Comparing Black-Scholes and Binomial Option Pricing Models

A Comparison of Option Pricing Models
Ekrem Kilic �
11.01.2005
Abstract
Modeling a nonlinear pay o¤ generating instrument is a
challenging work. The mod-
els that are commonly used for pricing derivative might divided
into two main classes;
analytical and iterative models. This paper compares the Black-
Scholes and binomial
tree models.
Keywords: Derivatives, Option Pricing, Black-Scholes,Binomial
Tree
JEL classi�cation:
1. Introduction
Modeling a nonlinear pay o¤ generating instrument is a
challenging work to
handle. If we consider a European option on a stock, what we
are trying to do is

estimating a conditional expected future value. In other words
we need to �nd out
the following question: what would be the expected future value
of a stock given
that the price is higher than the option�s strike price? If we
�nd that value we can
easily get the expected value of the option. For the case of the
American options
the model need to be more complex. For this case, we need to
check the path that
we reached some future value of the stock, because the buyer of
the option might
exercise the option at any time until the maturity date.
To solve the problem that summarized above, �rst we need to
model the move-
ment of the stock during the pricing period. The common model
for the change
of the stock prices is Geometric Brownian Motion. Secondly,
the future outcomes
of the model might have the same risk. Risk Neutrality
assumption provides that.
By constructing a portfolio of derivative and share makes
possible to have same
�E-mail address: [email protected]

A Comparison of Option Pricing Models 2
outcome with canceling out the source of the uncertainty.
The models that are commonly used for pricing derivative might
divided into
two main classes. The �rst classes is the models that provide
analytical formulae to
get the risk neutral price under some reasonable assumptions.
The Black-Scholes
formula is in this group. The formulae that we have to price the
derivatives
are quite limited. The reason is that we are trying to solve a
partial di¤erential
equation at the end of the day. But mathematician could manage
to solve just
someof thepartialdi¤erential equations; therefore,
weareboundedto some limited
solutions.
The second classes models provide numerical procedures to
price the option.
Binomial trees that �rst suggested by Cox, Ross and
Rubenstein, is in this group,

because we need to follow an iterative procedure called
�backwards induction�to
get option price. Monte Carlo simulations are another type of
models that belongs
to this class. Also �nite di¤erencing methods are a type of
numerical class.
In this paper, �rst I will introduce Black-Scholes and Binomial
Tree models for
option pricing. Second I will introduce the volatility estimation
methods I used
and calculate some option prices to compare models. Finally I
will conclude.
2. Option Pricing Models
2.1. Black-Scholes Model
Black-Scholes formula suggested by Fischer Black and Myron
Scholes at 1973.
The Black-Scholes (1973) option pricing formula prices
European put or call op-
tions on a stock that does not pay a dividend or make other
distributions. The
formula assumes the underlying stock price follows a geometric
Brownian motion

with constant volatility. The geometrics Brownian motion can
be shown as follows;
�S = �S�t+�S�z
where S is the current price of the stock, dS the change in the
stock price, � is
the expected rate of return, � is the volatility and dz is the part
follows a Wiener
process.
Then using Ito�s lemma the price of the option might be shown
as;
�f =
�
@f
@S
�S + @f
@t
+ 1
2
@2f
@S2
�2S2
�
�t+ @f
@S

�S�z
where df is the change in the option value.
This di¤erential equation formulates the movement of the option
price over
time. Now note that the source of the risk in the stock and the
option is exactly
same (dz). Then if we construct a risk neutral portfolio
consisting a short position
in the option and a long position in the stock as follows
derivative : �1
share : @f
@S
The pay o¤ function can be shown as;
� = �f + @f
@S
S
The change of the pay o¤ function is;
�� = ��f + @f
@S
�S
�� = �
�
@f
@S

�S + @f
@t
+ 1
2
@2f
@S2
�2S2
�
�t� @f
@S
�S�z + @f
@S
(�S�t+�S�z)
�� = �
�
@f
@t
+ 1
2
@2f
@S2
�2S2
�
�t

This portfolio is riskless because we can exclude the source of
uncertainty.
Since it is riskless it should yield risk free rate otherwise there
could be arbitrage
opportunities. Then we can write;
�� = r��t�
@f
@t
+ 1
2
@2f
@S2
�2S2
�
�t = r
�
f � @f
@S
S
�
�t
And
@f
@t
+ @f

@S
rS + 1
2
@2f
@S2
�2S2 = rf
This partial di¤erential equation is called Black Scholes
di¤erential equation.
The solution for this di¤erential equation is the Black-Scholes
formulae;
c = S0N (d1)�Ke�rTN (d2)
p = Ke�rTN (�d2)�S0N (�d1)
where
d1 =
ln(S0=K)+(r+�2=2)T
�
p
T
d2 =
ln(S0=K)+(r+�2=2)T
�
p
T
= d1 ��

p
T
Then the formula needs the current price of the stock S0 , the
stirke price K,
risk free short term interest rate r, time to maturity T and
volatility of the stock
�(N (:) is the cumulative normal distribution function).
The model is based on a normal distribution of underlying asset
returns which
is the same thing as saying that the underlying asset prices
themselves are log-
normally distributed. A lognormal distribution is right skewed
distirbution. The
lognormal distribution allows for a stock price distribution of
between zero and
in�nity (ie no negative prices) and has an upward bias
(representing the fact that
a stock price can only drop 100% but can rise by more than
100%).
In practice underlying asset price distributions often depart
signi�cantly from

the lognormal. For example historical distributions of
underlying asset returns
often have fatter left and right tails than a normal distribution
indicating that
dramatic market moves occur with greater frequency than would
be predicted by
a normal distribution of returns�ie more very high returns and
more very low
returns.
The main advantage of the Black-Scholes model is speed �it
lets you calculate
a very large number of option prices in a very short time.
The Black-Scholes model has one major limitation: it cannot be
used to ac-
curately price options with an American-style exercise as it only
calculates the
option price at one point in time �at expiration. It does not
consider the steps
along the way where there could be the possibility of early
exercise of an American
option.
As all exchange traded equity options have American-style
exercise (ie they

can be exercised at any time as opposed to European options
which can only be
exercised at expiration) this is a signi�cant limitation.
The exception to this is an American call on a non-dividend
paying asset. In
this case the call is always worth the same as its European
equivalent as there is
never any advantage in exercising early.
2.2. Binomial Model
The binomial model breaks down the time to expiration into
potentially a very
large number of time intervals, or steps. A tree of stock prices
is initially produced
working forward from the present to expiration. At each step it
is assumed that
the stock price will move up or down by an amount calculated
using volatility and
time to expiration. This produces a binomial distribution, or
recombining tree, of
underlying stock prices. The tree represents all the possible
paths that the stock

price could take during the life of the option.
At the end of the tree �ie at expiration of the option �all the
terminal option
prices for each of the �nal possible stock prices are known as
they simply equal
their intrinsic values.
Next the option prices at each step of the tree are calculated
working back from
expiration to the present. The option prices at each step are used
to derive the
option prices at the next step of the tree using risk neutral
valuation based on the
probabilities of the stock prices moving up or down, the risk
free rate and the time
interval of each step. Any adjustments to stock prices (at an ex-
dividend date) or
option prices (as a result of early exercise of American options)
are worked into
the calculations at the required point in time. At the top of the
tree you are left
with one option price.
For example we can consider a one step binomial tree. The one
step ahead

stock price has two di¤erent values in a binomial model; S0u or
S0d where u and
d are the upward and downward size respectively. Then if we
construct a risk
neutral portfolio by shorting the derivative and buying the
stock;
derivative : �1
share : �
Because the portfolio is risk neutral, whether stock price goes
up or down the
pay o¤ would be same, so;
S0u��fu = S0d��fd
� = fu�fd
S0u�S0d
The cost of settin up this portfolio is;
S0 �f
because we get a riskless position it should yield risk free rate.
Then;
S0 �f = e�rT (S0u��fu)

Finally option value is;
f = e�rT (pfu +(1�p)fd)
where
p = e
rT �d
u�d
To match this formulation with stock volatility, it is a popular
way to de�ne
u;d and p as follows;
u = e�
p
�t
d = 1=u
p = e
r�t�d
u�d
The big advantage the binomial model has over the Black-
Scholes model is
that it can be used to accurately price American options. This is
because with the
binomial model it�s possible to check at every point in an
option�s life (ie at every
step of the binomial tree) for the possibility of early exercise

(eg where, due to eg
a dividend, or a put being deeply in the money the option price
at that point is
less than the its intrinsic value).
Where an early exercise point is found it is assumed that the
option holder
would elect to exercise, and the option price can be adjusted to
equal the intrinsic
value at that point. This then �ows into the calculations higher
up the tree and
so on.
The binomial model basically solves the same equation, using a
computational
procedure that the Black-Scholes model solves using an analytic
approach and
in doing so provides opportunities along the way to check for
early exercise for
American options.
The main limitation of the binomial model is its relatively slow
speed. It�s

great for half a dozen calculations at a time but even with
today�s fastest PCs it�s
not a practical solution for the calculation of thousands of
prices in a few seconds.
3. Comparison of the Models
3.1. Volatility Estimation
For both models the volatility of the stock is the key factor.
Estimation of the
volatilty is another important topic. In this study I used three
di¤erent models for
Historical Average(HA), Exponentially Weighted Moving
Average (EWMA) and
Generalized Autoregressive Conditional Hetereoscedasitic
(GARCH) model.
3.1.1. Historical Average
If we assume that conditional expectation of the volatility is
constant and the
daily returns has zero mean, the proper estimate of the volatility
is;
� =

vuut 1
n�1
nX
i=1
r2
where � is the estimated volatility, n is the sample size and r is
the daily return.
The weakest point of this model is of course constant volatility
assumption.
This estimate of the volatility could not mimic the big changes
in the volatility
and remains nearly constant where the sample size increases.
3.1.2. Exponentially Weighted Moving Average
EWMA past observations with exponentially decreasing weights
to esitmate
volatility. Therefore this is a modi�ed version of historical
averaging. Instead of
equally weighting, in EWMA weights di¤er. The estimated
volatility can shown
as;
�2t = (1��)r2t +��2t�1
�t =
p

�2t
By repeated substitutions we can re-write the forecast as;
�2t = (1��)
nX
i=1
�i�1r2t
equation shows the volatility is equal to a weighting average.
The weights de-
crease geometrically. The value of , decay factor, estimated
simply by minimizing
the one week forecast errors.
3.1.3. GARCH
The Generalized ARCH model of Bollerslev (1986) de�ned
GARCH by;
rt = �+�t"t
�2t = �+
qX
i=1
�i (rt�i ��)
2

+
pX
j=1
�j�
2
t�j
GARCH imposes that the proper volatility estimate is based not
only on the
recent volatilities and also previous forecasts which include the
previous volatilities.
Then the GARCH model is a long memory model. The
parameters of the GARCH
can be estimated by a Maximum Likelihood procedure.
3.2. Some Hypothetic Options
3.2.1. An Stock Index Option on ISE-100
First let us consider a one-month european call option with
strike price 27000
(current price is taken as 25300). The price for several
combinations is as follows;
Black-Scholes CRR
Historical Average 2207.95 2206.60
EWMA 3068.71 3071.38

GARCH 2562.63 2560.39
The CRR results obtained with 100 steps. As we can see the
price of the
option changes according to the volatility estimates. The
volatility estimate of
the historical simulation leads us to lowest price (16%). The
volatility estimate of
EWMA and GARCH is 30% and 22% respectively.
Now let us consider the case we have a one-month european put
option with
strike price 27000. The prices are;
Black-Scholes CRR
EWMA 1294.48 1297.15
GARCH 788.40 786.16
This type option price change at a higher ratio, then this option
is more non-
linear with respective to volatility.
For the american call option (non-divident paying) the price is

same as we
calculated before. But for the american put option the value will
change and we
can not use the Black-Scholes formula for the american put
options. The table
shows the prices by the binomial model;
CRR
Historical Average 856.91
EWMA 1657.57
GARCH 1172.60
As we can see the price of the american option is higher than
the european
option.
3.2.2. An FX Option on USD/TRL
First let us start with a one-month european call option again
with strike price
1.38 (current price is taken as 1.38). The price for several
combinations is as
follows;

Black-Scholes CRR
EWMA 0.1633 0.1631
GARCH 0.1574 0.1572
The binomail model�s results obtained with 100 steps again.
The volatility
estimate of HA, EWMA and GARCH is 14% , 22 and 20%
respectively.
Another case, we have a one-month european put option with
strike price 1.38.
The prices are;
Black-Scholes CRR
EWMA 0.0320 0.0318
GARCH 0.0261 0.0259
The table shows the prices by the binomial model result for the
american
version of the last option;
CRR
Historical Average 0.0220

EWMA 0.0458
GARCH 0.0395
As we can see the price of the american option is higher than
the european
option again.
4. Conclusion
Due to its nonlienar pay o¤s pricing an option is quite di¢ cult
with respective
to other �nancial instruments like �xed income instruments.
One should consider
the possible future oturcomes of the underlying asset. If the
option is an exotic
that is e¤ected by more than one sources of the uncertainty it
becomes more
complicated. In this paper we just introduce an compare two
popular models.
The results show that the prices of the two models quite similar
where the
number of steps in the binomial. Since Black-Scholes model can
not calculate the

price of the american option one can use the binomial model
with high number
steps and can similar result as if Black-Scholes.
Another result shows that due to high volatility the option price
on the ISE
will be very high. For example a long position on one-month
american call with
strike price 27000 might have positive payo¤ if the prices goes
somewhere around
30000. On the other hand the price of the FX options is more
reasonable.
By the results of the paper we can see that the volatility
estimation is also as
important as option pricing. Because the volatilty changes the
price might change
exponentially.
Abstract—This paper presents the accuracy of binomial
model for the valuation of standard options with dividend yield

in the context of Black-Scholes model. It is observed that the
binomial model gives a better accuracy in pricing the American
type option than the Black-Scholes model. This is due to fact
that the binomial model considers the possibilities of early
exercise and other features like dividend. It is also observed
that the binomial model is both computationally efficient and
accurate but not adequate to price path dependent options.
Index Terms—American Option, Binomial Model, Black-
Scholes Model, Dividend Yield, European Option, Standard
Option
Mathematics Subject Classification 2010: 34K50, 35A09,
91B02, 91B24, 91B25
I. INTRODUCTION
Financial derivative is a contract whose value depends on
one or more securities or assets, called underlying assets.
An option is a contingent claim that gives the holder the
right, but not the obligation to buy or sell an underlying asset

for a predetermined price called the strike or exercise price
during a certain period of time. Options come in a variety of
"flavours". A standard option offers the right to buy or sell
an underlying security by a certain date at a set strike price.
In comparison to other option structures, standard options
are not complicated. Such options may be well-known in the
markets and easy to trade. Increasingly, however, the term
standard option is a relative measure of complexity,
especially when investors are considering various options
and structures. Examples of standard options are American
options which allow exercise at any point during the life of
the option and European options that allow exercise to occur
only at expiration or maturity date.
Black and Scholes published their seminar work on option
pricing [1] in which they described a mathematical frame
work for finding the fair price of a European option. They
used a no-arbitrage argument to describe a partial
differential equation which governs the evolution of the

option price with respect to the maturity time and the price
Manuscript received September 30, 2013; revised November
19, 2013.
Chuma Raphael Nwozo is with the Department of Mathematics,
University of Ibadan, Oyo State, Nigeria, phone:
+2348168208165; e-mail:
[email protected], [email protected]
Sunday Emmanuel Fadugba is with the Department of
Mathematical
Sciences, Ekiti State University, Ado Ekiti, Nigeria, email:
[email protected], [email protected]
of the underlying asset. Moreover, in the same year, [9]
extended the Black-Scholes model in several important
ways. The subject of numerical methods in the area of option
pricing and hedging is very broad, putting more demands on
computational speed and efficiency. A wide range of
different types of contracts are available and in many cases
there are several candidate models for the stochastic

evolution of the underlying state variables [12].
We present an overview of binomial model in the context
of Black-Scholes-Merton [1, 9] for pricing standard options
based on a risk-neutral valuation which was first suggested
and derived by [4] and assumes that stock price movements
are composed of a large number of small binomial
movements. Other procedures are finite difference methods
for pricing derivative by [3], Monte Carlo method for
pricing European option and path dependent options
introduced by [2] and a class of control variates for pricing
Asian options under stochastic volatility models considered
by [5]. The comparative study of finite difference method
and Monte Carlo method for pricing European option was
considered by [6]. Some numerical methods for options
valuation were considered by [10]. [11] Considered Monte
Carlo method for pricing some path dependent options. On
the accuracy of binomial model and Monte Carlo method for
pricing European options was considered by [7]. These

procedures provide much of the infrastructures in which
many contributions to the field over the past three decades
have been centered.
In this paper we shall consider only the accuracy of
binomial model for the valuation of standard options
namely; American and European options with dividend yield
in the context of Black-Scholes model.
II. BINOMIAL MODEL FOR THE VALUATION OF
STANDARD OPTIONS WITH DIVIDEND YIELD
This section presents binomial model for the valuation of
standard options with dividend yield.
A. Binomial Model
This model is a simple but powerful technique that can be
used to solve the Black-Scholes and other complex option-
pricing models that require solutions of stochastic
differential equations. The binomial option-pricing model
(two-state option-pricing model) is mathematically simple

and it is based on the assumption of no arbitrage.
The assumption of no arbitrage implies that all risk-free
investments earn the risk-free rate of return (zero dollars) of
On the Accuracy of Binomial Model for the
Valuation of Standard Options with Dividend
Yield in the Context of Black-Scholes Model
Chuma Raphael Nwozo
, Sunday Emmanuel Fadugba
IAENG International Journal of Applied Mathematics, 44:1
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ISSN: 1992-9978 (Print)
Page 33
mailto:[email protected]

investment but yield positive returns. It is the activity of
many individuals operating within the context of financial
markets that, in fact, upholds these conditions. The activities
of arbitrageurs or speculators are often maligned in the
media, but their activities insure that our financial market
work. They insure that financial assets such as options are
priced within a narrow tolerance of their theoretical values
[10].
1) Binomial Option Model
This is defined as an iterative solution that models the
price evolution over the whole option validity period. For
some types of options such as the American options, using
an iterative model is the only choice since there is no known
closed form solution that predicts price over time. There are
two types of binomial tree model namely
i. Recombining tree
ii. Non-recombining tree
The difference between recombining and non-

recombining trees is computational only. The recombining
n periods there can be n,...,3,2,1 ups) and a non-
recombining tree with n trading periods has
n
2 final nodes.
In binomial tree, the number of final nodes is the number of
rows of a worksheet when implementing the binomial model.
The Black-Scholes model seems to have dominated
option pricing, but it is not the only popular model, the Cox-
Ross-Rubinstein (CRR) “Binomial” model is also popular.
The binomial models were first suggested by [4] in their
paper titled “Option Pricing: A Simplified Approach” in
1979 which assumes that stock price movements are
composed of a large number of small binomial movements.
Binomial model comes in handy particularly when the
holder has early exercise decisions to make prior to maturity
or when exact formulae are not available. These models can
accommodate complex option pricing problems [11].

CRR found a better stock movement model other than the
geometric Brownian motion model applied by Black-
Scholes, the binomial models.
First, we divide the life time ],0[ T of the option into
N
T
(1)
Suppose that
0
S is the stock price at the beginning of a
given period. Then the binomial model of price movements
assumes that at the end of each time period, the price will
either go up to uS
0
with probability p or down to
dS
0

We recall that by the principle of risk neutral valuation,
the expected return from all the traded options is the risk-
free interest rate. We can value future cash flows by
discounting their expected values at the risk-free interest
rate. The parameters du, and p satisfy the conditions for
the risk-neutral valuation and lognormal distribution of the
stock price and we have the expected stock at time
T as )(
T
SE . An explicit expression for )(
T
SE is obtained
as follows:
Construct a portfolio comprising a long position in
when
portfolio riskless. If there is an up movement in the stock
price, the value of the portfolio at the end of the life of
option is
u

0
and if there is a down movement in
the stock price, the value becomes
d
0
. Since the
last two expressions are equal, then we have
du
du
du
ffduS
ffdSuS
fdSfuS
)(
0

00
00
)(
0
duS
ff
du
In the above case, the portfolio is riskless and must earn
the risk-
change in the option price to the change in the stock price as
we move between the nodes. If we denote the risk-free
interest rate by r , the present value of the portfolio
is
tr
u
efuS

0
. The cost of setting up the portfolio
0
, it follows that,
tr
u
efuSfS
00
tr
u
efuSSf
00
(3)
Substituting (2) into (3) and simplifying, then (3) becomes
))1((
du
tr

(4)
du
de
p
tr
(5)
-step binomial
model. Equations (4) and (5) become respectively
))1((
du
rT
(6)
du

de
p
rT
(6) and (7) enable an option to be priced using a one-step
binomial model.
Although we do not need to make any assumptions about
the probabilities of the up and down movements in order to
obtain (4).
The expression
du
from the option. With this interpretation of p , (4) then
states that the value of the option today is its expected future
value discounted at the risk-free rate. For the expected return
from the stock when the probability of an up movement is
assumed to be p , the expected stock price, )(
T

SE at time
T is given by
0 0 0 0 0
( ) (1 )
T
Therefore,
2014
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dSdupSSE
T 00
Substituting (7) into (8), yields
0 0

0 0 0 0
( ) ( )
rT
T
r t r t
e d
E S S u d S d
u d
S e S d S d S e
(9)
Now,
tr
tr
edppu

eSdSdupS
)1(
)(
000
Therefore,
dppue
tr
(10)
When constructing a binomial tree to represent the
movements in a stock price we choose the parameters u and
d to match the volatility of a stock price.
The return on the asset price
0

time is
t
Wt
S
S
0
0
(11)
asset price,
t
W = Standard Brownian motion and
)()( tWttWW
t
order two and above, it follows from (10) that the variance
of the return is
t

tt
tWEtEtE
tWttE
WtEWtE
S
S
E
S
S
E
t
t
tt

2
222
222222
222222
22
2
0
0
2
0
0
0
)()(2)(
)2(
))(()(

For the one period binomial model, we have that the
222
))1
To match the stock price volatility with the tree's
parameters, we must therefore have that
2222
Substituting (7) into (12), we have that
teuddue
tt
When terms in t
2

one solution to this equation is
t
ed
eu
(13)
The probability p obtained in (7) is called the risk neutral
probability. It is the probability of an upward movement of
the stock price that ensures that all bets are fair, that is it
ensures that there is no arbitrage. Hence (10) follows from
the assumption of the risk-neutral valuation.

2) Cox-Ross-Rubinstein Model
The Cox-Ross-Rubinstein model [8] contains the Black-
Scholes analytical formula as the limiting case as the number
of steps tends to infinity.
After one time period, the stock price can move up to
uS
0
with probability p or down to dS
0
with
Fig. 1 below.
Fig. 1: Stock and Option Prices in a General One-Step Tree
Therefore the corresponding value of the call option at

)0,max(
)0,max(
0
0
KdSf
KuSf
d
u
(14)
where
u
f and
d
f are the values of the call option after
upward and downward movements respectively.
We need to derive a formula to calculate the fair price of
the option. The risk neutral call option price at the present
time is
))1((
du

tr
(15)
To extend the binomial model to two periods, let
uu
upward stock movements,
ud
f for one upward and one
downward movement and
dd
f for two consecutive
downward movement of the stock price as shown in the Fig.
2 below.
Fig. 2: Binomial Tree for the respective Asset and Call
Price in a Two-Period Model
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Then we have
)0,max(
)0,max(
)0,max(
0
0

0
KddSf
KudSf
KuuSf
dd
ud
uu
(16)
))1((
))1((

ddud
tr
d
uduu
tr
u
fppfef
fppfef
(17)
Substituting (17) into (15), we have:
)))1(()1(
))1(((
ddud
tr
uduu
trtr

fppfep
fppfpeef
))1()1(2(
222
dduduu
tr
(18)
the numbers
2
p
2

neutral probabilities that the underlying asset prices Suu ,
udS
0
and ddS
0
respectively are attained.
We generate the result in (18) to value an option at
N
T
jNj
du
jNj
N
j
trN
fpp
j
N

0
)0,max()1(
0
0
KduSpp
j
N
ef
jNjjNj
N
j
trN

(19)
where
)0,max(
0
KduSf
jNj
du

!)!(
!
jjN
N
j
N
is the binomial coefficient. We
assume that m is the smallest integer for which the option's
intrinsic value in (19) is greater than zero. This implies
that KdSu
mNm
. Then (19) can be expressed as
jNj
N

mj
trN
jNjjNj
N
mj
trN
pp
j
N
Ke
dupp
j
N
eSf

)1(
)1(
0
(20)
(20) gives us the present value of the call option.

The term
trN
e
is the discounting factor that reduces
f to its present value. The first term of
jNj
pp
j
N
)1( is the binomial probability of j upward
movements to occur after the first N trading periods and
jNj
dSu
is the corresponding value of the asset after

j upward move of the stock price.
The second term is the present value of the option`s strike
price. Putting
tr
eR
jNj
N
mj
trN
jNjjNj
N
mj
N
pp
j
N
Ke
dupp
j

)1(
)1(
0
Therefore,
jNj
N
mj
trN
jNj
N
mj
pp
j
N
Ke
dpRpuR
j
N
Sf

))1(()(
11
0
(21)
function given by
N
mj
jNj
j
N
ppCpNm )1(),;( (22)
(22) is the probability of at least m success in
N independent trials, each resulting in a success with

Then let puRp
1
.
Consequently, it follows from (21) that
),;(),;(
0
pNmKepNmSf
rt
(23)
The model in (23) was developed by Cox-Ross-
Rubinstein [4] and we will refer to it as CRR model for the
valuation of European call option. The corresponding value
of the European put option can be obtained using call-put
parity of the form
0
SPKeC

E
rt
E
. We state a
lemma for CRR binomial model for the valuation of
European call option.
Lemma 1[7]: The probability of a least m success in
N independent trials, each resulting in a success with
probability p and in a failure with probability q is given by
N
mj
jNj
j
N
ppCpNm )1(),;(

Let puRp
1
, then it
follows that ),;(),;(
0
pNmKepNmSf
rt
B. Procedures for the Implementation of the Multi-
Period Binomial Model
When stock price movements are governed by a multi-
step binomial tree, we can treat each binomial step
separately. The multi-step binomial tree can be used for the
American and European style options.
Like the Black-Scholes model, the CRR formula in (23)

can only be used in the pricing of European style options
and is easily implementable in Matlab. To overcome this
problem, we use a different multi-period binomial model for
the American style options on both the dividend and non-
dividend paying stocks. The no-arbitrage arguments are used
and no assumptions are required about the probabilities of
the up and down movements in the stock price at each node.
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For the multi period binomial model, the stock price $S$
stock prices uS

0
and dS
0
are three possible stock prices uuS
0
, udS
0
and ddS
0
and
NjforduS
jNj
,...,2,1,0,
0
(24)
where N is the total number of movements and j is the total
number of up movements. The multi-period binomial model

can reflect numerous stock price outcomes if there are
numerous periods. The binomial option pricing model is
based on recombining trees, otherwise the computational
burden quickly become overwhelming as the number of
moves in the tree increases.
Options are evaluated by starting at the end of the tree at
time T and working backward. We know the worth of a call
and put at time T is
)0,max(
)0,max(
T
T
SK
KS

(25)
respectively. Because we are assuming the risk neutral
calculated as the expected value at time T discounted at rate
at
and so on. By working back through all the nodes, we obtain
the value of the option at time zero.
Suppose that the life of a European option on a non-
dividend paying stock is divided into N subintervals of
th
ji
f
,
as the value of the option at the ),( ji node. The stock
price at the ),( ji node is

jNj
dSu
. Then, the respective
European call and put can be expressed as
)0,max(
,
KdSuf
jNj
jN
(26)
,...2,1,0),0,max(
,
jfordSuKf
jNj
jN
(27)
There is a probability p of moving from the ),( ji node

neutral valuation is
ijNi
fppfef
jiji
tr
ji
0,10
])1([
,11,1,

(28)
For an American option, we check at each node to see
whether early exercise is preferable to holding the option for
account, this value of
ji
f
,
must be compared with the
option's intrinsic value [7]. For the American put option, we
have that
, , ,
,
i j i j i j
,11,10, jiji
trjij
ji
fppfeduSKf

(29)
(29) gives two possibilities:
jiji
fP
,,
jiji
fP
,,
C. Variations of Binomial Models
The variations of binomial models is of two forms namely
underlying stock paying a dividend or known dividend yield
and underlying stock with continuous dividend yield.
1) Underlying stock paying a dividend or known

dividend yield
The value of a share reflects the value of the company.
After a dividend is paid, the value of the company is reduced
so the value of the share.
1,...,2,1,0
,,...,2,1,0,
0
iN
NjforduS
jNj
(30)

1,
,,...,2,1,0,)1(
0
iiN
NjforduS
jNj
(31)
2) Underlying stock with continuous dividend yield
A stock index is composed of several hundred different
shares. Each share gives dividend away a different time so
the stock index can be assumed to provide a dividend
continuously.
We explored Merton's model, the adjustment for the
Black-Scholes model to carter for European options on
stocks that pays dividend. Here the risk-free interest rate is
dividend yield. We apply the same principle in our binomial

model for the valuation of the options. The risk neutral
probability in (5) is modified but the other parameters
remains the same.

ed
eu
du
de
p
tr
,
,
)(
(32)
These parameters apply when generating the binomial tree
of stock prices for both the American and European options
on stocks paying a continuous dividend and the tree will be
identical in both cases. The probability of a stock price
increase varies inversely with the level of the continuous
III. BLACK –SCHOLES EQUATION

Black and Scholes derived the famous Black-Scholes
partial differential equation that must be satisfied by the
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price of any derivative dependent on a non-dividend paying
stock. The Black-Scholes model can be extended to deal
with European call and put options on dividend-paying
stocks, this will be shown later. In the sequel, we shall
present the derivation of Black-Scholes model using a no-
arbitrage approach.
A. Black-Scholes Partial Differential Equation
We consider the equation of a stock price

tttt
)(tW follows a Wiener process on a filtered probability
space ))(,,,(
tt
tt
, where
t
B is the sigma-algebra
generated by }.0:{ TtS
t
),(),( StfStf
t
derivative contingent claim of the underlying asset price
S at time t . Assuming that ]),0[,(
1,2

the Ito’s lemma given below;
t
s
t
s

st
dW
x
U
d
x
UU
WsUWtU
2
2
),(),( (34)
We obtained the Black-Scholes partial differential equation
of the form
0
2
2
222

rf
S
fS
S
f
rS
t
(35)
Solving the partial differential equation above gives an
analytical formula for pricing the European style options.
These options can only be exercised at the expiration date.
The American style options are exercised anytime up to the

maturity date. Thus, the analytical formula we will derive is
not appropriate for pricing them due to this early exercise
privilege [8].
key boundary condition is
boundary condition is
(37)
B.
Solution
of the Black-Scholes Equation
We shall apply the boundary conditions for the European
call option to solve the Black-Scholes partial differential
equation. The payoff condition is

(38)
The lower and upper boundary conditions are given by
tEA
tTr
tEA
SKtSCKtSC
KeSKtSCKtSC
),,(),,,(

),,(),,,(
)(
These are the conditions that must be satisfied by the
partial differential equation.
is the expiration date and t is the

f
t
f
11 (39)
Substituting (39) into (35), yields
rf
S
fS
S
f
rS
f

f
SS
f
y
f
SS
y
y
f
S
f
Therefore,

Sy
f
SS
f
y
f
SS
f
(41)
We now introduce a new solution
yfeyw

(42)
Substituting (41) into (40), we have
rf
y
f
y
f
r
f

(43)
Also substituting (42) into (43), yields
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),(
2
2

Therefore,
2
222
22 y
w
y
w
r
w

(44)
(44) is called a diffusion equation which has a fundamental
solution as a normal function.

2
2
2
2
2
exp
2
1
),(
ry
y (45)
So,

2
2
2
2
2
exp
2
1
),(
ry
y (46)

y
7)
The payoff for call option becomes
)0,max(),0(),0(
)0,max(
)0,max(
)0,max(),(
Kewf
Ke
Ke
KSStf

(49)
We use the payoff condition in (48) and the fundamental
solution of (46) to obtain

2
2
exp)max(
2
1
),(
ry
dKeyw
K
(50)
We denote the distribution function for a normal variable
by )(xN
duexN
x u

2
2
1
)(
(51)
Then (51) becomes

2
2
ln
ln
2
2
exp
2
exp
2
1
),(
ry
de
K

2
2
ln
2
2
2
)(
exp
2
2
)(
exp
2
1

),(
(53)
We consider the second term in the right-hand side of

and the limits of (53) using (54) are given below
Kwhen
r
K
S
rSK
AK
z
whenz
ln,
2
ln
2
lnln
ln

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2
ln
2
2
r
K
S
d (55)
the right-hand side of (54) becomes

)(
22
2
22
2
2
2
2
2
dKNdze
K
dze
K
d z
d

z
(56)
The first integrand of the first term in (53) is expressed as

2
22
2
2
))((
exp
2
A
e
A
(57)
We use the definition of A to have

2
22
2
2
2
))((
exp
2
)(
exp
A
Se

A
e
r
(58)
Substituting (58) into the first term of (53), we have
K
r
d

By changing the variables as we did in the previous case,
we get
zd
r
z
r
dNSedzeSe )(
2

1
1
2
2
(59)
Where
zd z
dzedN 2
1

),(
2
2
r
K
S
KN
r
K
S
SNeyw
r
(61)
Recall that

E
r
, hence
)()(
21
dNKedSNC
r
E
This is the Black-Scholes formula for the price at time
zero of a European call option on a non-dividend paying

stock [1]. We can derive the corresponding European put
option formula for a non-dividend paying stock by using the
call-put parity given by
EE
KeSCP
The European put analytical formula is
)()(
12
dSNdNKeP
r
E

(63)
where )(1)(),(1)(
2211
The European call and put analytic formulae have gained
popularity in the world of finance due to the ease with which
one can use the formula for options valuation, the other
parameters apart from the volatility can easily be observed
from the market. Thus it becomes necessary to find
appropriate methods of estimating the volatility.
C. Dividend Paying Stock
We relax the assumption that no dividend are paid during

the life of the option and examine the effect of dividend on
the value of European options by modified the Black-
Scholes partial differential equation to carter for these
dividends payments.
Now we shall consider the continuous dividend yield
which is known. This means that the holder receives a
lowered after the payout of the dividend and so the expected
geometric Brownian motion model in (33) becomes
tttt
and the modified Black-Scholes partial differential equation

in (35) is given by
0
2
)(
2
222
rc

S
cS
S
c
Sr
t
the European call option for a dividend paying stock is given
by
)()(
21
dNKedNSeC
r

E
and the European put option is
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)()(
12
dSNedNKeP
r
E
(67)
where

1
2
ln
,
2
ln
d
r
K
S
d
r
K
S
d

(68)
The results in (66) and (67) can similarly be achieved by
considering the non-dividend paying stock formulae in (62)
and (63). The dividend payment lowers the stock price from
S to
Se and the risk-free interest rate which is the rate of
D. Boundary Condition for Black-Scholes Model
The boundary conditions for Black-Scholes model for
pricing a standard option are given by

)0,max(),(
),(lim
,0),0(
KStSf

StSf
ttf
c
c
S
c
(69)
We shall state below some theorems without proof as
follows:
Theorem 2 [7]: Under the binomial tree model for stock
pricing, the price of a European style option with expiration

T
T
r
fE
f
)1(
)(
*
0
Corollary 3 [7]: Under the binomial tree model for stock
pricing, the price of a European style option with expiration

T
T
j
jTjjTj
j
T
T
T
T
T
r
KduSppC

r
KSE
r
fE
f
)1(
)0,()1()(
)1(
))0,(max(
)1(
)(
0
0

(71)
where
*
E denotes expected value under the risk neutral
probability
*
p for stock price.
The above Theorem 2 can be written in words as “the
price of the option is equal to the present value of the
expected payoff of the option under the risk neutral
measure”.

Theorem 4: (Continuous Black Scholes Formula) [7]
Assume that
u
dutNT
1
that
t
eu
t
ed

rT
c
t
2
21
0
2
2
1
)(
)()(),(lim
(72)

where (.)N is the cumulative function of the standard
normal distributions.

1
2
ln
,
2
ln
(73)
Theorem 5: Let
CRRn,
delta of a standard European put option with extended tree
and
BS

2
ln
)),()((2))((
d
u
S
K
n
Tr
K
S
d
nndnf
y

(75)
and m is the largest integer which satisfies
KdSu
mnm
.
IV. NUMERICAL EXAMPLES AND RESULTS
This section presents some numerical examples and

results generated as follows;
A. Numerical examples
Example 1: Consider a standard option that expires in
three months with an exercise price of 65$ . Assume that the
underlying stock pays no dividend, trades at 60$ , and has a
volatility of %30 per annum. The risk-free rate is %8 per
annum.
We compute the values of both European and American
style options using Binomial model against Black-Scholes
model as we increase the number of steps with the following
parameters:
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The Black-Scholes price for call and put options are
1334.2 and 8463.5 respectively.
The results obtained are shown in Table I below.
Example 2: Binomial pricing results in a call price of

87.31$ and a put price of 03.5$ . The interest rate is %5 ,
the underlying price of the asset is 100$ and the exercise
price of the call and the put is 85$ . The expiration date is in
three years. What actions can an arbitrageur take to make a
riskless profit if the call is actually selling for 35$ ?

Comparing Black-Scholes and Binomial Option Pricing Models

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Comparing Black-Scholes and Binomial Option Pricing Models