Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
10.1
Introduction to
Binomial Trees
Chapter 10
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
10.2
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 = $18
Stock price = $20
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
10.3
Stock Price = $22
Option Price = $1
Stock Price = $18
Option Price = $0
Stock price = $20
Option Price=?
A Call Option (Figure 10.1,page 200)
A 3-month call option on the stock has a strike price of
21,
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
10.4
? Consider the Portfolio,long D shares
short 1 call option
? Portfolio is riskless when 22D – 1 = 18D or
D = 0.25
22D – 1
18D
Setting Up a Riskless Portfolio
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
10.5
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,5th edition ? 2002 by John C,Hull
10.6
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 )
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
10.7
Generalization (Figure 10.2,page 202)
? A derivative lasts for time T and is
dependent on a stock
S0
?u
S0d
?d
S0
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
10.8
Generalization
(continued)
? Consider the portfolio that is long D shares and short
1 derivative
? The portfolio is riskless when S0uD – ?u = S0d D – ?d or
D ? ??? u dfS u S d
0 0
S0 uD – ?u
S0dD – ?d
S0–f
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
10.9Generalization
(continued)
? Value of the portfolio at time T is
S0u D – ?u
? Value of the portfolio today is
(S0u D–?u )e–rT
? Another expression for the
portfolio value today is S0D – f
? Hence
? = S0D – (S0u D – ?u )e–rT
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
10.10
Generalization
(continued)
? Substituting for D we obtain
? = [ p ?u + (1 – p )?d ]e–rT
where
p e d
u d
rT
? ?
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
10.11Risk-Neutral Valuation
? ? = [ p ?u + (1 – p )?d ]e-rT
? The variables p and (1 – p ) can be interpreted as the
risk-neutral probabilities of up and down movements
? The value of a derivative is its expected payoff in a
risk-neutral world discounted at the risk-free rate
S0u
?u
S0d
?d
S0
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
10.12
Irrelevance of Stock’s Expected
Return
When we are valuing an option in terms
of the underlying stock the expected
return on the stock is irrelevant
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
10.13Original Example Revisited
? Since p is a risk-neutral probability
20e0.12 ′0.25 = 22p + 18(1 – p ); p = 0.6523
? Alternatively,we can use the formula
6523.09.01.1 9.00, 2 50, 1 2 ?? ????? ?edu dep rT
S0u = 22
?u = 1
S0d = 18
?d = 0
S0
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
10.14
Valuing the Option
The value of the option is
e–0.12′0.25 [0.6523′1 + 0.3477′0]
= 0.633
S0u = 22
?u = 1
S0d = 18
?d = 0
S0
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
10.15A Two-Step Example
Figure 10.3,page 205
? Each time step is 3 months
20
22
18
24.2
19.8
16.2
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
10.16Valuing a Call Option
Figure 10.4,page 206
? 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
20
1.2823
22
18
24.2
3.2
19.8
0.0
16.2
0.0
2.0257
0.0
A
B
C
D
E
F
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
10.17
A Put Option Example; K=52
Figure 10.7,page 208
50
4.1923
60
40
72
0
48
4
32
20
1.4147
9.4636
A
B
C
D
E
F
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
10.18
What Happens When an Option
is American (Figure 10.8,page 209)
50
5.0894
60
40
72
0
48
4
32
20
1.4147
12.0
A
B
C
D
E
F
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
10.19
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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
10.20
Choosing u and d
One way of matching the volatility is to
set
where s is the volatility and dt is the
length of the time step,This is the
approach used by Cox,Ross,and
Rubinstein
u e
d e
t
t
?
? ?
s d
s d