Two Alternative Binomial Option Pricing Model Approaches to
Derive Black-Scholes Option Pricing Model*
CHENG-FEW LEE
Department of Finance and Economics, Rutgers Business School
Rutgers University, New Brunswick
New Jersey, U.S.
CARL S. LIN
Department of Economics
Rutger University, New Brunswick
New Jersey, U.S.
Abstract
In this chapter, we review two famous models on binomial option pricing, Rendleman and
Barter (RB, 1979) and Cox, Ross, and Rubinstein (CRR, 1979). We show that the limiting
results of the two models both lead to the celebrated Black-Scholes formula. From our
detailed derivations, CRR is easy to follow if one has the advanced level knowledge in
probability theory but the assumptions on the model parameters make its applications limited.
On the other hand, RB model is intuitive and does not require higher level knowledge in
probability theory. Nevertheless, the derivations of RB model are more complicated and
tedious. For readers who are interested in the binomial option pricing model, they can
compare the two different approaches and find the best one which fits their interests and is
easier to follow.
1. Introduction
The main purpose of this chapter is to review two famous binomial option pricing
model: Rendleman and Barter (RB, 1979) and Cox, Ross, and Rubinstein (CRR,
1979). First, we will give an alternative detailed derivation of the two models and
show that the limiting results of the two models both lead to the celebrated
*
Section 3 of this chapter is essentially drawing from the paper by Lee et al.(2004).
1
Black-Scholes formula. Then we will make comparisons of the two different
approaches and analyze the advantages of each approach.
Hence, this chapter can help to understand the statistical aspects of option pricing
models for Economics and Finance professions. Also, it gives important financial and
economic intuitions for readers in statistics professions. Therefore, by showing two
alternative binomial option pricing models approaches to derive the Black-Scholes
model, this chapter is useful for understanding the relationship between the two
important optional pricing models and the Black-Scholes formula.
2. The Two-State Option Pricing Model of Rendleman and Bartter
In Rendleman and Bartter (1979), a stock price can either advance or decline during
the next period. Let H T and H T represent the returns per dollar invested in the
stock if the price rises (the + state) or falls (the - state), respectively, from time T-1 to
time T. And VT and VT the corresponding end-of-period values of the option.
Let R be the riskless interest rate, Rendleman and Bartter (1979) show that the
price of the option can be represented as a recursive form
PT 1 
V (1  R  HT )  VT ( HT  1  R)
( HT  HT )(1  R)
2
that can be applied at any time T-1 to determine the price of the option as a function of
its value at time T.
2.1 The Discrete Time Model
From the above equation, the value of a call option at maturing date T-1 is given by
WT 1 
WT (1  R  H  )  WT ( H   (1  R))
( H   H  )(1  R)
(2.1)
WT1 (1  R  H  )  WT1 ( H   (1  R))
( H   H  )(1  R)
(2.2)
Similarly,
WT 2 
Substituting (2.1) into (2.2) can get,
WT 2 
(WT (1  R  H  )  WT ( H   (1  R)))(1  R  H  )
( H   H  )2 (1  R)2

(WT (1  R  H  )  WT ( H   (1  R)))( H   (1  R))
( H   H  )2 (1  R)2
(2.3)
Noting that WT  WT , so (2.3) can be simplified as:
WT 2 
(WT (1  R  H  )2  2WT ( H   (1  R)))(1  R  H  )  WT ( H   (1  R)) 2
( H   H  )2 (1  R)2
(2.4)
We can use this recursive form to get W0 :
T 
Since after T periods, there are   ways that a sequence of (T) pluses can occur,
0
3
T 
T 
  ways that (T-1) pluses can occur,   ways that (T-2) pluses can occur, and so
1
2
on….
Hence, by Binomial Theorem W0 can be represented as:
W0 
T 
[  WT...... (1  R  H  )T ( H   (1  R ))0
0
T 
   WT...... (1  R  H  )T 1 ( H   (1  R ))1
1
T 
   WT..... (1  R  H  )T  2 ( H   (1  R )) 2
2
 T  ....

(1  R  H  )1 ( H   (1  R ))T 1
 WT
 T  1
T 
   WT...... (1  R  H  )0 ( H   (1  R ))T ]
T 
(2.5)
[( H   H  )(1  R )]T
Next to determine the value of the option at maturity. Suppose that stock increases i
T i
times and declines (T  i ) times, then the price of the stock will be S0 H  H 
i
on
the expiration date. So the option will be exercised if
T i
S0 H  H 
i
X
The maturity value of the option will be
WT  S0 H
i
4
T

H

i
X
(2.6)
Let a denote the minimum integer value of i
in (2.6) for which the inequality is
satisfied.
l n X
(
) T lHn (
S
0
a  1  INT 
 l nH   l H
n

 )



(2.7)
where INT [] is the integer operator.
i.e., taking natural logarithm of RHS of (2.6),
ln S0  i ln H   (T  i ) ln H   ln X
 i ln H   i ln H   ln(
X
)  T ln H 
S0
X
)  T ln H 
S0
i
ln H   ln H 
ln(
Hence, the maturing value of the option is given by
T

WT  S0 H H

i
WT  0
if
i
i
if i
X
a
(2.8)
a
Substituting (2.8) into (2.5), then the generalized option pricing equation for the
discrete time is
T
W0 
T 
  i (S H
i a
 
i
0
T
H   X ) (1 R  H 
i
( H   H  )T (1 R
2.2 The Continuous Time Model
For (2.9), we can write is as:
5
)H (   (R1
i
T
)
T i
))
(2.9)
T
T 
T 
i
T i
 i

T i
 i

T i
(
S
H
H
)(1

R

H
)
(
H

(1

R
))



  0
 X (1  R  H ) ( H  (1  R))
i
i
i a
i a  
W0   

 T
( H  H ) (1  R )T
T
T
T 
S0   [ H  (1  R  H  )]i [ H  ( H   1  R )]T i
ia  i 

( H   H  )T (1  R)T
T
T 
X   (1  R  H  )i ( H   (1  R))T i
ia  i 

( H   H  )T (1  R)T
i
 T   (1  R  H  ) H    ( H   1  R ) H  
 S0    


  


i  a  i   (1  R )( H  H )   ( H  H )(1  R ) 
T
X

(1  R)T
 T   (1  R  H  ) 

 

 
ia  i   ( H  H ) 
T
i
 ( H   1  R) 
 (H   H  ) 


T i
T i
(2.10)
Since
(1  R  H  ) H 
( H   1  R) H 
(1  R  H  ) ( H   1  R)
+
,

1

1 ,
(1  R)( H   H  ) ( H   H  )(1  R)
(H   H  )
(H   H  )
therefore, can interpret it as “pseudo probability”.
Let  
(1  R  H  ) H 
(1  R  H  )
and
, we can restate (2.10) as:


(1  R)( H   H  )
(H   H  )
W0  S0 B(a, T ,  ) 
X
B ( a, T ,  )
(1  R)T
(2.11)
where B (a, T , ()) is the cumulative binomial probability function, the number of
successes will fall between a and T after T trials.
As T   , W0
S0 N ( Z1 , Z1) 
X
N ( Z 2 , Z 2 )
(1  R)T
(2.12)
where N ( Z , Z  ) is the probability of a normally distributed random variable with zero
mean and variance 1 taking values between a lower limit Z and a upper limit Z  .
6
And by the property of binomial pdf,
Z1 
a  T
T  T
, Z1 
T  (1   )
T  (1   )
Z2 
a  T
T  T
, Z 2 
T  (1   )
T  (1   )
Thus, W0  S0 (lim Z1 , lim Z1) 
T 
T 
X
N (lim Z 2 , lim Z 2 )
T 
T 
lim(1  R)T
(2.13)
T 
Let 1  R  e
r
T
, then lim(1  R)T  er . And since lim Z1  lim Z 2   , the
T 
T 
T 
remaining things to be determined are lim Z1 and lim Z 2 .
T 
T 
From equation (10) and (11) of the text in the Rendleman and Bartter (1979),

 (
T
T

 (
T
T
H e
H e
)
)
(1 )


(1 )
substituting H  and H  into (2.7), so
Z1 
a  T

T  (1   )
 X

 ln( )     T
S0
1
1  INT 



T (1   )



  T



T  (1   )
In the limit, the term 1  INT [] will be simplify to [] . So,
7
ln(
Z1
X
)
S0

T (1   )
1

T  (1   )
X
)  T 
S0
1


 (1   )

T (1   )
 (1   )
ln(
ln(

X
)
S0
 (1   )
 (1   )
 T


 T

1
1
T  (1   )
T (1   )
1
T

T  (1   )
T  (1   )

T
1 
 T  (1   )
T  (1   )
T (1   )
X

)
T
T (1   )
S0
T
1




 (1   )
T  (1   )
T  (1   )

 (1   )
ln(
X
)
S0
T (   )

 (1   )
 (1   )

 (1   )
ln(
Substituting H  , H  and 1  R  e
(2.14)
r
T
into  ,
8
(1  R  H  ) H 

(1  R )( H   H  )
 
   
1
 r
(
)
(
)
T 1
T
T
T

T
e  e
e



1
 
 
r   
(
)
(
)
T
T 1
T

T
T
e e
e



 
 r
(
T
T
e  e


r  
(
)
T
e e T

 (
e



T
)

)
 (
e

e

(
T
T
)
T
T
)

1




e

1
e

 (
)
 (
)
T 1
T
T
1
)
1
)
 r
 eT
1


1
1
 (
e

T
)
1
 (
e

 (
e

T

(
(



T
)

1
T
)

1
1







   r  ( T )( 1   ) 

1
e T T





1



  ( T )

(
)

 e T 1 
 e
 

Now expanding in Taylor’s series in T,
9

1
 o( )
1
T

 1

1  1
 
1
 T (

)
(

)   o( )

1
2T 
1 
T

1

1
r     2(
)   2  O( )
2
1
T

 1

1  1
 
1
 T (

)
(

)   o( )

1
2T 
1 
T

 T
1

r     2(
)  2
1
2
1



T
1

1  2 1

1
(

) (
)(

)  o( )

1
2 T

1
T


1
1
 o( )
T
 1

1  1
 
1

) (
)(

)   o( )
(

1
2 T

1 
T

where,
 
H  1   
T
T

1 
  
3!  T
T
 1

T


T
1  1   
  
  2!  T
T
1 

 
2
3
1 
 
 
1


 2  1   1 

  o 
2T     T 
and
 
H  1   
T
T

1 
  
3!  T
T
 1

T


T
  1  

   
1    2!  T
T 1



2
3
 
 
1   

1

2     1 

  o 
2T  1     T 
1
1
where o( ) denotes a function tending to zero more rapidly than
.(when we
T
T
10
expanding in Taylor’s series in T, the rest of the terms tending to zero more rapidly
than
1
1
so regard them as a function o( ) .) Hence,
T
T

lim  
T 

1
1



1

1      (1   )
1 
1
 (1   )
 2

and,
11
lim T (   )
T 
1
 lim T [ 
T 
T
1

r     2(
)  2
2
1
1

1  2 1

1
(

) (
)(

)  o( )

1
2 T

1
T

1
1
 o( )]
T
 1

1  1
 
1

) (
)(

)   o( )
(

1
2 T

1 
T

1

r     2(
)  2
2
1


 lim[ T  
T 
1

1  2 1

1
(

) (
)(

)  o( )

1
2 T

1
T


T

1
1
 T o( )]
T
 1

1  1
 
1

) (
)(

)   o( )
(

1
2 T

1 
T


1


 T  (
 lim
T 
(

1

1  2 1

1 

) (
)(

)  o( )   T
1
2 T

1
T 
1



1  2 1

1
) (
)(

)  o( )
1
2 T

1
T
1

r     2(
)  2
1
2
1

 T o( )
2
T
1

1  1

1
(

) (
)(

)  o( )

1
2 T

1
T
1 2 1

1

 (

)  (r     2 (
)  2)
2

1


2
1



1

(

)

1
1 2 (1   ) 2   2
1

 (
)  r     2(
)  2
2
 (1   )
2
1

1

 (1   )
1 2 1  2
1

 (
)  r     2(
)  2
1
2
1
2
1

 (1   )
1 2 1 1 2 
1

 (
)  (
)  r     2(
)  2
1
2
1
2
1
2
1

 (1   )
12
After canceling terms,
1
  (1   )(r     2 )
2
lim T (   ) 

T 
Now substituting lim  for  and lim T (   ) for
T 
T 
ln(
lim Z1 
T 
ln(

X
)
S0
 (1   )

 (1   )

T (   ) into (2.14),
1
2
 (1   )(r     2 )
  (1   )
X
1
)r  2
S0
2

Similarly,
ln(
lim Z 2 
X
1
)r  2
S0
2

T 
Since N ( Z , )  N (,  Z ) , let D1   lim Z1 , D2   lim Z 2 , the continuous time
T 
T 
version of the two-state model is obtained:
w0  S0 N (, D1 )  Xe  r N (, D2 )
ln(
D1 
X
1
)r 2
S0
2
D2  D1  

The above equation is identical to the Black-Scholes model.
13
3. The Binomial Option Pricing Model of Cox, Ross and Rubinstein
In this section we will concentrate on the limiting behavior of the binomial option
pricing model proposed by Cox, Ross and Rubinstein (CRR, 1979).
3.1 The Binomial Option Pricing Formula of CRR
Let S be the current stock price, K the option exercise price, R  1 the riskless
rate. It is assumed that the stock follows a binomial process, from one period to the
next it can only go up by a factor of u with probability p or go down by a factor
of d with probability (1  p ) . After n periods to maturity, CRR showed that the
option price C is:
C
1
Rn
n
n!
 k!(n  k )! p
k
(1  p ) n  k Max[0, u k d n  k S  K ] .
(3.1)
k 0
An alternative expression for C, which is easier to evaluate, is
n!
u k d nk
K n
n!
p k (1  p) n k
]  n [
p k (1  p) n k ]
R
R k m k!(n  k )!
k  m k!( n  k )!
K
 SB(m; n, p )  n B(m; n, p).
R
n
C  S[ 
(3.2)
n
where B(m; n, p )   n C k p k (1  p) n  k and m is the minimum number of upward
k m
stock movements necessary for the option to terminate in the money, i.e., m is the
minimum value of k in (3.1) such that u m d n -m S - X  0.
14
3.2 Limiting Case
We now show that the binomial option pricing formula as given in Equation (3.2) will
converge to the celebrated Black-Scholes option pricing model. The Black-Scholes
formula is
C  S N (d1 )  e  rt X N (d1   t )
(3.3)
where
d1 
S
)
XR t   t
2
 t
log(
(3.4)
 2 = the variance of stock rate of return
t = the fixed length of calendar time to expiration date, such that h 
t
.
n
We wish to show that Equation (3.2) will coincide with Equation (3.3) when n   .
In order to show the limiting result that the binomial option pricing formula
converges to the continuous version of Black-Scholes option pricing formula, we
suppose that h represents the lapsed time between successive stock price changes.
Thus, if t is the fixed length of calendar time to expiration, and n is the total number
of periods each with length h, then h 
t
.
n
As the trading frequency increases, h
will get closer to zero. When h  0 , this is equivalent to n   .
15
Let R̂ be one plus the interest rate over a trading period of length h. Then, we
will have
Rˆ T  Rt
(3.5)
t
for any choice of n. Thus, Rˆ  R n , which shows that R̂ must depend on n for the
total return over elapsed time t to be independent of n. Also, in the limit, (1  r )  n
tends to e  rt as n   .
Let S* be the stock price at the end of the nth period with the initial price S.
If
there are j up periods, then
log
S*
u
 j log u  (n  j ) log d  j log( )  n log d
S
d
(3.6)
where j is the number of upward moves during the n periods.
Since j is the realization of a binomial random variable with probability of a
success being q, we have expectation of log (S*/S)
E (log
S*
u
)  [q log( )  log d ]n  ˆn,
S
d
(3.7)
and its variance
Var (log
S*
u
)  [log( )] 2 q(1  q)n  ˆ 2 n.
S
d
(3.8)
Since we divide up our original longer time period t into many shorter subperiods
16
of length h so that t  hn , our procedure calls for making n longer, while keeping the
length t fixed. In the limiting process we would want the mean and the variance of the
continuously compounded log rate of return of the assumed stock price movement to
coincide with that of actual stock price as n   .
̂ 2 n respectively.
Let the actual values of ̂ n and
Then we want to choose u, d, and q in such a manner
that ˆ n   t and ˆ 2 n   2 t as n   .
ue

d e
q

t
n
It can be shown that if we set
,
t
n
(3.9)
,
1 1   t
   ,
2 2   n
then ˆ n   t and ˆ 2 n   2 t as n   .
In order to proceed further, we need
the following version of the central limit theorem.
Lyapounov’s Condition.
Suppose X 1 , X 2 , are independent and uniformly
bounded with E ( X i )  0 , Yn  X 1    X n, and s 2  E (Yn2 )  Var (Yn ).
If lim
n
n
1
k 1
2
n
s
E Xk
2
 0 for some   0, then the distribution of
to the standard normal as n   .
Theorem 1.
If
17
Yn
converges
sn
p log u  ˆ  (1  p) log d  ˆ
3
ˆ 3 n
3
 0 as n  
(3.10)
then


S*
log(
)  ˆn


S
Pr 
 z  N ( z)
ˆ n




(3.11)
where N(z) is the cumulative standard normal distribution function.
Proof.
See Appendix.
It is noted that the condition (3.10) is a special case of the Lyapounov’s condition
which is stated as follows. When   1, we have the condition (3.10).
This theorem says that when the fixed length t is divided into many subperiods, the
log rate of return will approach to the normal distribution when the number of
subperiods approached infinity. For this theorem to hold, the condition stated in
Equation (3.10) has to be satisfied. We next show that this condition is indeed
satisfied.
We will next show that the binomial option pricing model as given in Equation (3.2)
will indeed coincide with the Black-Scholes option pricing formula as given in
Equation (3.3).
Observe that Rˆ  n is always equal to R t , as evidenced from
Equation (3.5). Thus, comparing the two option pricing formulae given in Equations
18
(3.2) and (3.3), we see that there are apparent similarities.
In order to show the
limiting result, we need to show that as n   ,
B(m; n, p )  N ( x) and B(m; n, p)  N ( x   t ).
In this section we will only show the second convergence result, as the same
argument will hold true for the first convergence.
From the definition of B (n, pm) ,
it is clear that
1  B(m; n, p)  Pr( j  m  1)
j  np
m  1  np
 Pr(

).
np(1  p)
np(1  p)
(3.12)
Recall that we consider a stock to move from S to uS with probability p and dS with
probability (1-p).
During the fixed calendar period of t=nh with n subperiods of
length h, if there are j up moves, then
S*
u
log
 j log( )  n log d .
S
d
(3.13)
The mean and variance of the continuously compounded rate of return for this
stock are ̂ P and ˆ P2 where
u
d
u
d
ˆ P  p log( )  log d and ˆ P2  [log( )] 2 p(1  p) .
From Equation (3.13) and the definitions for ̂ P and ˆ P2 , we have
19
j  np
np(1  p)
log(

S*
)  ˆ P n
S
.
ˆ P n
(3.14)
Also, from the binomial option pricing formula we have
X
)
n
Sd
m 1 

u
log( )
d
X
u
 [log  n log d ] / log   ,
S
d
log(
where is a real number between 0 and 1.
From the definitions of ̂ P and ˆ P2 , it is easy to show that
m  1  np
np(1  p)
log(

X
u
)  ˆ P n   log( )
S
d .
ˆ
P n
Thus from Equation (3.12) we have
log
1  B(m; n, p )  Pr(
S*
X
u
 ˆ P n log
 ˆ P n   log( )
S
S
d ).

ˆ P n
ˆ P n
(3.15)
We will now check the condition given by Equation (3.10) in order to apply the
central limit theorem.
p
Now recall that
rˆ  d
,
ud
t
n
with rˆ  r , and d and u are given in Equation (3.9).
20
We have
p
e
t
log r
n
t

n
e

t
n

t
n
e
e
3

t
t 1 2t
1  log r  [1  
  ]  O(n 2 )
n
n 2
n

3

t
t
1 
 [1  
]  O(n 2 )
n
n
1
log r   2
1 1
2 ] t  O( n 1 ).
  [
2 2

n
(3.16)
Hence, the condition given by Equation (10) is satisfied because
p | log u  ˆ p | 3 (1  p) | log d  ˆ p | 3
ˆ 3p n

(1  p) 2  p 2
np(1  p)
 0, as n  .
Finally, in order to apply the central limit theorem, we have to evaluate ̂ p n ,
u
d
̂ 2p n and log( ) as n  . It is clear that
1
2
u
d
ˆ p n  (log r   2 )t , ˆ 2p n   2 t and log( )  0 .
Hence, in order to evaluate the asymptotic probability in Equation (3.12), we have
log(
X
u
X
1
)  ˆ p n   log( )
log( )  (log r   2 )t
S
d z
S
2
.
ˆ p n
 t
Using the fact that 1  N ( z )  N ( z ) , we have, as n  
B(m; n, p)  N ( z )  N ( x   t ).
21
Similar argument holds for B (m; n, p ' ) , and hence we completed the proof that the
binomial option pricing formula as given in equation (3.2) includes the Block-Scholes
option pricing formula as a limiting case.
4. Comparison of the Two Approaches
From the results of last two sections, we show that both RB and CRR models lead to
the celebrated Black-Scholes formula. The following table shows the comparisons of
the necessary mathematical and statistical knowledge and assumptions for the two
models.
Model
Mathematical
and
Probability
Theory
Knowledge
Rendleman and Bartter (1979)
Cox, Ross and Rubinstein (1979)
Basic Algebra
Taylor Expansion
Binomial Theorem
Central Limit Theorem
Properties of Binomial Distribution
Basic Algebra
Taylor Expansion
Binomial Theorem
Central Limit Theorem
Properties of Binomial Distribution
Lyapounov’s Condition
1. The distribution of returns of the The
Assumption
Advantage
stock
follows
a
binomial
stock is stationary over time and the process from one period to the next
stock pays no dividends.(Discrete it can only go up by a factor of “u”
Time Model)
with probability “p” or go down by a
factor of “d” with probability “1-p”.
2. The mean and variance of In order to apply the Central Limit
logarithmic returns of the stock are Theorem, “u”, “d”, and “p” are
held constant over the life of the needed to be chosen.
option.(Continuous Time Model)
1. Readers who have undergraduate 1. Readers who have advanced level
22
and
level training in mathematics and knowledge in probability theory can
Disadvantage probability theory can follow this
approach.
2. The approach of RB is intuitive.
But the derivation is more
complicated and tedious than the
approach of CRR.
follow this approach; but for those
who don’t, CRR approach may be
difficult to follow.
2. The assumption on the parameters
“u”, “d”, “p” makes CRR approach
more restricted than RB approach.
Hence, like we indicate in the table, CRR is easy to follow if one has the advanced
level knowledge in probability theory but the assumptions on the model parameters
make its applications limited. On the other hand, RB model is intuitive and does not
require higher level knowledge in probability theory. However, the derivation is more
complicated and tedious.
For readers who are interested in the binomial option pricing model, they can
compare the two different approaches and find the best one which fits their interests
and is easier to follow.
Appendix
The Binomial Theorem
n n
( x  y )n   k 0   x k y n k
k 
Lindberg-Levy Central Limit Theorem
23
If x1 ,..., xn are a random sample from a probability distribution with finite mean 
1
n
and finite variance  2 and x  ( ) i 1 xi , then
n
d
n ( xn   ) 
 N 0,  2 
Proof of Theorem 1.
Since
3
p log u  ˆ  p log u  p log
3
u
u
 log d  p (1  p )3 log
d
d
3
And
3
3
u
u
(1  p) log d  ˆ  (1  p ) log d  p log  log d  p 3 (1  p ) log
,
d
d
3
We have p log u  ˆ  (1  p) log d  ˆ  p(1  p)[(1  p) 2  p 2 ] log
3
3
u
d
3
.
Thus
p log u  ˆ  (1  p ) log d  ˆ
3
3
ˆ 3 n
p (1  p )[(1  p ) 2  p 2 ] log

u
d
3
u
( p (1  p ) log( ))3 n
d
2
2
(1  p )  p

 0 as n  .
np (1  p )
Hence the condition for the theorem to hold as stated in Equation (3.10) is satisfied.
24
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