Chapter 3 Brownian Motion 報告者：何俊儒. 3.1 Introduction Define Brownian motion ． Provide in section 3.3 Develop its basic properties ． section 3.5-3.7 develop.

Chapter 3

Brownian Motion

報告者：何俊儒

3.1 Introduction

• Define Brownian motion

． Provide in section 3.3

• Develop its basic properties

． section 3.5-3.7 develop properties of

Brownian motion we shall need later

The most important properties of Brownian motion

• It is a martingale

• It accumulates quadratic variation at rate one per unit time

3.2 Scaled Random Walks

• 3.2.1 Symmetric Random Walk

• 3.2.2 Increments of the Symmetric

Random Walk

• 3.2.3 Martingale Property for the

Symmetric Random Walk

• 3.2.4 Quadratic Variation of the Symmetric

Random Walk

3.2 Scaled Random Walks

• 3.2.5 Scaled Symmetric Random Walk

• 3.2.6 Limiting Distribution of the Scaled

Random Walk

• 3.2.7 Log-Normal Distribution as the Limit

of the Binomial Model

3.2.1 Symmetric Random Walk

• To create a Brownian motion, we begin with a symmetric random walk, one path of which is shown in Figure 3.2.1.

Construct a symmetric random walk

• Repeatedly toss a fair coin

． p, the probability of H on each toss

． q = 1 – p, the probability of T on each

toss

• Because the fair coin 1

= = 2

p q

• Denote the successive outcomes of the tosses by

• is the infinite sequence of tosses

• is the outcome of the nth toss

• Let

1 2 3 = ....

n

1 if H, =

1 if = T, j

jj

X

• Define = 0,

• The process , k = 0,1,2,…is a symmetric random walk

• With each toss, it either steps up one unit or down one unit, and each of the two probabilities is equally likely

0M

1

= , 1, 2,...k

k jj

M X k

kM

3.2.2 Increments of the Symmetric Random Walk• A random walk has independent increments

． If we choose nonnegative integers

0 = , the random variables

are independent

• Each of these rvs.

is called an increment of the random walk

0 1 mk k k

1 1 0 2 1 1( ), ( ), , ( )

m mk k k k k k kM M M M M M M

1

11

i

i i

i

k

k k jj k

M M X

• Increments over nonoverlapping time intervals are independent because they depend on different coin tosses

• Each increment has expected value 0 and variance

1i ik kM M

1i ik k

Proof of the

1 1

1

1 1

1 1

1 1

1 1 = 1 ( 1) (0)

2 2

= 0

i i

i i

i i

i i

i i

k k

k k j jj k j k

k k

j k j k

M M X X

1 1

& Vari i i ik k k kM M M M

Proof of the 1 1

& Vari i i ik k k kM M M M

22

22 2

Var( )= ( )-

1 1 = 1 ( 1) - 0

2 2

=1

j j jX X X

1

1

1 1

1

11 1

Var Var ( )

Var 1

i

i i

i

i i

i i

k

k k j i jj k

k k

j i ij k j k

M M X X X i j

X k k

Q

3.2.3 Martingale Property for the Symmetric Random Walk• To see that the symmetric random walk is

a martingale, we choose nonnegative integers k < l and compute

( ) = [( ) ]

= [ ] + [ ]

= [ ] + (( ) )

= [ ] + =

l l k kk k

l k kk k

l k k l k kk

l k k k

M F M M M F

M M F M F

M M F M M M F

M M M M

3.2.4 Quadratic Variation of the Symmetric Random Walk• The quadratic variation up to time k is define

d to be

• Note :

． this is computed path-by-path and

． by taking all the one-step increments

along that path, squaring

these increments, and then summing them

211

, =k

j jkj

M M M M k

(3.2.6)

1j jM M

How to compute the

• Note that is the same as , but the computations of these two quantities are quite different

• is computed by taking an average over all paths, taking their probabilities into account

• If the random walk were not symmetric, this would affect

,k

M M Var( )kM

Var( )kM

Var( )kM

Var( )kM

How to compute the

• is computed along a single path

• Probabilities of up and down steps don’t enter the computation

,k

M M

,k

M M

The difference between computing variance and quadratic variation

• Compute the variance of a random walk only theoretically because it requires an average over all paths

• From tick-by-tick price data, one can compute the quadratic variation along the realized path rather explicitly

3.2.5 Scaled Symmetric Random Walk• To approximate a Brownian motion, we

speed up time and scale down the step size of a symmetric random walk

• We fix a positive integer n and define the scaled symmetric random walk

( ) 1( ) = n

ntW t Mn

(3.2.7)

Note

• nt is an integer

• If nt isn’t an integer, we define

by linear interpolation between its value at the nearest points s and u to the left and right of t for which ns and nu are integers

• Obtain a Brownian motion in the limit

as

( ) ( )nW t

n

• Figure 3.2.2 shows a simulated path of

up to time 4; this was generated by 400 coin tosses with a step up or down of size 1/10 on each coin toss

(100)W

• The scaled random walk has independent increments

• If 0 = are such that each

is an integer, then

are independent

• If are such that ns and nt are integers, then

0 1 mt t t jnt

( ) ( ) ( ) ( ) ( ) ( )1 0 2 1 1( ) - ( ) , ( ) - ( ) , , ( ) - ( )n n n n n n

m mW t W t W t W t W t W t

0 s t

( ) ( ) ( ) ( )( ) - ( ) 0, Var ( ) - ( )n n n nW t W s W t W s t s

• Let be given, and decompose

as

• If s and t are chosen so that ns and nt are integers

0 s t ( ) ( )nW t

( ) ( ) ( ) ( )( ) - ( ) ( )n n n nW t W t W s W s

( ) ( )

( )

( ) - ( )

the -algebra of information available at time s

( ) is measurable

n n

n

W t W s F s

W s F s

Prove the martingale property for scaled random walk

( ) ( )( ) ( )n nW t F s W s

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) (

n n n n

n n n

n n n

n n n n

W t F s W t W s W s F s

W t W s F s W s F s

W t W s F s W s

W t W s W s W s

)

Proof:

An example of the quadratic variation of the scaled random walk

• For the quadratic variation up to a time, say 1.37, is defined to be

(100)W

2137

(100) (100) (100) (100)

1

2137 137

1 1

1, 1.37

100 100

1 1 = 1.37

10 100

j

jj j

j jW W W W

X

2

( ) ( ) ( ) ( )

1

2

1 1

In general, for 0 such that is an integer

1,

1 1 =

ntn n n n

j

nt nt

jj j

t nt

j jW W t W W

n n

X tnn

3.2.6 Limiting Distribution of the Scaled Random Walk• We have fixed a sequence of coin tosses and drawn the path of the

resulting process as time t varies• Another way to think about the scaled

random walk is to fix the time t and consider the set of all possible paths evaluated at that time t

• We can fix t and think about the scaled random walk corresponding to different values of , the sequence of coin tosses

1 2 3 = ....

Example

• Set t = 0.25 and consider the set of possible values of

• This r.v. is generated by 25 coin tosses, and the unscaled random walk can take the value of any odd integer between -25 and 25, the scaled random walk

can take any of the values

-2.5,-2.3,…,-0.3,-0.1,0.1,0.3,…2.3,2.5

(100)25

10.25

10W M

(100) 0.25W

• In order for to take value 0.1, we must get 13H and 12T in the 25 coin tosses

• The probability of this is

• We plot this information in Figure 3.2.3 by drawing a histogram bar centered at 0.1 with area 0.1555

(100) 0.25W

25

(100) 25! 10.25 0.1 0.1555

13!12! 2W

The bar has width 0.2, its height must be 0.1555 / 0.2 = 0.7775

The limiting distribution of

• Superimposed on histogram in Figure 3.2.3 is the normal density with mean = 0 and variance = 0.25

• We see that the distribution of

is nearly normal

(100)

(100)

0.25 0

Var 0.25 0.25

W

W

(100) 0.25W

(100) 0.25W

• Given a continuous bounded function g(x)• Asked to compute• We can obtain a good approximation

• The Central Limit Theorem asserts that the approximation in (3.2.12) is valid

(100) 0.25g W

2(100) 220.25 ( )

2xg W g x e dx

(3.2.12

)

Theorem 3.2.1 (Central limit)

• Outline of proof:

藉由 MGF 的唯一性來判斷 r.v. 屬於何種分配

( )

0. ,

( )

0

n

Fix t As n the distribution of the scaled

random walk W t evaluated at time t converges

to the normal distribution with mean and variance t

• For the normal density f(x) with E(x) = 0, Var(x) = t 2

21( )

2

x

tf x et

2 2

22

2

2

2

10

2 2

,

,

0,

uu

x

t uu u t

x

In general M u e

for mean variance

now t

M u e e

(3.2.13)

If t is such that nt is an integer, then the m.g.f. for is ( )nW t

( )

11

1

1

exp

exp exp

exp

1 1 1 1

2 2 2 2

nuW tn nt

nt nt

j jjj

nt

j i jj

ntu u u untn n n n

j

uu e M

n

u uX X

n n

uX X X i j

n

e e e e

To show that

21

2

1 1lim lim

2 2

ntu u

n nn

n n

u t

x

u e e

M u e

Proof:

2

1 1lim ln lim ln

2 2

1 ln

2

u u

n nn

n n

x

u nt e e

u t M u

1: key Let x

n

20

0 0

0

1 1ln

02 2lim ln lim ( L'Hopital's rule)

0

02 2lim lim ( ) 1 1 2 ( ) 02 ( )2 2

1lim

2 ( ) 2 1 1 2

ux ux

nn x

ux uxux ux

ux uxx xux ux

ux ux

ux ux ux uxx

e eu t

xu u

e e tu e et

x e ex e e

tu ue ue tu u u

e e x ue ue

型

型

2u t

3.2.7 Log-Normal Distribution as the Limit of the Binomial Model• The limit of a properly scaled binomial

asset-pricing model leads to a stock price with a log-normal distribution

• Present this limiting argument here under the assumption that the interest rate r is 0

• Results show that the binomial model is a discrete-time version of geometric Brownian motion model

• Let us build a model for a stock price on the time interval from 0 to t by choosing an integer n and constructing a binomial model for the stock price that takes n steps per unit time

• Assume that n and t are chosen so that nt is an integer

• Up factor to be

• Down factor to be

• is a positive constant

1nun

1ndn

The risk-neutral probability

• See (1.1.8) of Chapter 1 of Volume I

° 1 / 1

22 /

1 / 1

22 /

n

n n

n

n n

r d np

u d n

u r nq

u d n

%

• The stock price at time t is determined by the initial stock price S(0) and the result of first nt coin tosses

• : the sum of the number of heads

• : the sum of the number of tails ntH

ntT

nt ntnt H T

• The random walk is the number of heads minus the number of tails in these nt coin tosses

ntM

1=

21

2

nt nt nt

nt nt

nt nt

nt nt

M H T

nt H T

H nt M

T nt M

求 的極限分配 nS t

1 1

2 2

(0)

(0) 1 1

nt nt

nt nt

H Tn n n

nt M nt M

S t S u d

Sn n

•To identify the distribution of this r.v. as n

(3.2.15)

Theorem 3.2.2

2

, (3.2.15)

1 ( ) (0) exp ( )

2

( ) . .

nAs n the distribution of S t in converges

to the distribution of

S t S W t t

where W t is a normal r v with mean zero and variance t

Proof of theorem 3.2.2

2

To show that the distribution of

ln ( )

1 1ln (0) ln 1 ln 1

2 2

converges to the distribution of

1ln ( ) ln (0) ( )

2

n

nt nt

S t

S nt M nt Mn n

S t S W t t

(3.2.17)

Review of Taylor series expansion

2 3

2 3

2 3

ln(1 ) 0

( ) ln(1 )

1'( )

1

1

'( ) 1

1 ( )

2

x x

Let f x x

f xx

x x x

f x dx x x x dx

x x O x

在 附近展開

322

322

3 322 2

ln(1 )

1ln ln 0 ( )

2 2

1 ( )

2 2

ln 0 ( ) ( )2

n nt

nt

nt

x x xn n

S t S nt M O nnn

nt M O nnn

S nt O n M O nn n

將 和 代入 中

3

2 121

ln 0 ( ) ( ) ( ) ( )2

n nS t O n W t O n W t

By CLT is a normal distribution

and converge to

with r.v.

ln ( )S t

21ln 0 ( )

2S t W t

( )W t

32 2

1

2

1lim ln lim ln 0 ( )

2

( ) ( ) ( )

. ~ (0, ) lim 0 . .

1 ln 0 ( )

2

nn n

n n

n

S t S t O n

W t O n W t

xie x N t a s

n

S t W t