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Gurzuf, Crimea, June 2001 1
Option Pricing:The Multi Period Binomial Model
Henrik Jönsson
Mälardalen University
Sweden
Gurzuf, Crimea, June 2001 2
Contents
• European Call Option
• Geometric Brownian Motion
• Black-Scholes Formula
• Multi period Binomial Model
• GBM as a limit
• Black-Scholes Formula as a limit
Gurzuf, Crimea, June 2001 3
European Call Option
• C - Option Price• K - Strike price• T - Expiration day• Exercise only at T• Payoff function, e.g.
400 420 440 460 480 500 520 540 560 580 6000
10
20
30
40
50
60
70
80
90
100
s
g(s)
K=
KsKssg ,0max][)(
Gurzuf, Crimea, June 2001 4
Geometric Brownian Motion
S(y), 0y<t, follows a geometric Brownian motion if
• independent of all prices up to time y
•
)(
)(
yS
ytS
ttNyS
ytS 2,~)(
)(ln
Gurzuf, Crimea, June 2001 5
Black-Scholes Formula
The price at time zero of a European call
option (non-dividend-paying stock):
where
)()()0( tKeSC rt
t
SKtrt
)0(ln22
Gurzuf, Crimea, June 2001 6
The Multi Period Binomial Model
i
i
i
i
i p
pprob
dS
uSS
11
1
0S
0dS
0uS
0udS
0
2Su
0
2Sd
0
2dSu
0
3Su
0
2Sud
0
3Sd
i
S
i=1,2,…
Note:
• u and d the same for all moments i
• d < 1+r < u, where r is the risk-free interest rate
Gurzuf, Crimea, June 2001 7
The Multi Period Binomial Model
• Let
• Let (X1, X2,…, Xn) be the vector describing the outcome after n steps.
• Find the set of probabilities P{X1=x1, X2 =x2,…, Xn =xn}, xi=0,1, i=1,…,n, such that there is no arbitrage opportunity.
1
1
0
1
ii
ii
dSSif
uSSifi
X i=1,2,…
Gurzuf, Crimea, June 2001 8
The Multi Period Binomial Model
• Choose an arbitrary vector (1, 2, …, n-1) • If A={X1= 1, X2= 2, …, Xn-1= n-1} is true
buy one unit of stock and sell it back at moment n
• Probability that the stock is purchased qn-1=P{X1= 1, X2= 2, …, Xn-1= n-1}
• Probability that the stock goes up pn= P{Xn=1| X1= 1, …, Xn-1= n-1}
Gurzuf, Crimea, June 2001 9
The Multi Period Binomial Model
i
i
i
i
i p
pprob
dS
uSS
11
1
}1,1,01{
}1,1,0{
)1,1,0(
32144
3213
XXXXPp
XXXPq
Example: 0S
0uS
0dS
0
2dSu
0
3Su
0
2Su
0
2Sud
0udS
0
3Sd
0
2Sd
i
S
1 2 3 n=4
Gurzuf, Crimea, June 2001 10
The Multi Period Binomial Model
• Expected gain =
• No arbitrage opportunity implies
qn-1[pn(1+r)-1uSn-1+(1- pn) (1+r)-1dSn-1-Sn-1]
du
drpn
1
r = risk-free interest rate
Gurzuf, Crimea, June 2001 11
The Multi Period Binomial Model
• (1, 2, …, n-1) arbitrary vector
• No arbitrage opportunity
X1,…, Xn independent with
P{Xi=1}=p, i=1,…,n
du
drp
1 Risk-free interest rate r the
same for all moments i
Gurzuf, Crimea, June 2001 12
The Multi Period Binomial Model
Limitations:• Two outcomes only • The same increase &
decrease for all time periods
• The same probabilities
Qualities:• Simple mathematics• Arbitrage pricing• Easy to implement
Gurzuf, Crimea, June 2001 13
Geometric Brownian Motion as a Limit
The Binomial process:
rateinterest freerisk period one n
rt1, and
du
dp
ed
eu nrt
n
t
n
t
n 2,..., 1,j ,1
))1(()(
p
pprob
d
u
n
tjS
n
tjS
Gurzuf, Crimea, June 2001 14
0S
0dS
0uS
0udS
0
2Su
0
2Sd
0
2dSu
0
3Su
0
2Sud
0
3Sd
S
in
tn
t2 tnt3
The Binomial Process
Gurzuf, Crimea, June 2001 15
GBM as a limit
Let
and , Y ~ Bin(n,p)
n
jjXY
1
))1(()(0
))1(()(1
n
tjdS
n
tjSif
n
tjuS
n
tjSif
jX
Gurzuf, Crimea, June 2001 16
GBM as a Limit
The stock price after n periods
where
W
ntY
n
t
n
Y
YnY
eS
eeS
dd
uS
duStS
)0(
)0(
)0(
)0()(
2
ntYn
tW 2
Gurzuf, Crimea, June 2001 17
GBM as a Limit
Taylor expansion
gives
n
t
n
ted
n
t
n
teu
n
t
n
t
21
21
2
2
422
11 nt
nt
nrt r
du
dp
Gurzuf, Crimea, June 2001 18
GBM as a limit
Expected value of W Variance of W
tr
rnt
pnt
ntEYn
tEW
nt
nt
)2
(
)42
(2
)2
1(2
2
2
tpnpn
t
VarYn
tVarW
22
2
)1(4
2
EY = np
VarY = np(1-p)
Gurzuf, Crimea, June 2001 19
GBM as a limit
By Central Limit Theorem
nasttrNS
tSW 2
2
,)2
(~)0(
)(ln
ntYn
tweStS W 2,)0()(
Gurzuf, Crimea, June 2001 20
GBM as a limit
The multi period Binomial model becomes geometric Brownian motion when n → ∞, since
• are independent
•
,,...,1,)1(
nj
n
tjS
n
tjS
ttrN
S
tS 22
,)2
(~)0(
)(ln
Gurzuf, Crimea, June 2001 21
B-S Formula as a limit
• Let , Y ~ Bin(n,p)
• The value of the option after n periods = where S(t)= uY dn-Y S(0)
n
iiXY
1
K]-E[S(t))(1 C n-
n
rt
max[S(t)-K,0] = [S(t)-K]+
• No arbitrage
Gurzuf, Crimea, June 2001 22
B-S formula as a limit
The unique non-arbitrage option price
As n → ∞
ntYwKeSEn
rt
Kdd
uSE
n
rtC
ntW
n
n
Yn
2)0(1
)0(1
,)0(e)
2(rt-
2
KeSEC
Xttr
X~N(0,1)
Gurzuf, Crimea, June 2001 23
B-S formula as a limit
where X~N(0,1) and
AX
xttr
Xttr
dxxfKeS
KeSEC
)()0(e
)0(e
)2
(rt-
)2
(rt-
2
2
)0(
ln2
1:
2
AS
Krtt
txx
Gurzuf, Crimea, June 2001 24
B-S formula as a limit
21
rt-
X
)2
(rt-
)2
(rt-
)0(
)(e-(x)dxf)0(e
)()0(e
2
2
KIeIS
dxxfKeS
dxxfKeS
rt
A AX
xttr
AX
xttr
Gurzuf, Crimea, June 2001 25
B-S formula as a limit
)ln( where)(
)}ln(:{
and where
)}ln(:{ where
)0(21
)0(21
221
)0(2122
21
1
2
2
2
2
22
SK
t
SK
t
B
y
SK
tA
xxtt
trt
trttyyB
txydxe
rttxxAdxeI
(·) is the N(0,1) distribution function
Gurzuf, Crimea, June 2001 26
B-S formula as a limit
)ln( where)(
)}ln(:{ where
)0(21
)0(212
21
2
2
2
2
SK
t
SK
tA
x
trtt
rttxxAdxeI
Gurzuf, Crimea, June 2001 27
B-S formula as a limit
nastKeSC rt )()()0(
t
SKtrt
)0(ln22
where