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Binomial Trees
Chapter 11
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
11.1
A Simple Binomial Model


A stock price is currently $20
In three months it will be either $22 or $18
Stock Price = $22
Stock price = $20
Stock Price = $18
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
11.2
A Call Option (Figure 11.1, page 242)
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
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
11.3
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
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
11.4
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
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
11.5
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
f=0.633 (= 5.000 – 4.367 )
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
11.6
Generalization (Figure 11.2, page 243)
A derivative lasts for time T and is
dependent on a stock
S0
ƒ
S0u
ƒu
S0d
ƒd
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
11.7
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
Δ is the ratio of the change in the option price to the
change in the stock price
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
11.8
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
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
11.9
Generalization
(continued)

Substituting for D we obtain
ƒ = [ pƒu + (1 – p)ƒd ]e–rT
where
e d
p
ud
rT
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
11.10
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
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
11.11
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 riskfree 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
risk-free rate and discount at the risk-free rate
This is known as using risk-neutral valuation
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
11.12
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
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
11.13
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
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
11.14
Irrelevance of Stock’s Expected
Return


When we are valuing an option in terms of the
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
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
11.15
A Two-Step Example
Figure 11.3, page 246
24.2
22
19.8
20
18
16.2


Each time step is 3 months
K=21, r=12%
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
11.16
Valuing a Call Option
Figure 11.4, page 247
D
22
20
1.2823
A
B
2.0257
18
E

19.8
0.0
C
0.0

24.2
3.2
F
16.2
0.0
Value at node B
= e–0.120.25(0.65233.2 + 0.34770) = 2.0257
Value at node A
= e–0.120.25(0.65232.0257 + 0.34770)
= 1.2823
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
11.17
Generalization
D
S0u
S0
f
A
fu
S0d
S0uu
fuu
B
E
S0ud
fud
C
fd
S0dd
fdd
F
fu = e-rT (p fuu + (1-p) fud)
fd = e-rT (p fud + (1-p) fdd)
f = e-rT (p fu + (1-p) fd), and substituting we have
f = e-2rT [p2 fuu + 2p(1-p) fud + (1-p)2 fdd]
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
11.18
A Put Option Example; K=52
Figure 11.7, page 250
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
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
32
20
11.19
What Happens When an Option is
American (Figure 11.8, page 251)
D
60
50
5.0894
A
B
1.4147
40
72
0
48
4
E
C
At each node the value of the option is the 12.0
greater of:
F
32
20
1. The value given by ƒ = [ pƒu + (1 – p)ƒd ]e–rT
2. The payoff from early exercise
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
11.20
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
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
11.21
The Probability of an Up Move
p
ad
ud
a  e rDt for a nondividen d paying stock
a  e ( r  q ) Dt for a stock index wher e q is the dividend
yield on the index
ae
( r  r f ) Dt
for a currency w here r f is the foreign
risk - free rate
a  1 for a futures contract
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005
11.22