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  j  H,
Xj = 
1 if  j = T,
3.2.1 Symmetric Random Walk
• Define 𝑀0 = 0
• 𝑀𝑘 =
𝑘
𝑗=1 𝑋𝑗
k=1,2,…..
,
3.2.2 Increments of the Symmetric Random Walk
• A random walk has independent increments
．If we choose nonnegative integers
0 = k  k    k , the random variables are independent
0
1
m
• Each M ki1  M ki 
walk
ki 1

j  ki 1
X j is called increment of the random
3.2.2 Increments of the Symmetric Random Walk
• Each increment M k  M khas expected value 0 and variance ki 1  ki
i 1
i

 M ki1  M ki

 ki1
 ki1
   X j     X j 
 j  ki 1  j  ki 1
ki 1
ki 1
1
1


=   1   ( 1)    (0)
2
 j  ki 1
j  ki 1  2
=0
3.2.2 Increments of the Symmetric Random Walk
Var ( M ki1  M ki )  Var (

ki 1
X
j  ki 1
j
ki 1
Var ( X
j  ki 1
(X j  X i , i  j)
)
j
)
ki 1
1  k
j  ki 1
i 1
 ki
3.2.3 Martingale Property for the Symmetric
Random Walk
• Choose nonnegative integers k < l , then
(M l F k ) = [( M l  M k )  M k F k ]
= [ M l  M k F k ] +[M k F k ]
= [M l  M k F k ] + M k
= [M l  M k ] + M k = M k
(( M l  M k )  Fk )
3.2.4 Quadratic Variation for the Symmetric
Random Walk
• The quadratic variation up to time k is defined to be
 M , M k =  M j  M j 1 
k
j 1
• Note :
．this is computed path-by-path and
．by taking all the one-step increments
M j  M j 1 along that path, squaring
these increments, and then summing them
2
k
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
W
(n)
1
(t ) =
M nt
n
• If nt is not an integer, we define 𝑊
•𝑊
𝑛
𝑛
𝑡 by linear interpolation
𝑡 is a Brownian motion as 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 = t0  t1    tm are such that each nt j
is an integer, then
(n)
(n)
(n)
( n)
( n)
( n)
W
(
t
)
W
(
t
)
,
W
(
t
)
W
(
t
)
,

,
W
(
t
)
W
(tm1 ) 
 1
 m
0  
2
1 
are independent
• If 0  s  t are such that ns and nt are integers, then
 W ( n ) (t ) - W ( n ) (s)   0, Var W ( n ) (t ) - W ( n ) (s)   t  s
3.2.5 Scaled Symmetric Random Walk
• Scaled Symmetric Random Walk is Martingale  W ( n ) (t ) F  s    W ( n ) (s )
• Let 0  s  t be given and s , t are chosen so that ns and nt are
integers


  W   (t )  W   ( s)  F  s     W   ( s) F  s  


  W   (t )  W   ( s)  F  s    W   ( s)


n
n
n
n



 W (t ) F  s    W (t )  W (s)  W (s) F  s 


n
n
n
n
n
n
n
n 
n
n

  W (t )  W ( s)   W ( s)  W ( s )
3.2.5 Scaled Symmetric Random Walk
• Quadratic Variation
In general, for t  0 such that nt is an integer
W
 (n)  j 
(n)  j  1 
  t    W    W 

n
 n 
j 1 
nt
(n)
,W
(n)
2
nt
1
 1

= 
Xj  t
j 1  n
j 1 n

nt
2
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
1
• Set t = 0.25 and consider the set of possible values of W (100)  0.25  M 25
10
• We have values:
-2.5,-2.3,…,-0.3,-0.1,0.1,0.3,…2.3,2.5
• The probability of this is
25
25!  1 
 W (100)  0.25  0.1 
   0.1555
13!12!  2 
3.2.6 Limiting Distribution of the Scaled Random Walk
• The limiting distribution of
W (100)  0.25
 W (100)  0.25   0
Var W (100)  0.25   0.25
• Converges to Normal
3.2.6 Limiting Distribution of the Scaled Random Walk
• Given a continuous bounded function g(x)
  g W
(100)
 0.25  
2
2



g ( x )e
2 x 2
dx
3.2.6 Limiting Distribution of the Scaled Random Walk
• Theorem 3.2.1 (Central limit)
Fix t  0. As n  , the distribution of the scaled
random walk W ( n ) (t ) evaluated at time t converges
to the normal distribution with mean 0 and variance t
藉由MGF的唯一性來判斷r.v.屬於何種分配
3.2.6 Limiting Distribution of the Scaled Random Walk
• Let f(x) be Normal density function with mean=0, variance=t
1
f ( x) 
e
2 t
x2

2t
In general,  x (u )  e
u 
 2u 2
Now, we have  x (u )  e
2
u 2t 2
2
3.2.6 Limiting Distribution of the Scaled Random Walk
• If t is such that nt is an integer, then the m.g.f. for W  n (t ) is

n  u    e
n
uW   ( t )

u


  exp
M nt 
n



 u nt

 nt

u
  exp 
X j      exp
Xj

n

 j 1

 n j 1
 
u


  exp
Xj
n
j 1 

nt
1
  e

j 1  2
nt
u
n
1
 e
2
u

n

X i  X j i  j 
 1
   e
 2
u
n
1
 e
2
u

n



nt
3.2.6 Limiting Distribution of the Scaled Random Walk
• To show that
1
lim n (u)  lim( e
n
n 2
u
n
1
 e
2

u
n
) nt   (u)  e
1 2
u t
2
• Then,
1
lim lnn (u)  lim nt ln( e
n 
n 
2
u
n
1
 e
2

u
n
1 2
) u t
2
3.2.6 Limiting Distribution of the Scaled Random Walk
1
key : Let x 
n
 1 ux 1 ux 
ln  e  e 
0
2
2


lim ln n  u   lim t
( 型  L'Hopital's rule)
2
n 
x 0
x
0
u ux u ux
e  e
ux
 ux
tu
e

e
0
2
2
 lim t
 lim ux ux
( 型)
x 0
x 0 x (e  e
1 ux 1 ux
2
)
0
2 x( e  e )
2
2
tu
ueux  ueux
tu u  u 1 2
 lim ux ux
 
 ut
ux
ux
2 x0 e  e  x(ue  ue ) 2 1  1 2
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
un  1 
• Down factor to be

n

dn  1 
n
•  is a positive constant
3.2.7 Log-Normal Distribution as the Limit of the
Binomial Model
• The risk-neutral probability and we assume r=0

1  r  dn
1
n
p


2
un  d n
2
n
～

un  1  r
1
n
q


2
un  d n
2
n
～
3.2.7 Log-Normal Distribution as the Limit of the
Binomial Model
• The stock price at time t is determined by the initial stock price S(0)
and the result of first nt coin tosses
• H nt: the sum of the number of heads
• Tnt : the sum of the number of tails
 nt  H nt  Tnt
3.2.7 Log-Normal Distribution as the Limit of the
Binomial Model
• The random walk M nt is the number of heads minus the number of
tails in these nt coin tosses
M nt  H nt  Tnt
nt
 H nt  Tnt
1

 H nt = 2  nt  M nt 

 T  1  nt  M 
nt
 nt 2
3.2.7 Log-Normal Distribution as the Limit of the
Binomial Model
• We wish to identify the distribution of this random variables as
S n  t   S (0)u d
H nt
n
n 
Tnt
n
 

 S (0) 1 

n

1
 nt  M nt 
2
 

1 

n

1
 nt  M nt 
2
1 2
S (t )  S (0) exp{W (t )   t} as n  
2
• Where W(t) is a normal random variable with mean 0 amd variance t
3.2.7 Log-Normal Distribution as the Limit of the
Binomial Model
• We take log for equation
1

1

log S n (t )  log S (0)  (nt  M nt ) log(1 
)  (nt  M nt ) log(1 
)
2
n 2
n
• To show that it converges to distribution of
1 2
log S (t )  log S (0)  W (t )   t
2
3.2.7 Log-Normal Distribution as the Limit of the
Binomial Model
• Taylor series expansion

f ( x)  
f
n 0
(n)
(a)
n
( x  a)
n!
• Expansion at 0
1
2
3
f ( x)  f (0)  f ' (0) x  f " (0) x  O( x )
2
• Let log(1+x)=f(x)
1 2
3
log(1  x)  x  x  O( x )
2
3.2.7 Log-Normal Distribution as the Limit of the
Binomial Model
1
 
log S (t )  log S (0)  (nt  M nt )(

 O(n ))
2
n 2n
2

3
2
1
 
 (nt  M nt )(

 O (n ))
2
n 2n
2
 log S (0)  nt(
Note : W
(n)
1
(t ) 
M nt
n

2
2n

3
2
 O(n ))  M nt (

3
2


n
1 2
(n)
 log S (0)   t  ntO(n )  W (t )
2
)
3
2
3.2.7 Log-Normal Distribution as the Limit of the
Binomial Model
• Then
3

2
1 2
( n)
lim log S n (t )  lim log S (0)   t  W (t )  O(n )
n 
n 
2
1 2
 log S (0)   t  W (t )
2
• Hence
1 2
S (t )  S (0) exp{W (t )   t}
2