Option Pricing: The Multi Period Binomial Model
Contents
European Call Option
Geometric Brownian Motion
Black-Scholes Formula
The Multi Period Binomial Model
The Multi Period Binomial Model
The Multi Period Binomial Model
The Multi Period Binomial Model
The Multi Period Binomial Model
The Multi Period Binomial Model
The Multi Period Binomial Model
Geometric Brownian Motion as a Limit
GBM as a limit
GBM as a Limit
GBM as a Limit
GBM as a limit
GBM as a limit
GBM as a limit
B-S Formula as a limit
B-S formula as a limit
B-S formula as a limit
B-S formula as a limit
B-S formula as a limit
B-S formula as a limit
B-S formula as a limit
189.00K
Category: mathematicsmathematics

The binomial model for option pricing

1. Option Pricing: The Multi Period Binomial Model

Henrik Jönsson
Mälardalen University
Sweden
Gurzuf, Crimea, June
1

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
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2

3. European Call Option

C - Option Price
K - Strike price
T - Expiration day
Exercise only at T
Payoff function, e.g.
90
80
70
60
g(s)
100
50
40
30
20
10
0
400
420
440
g (s) [s K ] max 0, s K
460
480 K= 500
s
520
540
560
580
600
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3

4. Geometric Brownian Motion

S(y), 0 y<t, follows a geometric Brownian
motion if
S (t y )
S(y )
independent of all prices up to time y
S (t y )
2
ln
~
N
t
,
t

S( y )
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4

5. Black-Scholes Formula

The price at time zero of a European call
option (non-dividend-paying stock):
C S(0) ( ) Ke ( t )
rt
where
2
rt t ln K S(0)
2
t
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5

6. The Multi Period Binomial Model

S
u 3S 0
u 2S 0
uS 0
S0
2
u dS 0
pi
uS i 1
Si
prob
1 pi
dS i 1
udS 0
i=1,2,…
dS 0
2
ud S 0
Note:
d 2S 0
3
d S0
i
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• u and d the same for all
moments i
• d < 1+r < u, where r is
the risk-free interest rate
6

7. The Multi Period Binomial Model

• Let
1 if S i uS i 1
Xi
0 if S i dS i 1
i=1,2,…
• 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.
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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}
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9. The Multi Period Binomial Model

S
u 3S 0
pi
uS i 1
Si
prob
dS
1 pi
i 1
u 2S 0
uS 0
S0
u 2 dS 0
udS 0
dS 0
Example:
ud 2 S 0
(0,1,1)
q 3 P{X1 0, X 2 1, X 3 1}
d 2S 0
p 4 P{X 4 1 X1 0, X 2 1, X 3 1}
3
d S0
1
2
i
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n=4
9

10. The Multi Period Binomial Model

• Expected gain =
qn-1[pn(1+r)-1uSn-1+(1- pn) (1+r)-1dSn-1-Sn-1]
r = risk-free interest rate
• No arbitrage opportunity implies
1 r d
pn
u d
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10

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
1 r d
p
u d
Risk-free interest rate r the
same for all moments i
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11

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
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13. Geometric Brownian Motion as a Limit

The Binomial process:
p
t
t u
S( j ) S(( j 1) ) prob
,
1 p
n
n d
t
n
rt
1
u e
n d
p
t ,
u
d
n
d e
and
j 1, 2,..., n
rt
one period risk free interest rate
n
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14.

The Binomial Process
S
u 3S 0
u 2S 0
uS 0
u 2 dS 0
S0
udS 0
dS 0
ud 2 S 0
d 2S 0
d 3S 0
t
n
2t
n
3t
n
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t
i
14

15. GBM as a limit

Let
t
t
1 if S( j n ) uS (( j 1) n )
Xj
t
t
0 if S( j ) dS (( j 1) )
n
n
n
and Y X j , Y ~ Bin(n,p)
j 1
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16. GBM as a Limit

The stock price after n periods
S(t ) S(0)u Y d n Y
Y
u
S(0) d n
d
2
S(0)e
S(0)e W
where
W 2
t
Y
n
e
nt
t
Y nt
n
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17. GBM as a Limit

Taylor expansion
t
n
t
2 t
u e
1
n
2n
t
t
2 t
n
d e
1
n
2n
gives
1 rtn d 1 r nt nt
p
u d
2 2
4
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17

18. GBM as a limit

Expected value of W
t
EY nt
n
1
2 nt (p )
2
r nt nt
2 nt (
)
2
4
2
(r )t
2
EW 2
Variance of W
2
VarW
t
2 VarY
n
t
4 2 np (1 p)
n
2t
EY = np
VarY = np(1-p)
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19. GBM as a limit

S(t ) S(0)e W ,
w 2
t
Y nt
n
By Central Limit Theorem
S (t )
2
2
W ln
~ N (r )t, t as n
2
S ( 0)
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20. GBM as a limit

The multi period Binomial model becomes
geometric Brownian motion when n → ∞,
since
t
S j
n
, j 1,..., n,
t
S ( j 1)
n
are independent
2
S (t )
2
ln
~ N (r )t, t
2
S(0)
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20

21. B-S Formula as a limit

n
• Let Y X i , Y ~ Bin(n,p)
i 1
• The value of the option after n periods =
max[S(t)-K,0] = [S(t)-K]+
where S(t)= uY dn-Y S(0)
• No arbitrage
C (1
rt - n
) E[S(t) - K]
n
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22. B-S formula as a limit

The unique non-arbitrage option price
Y
rt
u n
C 1 E S(0) d K
n
d
n
rt
1 E S(0)e W K
n
n
As n → ∞
C e E S(0)e
- rt
2
( r ) t tX
2
w 2
t
n
Y nt
K ,
Gurzuf, Crimea, June
X~N(0,1)
22

23. B-S formula as a limit

C e E S(0)e
- rt
e S(0)e
A
- rt
2
( r ) t tX
2
2
( r ) t tx
2
where X~N(0,1) and A
K
K f X (x)dx
1 2
K
t rt ln
x : x
t 2
S(0)
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24. B-S formula as a limit

e S(0)e
A
- rt
e - rt S(0)e
2
( r ) t tx
2
(r
2
) t tx
2
A
K f X (x)dx
f X (x)dx - e - rt K f X (x)dx
A
S(0)I 1 e rt KI 2
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25. B-S formula as a limit

I1
A
B
1
2
1
2
e
e
2
x2
t tx
2
2
y2
2
dx
dx where A {x : x t ( t rt ln SK( 0 ) )}
2
2
1
where y x t and
B {y : y 1 t ( 2 t rt ln SK( 0 ) t )}
where 1 t (rt 2 t ln SK( 0 ) )
2
( )
2
(·) is the N(0,1) distribution
function
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26. B-S formula as a limit

I2
A
1
2
e
x2
2
dx
where A {x : x 1 t ( 2 t rt ln SK( 0 ) )}
2
( t ) where t (rt t ln SK( 0 ) )
1
Gurzuf, Crimea, June
2
2
26

27. B-S formula as a limit

C S(0) ( ) e rt K ( t ) as n
where
2
rt t ln K S(0)
2
t
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