Binomial Trees
Chapter 11
Options, Futures, and Other
Derivatives, 7th Edition, Copyright ©
John C. Hull 2008
1
A Simple Binomial Model
A stock price is currently $20
 In 3 months it will be either $22 or $18

Stock Price = $22
Stock price = $20
Stock Price = $18
Options, Futures, and Other Derivatives
7th Edition, Copyright © John C. Hull
2008
2
A Call Option (Figure 11.1, page 238)
A 3-month call option on the stock has a strike
price of 21.
Stock Price = $22
Option Price = $1
Stock price = $20
Option Price=?
Stock Price = $18
Option Price = $0
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7th Edition, Copyright © John C. Hull
2008
3
Setting Up a Riskless Portfolio

Consider the Portfolio: long D shares
short 1 call option
22D – 1
18D

Portfolio is riskless when 22D – 1 = 18D
or
D = 0.25
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7th Edition, Copyright © John C. Hull
2008
4
Valuing the Portfolio
(Risk-Free Rate is 12%)
The riskless portfolio is:
long 0.25 shares
short 1 call option
 The value of the portfolio in 3 months is
22  0.25 – 1 = 4.50
 The value of the portfolio today is
4.5e – 0.120.25 = 4.3670

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7th Edition, Copyright © John C. Hull
2008
5
Valuing the Option
The portfolio that is
long 0.25 shares
short 1 option
is worth 4.367
 The value of the shares is
5.000 (= 0.25  20 )
 The value of the option is therefore
0.633 (= 5.000 – 4.367 )

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7th Edition, Copyright © John C. Hull
2008
6
Generalization (Figure 11.2, page 239)
A derivative lasts for time T and is
dependent on a stock
S0
ƒ
S0u
ƒu
S0d
ƒd
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7th Edition, Copyright © John C. Hull
2008
7
Generalization
(continued)

Consider the portfolio that is long D shares and
short 1 derivative
S0uD – ƒu
S0dD – ƒd

The portfolio is riskless when S0uD – ƒu = S0dD – ƒd
or
ƒu  f d
D
S 0u  S 0 d
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2008
8
Generalization
(continued)
Value of the portfolio at time T is
S0uD – ƒu
 Value of the portfolio today is
(S0uD – ƒu)e–rT
 Another expression for the portfolio
value today is S0D – f
 Hence
ƒ = S0D – (S0uD – ƒu )e–rT

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7th Edition, Copyright © John C. Hull
2008
9
Generalization
(continued)

Substituting for D we obtain
ƒ = [ pƒu + (1 – p)ƒd ]e–rT
where
e d
p
ud
rT
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2008
10
p as a Probability


It is natural to interpret p and 1-p as probabilities
of up and down movements
The value of a derivative is then its expected
payoff in a risk-neutral world discounted at the
risk-free rate
S0
ƒ
S0u
ƒu
S0d
ƒd
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7th Edition, Copyright © John C. Hull
2008
11
Risk-Neutral Valuation




When the probability of an up and down
movements are p and 1-p the expected stock price
at time T is S0erT
This shows that the stock price earns the risk-free
rate
Binomial trees illustrate the general result that to
value a derivative we can assume that the
expected return on the underlying asset is the riskfree rate and discount at the risk-free rate
This is known as using risk-neutral valuation
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7th Edition, Copyright © John C. Hull
2008
12
Original Example Revisited
S0u = 22
ƒu = 1
S0
ƒ


S0d = 18
ƒd = 0
Since p is the probability that gives a return on the
stock equal to the risk-free rate. We can find it from
20e0.12 0.25 = 22p + 18(1 – p )
which gives p = 0.6523
Alternatively, we can use the formula
e rT  d e 0.120.25  0.9
p

 0.6523
ud
1.1  0.9
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7th Edition, Copyright © John C. Hull
2008
13
Valuing the Option Using RiskNeutral Valuation
S0u = 22
ƒu = 1
S0
ƒ
S0d = 18
ƒd = 0
The value of the option is
e–0.120.25 (0.65231 + 0.34770)
= 0.633
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7th Edition, Copyright © John C. Hull
2008
14
Irrelevance of Stock’s Expected
Return


When we are valuing an option in terms
of the price of the underlying asset, the
probability of up and down movements in
the real world are irrelevant
This is an example of a more general
result stating that the expected return on
the underlying asset in the real world is
irrelevant
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7th Edition, Copyright © John C. Hull
2008
15
A Two-Step Example
Figure 11.3, page 242
24.2
22
19.8
20
18
16.2
Each time step is 3 months
 K=21, r=12%

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7th Edition, Copyright © John C. Hull
2008
16
Valuing a Call Option
Figure 11.4, page 243
D
22
20
1.2823
A
B
2.0257
18
24.2
3.2
E
19.8
0.0
C
0.0
F
16.2
0.0
Value at node B is
e–0.120.25(0.65233.2 + 0.34770) = 2.0257
 Value at node A is
e–0.120.25(0.65232.0257 + 0.34770) = 1.2823

Options, Futures, and Other Derivatives
7th Edition, Copyright © John C. Hull
2008
17
A Put Option Example
Figure 11.7, page 246
K = 52, time step =1yr
r = 5%
D
60
50
4.1923
A
B
1.4147
40
72
0
48
4
E
C
9.4636
F
32
20
Options, Futures, and Other Derivatives
7th Edition, Copyright © John C. Hull
2008
18
What Happens When an Option
is American (Figure 11.8, page 247)
D
60
50
5.0894
A
72
0
B
1.4147
40
48
4
E
C
12.0
F
32
20
Options, Futures, and Other Derivatives
7th Edition, Copyright © John C. Hull
2008
19
Delta
Delta (D) is the ratio of the change in the
price of a stock option to the change in
the price of the underlying stock
 The value of D varies from node to node

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7th Edition, Copyright © John C. Hull
2008
20
Choosing u and d
One way of matching the volatility is to set
u  es
Dt
d  1 u  e s
Dt
where s is the volatility and Dt is the length
of the time step. This is the approach used
by Cox, Ross, and Rubinstein
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2008
21
The Probability of an Up Move
p
ad
ud
 a  e rDt for a nondividend paying stock
 a  e ( r  q ) Dt for a stock index w here q is the dividend
yield on the index
a  e
( r  r f ) Dt
for a currency where rf is the foreign
risk - freerate
 a  1 for a futurescontract
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2008
22