Chapter 11
Binomial Option Pricing: II
1
Understanding Early Exercise
 Options may be rationally exercised prior to
expiration.
 By exercising a Call option, the option holder

receives the stock and thus receives dividends,

pays the strike price prior to expiration (this has
an interest cost),

loses the insurance/flexibility implicit in the call.
2
Understanding Early Exercise
 By exercising a Put option, the option holder

receives the strike price and thus collects
interest on it going forward,

gives the stock away and thus stops
receiving dividends

loses the insurance/flexibility implicit in the put.
3
Call Option Without Dividends




Put-Call Parity says: C=P+S-Ke-rT
Since P>0, C>S-Ke-rT
And since K>Ke-rT , we have C>S-K
Therefore the Call price C is always
greater than the intrinsic value S-K (which
is what you would get if you exercised).
 Thus it is never optimal to exercise an
American Call option on a non-dividend
paying stock!!!
4
Call Option With Dividends
 Put-Call Parity says: C=P+S-Ke-rT-D
 (where D is the present value of all future dividends to be
received)
 So C = S-K + P + K-Ke-rT – D
 Thus C = S-K + P + K(1-e-rT) – D
 Therefore, when S is high and thus P close to zero, the Call
price C can be greater or lower than the intrinsic value S-K
(which is what you would get if you exercised) depending on
whether K(1-e-rT) is greater or lower than D, i.e. whether the
interest saved by delaying the payment K is greater or lower
than the lost dividends.
 Thus it can be optimal to exercise an American Call option
on a dividend-paying stock.
5
Put Option Without Dividends
 Put-Call Parity says: P=C+Ke-rT-S
 Thus P = K-S + C-K(1-e-rT)
 Therefore the Put price P can be greater
or lower than the intrinsic value K-S
depending on whether C is greater or lower
than K(1-e-rT).
 Thus even on a non-dividend paying stock,
it may be optimal to exercise a Put option !
(It usually happens for low values of S,
when C is very small)
6
Understanding
Risk-Neutral Pricing

A risk-neutral investor is indifferent between a sure thing and a
risky bet with an expected payoff equal to the value of the sure
thing.
 p* is the risk-neutral probability that the stock price will go up.
 The option pricing formula can be said to price options as if
investors are risk-neutral.
 Note that we are not assuming that investors are actually
risk-neutral, and that risky assets are actually expected to
earn the risk-free rate of return.
7
Pricing an option using risk-neutral
probabilities
 Consider a world populated by risk-neutral investors.
 Investors would only be concerned with expected returns, and
not about the level of risk.
 Hence investors would not “charge” or require a premium for
risky securities.
 Therefore risky securities would have the same expected rate of
return as riskless securities. In other words, every security is
returning the risk-free rate.
8
Fine point to think about:
 Now consider the following scenario: suppose a risky security is
expected to achieve great growth and great future profits. (You
can even assume that the security actually delivers on its
promises later, if you wish.)
 Even though risk-neutral investors might only usually require the
risk-free rate of return, doesn’t this mean that the expected rate
of return on this security will be much higher than the risk-free
rate? And if yes, doesn’t that invalidate what we’ve just
discussed?
 WHAT IS GOING ON HERE ? ANY POSSIBLE
RECONCILIATION OF THE TWO IDEAS?
9
Pricing an option using risk-neutral
probabilities
 If the stock is thus expected to earn the risk-free rate r and
pays out a continuous dividend yield d, then the risk-neutral
probability p* that the stock will go up must satisfy:
p*u edhS + (1 – p*)d edhS = erhS
 Solving for p* gives us
e( r d )h  d
p* 
u d
10
Pricing an option using real
probabilities
 Is option pricing consistent with standard discounted cash flow
calculations?
Yes. However, discounted cash flow is not used in practice to
price options.
 This is because it is necessary to compute the option price in
order to compute the correct discount rate.
11
Pricing an option using real
probabilities
 Suppose that the continuously compounded expected return
on the stock is  and that the stock does not pay dividends.
 If p is the true probability of the stock going up, p must be
consistent with u, d, and :
puS + (1 – p)dS = ehS
 Solving for p gives us
eh  d
p
ud
(11.3)
(11.4)
12
Pricing an option using real
probabilities
 Using p, the actual expected payoff to the option one
period hence is
eh  d
u  eh
pCu  (1  p)Cd 
Cu 
Cd
ud
ud
(11.5)
 At what rate do we discount this expected payoff?

It is not correct to discount the option at the
expected return on the stock, , because the
option is equivalent to a leveraged investment in
the stock and hence is riskier than the stock.
13
Pricing an option using real
probabilities
 Let us denote the appropriate per-period discount
rate for the option as .
 Since an option is equivalent to holding a portfolio
consisting of  shares of stock and B bonds, the
expected return on this portfolio is
S h
B
e 
e 
erh
S  B
S  B
h
 And since an option is equivalent to holding a
(11.6)
portfolio consisting of  shares of stock and B bonds,
the denominator is indeed the option price. This
confirms that in order to compute the discount rate ,
14
one needs to have the price of the option first.
Pricing an option using real
probabilities
 We can nevertheless now compute the option price
as the expected option payoff, discounted at the
appropriate discount rate, given by equation (11.6).
We thus need to compute:
h
h


e

d
u

e
h
Ce 
Cu 
Cd  (11.7)
ud
 ud


It turns out that this gives us the same option price as
performing the risk-neutral calculation.
15
Application: one-period example
 Assume the following information:
 =15%, S=$41, K=$40, S=0.30, r=8%, T=1,
h=1, and d=0. The “up-price” for the stock is
$59.954 and the “down-price” is $32.903.
 Compute the price of a European call option
by using:
• True probabilities
• Risk-neutral probabilities
Are the results identical?
16
Application: one-period example
 The true up probability is (with u = 59.954 / 41
= 1.4623 and d = 32.903 / 41 = 0.8025):
 p = [e0.15-0.8025] / [1.4623-0.8025] = 0.5446.
 The expected option payoff therefore is:
 0.5446($19.954) + (1-0.5446)($0) = $10.867
 We now need to compute the discount rate 
in order to get the option price today.
17
Application: one-period example
 For , we need the values of  and B.
  = (19.954-0)/(59.954-32.903) = 0.738
 B is the present value of loan needed to match cash
flows of S and option (pick the case where stock
goes down, since easier – but works for both):
 B = -e-0.08[(0.738)(32.903)-0] = - $ 22.405
 We thus have:
(41)0.738
(-22.405)
0.15
e 
e 
e0.08
(41)0.738  (-22.405)
(41)0.738  (-22.405)
h
 Or  = ln(1.386) = 32.64%
18
Application: one-period example
Finally, armed with , we can
compute the discounted
expected option value as:
C = e-.3264(10.867) = $ 7.839
19
Application: one-period example
 The risk-neutral probability of the stock going up is:
 p* = [e0.08-0.8025] / [1.4623-0.8025] = 0.4256.
 The call option price therefore is:
 C = e-0.08[(0.4256)(19.954)+(1- 0.4256)(0)] = $ 7.839
 This is exactly the price we obtained before.
20
The Binomial Tree and
Lognormality
 The usefulness of the binomial pricing model hinges on the
binomial tree providing a reasonable representation of the stock
price distribution.
 The binomial tree approximates a lognormal distribution.
21
The random walk model
 It is often said that stock prices follow a random walk.
 Imagine that we flip a coin repeatedly.




Let the random variable Y denote the outcome of the flip.
If the coin lands displaying a head, Y = 1; otherwise, Y =
– 1.
If the probability of a head is 50%, we say the coin is fair.
After n flips, with the ith flip denoted Yi, the cumulative
n
total, Zn, is
Z 
Y
n
i 1
i
(11.8)
 It turns out that the more times we flip, on average the
farther we will move from where we started.
22
The random walk model
 We can represent the process followed by Zn in term of the
change in Zn:
Zn – Zn-1 = Yn
or
Heads: Zn – Zn-1 = +1
Tails: Zn – Zn-1 = –1
23
The random walk model
 A random walk, where with heads, the change in Z is 1, and
with tails, the change in Z is – 1:
24
The random walk model
 The idea that asset prices should follow a random walk was
articulated in Samuelson (1965).
 In efficient markets, an asset price should reflect all available
information. In response to new information the price is
equally likely to move up or down, as with the coin flip.
 The price after a period of time is the initial price plus the
cumulative up and down movements due to new information.
25
Modeling prices as a random walk.
 The above description of a random walk is not a satisfactory
description of stock price movements. There are at least three
problems with this model:
1. If by chance we get enough cumulative down movements,
the stock price will become negative.
2. The magnitude of the move ($1) should depend upon how
quickly the coin flips occur and the level of the stock price.
3. The stock, on average, should have a positive return.
However, the random walk model taken literally does not
permit this.
 The binomial model is a variant of the random walk model that
solves all of these problems.
26
Continuously compounded returns
 The binomial model assumes that continuously compounded
returns are a random walk.
 Some important properties of continuously compounded returns:




The logarithmic function computes returns from prices.
The exponential function computes prices from returns.
Continuously compounded returns are additive.
Continuously compounded returns can be less than –100%.
27
The standard deviation of returns
 Suppose the continuously compounded return over month i is
rmonthly,i. The annual return is
rannual  
12
r
i 1 monthly ,i
 The variance of the annual return is
Var (rannual )  Var

12
r
i 1 monthly,i

(11.14)
28
The standard deviation of returns
 Suppose that returns are uncorrelated over time and that each
month has the same variance of returns. Then from equation
(11.14) we have
2 = 12  2monthly ,
where 2 is the annual variance.
The annual standard deviation is
 monthly


12
 If we split the year into n periods of length h (so that h = 1/n), the
standard deviation over the period of length h is
(11.15)
h   h
29
The binomial model
 The binomial model is
 Taking logs, we obtain
St  h  St e( r  d ) h  
h
(11.16)
ln( St  h / St )  (r  d)h   h


Since ln (St+h/St) is the continuously compounded return from
t to t+h, the binomial model is simply a particular way to
model the continuously compounded return.
That return has two parts, one of which is certain, (r–d)h, and
the other of which is uncertain, h.
30
The binomial model
 Equation (11.6) solves the three problems in the random walk:
1. The stock price cannot become negative.
2. As h gets smaller, up and down moves get smaller.
3. There is a (r – d)h term, and we can choose the probability of
an up move, so we can guarantee that the expected change
in the stock price is positive.
31
Lognormality and the binomial
model
 The binomial tree approximates a lognormal distribution, which
is commonly used to model stock prices.
 The lognormal distribution is the probability distribution that
arises from the assumption that continuously compounded
returns on the stock are normally distributed.
 With the lognormal distribution, the stock price is positive, and
the distribution is skewed to the right, that is, there is a chance
of extremely high stock prices.
32
Lognormality and the binomial
model
 The binomial model implicitly assigns probabilities to the
various nodes.
33
Lognormality and the binomial
model
 The following information is used to draw the graphs on the next slides:
initial stock price S = 100, volatility=30%, E(RS)=10%, T=1year, and we
divide the 1-year period into 25 periods (h=1/25). Note that n=25.
 The probability of the stock going up from one period to the next is
p=[eRh-d]/[u-d]
 Use u=e sqrt(h) and d=e sqrt(h) .
 Proba of reaching ith node =(number of ways to reach ith node) pn-i(1-p)i
where number of ways to reach ith node = n!/[(n-i)!i!]
34
Lognormality and the binomial
model
 The following graph compares the probability distribution for a
25-period binomial tree with the corresponding lognormal
distribution:
35
Lognormality and the binomial
model
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
node
p*
0.518384556
1
25
300
2300
12650
53130
177100
480700
1081575
2042975
3268760
4457400
5200300
5200300
4457400
3268760
2042975
1081575
480700
177100
53130
12650
2300
300
25
1
number of ways to reach ith node
448.168907
397.4901627
352.5421487
312.6768365
277.3194764
245.9603111
218.1472265
193.4792334
171.6006862
152.1961556
134.9858808
119.7217363
106.1836547
94.17645336
83.52702114
74.08182207
65.70468198
58.27482524
51.68513345
45.84060113
40.65696597
36.05949402
31.98190218
28.36540265
25.15785531
22.31301601
terminal stock price
0.00%
0.00%
0.00%
0.01%
0.07%
0.27%
0.84%
18.00%
2.11%
4.41%
16.00%
7.74%
11.51%
14.00%
14.59%
12.00%
15.81%
14.69%
10.00%
11.70%
8.00%
7.97%
4.63%
6.00%
2.28%
0.94%
4.00%
0.32%
2.00%
0.09%
0.02%
0.00%
0.00%
0
0.00%
0.00%
0.00%
probability to reach ith node
u=
d=
100
200
1.061837
0.941765
300
400
36
500
Lognormality and the binomial
model: Exercise
 Use the following information to draw a probability
distribution graph: initial stock price S = 100, volatility=30%,
E(RS)=10%, T=1year, and we divide the 1-year period into
3 periods (h=1/3). Note that n=3.
 The probability of the stock going up from one period to the
next is p=[eRh-d]/[u-d]
 Use u=e sqrt(h) and d=e sqrt(h) .
 Proba of reaching ith node =(number of ways to reach ith
node) x pn-i(1-p)i
where number of ways to reach ith node = n!/[(n-i)!i!]
37
Lognormality and the binomial
model: Solution
1
A
B
C
D
2
0
1
168.1380601
17.02%
3
1
3
118.9109944
41.07%
4
2
3
84.09651314
33.05%
5
3
1
59.47493384
8.86%
6
node
number of ways
terminal stock price probability to reach ith node
7
to reach ith node
45.00%
8
p*
9 0.5541659
40.00%
10
35.00%
11
=FACT(3)/(FACT(3-A3)*FACT(A3))
30.00%
12
13
=(G$3^(3-A5)*(G$4)^A5)*100 25.00%
=(EXP(0.1/3)-G3)/(G2-G3)
20.00%
E F
G
u= 1.18911
d= 0.840965
H
==>EXP(0.3*SQRT(1/3))
==>EXP(-0.3*SQRT(1/3))
=A$9^(3-A3)*(1-A$9)^A3*B3
15.00%
10.00%
5.00%
0.00%
0
50
100
150
200
38
Alternative binomial trees
 There are other ways besides equation (11.6) to construct a
binomial tree that approximates a lognormal distribution.
 An acceptable tree must match the standard deviation of
the continuously compounded return on the asset and
must generate an appropriate distribution as h  0.
 Different methods of constructing the binomial tree will
result in different u and d stock movements.
 No matter how we construct the tree, to determine the
risk-neutral probability, we use
e( r  d ) h  d
p* 
ud
and to determine the option value, we use
C = e–rh [p* Cu + (1 – p*) Cd]
39
Alternative binomial trees
 The Cox-Ross-Rubinstein binomial tree:

The tree is constructed as
u  e h
d  e h
(11.18)

A problem with this approach is that if h is large or  is small,
it is possible that erh > eh. In this case, the binomial tree
violates the restriction of u > e(r–d)h > d.

In practice, h is usually small, so the above problem does
not occur.
40
Alternative binomial trees
 The lognormal tree:

The tree is constructed as
( r  d  0.5 2 ) h   h
ue
(11.19)
( r d  0.5 2 ) h   h
d e
 Although the three different binomial models give different
option prices for finite n, as n   all three binomial trees
approach the same price.
41
Is the binomial model realistic?
 The binomial model is a form of the random walk model,
adapted to modeling stock prices. The lognormal random walk
model here assumes that
 volatility is constant,
 “large” stock price movements do not occur,
 returns are independent over time.
 All of these assumptions appear to be violated in the data.
42
Estimated Volatility
 We need to decide what value to assign to , which we
cannot observe directly.
 One possibility is to measure  by computing the standard
deviation of continuously compounded historical returns.

Volatility computed from historical stock returns is
historical volatility.

This is a reasonable way to estimate volatility when
continuously compounded returns are independent and
identically distributed.

If returns are not independent—as with some
commodities— volatility estimation becomes more
complicated.
43
Stock Paying Discrete Dividends
 Suppose that a dividend will be paid between times t and t+h
and that its future value at time t+h is D.
 The time t forward price for delivery at t+h is
Ft,t+h = St erh – D
 Since the stock price at time t+h will be ex-dividend, we create
the up and down moves based on the ex-dividend stock price:
Stu  ( St erh  D)e
h
Std  ( St erh  D)e 
(11.20)
h
44
Stock Paying Discrete Dividends
 When a dividend is paid, we have to account for the fact that the
stock earns the dividend.
(Sut + D) + erh B = Cu
(Sdt + D) + erh B = Cd
 The solution is
Cu  Cd
 u
St  Std
B

u
 rh  St Cd
e 
u
S
t

 Std Cu 
 rh


De

d
 St

Because the dividend is known, we decrease the bond
position by the PV of the certain dividend.
45
Problems with the discrete
dividend tree
1. The conceptual problem with equation (11.20) is that the stock price
could in principle become negative if there have been large
downward moves in the stock prior to the dividend.
2. The practical problem is that the tree does not completely
recombine after a discrete dividend.
The following tree, where a $5 dividend is paid between periods 1
and 2, demonstrates that with a discrete dividend, the order of up
and down movements affects the price.

In the third binomial period, there are six rather than four
possible stock prices.
46
Problems with the discrete
dividend tree
47
A binomial tree using the prepaid
forward
 Hull (1997) presents a method of constructing a tree for a
dividend-paying stock that solves both problems.
 If we know for certain that a stock will pay a fixed dividend, then
we can view the stock price as being the sum of two
components:
1. the dividend, which is like a zero-coupon bond with zero
volatility, and
2. the PV of the ex-dividend value of the stock, i.e., the prepaid
forward price.
48
A binomial tree using the prepaid
forward
 Suppose we know that a stock will pay a dividend D at time
TD < T, where T is the expiration date of the option.

We base stock price movements on the prepaid forward
price
Ft,PT  St  De r ( T
D
t )

The one-period forward price for the prepaid forward is

This gives us up and down movements of

Ft .t  h  Ft,PT erh
u  erh
d  erh 
h
h
However, the actual stock price at each node is given by
St = Ft,PT  De r ( T
D
t )
49
A binomial tree using the prepaid
forward
 A binomial tree constructed using the prepaid-forward method:
50