Download - Chapter 3 Brownian Motion

Chapter 3 Brownian Motion

3.2 Scaled random Walks

3.2.1 Symmetric Random Walk• To construct a symmetric random walk, we toss a fair coin (p, the

probability of H on each toss, and q, the probability of T on each toss)

1 if H, =

1 if = T, j

jj

X

3.2.1 Symmetric Random Walk• Define • , k=1,2,…..

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 is called increment of the random walk

1

11

i

i i

i

k

k k jj k

M M X

0 1 mk k k

3.2.2 Increments of the Symmetric Random Walk• Each increment has expected value 0 and variance

1i ik kM M

1i ik k

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

3.2.2 Increments of the Symmetric Random Walk

ii

k

kj

k

kjj

k

kjjkk

kkXVar

XVarMMVar

i

i

i

i

i

i

ii

111

ij1

11

1

1

1)(

)ji , XX( )()(

3.2.3 Martingale Property for the Symmetric Random Walk• Choose nonnegative integers k < l , then

( ) = [( ) ]

= [ ] + [ ]

= [ ] + (( ) )

= [ ] + =

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 for the Symmetric Random Walk• The quadratic variation up to time k is defined 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

2

11

, =k

j jkj

M M M M k

1j jM M

3.2.5 Scaled Symmetric Random Walk• To approximate a Brownian motion• Speed up time of a symmetric random walk• Scale down the step size of a symmetric random walk

• Define the Scaled Symmetric Random Walk

• If nt is not an integer, we define by linear interpolation• is a Brownian motion as

( ) 1( ) = nntW t M

n

n

3.2.5 Scaled Symmetric Random Walk• Consider • n=100 , t=4

3.2.5 Scaled Symmetric Random Walk• 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

3.2.5 Scaled Symmetric Random Walk• Scaled Symmetric Random Walk is Martingale• Let be given and s , t are chosen so that ns and nt are

integers

0 s t ( ) ( )( ) ( )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

)

3.2.5 Scaled Symmetric Random Walk• Quadratic Variation

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 Wn n

X tnn

3.2.6 Limiting Distribution of the Scaled Random Walk

• We fix the time t and consider the set of all possible paths evaluated at that time t• Example• Set t = 0.25 and consider the set of possible values of • We have values:

-2.5,-2.3,…,-0.3,-0.1,0.1,0.3,…2.3,2.5 • The probability of this is

(100)25

10.2510

W M

25

(100) 25! 10.25 0.1 0.155513!12! 2

W


• The limiting distribution of

• Converges to Normal

(100) 0.25W

(100)

(100)

0.25 0

Var 0.25 0.25

W

W


• Given a continuous bounded function g(x)

2(100) 220.25 ( )2

xg W g x e dx


• Theorem 3.2.1 (Central limit)

( )

0. ,

( ) 0

n

Fix t As n the distribution of the scaled

random walk W t evaluated at time t convergesto the normal distribution with mean and variance t

藉由 MGF的唯一性來判斷 r.v.屬於何種分配


• Let f(x) be Normal density function with mean=0, variance=t 2

21( )2

xtf x e

t

2

2

22

22

)( have weNow,

)( general,In tu

x

uu

x

eu

eu


• 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 Mn

u uX Xn n

u X X X i jn

e e e e


• To show that

• Then,

tuntnu

nu

nnneueeu

2

21

)()21

21(lim)(lim

tu21)e

21e

21ln(nt lim(u)ln lim 2n

un

u

nnn


1: key Let xn

20

0 0

0

1 1ln02 2lim ln lim ( L'Hopital's rule)0

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

1lim2 ( ) 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 ue e tu e et

x e ex e e

tu ue ue tu u ue e x ue ue

型

型

2u t

3.2.7 Log-Normal Distribution as the Limit of the Binomial Model

• The Central Limit Theorem, (Theorem3.2.1), can be used to show that the limit of a properly scaled binomial asset-pricing model leads to a stock price with a log-normal distribution• 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 and we assume r=0

21

21

21

21

n

ndu

ruq

n

ndudrp

nn

n～

nn

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=212

nt nt nt

nt nt

nt nt

nt nt

M H Tnt H T

H nt M

T nt M


• We wish to identify the distribution of this random variables as

• Where W(t) is a normal random variable with mean 0 amd variance t

n

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

n as }21)(exp{)0()( 2ttWStS


• We take log for equation

• To show that it converges to distribution of

)1log()(21)1log()(

21)0(log)(log

nMnt

nMntStS ntntn

ttWStS 2

21)()0(log)(log


• Taylor series expansion

• Expansion at 0

• Let log(1+x)=f(x)

n

n

n

axn

afxf )(!

)()(0

)(

)()0("21)0(')0()( 32 xOxfxffxf

)(21)1log( 32 xOxxx


ntn M

ntWNote 1)(: )(

)()(21)0(log

)())(2

()0(log

))(2

)((21

))(2

)((21)0(log)(log

)(23

2

232

232

232

tWnntOtS

nMnO

nntS

nOnn

Mnt

nOnn

MntStS

n

nt

nt

nt


• Then

• Hence

)()(21)0(loglim)(loglim 2

3)(2

nOtWtStS n

nnn

)(21)0(log 2 tWtS

}21)(exp{)0()( 2ttWStS