chap4

CHAPTER 4. BROWNIAN MOTION 50

Chapter 4

Brownian motion

The binomial-tree model can be mathematically described as a stochastic pro-

cess.1 A stochastic process is a system which evolves in time while undergoing

chance fluctuations. The time can change discretely or continuously. The variable

can have discrete values or continuous values. The binomial model assumed a

discrete-valued and discrete-time stochastic process.2 In Financial Mathematics,

the stochastic behaviour of share prices is studied under the Markovian approx-

imation. A Markovian process is a process whose future does not depend on its

past history. The prediction of a future event depends only on the present state

for a Markovian process.

The Markov process is related to the weak form of the efficient-market hypoth-

esis3.

1The word stochastic is a jargon for random.2We will see that Black and Scholes assume a continuous-valued and continuous-time stochas-

tic process for the asset movement.3When economists say that the securities market is efficient, they mean that information is

widely and cheaply available to investors and that all relevant and ascertainable information

is already reflected in security prices. The efficient-market hypothesis comes in three different

flavours. The weak form of the hypothesis states that prices efficiently reflect all the information

contained in the past series of stock prices. In this case it is impossible to earn superior returns

simply by looking for patterns in stock prices as price changes are random. The semi-strong

form of the hypothesis states that prices reflect all published information. That means it is


4.1 Brownian motion

The observation that, when suspended in water, small pollen grains are found

to be in a very animated and irregular state of motion, was first systematically

investigated by Robert Brown in 1827, and the observed phenomenon took the

name Brownian motion because of his fundamental pioneering work.

Brownian motion, which is a limiting process of a random walk, is a Markov

process with a continuous state space and a continuous time set. The process

and its many generalisations occupy a central role in an option pricing model.

We can derive the diffusion equation underlying the Brownian motion process.

4.1.1 Einstein’s Brownian motion

Brownian motion was mathematically formulated by Einstein. He found the

following equation to solve the probability u(x, t) of particles being found in x at

t for one-dimensional Brownian motion:

∂u

∂t=

σ2

2

∂2u

∂x2. (4.1)

Here, σ2 is the diffusion coefficient. The formal solution of Eq.(4.1) is a Gaussian

function or a normal function

u(x, t) =1√

2πσ2te−x2/2σ2t. (4.2)

This is a Gaussian function. The expectation value of x is zero and the variance

is determined by the diffusion coefficient:

E[x] = 0 and var[x] = σ2t. (4.3)

impossible to make consistently superior returns just by reading the newspaper, looking at the

company’s annual accounts and so on. The strong form of the hypothesis states that stock

prices effectively impound all available information. It tells us that inside information is hard to

find because in pursuing it you are in competition with thousands, perhaps millions of active,

intelligent and greedy investors.


The expectation value and variance of x are found as

E[x] =

∫ ∞

−∞xu(x)dx

var[x] =

∫ ∞

−∞x2u(x)dx − E[x]2.

This is the definition of the expectation value and the variance

when the probability function u(x) is continuous (See Section 2.6

for the discrete case.)

Gaussian integrals

The Gaussian function is so important in statistics and it is useful to know its

integrals.∫ ∞

0x2ne−px2

dx =(2n − 1)!!

2(2p)n

√

π

p

and∫ ∞

0x2n+1e−px2

dx =n!

2pn+1

where p > 0 and n = 0, 1, 2 · ·· and (2n − 1)!! = (2n − 1)(2n − 3)(2n − 5) · · · 1.Beware the range of integral. Simple cases are

∫ ∞

−∞e−px2

dx =

√

π

p∫ ∞

−∞xe−px2

dx = 0

∫ ∞

−∞x2e−px2

dx =1

2p

√

π

p.

4.1.2 Random walk

We consider one-dimensional discrete random walk which yields Brownian mo-

tion4 under the continuum limit. Let {Xi} be random variables with

P (xi = k) = p and P (xi = −k) = q, (4.4)

4When the mean displacement is not zero, the system is not the Brownian motion by standard

definition but we will conventionally call it a Brownian motion.


where k denotes the size of the ith step, with probability p that the walk is toward

the positive direction and with probability q toward the negative direction. It is

easy to verify that the expected value and the variance are

E[xi] = (p − q)k and var[xi] = 4pqk2 (4.5)

For n = 1, 2, · · ·, let Xn = x1 + · · · + xn. Then Xn denotes the position of the

random walk after n steps on an Xn plane. The stochastic process {Xn, n ≥ 0}is called a random walk process. The expectation value and variance of Xn are

E[Xn] = (p − q)nk and var[Xn] = 4pqk2n. (4.6)

We now extend this idea to the continuum limit. Suppose there are r random

walks per unit time, then λ = 1/r is the time interval between two random walks.

In the continuum limit, we take

r → ∞ (λ → 0) and nλ = t. (4.7)

We denote the mean displacement and variance per unit time by µ and σ2, re-

spectively. Using Eq.(4.6),

µ =E[Xn]

nλ= (p − q)

k

λ(4.8)

σ2 =var[Xn]

nλ= 4pq

k2

λ. (4.9)

Using p + q = 1 and Eq.(4.8), we find that

p =1

2

(

1 + µλ

k

)

and q =1

2

(

1 − µλ

k

)

. (4.10)

With use of Eqs.(4.9) and (4.10), we find

σ2 =k2

λ− µ2λ ≈ k2

λ, (4.11)

where λ small has been assumed.

Let u(x, t) denote the probability that the particle takes the position x at time

t, then

u(x, t) = P (Xn = x) at t = nλ. (4.12)


The probability function satisfies the recurrence relation

u(x, t + λ) = pu(x − k, t) + qu(x + k, t). (4.13)

The Taylor expansion of Eq.(4.13) is

u(x, t) + λ∂u(x, t)

∂t+ O(λ2) = u(x, t) + k(q − p)

∂u(x, t)

∂x+

k2

2

∂2u(x, t)

∂x2+ O(k3)

⇒ ∂u(x, t)

∂t=

[

(q − p)k

λ

]

∂u(x, t)

∂x+

k2

2λ

∂2u(x, t)

∂x2(4.14)

under the assumption λ, k → 0. With use of Eqs.(4.11) in Eq.(4.14), we obtain

the partial differential equation

∂u(x, t)

∂t= −µ

∂u(x, t)

∂x+

σ2

2

∂2u(x, t)

∂x2(4.15)

This is called the forward Kolmogorov equation for the drift rate µ and the diffu-

sion rate σ2. The formal solution of Eq.(4.15) is

u(x, t) =1√

2πσ2texp

[

−(x − µt)2

2σ2t

]

. (4.16)

This is again a Gaussian function of its peak at x = µt and its width σ2t. Note

that the peak and the width are time dependent. They grow as time goes.

From Eq.(4.11) we find that ∆X = σ√

∆t. Without drift, when ∆t goes to 0,

(dX(t))2 = σ2dt (4.17)

thus the generalised Brownian motion with drift is written as

dX(t) = µdt + σdZ (4.18)

where (dZ)2 = dt.

Taylor expansion: In the vicinity of a point x0, the value of a

function f(x) is Taylor expanded as follows:

f(x) = f(x0) + (x − x0)

[

∂f(x)

∂x

]

x=x0

+1

2(x − x0)

2

[

∂2f(x)

∂x2

]

x=x0

+ · · ·


4.2 Geometric Brownian motion

When X(t) denotes the Brownian motion with drift rate µ and variance rate σ2,

the stochastic process defined by

y(t) = ex(t) (4.19)

is called the Geometric Brownian motion. The probability density function for

y(t) is

u(y, t) =1

yσ√

2πtexp

[

−(ln y − µt)2

2σ2t

]

, y > 0 (4.20)

The mean and variance of y(t) are, respectively,

E[y(t)|y(0) = yo] = y0 exp

(

µt +σ2t

2

)

var[y(t)|y(0) = yo] = y20e

2µt(

eσ2t − 1)

. (4.21)

4.3 Normal and lognormal distributions

The density function of a normally distributed random variable x with mean µ

and variance σ2 is given by

u(x) =1√

2πσ2exp

[

−(x − µ)2

2σ2

]

. (4.22)

The sum of n independent normally distributed variables Xi, i = 1, 2, ··, N , is also

normally distributed. Let Y = X1 + X2 + · · +Xn, then the mean and variance

of Y are respectively

E[Y ] = E[X1] + E[X2] + · · · + E[Xn] and

var[Y ] = var[X1] + var[X2] + · · · + var[Xn] (4.23)

If the normal variable has its mean zero and its variance unity, it is said to

be the standard normal random variable whose density n(x) and distribution


functions N(x) are

n(x) =1√2π

e−x2/2 and

N(x) =1√2π

∫ x

−∞e−t2/2dt. (4.24)

The lognormal density function for z = ex is

g(z) =1√

2πσxzexp

[

−(ln z − µx)2

2σ2x

]

(4.25)

where the mean and variance of x are denoted by µx and σx. The truncated

mean of z, defined as E[z; z > a], is

E[z; z > a] =

∫ ∞

azg(z)dz

=1

σexp

[

µx +σ2

x

2

]

N

[

µx − ln a

σx+ σx

]

. (4.26)

4.4 Ito’s lemma

A prominent generalisation of Brownian motion is the class of processes known

as Ito processes. Since publication of the seminal paper of Black and Scholes in

1973, Ito processes have remained in the centre stage of continuous-time finance.

If X follows an Ito process:

dX(t) = a(X, t)dt + b(X, t)dZ(t) (4.27)

where the parameters a and b are functions of the value of the underlying variable,

X, and t and dZ(t) is a Wiener process. Ito’s lemma shows that a function, Y ,

of X and t follows the process

dY =

(

∂Y

∂Xa +

∂Y

∂t+

1

2

∂2Y

∂X2b2

)

dt +∂Y

∂XbdZ. (4.28)

Thus Y also follows an Ito process with the drift rate

∂Y

∂Xa +

∂Y

∂t+

1

2

∂2Y

∂X2b2 (4.29)

and the variance rate(

∂Y

∂X

)2

b2. (4.30)


Proof

By the Taylor expansion of ∆Y , which is a function of X and t, we obtain

∆Y =∂Y

∂X∆X +

∂Y

∂t∆t +

1

2

(

∂2Y

∂X2∆X2 + 2

∂2Y

∂X∂t∆X∆t +

∂2Y

∂t2∆t2

)

+1

3!

(

∆X∂

∂X+ ∆t

∂

∂t

)3

Y + · · ·. (4.31)

The discretised form of Eq.(4.27) is

∆X = a(X, t)∆t + b(X, t)√

∆t. (4.32)

In the limit of ∆t → 0, we realise that ∆X2 ≈ b2(X, t)∆t. Substituting this into

Eq.(4.31) we obtain

dY =∂Y

∂XdX +

∂Y

∂tdt +

1

2

∂2Y

∂x2b2(X, t)dt

=

(

∂Y

∂Xa +

∂Y

∂t+

1

2

∂2Y

∂X2b2

)

dt +∂Y

∂XbdZ. (4.33)

The second line in the right-hand side has been obtained by using Eq.(4.27).

Application to Geometric Brownian motion

Let us assume the following Brownian motion

dM = rdt + σdZ (4.34)

Then the governing equation for the variable S = eM is

dS = S

(

r +σ2

2

)

dt + SσdZ

⇒ dS

S=

(

r +σ2

2

)

dt + σdZ (4.35)

which has been obtained using Ito’s lemma and

∂S

∂M= eM = S ,

∂S

∂t= 0 ,

∂2S

∂M2= eM = S. (4.36)


Stock price

If there is no uncertainty in the stock market, when the expected rate of return

on the stock is µ, the stock price at time T is Steµ(T−t) which is a solution of the

equationdS

S= µdt (4.37)

If the uncertainty of the stock price is described by the geometric Brownian

motion,

dS

S= µdt + σdZ

⇒ dS = µSdt + σSdZ (4.38)

where σ is the market volatility. Comparing Eq.(4.36) with Eq.(4.38) we can see

that the stock price undergoes geometric Brownian motion.

Application to Forward contracts

Let the risk-free interest rate equal to r, the forward price is F = Ser(T−t).

Assume that the share price S follows geometric Brownian motion as shown in

Eq.(4.38) with expected return µ and volatility σ. As

∂F

∂S= er(T−t) ,

∂2F

∂S2= 0 ,

∂F

∂t= −rSer(T−t), (4.39)

using Ito’s lemma (4.28),

dF = [er(T−t)µS − rSer(T−t)]dt + er(T−t)σSdZ

= (µ − r)Fdt + σFdZ. (4.40)

The second line in the right-hand side shows that F also follows geometric Brow-

nian motion.

chap4

Documents