Appendix 17A
Application of the Binomial
Distribution to Evaluate Call
Options
Appendix 17A: Applications of the Binomial Distribution
to Evaluate Call Options
• In
this appendix, we show how the binomial
distribution is combined with some basic
finance concepts to generate a model for
determining the price of stock options.
• What is an option?
• The simple binomial option pricing model
• The Generalized Binomial Option Pricing
Model
2
Appendix 17A: Applications of the Binomial Distribution
to Evaluate Call Options
Cu  Max(0, uS  X )
(17A.1)
Cd  Max(0, dS  X )
(17A.2)
h(uS )  Cu  h(dS )  Cd
Cu  Cd
h
(u  d ) S
3
(17A.3)
(17A.4)
Appendix 17A: Applications of the Binomial Distribution
to Evaluate Call Options
(1  r )(hS  C )  h(uS )  Cu  h(dS )  Cd
 R  d 
u  R 
C  
Cu  
C d  R
ud  
 u  d 
Rd
p
ud
u  R 
so 1  p  

u  d 
C   pCu  (1  p)Cd  R
4
(17A.5)
(17A.6)
(17A.7)
(17A.8)
Appendix 17A: Applications of the Binomial Distribution
to Evaluate Call Options
Table 17A.1 Possible Option Value at Maturity
Today
Stock (S)
Option (C)
Next Period (Maturity)
uS = $110
$100
Max (0,uS – X)
=
Max (0,110 – 100)
=
Max (0,10)
=
$ 10
Cd =
Max (0,dS – X)
=
Max (0,90 – 100)
=
Max (0, –10)
=
$0
C
dS = $ 90
5
Cu =
Appendix 17A: Applications of the Binomial Distribution to
Evaluate Call Options
• Let
•
X = $100
•
S = $100
•
u = (1.10). so uS = $110
•
d = (.90), so dS = $90
•
R = 1 + r = 1 + 0.07 = 1.07
• Next we calculate the value of p as indicated in Equation
(17A.7):
p
•
•
1.07  .90
 .85
1.10  .90
1 p 
1.10  1.07
 .15
1.10  .90
Solving the binomial valuation equation as indicated in
Equation (17A.8), we get
C = [.85(10) + .15(0)] / 1.07 = $7.94.
Appendix 17A: Applications of the Binomial Distribution
to Evaluate Call Options
CT = Max [0, ST – X]
Cu = [pCuu + (1 – p)Cud] / R
Cd= [pCdu + (1 – p)Cdd] / R

(17A.9)
(17A.10)
(17A.11)

C  p 2 Cuu  2 p(1  p)Cud  (1  p) 2 C dd R 2
(17A.12)
C   p3Cuuu  3p2(1 p)Cuud 3p(1 p)2Cudd  (1 p)3Cddd  R3


(17A.13)
7
Example 17A.1
Figure 17A.1
Price Path of
Underlying
Stock and Value
of Call Option
Source: R.J.Rendelman,
Jr., and B.J.Bartter
(1979), “Two-State
Option Pricing,”
Journal of Finance
34 (December), 1906.
8
Example 17A.1
•
In addition, we can Equation (17A.13) to determine the value of the
call option.
p
R  d (1.16)  1.15

 0.4
u  d 1.175 1.15
u  R 1.175  1.16
1 p 

 0.6
u  d 1.175 1.15
C   p3Cuuu  3 p2 (1 p)Cuud  3 p(1  p)2 Cudd  (1  p)3Cddd  R3


3
2
3
  0.4 52.22  3 0.4 (0.6)  7.3  3 0.4 (0.6)2  0  (0.6)3  0 1.16


3
2
3
  0.4 52.22  3 0.4 (0.6)  7.3 1.16


 3.4880
•
The cumulative binomial density function can be defined as:
B  n, p  
•
n
n!
 k !(n  k )! p n  k (1  p)k
k 0
•
(17A.14)
where n is the number of periods,k is the number of successful trials,
n!  n  n 1 n  2
1
k !  k  k 1 k  2
1 n  k !  n  k n  k 1n  k  2 1
Appendix 17A: Applications of the Binomial Distribution
to Evaluate Call Options
1
C n
R
n
n!
k
nk
k nk
p
(
1

p
)
Max
[
0
,
u
d S  X ] (17A.15)

k  0 k!( n  k )!
C1 = Max [0, (1.1)3(.90)0(100) – 100] = 33.10
C2 = Max [0, (1.1)2(.90) (100) – 100] = 8.90
C3 = Max [0, (1.1) (.90)2(100) – 100] = 0
C4 = Max [0, (1.1)0(.90)3(100) – 100] = 0
10
Appendix 17A: Applications of the Binomial Distribution
to Evaluate Call Options
1  3!
3!
0
3
1
2
C
(.85)
(.15)
X
0

(.85)
(.15)
X0
3 
(1.07)  0!3!
1!2!
3!
3!

2
1
3
0

(.85) (.15) X 8.90 
(.85) (.15) X 33.10 
2!1!
3!0!

1 
3X 2 X1
3X 2 X1

0

0

(.7225)(.15)(8.90)

X
(.61413)(1)(33.10)

1.225 
2 X 1X 1
3 X 2 X 1X 1
1

(.32513 X 8.90)  (.61413 X 33.10)]
1.225
 $18.96

11
Appendix 17A: Applications of the Binomial Distribution
to Evaluate Call Options
k nk
 n
 X  n
n!
n!
K
nk u d
k
nk 
C  S 
p (1  p)
p (1  p) 
  n 
n
R  R k m k!(n  k )!

k m k!(n  K )!
P (1  p)
k
nk
k
nk
u d
Rn
(17A.16)
 p rk (1  p ) nk
X
C  SB1 (n, p , m)  n B2 (n, p, m)
R
n
B1 (n, p , m)   C kn p  k (1  p ) n  k
k m
n
B2 (n, p, m)   C kn p k (1  p ) n  k
k m
12
(17A.17)
Appendix 17A: Applications of the Binomial Distribution
to Evaluate Call Options
16
 Pi
190.61  137.89  . . .  52.20
P

16
16
i 1
 $105.09
12
 (190.61  105.09)  . . .  (52.20  105.09) 
P  

16


2
 $34.39
13
2