Introduction to Binomial
Trees
Chapter 11
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright © John C. Hull 2013
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
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright © John C. Hull 2013
2
A Call Option (Figure 12.1, page 274)
A 3-month call option on the stock has a strike price of
21.
Up
Move
Stock Price = $22
Option Price = $1
Stock price = $20
Option Price=?
Down
Move
Stock Price = $18
Option Price = $0
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright © John C. Hull 2013
3
Setting Up a Riskless Portfolio


For a portfolio that is long D shares and a
short 1 call option values are
Up
Move
22D – 1
Down
Move
18D
Portfolio is riskless when 22D – 1
= 18D or D = 0.25
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright © John C. Hull 2013
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
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright © John C. Hull 2013
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 )
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright © John C. Hull 2013
6
Generalization (Figure 12.2, page 275)
A derivative lasts for time T and is
dependent on a stock
S
ƒ
Up
Move
Su
ƒu
Down
Move
Sd
ƒd
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright © John C. Hull 2013
7
Generalization (continued)


Value of a portfolio that is long D shares and short 1
derivative:
Up
Move
S0uD – ƒu
Down
Move
S0dD – ƒd
The portfolio is riskless when S0uD – ƒu = S0dD – ƒd or
ƒu  f d
D
S 0u  S 0 d
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright © John C. Hull 2013
8
Generalization
(continued)




Value of the portfolio at time T is
Su D – ƒu
Value of the portfolio today is
(Su D – ƒu )e–rT
Another expression for the
portfolio value today is S D – f
Hence
ƒ = S D – (Su D – ƒu )e–rT
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright © John C. Hull 2013
9
Generalization
(continued)
Substituting for D we obtain
ƒ = [ pƒu + (1 – p)ƒd ]e–rT
where
e rT  d
p
ud
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright © John C. Hull 2013
10
p as a Probability


It is natural to interpret p and 1−p as the probabilities
of up and down movements
The value of a derivative is then its expected payoff in
discounted at the risk-free rate
S0u
ƒu
S0
ƒ
S0d
ƒd
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright © John C. Hull 2013
11
Original Example Revisited
Su = 22
ƒu = 1
S
ƒ
Sd = 18
ƒd = 0
e rT  d e 0.120.25  0.9
p

 0.6523
ud
1.1  0.9
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright © John C. Hull 2013
12
Valuing the Option Using RiskNeutral Valuation
Su = 22
ƒu = 1
S
ƒ
Sd = 18
ƒd = 0
The value of the option is
e–0.120.25 [0.65231 + 0.34770]
= 0.633
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright © John C. Hull 2013
13
A Two-Step Example
Figure 12.3, page 280
24.2
22
19.8
20
18
16.2


Each time step is 3 months
K=21, r =12%
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright © John C. Hull 2013
14
Valuing a Call Option
Figure 12.4, page 280
D
22
20
1.2823
A
B
2.0257
18
E
F

19.8
0.0
C
0.0

24.2
3.2
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
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright © John C. Hull 2013
15
A Put Option Example
Figure 12.7, page 283
50
4.1923
60
1.4147
40
9.4636
72
0
48
4
32
20
K = 52, time step =1yr
r = 5%, u =1.32, d = 0.8, p = 0.6282
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright © John C. Hull 2013
16
What Happens When the Put
Option is American (Figure 12.8, page 284)
72
0
60
50
5.0894
The American feature
increases the value at node
C from 9.4636 to 12.0000.
48
4
1.4147
40
C
12.0
32
20
This increases the value of
the option from 4.1923 to
5.0894.
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright © John C. Hull 2013
17
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
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright © John C. Hull 2013
18
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 (1979)
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright © John C. Hull 2013
19
Assets Other than Non-Dividend
Paying Stocks

For options on stock indices, currencies
and futures the basic procedure for
constructing the tree is the same except
for the calculation of p
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright © John C. Hull 2013
20
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
( r  r ) Dt
ae f
for a currency where r f is the foreign
risk - free rate
a  1 for a futures contract
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright © John C. Hull 2013
21
Increasing the Time Steps


In practice at least 30 time steps are
necessary to give good option values
DerivaGem allows up to 500 time steps to
be used
Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright © John C. Hull 2013
22