Chapter 3 Brownian Motion 3.2 Scaled random Walks
Feb 22, 2016
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
3.2.6 Limiting Distribution of the Scaled Random Walk
• The limiting distribution of
• Converges to Normal
(100) 0.25W
(100)
(100)
0.25 0
Var 0.25 0.25
W
W
3.2.6 Limiting Distribution of the Scaled Random Walk
• Given a continuous bounded function g(x)
2(100) 220.25 ( )2
xg W g x e dx
3.2.6 Limiting Distribution of the Scaled Random Walk
• 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
3.2.6 Limiting Distribution of the Scaled Random Walk
• 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
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 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
3.2.6 Limiting Distribution of the Scaled Random Walk
• 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
3.2.6 Limiting Distribution of the Scaled Random Walk
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
3.2.7 Log-Normal Distribution as the Limit of the Binomial Model
• The risk-neutral probability and we assume r=0
21
21
21
21
n
ndu
ruq
n
ndudrp
nn
n~
nn
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• : the sum of the number of heads• : the sum of the number of tails
ntH
ntT
nt ntnt H T
3.2.7 Log-Normal Distribution as the Limit of the Binomial Model
• 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
3.2.7 Log-Normal Distribution as the Limit of the Binomial Model
• 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
3.2.7 Log-Normal Distribution as the Limit of the Binomial Model
• 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
3.2.7 Log-Normal Distribution as the Limit of the Binomial Model
• 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
3.2.7 Log-Normal Distribution as the Limit of the Binomial Model
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
3.2.7 Log-Normal Distribution as the Limit of the Binomial Model
• Then
• Hence
)()(21)0(loglim)(loglim 2
3)(2
nOtWtStS n
nnn
)(21)0(log 2 tWtS
}21)(exp{)0()( 2ttWStS