σ^{2}) t + σ W (t)) .

At each time the Geometric Brownian Motion has lognormal distribution with parameters $(ln (z_{0}) + r t - (1 ∕ 2) σ^{2} t)$ and $σ \sqrt{t}$ . The mean of the Geometric Brownian Motion is $E [X (t)] = z_{0} exp (r t)$ . The variance of the Geometric Brownian Motion is

Var [X (t)] = z_{0}^{2} exp (2 r t) [exp (σ^{2} t) - 1]

If the primary object is the Geometric Brownian Motion

X (t) = z_{0} exp (μ t + σ W (t)) .

then by Itô’s formula the SDE satisfied by this stochastic process is

dX = (μ + (1 ∕ 2) σ^{2}) X (t) dt + σ X (t) dW X (0) = z_{0} .

At each time the Geometric Brownian Motion has lognormal distribution with parameters $(ln (z_{0}) + μ t)$ and $σ \sqrt{t}$ . The mean of the Geometric Brownian Motion is $E [X (t)] = z_{0} exp (μ t + (1 ∕ 2) σ^{2} t)$ . The variance of the Geometric Brownian Motion is

z_{0}^{2} exp (2 μ t + σ^{2} t) [exp (σ^{2} t) - 1] .

Ruin and Victory Probabilities for Geometric Brownian Motion

Because of the exponential-logarithmic connection between Geometric Brownian 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

T_{A, B} = min {t \geq 0 : \frac{z_{0} exp (μ t + σ W (t))}{z_{0}} = A, \frac{z_{0} exp (μ t + σ W (t))}{z_{0}} = B}

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

ℙ [\frac{z_{0} exp (μ t_{A, B} + σ W (T_{A, B}))}{z_{0}} = B] = \frac{1 - A^{1 - (2 μ - σ^{2}) ∕ σ^{2}}}{B^{1 - (2 μ - σ^{2}) ∕ σ^{2}} - A^{1 - (2 μ - σ^{2}) ∕ σ^{2}}}

Quadratic Variation of Geometric Brownian Motion

The quadratic variation of Geometric Brownian Motion may be deduced from Ito’s formula:

dX = (μ - σ^{2} ∕ 2) X dt + σ X dW

so that

{(dX)}^{2} = {(μ - σ^{2} ∕ 2)}^{2} X^{2} {dt}^{2} + (μ - σ^{2} ∕ 2) X^{2} σ dt dW + σ^{2} X^{2} {(dW)}^{2} .

Operating on the principle that terms of order ${(dt)}^{2}$ and $dt \cdot dW$ are small and may be ignored, and that ${(dW)}^{2} = dt$ , we obtain:

{(dX)}^{2} = σ^{2} X^{2} dt .

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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Problems to Work

Problems to Work for Understanding

Differentiate $\int_{- \infty}^{(ln (x ∕ z_{0}) - μ t) ∕ (σ \sqrt{t})} \frac{1}{\sqrt{2 π}} exp (- y^{2} ∕ 2) d y$
to obtain the p.d.f. of Geometric Brownian Motion.
What is the probability that Geometric Brownian Motion with parameters $μ = - σ^{2} ∕ 2$ 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?
What is the probability that Geometric Brownian Motion with parameters $μ = 0$ 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?
Derive the probability of ruin (the probability of Geometric Brownian Motion hitting $A < 1$ before hitting $B > 1$ ).

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Books

Reading Suggestion:

References

[1] S. Karlin and H. Taylor. A Second Course in Stochastic Processes. Academic 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.

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Links

Outside Readings and Links:

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Email to Steve Dunbar, sdunbar1 at unl dot edu

Last modified: Processed from LATEX source on June 29, 2011