Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.1
Introduction to
Binomial Trees
Chapter 9
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.2
A Simple Binomial Model
of Stock Price Movements
In a binomial model,the stock price
at the BEGINNING of a period
can lead to only 2 stock prices
at the END of that period
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.3
Option Pricing
Based on the Assumption of No Arbitrage Opportunities
Procedures:
Establish a portfolio of stock and option
Value the Portfolio
no arbitrage opportunities
no uncertainty at maturity
no risk with the portfolio
risk-free interest earned
Value the option
Risk-free interest = value of portfolio today
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.4A Simple Binomial Model:
Example
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,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.5
Stock Price = $22
Option Price = $1
Stock Price = $18
Option Price = $0
Stock price = $20
Option Price=?
A Call Option
A 3-month call option on the stock has a strike
price of $21,
Figure 9.1 (P.202)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.6
Consider the Portfolio,LONG D shares
SHORT 1 call option
Figure 9.1 becomes
Portfolio is riskless when 22D – 1 = 18D
or D = 0.25
22D – 1
18D
Setting Up a Riskless Portfolio
S0 = 20
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.7
Valuing the Portfolio
( with Risk-Free Rate 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 = 18 * 0.25
The value of the portfolio today is
4.50e-0.12*0.25=4.3670
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.8
Valuing the Option
The portfolio that is,LONG 0.25 shares
SHORT 1 call 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,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.9
Generalization
Consider a derivative
that lasts for time T and
that is dependent on a stock
Figure 9.2 (P.203) S
0u
u
S0d
d
S0
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.10
Generalization (continued)
Consider the portfolio that is,LONGD shares
SHORT 1 derivative
Figure 9.2 becomes
The portfolio is riskless when S0uD –?u = S0d D –?d
or when
dSuS
ff du
00
D
S0uD –?u
S0 dD –?d
DS0 - f
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.11
Generalization (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
S0 D – f
Hence,
= S0 D – (S0u D –?u )e–rT
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.12
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,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.13Generalization
(continued),
Proof with an Example
This is known as the No Arbitrage methodology
In our earlier example f=0.633 and D=0.25
If f<0.633,e.g,f=0.60
==> D S0-f=0.25*20-0.6=4.4>4.367
t = 0 ST=18 ST=22
Buy call -0.600 0 1
Sell D Shares 5.000 -18*0.25=-4.50 -22*0.25=-5.50
Lend 4.367 at r -4.367 4.50 4.50
Net Flows 0.033 0 0
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.14Generalization
(continued),
Proof with an Example
If f > 0.633,e.g,f=0.65
==> D S0-f=0.25*20-0.65=4.35<4.367
t = 0 ST=18 ST=22
Buy D Shares -5.000 18*0.25=4.50 22*0.25=5.50
Borrow 4.367 at r 4.367 -4.50 -4.50
Sell call 0.650 0 -1
Net Flows 0.017 0 0
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.15Irrelevanceof a Stock’s
Expected Return
When we are valuing an option in terms of the
underlying stock the expected return on the
stock is irrelevant
This is because in our formula
f = S0D - (S0uD-fu)e-rT
f does not involve the probability of the stock
moving up or down
It does not matter if we say the probability of an
increase is 50% or 80% we get the same result
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.16Irrelevance of a Stock’s E(R)
Proof,(continued)
Let’s call pu the probability of an increase in the stock
price and pd=1- pu the probability of a stock decrease
S0D - f = [pu(S0uD-fu)+ pd(S0dD-fd)] e-kT
where k is the appropriate rate for the risk involved
However,D is chosen such that
S0uD-fu= S0dD-fd and we know that pd=1- pu
Substituting,
S0D - f = [pu(S0uD-fu)+ (1- pu)(S0uD-fu)] e-kT
= (S0uD-fu)e-rT as since this is risk-free,k = r
No pu’s or pd’s left,thus probability of stock increase is
irrelevant
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.17
Irrelevance of a Stock’s E(R)
(continued)
The probability of an increase in the stock price is
irrelevant because options are redundant securities
In our two-step models,we form a risk-less portfolio
with stock and the option
Thus,the return/pay-off from the option is offset by
the return on the stock and the portfolio return is the
same in both states
Thus,no matter what the probability of a stock
increase,the answer is the same
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.18
Pseudo-Probabilities
However,we can solve for an implied or pseudo-
probability that will give us the correct answer
Using
in f = S0D - (S0uD-fu)e-rT we obtain
where
dSuS
ff du
00?
D
rT
du fppff
e])1([
du
dp rT
e
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.19Stock Expected Return using
Pseudo-Probabilities
What is the stock’s expected return using these
pseudo-probabilities?
Thus,the expected return on the stock is the
risk-free rate even though the return has some
variance,Hence,we are in a risk-neutral world.
rt
rtrt
rt
T
S
dSdSSdSSd
dSduS
du
d
dSdupSdSdpSupS
dpSdSupSdSpupSSE
e
e)e(
)(
e
)(
)1()(
0
00000
00
00000
00000




du dp
rt
e
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.20
Risk-Neutral Valuation
The variable? does NOT appear in the solution to
the option value using the Binomial Method
Thus,the solution is independent of all variables
affected by risk preference
The solution is therefore the same in a risk-free
world as it is in the real world
Hence,we can assume that the world is risk-
neutral
This leads to the principle of risk-neutral valuation
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.21Risk-Neutral Valuation
(Continued)
= [ 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
Figure 9.2 becomes S
0u
u
S0d
d
S0
p
1-p
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.22
Original Example Revisited
Figure 9.2 becomes
Since p is a risk-neutral probability
20e0.12*0.25 = 22p + 18(1-p) p = 0.6523
Alternatively,we can use the formulap
d
u d
rT
e e 0,1 2 0,2 5 0 9
1 1 0 9
0 6523
.
.,
.
S0u = 22
u = 1
S0d = 18
d = 0
S0
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.23
Original Example Revisited
Figure 9.2 becomes
S0u = 22
fu = 1
S0=20
f
S0d = 18
fd = 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,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.24
A Two-Step Example
20
22
18
24.2
19.8
16.2
6523.09.01.1 9.0ee
0,2 5*0,1 2

D
du
dp TrNote the change in the formula for p with a multi-step tree
Figure 9.3
Each time step is 3 months,r is still 12%.
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.25Valuing a Call Option
Figure 9.4,X=21
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,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.26
Generalization
Figure 9.6 (P.209)
S0
f
S0u
fu
S0d
fd
S0u2
fuu
S0d2
fdd
S0ud
fud
])1()1(2[
])1([
])1([
])1([
222
dduduu
tr
du
tr
ddud
tr
d
uduu
tr
u
fpfppfpef
fppfef
fppfef
fppfef




D?
D?
D?
D?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.27
A Put Option Example
Figure 10.7,X = 52,u = 1.2,d = 0.8,
r = 5%,and T = 2
f = e-0.05*2(0.62822*0+2*0.6282*0.3718*4+0.37182*20)
= 4.1923
72
0
48
4
32
20
60
1.4147
40
9.4636
50
4.1923 A
B
C
D
F
E 6282.08.02.1
8.0ee 1.0*0.05?

D
du
dp Tr
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.28
Figure 9.8,X = 52,u = 1.2,
d = 0.8,r = 5%,and T = 2
12 > 9.46376
What Happens When an
Option is American?
72
0
48
4
32
20
60
1.4147
40
12
50
5.0894 A
B
C
D
F
E
6282.08.02.1 8.0ee
1.0*0.05

D
du
dp Tr
Rule:
The value of the option at the final nodes is
the same for the European option
At earlier nodes it is the greater of
-- The value given by (9.2)
-- The payoff from early exercise
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.29
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
dSuS
ff du
00?
D
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.30
Using Binomial Trees in Practice
Realistically,only 1 or 2 time steps is not nearly
enough,Practitioners usually use 30 or more.
The values for u and d are usually determined
from the stock’s volatility
If stock prices are assumed to be lognormal (then
geometric returns are normal),then
u d
u
p
d
u d
t
r t

e
e
D
D
1
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.31
Importanceof a Stock’s Volatility
Let’s look at two examples,both as 3 month calls
with X=21 and where r = 0
Case I,S0u = 22 Case II,S0u = 26
fu = 1 fu = 5
S0=20 S0=20
f =0.5 f =2.5
S0d = 18 S0d = 14
fd = 0 fd = 0
In both cases,p=0.5
5.06.0 3.07.03.1 7.017.03.1 7.0ee
5.02.0 1.09.01.1 9.019.01.1 9.0ee
12/3*0
2
12/3*0
1


D
D
du
dp
du
dp
tr
tr
Vo
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
9.32
Assignment
9.11,9.4,9.5,9.6,9.8,9.9,9.10,9.11,9.12,
9.13
Assignment Questions