1. Binomial Option Pricing
Hien Truong
ABSTRACT. This paper introduces the reader to a useful and very
popular technique for pricing an option by constructing a binomial
tree. An option, just like a stock or bond, is a security. A binomial
tree diagram represents different paths that might be followed by
the stock price over the life of an option. The option pricing model
introduces a very important principle known as risk-neutral
valuation by providing a power tool to understand arbitrage pricing
theory and probability. Hence, there are no arbitrage opportunities
in the market if, and only if, a risk-neutral world is applied.

2. CONTENT
1 Introduction 1
2 Preliminaries 1
2.1 Binomial Distribution aka Bernoulli Distribution 1
2.2 Key Words and Definition 2
2.3 No-Arbitrage Principle 3
2.4 Risk-Neutral Principle 3
3 Binomial Option Pricing Models 4
3.1 Binomial Model 5
3.2 Trinomial Model 10
3.3 Examples 11
3.4 The Limiting Model 12
4 Applications and Extensions 13
4.1 Control Variate Technique 13
4.2 Time Dependent Parameters 14
5 References 16

3. Page 1
1 Introduction
An option is a contract that gives the buyer the right, but not the obligation, to buy
or sell an underlying asset at a specific price on or before a certain date. An option,
just like a stock or bond, is a security. A security is any proof of ownership or debt
that has been assigned a value and may be sold. Thus, it represents an investment as
an owner, creditor or rights to ownership on which an individual hopes to gain
profit. An investor will use options to speculate and to hedge. To speculate, one
chooses to bet on the movement of the security. The advantage of options is that
there is no limit to making profit only when the market goes up. Because of the
versatility of options, money can be earned when the market goes down or even
sideways. To hedge, one would use options as insurance policies in order to reduce
the risk of adverse price movements in the underlying asset.
As useful as options are, how does one know how much they should be sold for
when the underlying asset has an unknown future price? This is where the binomial
option pricing model is exercised. It is a simple but powerful technique that can be
used to solve many complex option-pricing problems. Foundational, the model is
based on the assumption of no-arbitrage and risk-neutral valuation principle.
2 Preliminaries
2.1 Binomial Distribution aka Bernoulli Distribution
a. Combination Counting
Consider the set {a, b, c, d} containing the four different letters. We want to count
the number of distinct subsets of size two. In this case, we can list all of the subsets
of size two: {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, and {c, d}. We see that there are six
distinct subsets of size two.
For much larger sets, it would be tedious, if impossible, to enumerate all of the
subsets of a given size. However, there is a connection between counting subsets
and counting permutations that will allow us to derive the general formula for the
number of subsets.
Suppose that there is a set of n distinct elements from which we desire to choose a
subset containing k elements (1 ≤ k ≤ n). Next, we determine the number of
different subsets that can be chosen, in which, the arrangement of the elements in a
subset is irrelevant and each subset is treated as a unit. Each subset of size k chosen

4. Page 2
is called a combination of n elements taken k at a time, denoted as Cn,k, aka Binomial
Coefficient
,
!
!( )!
n k
n n
C
k k n k
 
  
 
b. Probability Distribution
The binomial distribution gives the discrete probability of obtaining exactly n
successes out of N Bernoulli trials, where the result of each Bernoulli trial is true
with probability p, and false with probability q = 1 – p. Note that the N trials are
independent. Knowing the result of one observation tells us nothing about the other
observations.
(n;N,p) n N nN
b p q
n
 
  
 
c. Mean, Variance, Standard deviation of Binomial Distribution
The binomial distribution has the following properties:
 Mean of the distribution = μ = np
 Variance of the distribution = σ2 = np(1-p) = npq
 Standard deviation of the distribution = σ = npq
2.2 Keys Words and Definitions
Before describing the model, it is important to understand some terminology.
 An American option is an option that can be exercised anytime during its life.
It allows its holders to exercise the option at any time prior to and including
its maturity date. The majority of exchange-traded options are American.
 A European option is an option that can only be exercised at its maturity. It
tends to sometimes trade at a discount to its comparable American option.
 The current stock price, S0, is the market value at which stocks are currently
being sold in the market.
 The strike price, K, is the price at which an option is exercised. In the case of a
call, it’s the fixed price at which the owner of the option has the choice to buy,
regardless of current market price. Or in the case of a put, it’s the fixed price
at which owner has the choice to sell, regardless of current market price.
 The time to expiration, T, is the time after which options contracts are no
longer valid.
 The volatility of the stock price, σ, is a variable showing the extent to which
the return of the underlying asset will fluctuate between now and the

5. Page 3
option’s expiration. Volatility, as expressed as a percentage coefficient within
option-pricing formulas, arises from daily trading activities.
 The risk-free rate, r, is the theoretical rate of return of an investment with no
risk of financial loss. Therefore, we assume there’s a frictionless financial
market in which there exists a risk free asset (zero coupon bond) and a risky
asset (a stock). We assume the market is free of arbitrage.
2.3 No-Arbitrage Principle
A situation, in which, all relevant assets are priced appropriately and there is no way
for one’s gains to outpace market gains without taking on more risk is one without
arbitrage. This principle is often informally restated as the phrases ‘there is no such
thing as free money’ or ‘one cannot get something for nothing’.
a. Monotonicity theorem
If portfolios A and B are such that in every possible state of the market at time T,
portfolio A is worth at least as much as portfolio B, then at any time t < T, portfolio
A is worth at least as much as portfolio B. If in addition, portfolio A is worth more
than portfolio B in only some states of the world, likewise at any time t < T, portfolio
A is worth more than portfolio B.
Proof. The proof of this theorem follows simply by applying the no-arbitrage
principle to a portfolio C, such that
the value of portfolio C = the value of (portfolio A – portfolio B)
This assumes portfolio C then has a non-negative value in all world states at time T
and at time t < T.
If portfolio A > portfolio B at time T then portfolio C can have a positive value at
time T and at time t < T. Otherwise, there would be the possibility of making money
from a portfolio of zero cost with no risk.
b. Arbitrage violations
Financial markets in practice have a tendency to challenge foundational
assumptions, and key challenges for the field have often arisen from occasions when
no-arbitrage arguments were violated. [Blyth, 2014]
2.4 Risk-Neutral Principle
The risk-neutral valuation principle states that an option, or other derivatives, can
be valued on the assumption that the world is risk neutral. This means that for
valuation purposes, we can use the following procedure:

6. Page 4
1. Assume that the expected return from all traded assets is the risk-free interest
rate
2. Value payoffs from the derivative by calculating their expected values and
discounting at the risk-free interest rate
a. Risk-Neutral World vs. Real World
In a risk-neutral world, the free simulation parameters like volatility are estimated
in a way that the theoretical price and the traded prices match. This way, we obtain
market consistent option prices for similar options, which are not traded. On the
contrary, in the real world, we simulate market values, which create a realistic
behavior. For example, one can match the historical drift, volatility and correlation
of the simulated asset to use for testing trading strategies, for optimizing portfolios
and minimizing hedging errors.
b. Application
 The real world simulation is used in risk-management, back-testing, and
portfolio optimization.
 The risk-neutral world simulation is used for market consistent option
pricing.
 A simulation based market risk management of an options portfolio requires
a real world simulation and within this simulation, a nested risk-neutral
valuation is implemented. [Grau, 2012]
3 Binomial Option Pricing Models
We will use the no-arbitrage and risk-neutral principles together with a simple
discrete time model of a stock price process, the binomial model, to price options.
Although this model is simple, it is very useful because the techniques we will
describe below carry through to more complicated models for a stock price process
under continuous time models, conditional expectations, martingales and etc.
Additionally, we follow these assumptions [Dickson, Hardy, and Waters, 2013]:
 The financial market is modelled in discrete time. Trades occur only at
specified time points. Changes in asset prices and exercise date for an option
can occur only at these same times.
 In each unit of time, the stock price either moves up by a predetermined
amount or moves down likewise.
 Investors can buy and sell assets without cost. These trades do not impact
the prices.

7. Page 5
 Investors can short sell assets, so that they can hold a negative amount of an
asset. This is achieved by selling an asset they don’t own, so the investor
owes the asset to the lender. Furthermore, we say that an investor is long in
an asset if the investor has a positive holding of the asset, and is short in the
asset if the investor has a negative holding.
3.1 Binomial Model
a. One-Step Binomial Model
The binomial model for option pricing is based upon a special case in which the
price of a stock over a period can either go up by u percent or down by d percent.
Assume no other outcomes are possible over the next period for the stock price.
Since S0 is the current price then the next period price will be either,
Su = S0 (1+u)
S0
Sd = S0 (1+d)
T0 T1
We will assume that u > d. If u = d then the stock price at T0 is not random and the
model will be uninteresting. Furthermore, we will start considering our risk-free
rate, r, and implement an important assumption of 0 < d < 1+r < u to rule out
arbitrage.
If d ≥ 1 + r, then everyone will begin with zero wealth at T0 and therefore borrow
money from the market to buy stock. Even in the worst case, the stock at T1 will be
worth enough to pay off debt and has a positive probability of being worth strictly
more since u > d ≥ 1 + r.
If u ≤ 1 + r, then everyone will begin by selling the stock short and invest the
proceeds in the money market. Even in best case, the cost of replacing it at T1 will be
less than or equal to the value of investment and since d < u ≤ 1 +r, there’s a
positive probability that the cost of replacing the stock will be strictly less than
value of investment.
Suppose that at T0, there’s a portfolio with h shares of the stock where the owner of
the portfolio sells one call with an expiration date of one period.
Thus, the payoffs on an option will be
max( ,0)
max( ,0)
u u
d d
P S K
P S K
 
 

8. Page 6
If the stock price goes up or down, the portfolio has a value of
0
0
(1 )
(1 )
u u
d d
V hS u P
V hS d P
  
  
Suppose h is chosen so that the portfolio has the same price whether the stock price
goes up or down. The value of h that achieves this condition is given by
0 0(1 ) (1 )
max( ,0) max( ,0)
u d
u d u d
u d u d
hS u P hS d P
P P S K S K
h
S S S S
    
   
 
 
Thus, given only S0, K, u, d, the ratio h can be determined. In particular, it doesn’t
depend upon the probability of a rise or fall. The value that h that make the value of
the portfolio independent of the stock price is called the hedge ratio. A portfolio that
is perfectly hedged is a risk-free portfolio so its value should grow at the risk-free
rate.
The current value of the hedged portfolio is the value of the stocks less the liability
involved with having written the call. If C represents the value of owning the call
then the liability involved with having written the call is –C. Therefore, the value of
the portfolio is 0hS C . After one period of growing at the risk-free rate, its value
will be  01 r hS C  , which is the same as      0 01 1u dhS u C hS d C     .
Solving for C gets
0
0
0
0
0
0
0
( (1 ) )
1
(1 )
1 1
1 (1 )
1 1
( )
1
(u r)
1
u
u
u
u
u
hS u P
C hS
r
hS u P
hS
r r
Pu
hS
r r
hS r u P
r
hS P
r
 
 


  
 
  
    
 


  


Noting that
0 0
0
(1 ) (1 )
(u d)
u d
u d
P P
h
S u S d
P P
S


  



,

9. Page 7
(P )(r u)
1
(r u) (r u)
1
( ) (u d)
1
(r d) (u r)
(u d) ( )
1
u d
u
u d
u d
P
P
u d
C
r
P P
u d
r
P P
u d
r
  
  

   
      

  
   

If
(r d)
(u d)


is denoted as p, then
(u )
1
(u d)
r
p

 

So, we can express the option price as
[p (1 p)P ]
1
u dP
C
r
 


(3.1)
The result represents the fair price of the option, given the evolution in the price of
the underlying asset. It is the value if it were to be held-as opposed to being
exercised.
b. Two-Step Binomial Model
We can extend the analysis to a two-step binomial tree. Based off of the one-step
binomial model, an additional time node is added
Su = S0 (1+u) Suu = S0 (1+u)2
S0 Sud = S0 (1+u)(1+d)
Sd = S0 (1+d) Sdd = S0 (1+d)2
T0 T1 T2

10. Page 8
The payoffs are
max( ,0)
max( ,0)
max( ,0)
max( ,0)
max( ,0)
u u
d d
uu uu
dd dd
ud ud
P S K
P S K
P S K
P S K
P S K
 
 
 
 
 
The stock price is still initially S0. During each time step, it either moves up to 1+ u
times its initial value or moves down to 1 + d times its initial value. We suppose that
the risk-free interest rate is still r and the length of the time step is now t years.
[p (1 p)P ] r t
u dC P e 
   (3.2)
 
 
r t
e d
p
u d




Repeated application of Equation (3.2) gives
[p (1 p)P ] r t
u uu udP P e 
   (3.3)
[p (1 p)P ] r t
d ud ddP P e 
   (3.4)
[p (1 p)P ] r t
u dC P e 
   (3.5)
Substituting from Equations (3.3) and (3.4) into (3.5), we get
 2 2 2
[p 2 1 (1 p) P ] r t
uu ud ddC P p p P e 
    
c. Multiple Time Steps
So far, we have seen one-step and two-step binomial trees. The results were:
For one-step, the price of an option is
[p (1 p)P ] r t
u dC P e 
  
For two-step, the price of an option is
 2 2 2
[p 2 1 (1 p) P ] r t
uu ud ddC P p p P e 
    

11. Page 9
If we continue by adding another step, we find
   
33 2 2 3
[p 3 1 3 (1 p) P 1 P ] r t
uuu uud udd dddC P p p P p p e 
      
Notice all steps contain
 
 
r t
e d
p
u d




We can prove that
 
0
1 n k k
n
knr t n k
u d
k
n
C e p p P
k

  

 
  
 

The binomial model with just one or two steps is unrealistically simple. However, if
we take a larger number of steps, say more than 50 steps, we can actually get a
reasonable model. Traders use software for the calculations. The most delicate point
in the procedures is the assumption that we know only certain values of the stocks
at time T will occur. The price of the option will depend on the choice. See [Hull,
2012] for more information.
3.2 Trinomial Model
a. Definition
Trinomial trees provide an effective method of numerical calculation of option
prices within Black-Scholes share pricing model [Hull, 2012]. Trinomial trees can be
built in a similar way to the binomial tree. To create the jump sizes u and d and the
transitional probabilities pu and pd in a binomial model we aim to match these
parameters to the first two moments of the distribution of our geometric Brownian
motion. Furthermore, there will be a third transition probability called pm that leads
toward the middle of the trinomial tree where the current stock price remains the
same.
The trinomial tree model is defined by
with probability pu
with probability 1 - pu - pd
with probability pd
 
 
 
 
S t u
S t t S t
S t d


   



12. Page 10
By applying the no-arbitrage principle, we derive our risk free rate as
1 r t
u d u dp p p u p d e 
    
Thus the value of the option will be
 r t
u u m m d dC e p P p P p P 
  
The trinomial model is the improved version of the binomial model that allows a
stock price to go up, down or stay the same with certain probabilities. It’s perceived
as more relevant to real life situations, as it is possible that the value of the stock
may not change over a time period, such as a month or a year. [Hull, 2012]
b. Figure
Su = S0 (1+u)
S0 Sm
Sd = S0 (1+d)
3.3 Examples
a. One –Step Binomial Model
Let u = +0.1, d = -0.1, r = 0.05, S0 = 100 and K = 95. Our model would look like
110 = S0 (1+u) = 100 (1.1)
100
90= S0 (1+d) = 100 (0.9)
T0 T1
The payoffs will be 15uP  and 0dP 

13. Page 11
Therefore
 
 
  
  
 
15 0
0.75
110 90
0.05 0.1 0.15
0.75
0.200.1 0.1
0.75 15 0.25 0 11.5
$10.71
1.05 1.05
h
p
C

 

 
  
 
  
  
b. Two-Step Binomial Model
Suppose we have a 6 months option with strike price, K = $21, current stock price,
S0 = 20. In two time steps of 3 months, the stock can go up or down by 10%. Lastly,
let the risk-free rate, r = 12%. Our model would look like
22 = S0 (1+u) 24.2 = S0 (1+u)2
20 19.8 = S0 (1+u)(1+d)
18 = S0 (1+d) 16.2 = S0 (1+d)2
T0 T1 T2
The payoffs at T2 will be
 
 
 
max 24.2 21,0 3.2
max 19.8 21,0 0
max 16.2 21,0 0
uu
ud
dd
P
P
P
  
  
  
Solving for p
 
 
30.12
12
0.9
0.07045
0.3522
1.1 0.9 0.20
e
p
 

  


14. Page 12
Therefore
 
       
2 2 2
32 0.122 2 12
[p 2 1 (1 p) P ]
[0.3522 3.2 2 0.3522 1 0.3522 0 (1 0.3522) 0 ]
$0.3738
r t
uu ud ddC P p p P e
C e
C
 
  
    
    

3.4 The Limiting Model
Since the no-arbitrage principle is very important, we need to derive upper and
lower bounds for option prices. If an option price is above the upper bound or below
the lower bound, then there will be profitable opportunities for arbitrageurs.
a. Upper bounds
As we recall, a American or European call option gives the holder the right to buy
one share of a stock for a certain price. No matter what happens, the option can
never be worth more than the stock. Therefore, the stock price is an upper bound to
the option price.
0C S
American put option gives the holder the right to sell one share of a stock for strike
price, K. No matter how low the stock price becomes, the option can never be worth
more than K.
C K
Because European option is exercised at maturity, the price cannot be worth more
than K. It follows that it cannot be worth more than the present value of K, and so
we discount K by using the risk-free rate r
rT
C Ke

If all of these relationships were not true, an arbitrageur could easily make a riskless
profit by buying the stock and selling the call option or by writing the option and
investing the proceeds of the sale at the risk-free interest rate.

15. Page 13
b. Lower bounds
Lower bounds only apply to European call options and put options on non-dividend-
paying stocks
For European call options,
0
rT
S Ke

For European put options,
0
rT
Ke S

4 Applications and Extensions
4.1 Control Variate Technique
a. Brief explanation of Black-Scholes-Merton model
Black-Scholes-Merton is another model of price variation over time of stocks to
determine the price of a European call option. The model assumes that the price of
heavily traded assets follow a geometric Brownian motion with constant drift and
volatility. When applied to a stock option, the model incorporates the constant price
variation of the stock, the time value of money, the option’s strike price and the time
to the option’s expiration for consistency.
One way of deriving the model is by allowing the number of time steps in a binomial
tree to approach infinity. Suppose that a tree with n time steps is used to value a call
option with initial stock price, S0, strike price, K, and life, T. Each step is of length
T/n. There will be j upward and n – j downward movements on the tree, where u is
the proportional up movement and d is the proportional down movement. The
payoff will be
 0max ,0j n j
S u d K

From the properties of the binomial distribution, the probability of exactly j upward
and n – j downward movements is given by
 
 
!
1
! !
n jjn
p p
n j j




16. Page 14
Hence, the expected payoff will be
 
   0
0
!
1 max ,0
! !
n
n jrT j j n j
j
n
P e p p S u d K
n j j
 

  


Then apply the risk-free rate and limiting bounds. Also, since u =
Tr
n
e and
d =
Tr
n
e

, our option price formulas are
   0 1 2
rT
C S N d Ke N d
 
where
 2
0
1
ln
2
S
r T
K
d
T


    
 
and
 2
0
2 1
ln
2
S
r T
K
d d T
T



    
   
The function N(x) is the cumulative probability distribution function for a
standardized normal distribution. For more information, see [Hull, 2012]
b. Explanation of the Technique
The control variate technique is used to improve the accuracy when pricing an
American option. It involves using the binomial tree to calculate the value of both
the American option, CA and the European option, CE. The Black-Scholes-Merton
price of the European option, CBS, mentioned above is also calculated. The error
when the tree is used to price the European option, CBS – CE, is assumed equal to the
error when the tree is used to price the American option. This gives the estimate of
the price of the American option as
 A BS E AOldC C C NewC  
For example, consider a 5 time steps binomial model problem that solves for CE =
$4.32, CBS = $4.08 and CA =$4.49. By using the control variate formula, we solve for
a more accurate result for CA
 $4.49 $4.08 $4.32 $4.25  

17. Page 15
4.2 Time Dependent Parameters
Up to now, we have assumed that the interest rate, risk-free rate and volatility of the
stock price are constants. In reality, they are usually assumed to be time dependent.
The values of these variable between times t and t + t are assumed to be equal to
their forward values. First, make the risk-free rate a function of time
   f t g t t
a e
   

for nodes at time t, where f(t) is the forward interest rate and g(t) is the forward
value of the risk-free interest rate between times t and t + t . This doesn’t change
the geometry of the tree because u and d don’t depend on a. These probabilities on
the branches stem from nodes at time t are
   
   
1
f t g t t
f t g t t
e d
p
u d
u e
p
u d
   
   




 

The rest of the way that we use the tree is the same as before, except that when
discounting between times t and t + t , we use f(t).
Making the volatility of the stock price a function of time in a binomial tree is more
challenging. One approach is to make the lengths of time steps inversely
proportional to the variance rate. Consequently, the values of u and d will always be
the same and the tree recombines. For more discussion, see [Hull, 2012]

18. Page 16
5 References
Blyth, Stephen. An Introduction to Quantitative Finance. New York: Oxford University
Press, 2014. pdf.
Brown, Angus. “A Risky Business: How to Price Derivatives.” +plus magazine. University of
Cambridge, 1 Dec 2008. Web. 20 Apr 2015.
Clifford, Paul and Oleg Zaboronski. “Pricing Options Using Trinomial Trees.” 17 Nov. 2008.
Web. 29 Apr 2015.
Conroy, Robert. “Binomial Option Pricing.” Darden. University of Virginia, 2003. Web. 29
Apr 2015.
Dickson, David C.M., Mary R. Hardy, and Howard R. Waters. Actuarial Mathematics for Life
Contigent Risks. New York: Cambridge University Press, 2013. Print.
Grau, Andreas. “What is the Difference Between Risk-Neutral Valuation and Real-World
Valuation?” Computer Aided Finance. 26 Mar 2012. Web. 18 Apr 2015.
Joshi, M. S. The Concepts and Practice of Mathematical Finance. Cambridge, UK: Cambridge
University Press, 2003. pdf.
Hull, John C. Options, Futures, and Other Derivatives. USA: Prentice Hall, 2012. pdf.
Investopedia, LLC. Investopedia. 2015. Web. 1 Apr 2015.