Chapter 12
Binomial Trees
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
2
A Call Option
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
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
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
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 (4.367=5.000-0.633)
6
Generalization
A derivative lasts for time T and is
dependent on a stock
S0
ƒ
S0u
ƒu
S0d
ƒd
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
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
S0D – f= (S0uD – ƒu)e–rT
Option price: ƒ( = S0D – (S0uD – ƒu )e–rT
9
Generalization
(continued)

Substituting for D we obtain
ƒ = [ qƒu + (1 – q)ƒd ]e–rT
where
e d
q
ud
rT
10
q as a Probability




As long as d=<erT =<u : 0=<q=<1.
It is natural to interpret q and 1-q as probabilities of up and
down movements. Actually, q is called risk-neutral
probability
To value a derivative we can also discount the expected
return on the underlying asset at the risk-free rate.
However, the expectation shall be calculated using the
risk-neutral probability q.
S0
ƒ
S0u
ƒu
S0d
ƒd
11
Risk-Neutral Valuation




The real world is not risk neutral. In the real world,
investors are in general risk-averse.
If p is the real-world probability of an up movement, we
have: S0 erT < Ep(ST ) =pS0u+(1-p)S0d.
Thus, risk preferences will affect the price of the
underlying asset.
Each probability measure defines a world. In the real
world, the measure is P(p for price up and 1-p for price
down)
12
Risk-Neutral Valuation



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
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Recall, investors are not risk-neutral in the real world.
Can we construct another world? Where investors are risk-neutral
but stock price and future scenarios of stock prices do not change?
We could distort the real-world probability measure (P).
Denote q as the probability of price up-move in the new world, if
investors are risk-neutral in this world, we must have:
Eq(ST )=qS0u+(1-q)S0d=qS0(u-d)+S0d=S0erT
That is, stock price earns the risk-free rate in the new world.
This gives us q=(erT –d )/(u-d)
This world defined by Q (q and 1-q) is called the risk-neutral world. Q is
called risk-neutral probability.
13
Risk-Neutral Valuation



If investors are risk-averse, for a risky asset, current
value is its expected future value discounted by: risk free
rate+risk premium
If stock price and probability of future stock prices are
given, we can back out its risk premium.
An option is also a risky asset. However the risk
premium for an option is not the same as the one for the
underlying stock. How to price an option?
14
Risk-Neutral Valuation


If investors are risk-neutral, life is easy: the price of any
asset is the its expected future value discounted by riskfree rate.
Thus, in a risk-neutral world, we can price an option by:
 Calculated its expected future payoff using risk
neutral probability
 Discount the expected future payoff by risk-free rate

How to find the risk-neutral world? Using the
given stock price: qS0(u-d)+S0d=S0erT

Option price: ƒ = [ qƒu + (1 – q)ƒd ]e–rT
15
Risk-Neutral Valuation





Application of risk-neutral valuation:
First step, find out probabilities (q and 1-q) in the riskneutral world:
Eq(ST )=qS0u+(1-q)S0d=qS0(u-d)+S0d=S0erT
This gives us q=(erT –d )/(u-d)
Then, discount expected future payoff of the derivatives
using the risk-free rate in the risk-neutral world to get the
option price
ƒ =Eq (option payoff) e–rT = [ qƒu + (1 – q)ƒd ]e–rT
16
Original Example Revisited
S0u = 22
ƒu = 1
S0
ƒ


S0d = 18
ƒd = 0
Since q 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 = 22q + 18(1 – q )
which gives q = 0.6523
Alternatively, we can use the formula
e rT  d e 0.120.25  0.9
q

 0.6523
ud
1.1  0.9
17
Valuing the Option Using Risk-Neutral
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
18
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
19
A Two-Step Example
24.2
22
19.8
20
18
16.2


Each time step is 3 months
K=21, r=12%
20
A Two-Step Example

Find out q in each step
erT  d 2 e0.120.25  0.9
q2 

 0.6523
u2  d 2
1.1  0.9
erT  d1 e0.120.25  0.9
q1 

 0.6523
u1  d1
1.1  0.9
21
Valuing a Call Option
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(0.57613.2 + 0.42390) = 2.0257
 Value at node A is
e–0.120.25(0.57612.0257 + 0.42390) = 1.2823

22
A Put Option Example
K = 52, time step =1yr
r = 5%
D
60
50
4.1923

A
Value at node B is
e–0.05
(0.62820+0.37184)=1.4147
 Value at node C is:
e–0.05(0.62824 + 0.371820)
=9.4636
B
1.4147
40
72
0
48
4
E
C
9.4636
F
32
20
23
A Generalization
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Length of the time step is Dt years.
Values of option after first step: fu fd
Values of option after second step fuu fud fdd
q=(erDt – d)/(u-d)
fu = e–rDt[ qƒuu + (1 – q)ƒud ]
fd = e–rDt[ qƒud + (1 – q)ƒdd ]
ƒ = e–rDt[ qƒu + (1 – q)ƒd ]
Or: f= e–2rDt (q^2ƒuu + 2q(1-q) ƒud +(1-p)^2 ƒdd )
24
What Happens When the Put
Option is American
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
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At node B, the value of the option without early exercise is 1.4147,
whereas the payoff from early exercise is negative (=-8).
Early exercise is not optimal at node B, and the value of the option
at this node is 1.4147.
At node C, the value of the option without early exercise is 9.4646,
whereas the payoff from early exercise is 12.
Early exercise is optimal in this case and the value of the option at
this node is 12.
At the initial node A, the value of the option without early exercise is
e-0.05*1 (0.6282*1.4147+0.3718*12)=5.0894
The payoff from early exercise is 2. Early exercise is not optimal.
The value of the option is therefore 5.0894.
25
What Happens When the Put Option is
American (Figure 12.8, page 264)
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.
26
Delta

Delta (D) is the ratio of the change in the price
of a stock option to the change in the price of
𝑓 −𝑓
the underlying stock: 𝑢 𝑑
𝑆0 𝑢−𝑆0 𝑑


It is the number of units of the stocks we should
hold for each option shorted in order to create a
riskless profit.
The value of D varies from node to node
27
Delta of a Call Option
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
At node B: D(3.20)/(24.219.8)0.7273
At node C: D =(0-0)/(19.8-16.2)=0
At node A: D =(2.0257-0)/(22-18)=0.5056
28
Delta of a Put Option
K = 52, time step =1yr
r = 5%
D
60
50
4.1923
A

At node B: D(04)/(7248)0.1667
At node C: D =(4-20)/(48-32)=-1
At node A: D (6040)/(1.4147
9.4636)2.4848
B
1.4147
40
72
0
48
4
E
C
9.4636
F
32
20
29