Outline
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A One-Step Binomial Model
Risk-Neutral Valuation
Two-Step Binomial Model
American Options
Delta
Matching Volatility With u and d
Options On Other Assets
Binomial Trees

Binomial Tree representing different possible
paths that might be followed by the stock
price over the life of an option

In each time step, it has a certain probability
of moving up by a certain percentage amount
and a certain probability of moving down by a
certain percentage amount
A one-step binomial model
A Simple Binomial Model
 A 3-month call option on the stock has a
strike price of 21.
 A stock price is currently $20
 In three months it will be either
$22 or
$18
Stock Price
= $22
Option Price = $1
Stock price = $20
Stock Price = $18
Option Price = $0
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 => D = 0.25
A riskless portfolio is therefore=> Long : 025 shares
Short : 1 call option
Valuing the Portfolio



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 = 4.5 or 18 ×0.25=4.5
The value of the portfolio today is (if
Rf=12%)
4.5e – 0.12×0.25 =
4.367
Valuing the Option

Stock price today = $20
Suppose the option price = f

the portfolio today is

0.25 × 20 – f = 5 – f
It follows that
5 – f =4.367
So
f=0.633 ---- the current
Generalization
S0 = stock price
S0u
u= percentage increase in
ƒu
S0
the stock price
S0d
d= percentage decrease in ƒ
ƒd
the stock price
ƒ= option on stock price whose current price
ƒu = payoff from the option(when price moves up)
ƒd= payoff from the option(when price moves down)
T= the duration of the option
Generalization (continued)

Consider the portfolio that is long D shares
and short 1 call option
S0uD – ƒu
S0dD – ƒd

The portfolio is riskless when S0uD – ƒu = S0dD –
ƒd or
ƒu  fd
D=
S 0u  S 0 d
Generalization (continued)




Value of the portfolio at time T is (S0uD –
ƒu)e–rT
The cost of setting up the portfolio is S0D –
f
–rT
Hence S0D – ƒ = (S
uD
–
ƒ
)e
ƒ u0  f d u
D=
ƒ = S0D –S 0(S
u 0uD
S 0 d– ƒu )e–rT
rT
e d
p=
ud
Substituting for
we obtain
Generalization (continued)
Ex. (see Figure11.1)
u=1.1, d=0.9,r=0.12,T=0.25,f
u=1, ƒd=0
0.123 / 12
e
 0.9
rT
= 0.6523
e d
p=
= 1.1  0.9
ud
ƒ = [ pƒu + (1 – p)ƒd ]e–rT
= [ 0.6523×1 + 0.3477×0 ]e–0.12×0.25
= 0.633


The option pricing formula in equation(11.2)
does not involve the probabilities of stock price
moving up or down.
The key reason is that we are not valuing the
option in absolute terms. We are calculating
its value in terms of the price of the underlying
stock. The probabilities of future up or down
movements are already incorporated into the
stock price.
Risk-Neutral Valuation


We assume p and 1-p as probabilities of up and
down movements.
Expected option payoff = p × ƒu + (1 – p ) × ƒd
The expected stock price at time T is
E(ST) = pSp0=ue+ (1-p)
S0d = pS0 (u-d) + S0d
d
ud
substituting
=> E(ST)=S0erT

rT
From this equation, we can see that the stock price grows
on average
at the risk-free rate. Because setting the probability of the
up
Risk-Neutral Valuation (continued)

In a risk-neutral world all individuals are
indifferent to risk. In such a world , investors
require no compensation for risk, and the
expected return on all securities is the riskfree interest rate.

Risk-neutral valuation states that we can
with complete impunity assume the world is
risk neutral when pricing options.
Original Example Revisited
* European 3-month call option
*Rf=12%
S0=20
ƒ
S0u = 22
ƒu = 1
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
E(ST)=S0erT => 22p + 18(1 – p ) = 20e0.12 ×0.25 => p =
0.6523
At the end of the three months, the call option
has a
0.6523 probability of being worth 1 and a 0.3477
probability of being worth zero. So the expect

Real world
compare with
Risk-Neutral world

It is not easy to know the correct discount
rate to apply to the expected payoff in the
real world.

Using risk-neutral valuation can solve this
problem because we know that in a riskneutral world the expected return on all
Two-Step Binomial Model



Stock price=$20 , u=10% , d=10%
Each time step is 3 months
r=12%, K=21
(Figure 11.3 Stock prices in a two-step tree)
22
24.2
19.8
20
18
16.2
Valuing a Call Option
(Figure 11.4)
D
22
20
1.2823 A
e
0.123 / 12
 0.9
= 0.6523
1.1  0.9
B
2.0257
18
24.2
3.2=max{24.2-21,0}
E
19.8
0=max{19.8-21,0}
F
16.2
0=max{16.2-21,0}
C
0

p=

Value at node B
= e–0.12×0.25(0.6523×3.2 + 0.3477×0) =
2.0257
Value at node A
rT
–0.12×0.25
e
=e
(0.6523×2.0257 +p =  d
ud
0.3477×0)

Generalization
Figure11.6 Stock and option prices in general two-step tree
S0
ƒ
S0u
ƒu
S0d
ƒd
S 0u 2
ƒuu
S0ud
ƒud
S 0d 2
ƒdd
Generalization (continued)
*The length of time step is Dt years

ƒ = e–r Dt[ pƒu + (1 – p)ƒd ]--------------(1)
(11.2)
e rDt  d
p=
ud
(11.3)



ƒu = e–r Dt[ pƒuu + (1 – p)ƒud ]-----------(2)
ƒd= e–r Dt[ pƒud + (1 – p)ƒdd ]------------(3)
ƒ = e–2rDt[ p2ƒuu +2p (1 – p)ƒud + (1 –
p)2ƒ ]
A Put Example (Figure 11.7)
K = 52, duration = 2yr, current price =
72
$50
D
0
u=20%, d=20%, r = 5%
60
=max{52-72,0}
B
50
4.1923
A
e0.051  0.8
p=
= 0.6282
1.2  0.8
1.4147
40
48
E
4=max{52-48,0}
C
9.4636
F
32
20=max{52-32,0}
ƒ = e–2rDt[ p2ƒuu +2p (1 – p)ƒud + (1 – p)2ƒdd ]
= e–2*0.05*1 [ 0.62822 ×0 + 2× 0.6282(1 – 0.6282) ×4 +
(1 –0.6282)2× 20]
= 4.1923
American Options

American options can be valued using a
binomial tree

The procedure is to work back through the
tree from the end to the beginning, testing at
each node to see whether early exercise is
optimal
American Options(Figure 11.8)
American Put option
K = 52, duration = 2yr, current price = $50,u=20%,
72
d=20%, r = 5%
D
0=max{52-72,0}
60
B
50
5.0894
1.4147
A
max{5.0894,52-50}
e0.051  0.8
p=
= 0.6282
1.2  0.8
max{1.4147,52-60}
40
48
4=max{52-48,0}
E
C
12.0
max{9.4636,52-40}
F
32
20=max{52-32,0}
Value at node B = e–-0.05×1(0.6282×0 +0.3718×4)=1.4147
Value at node C = e–-0.05×1(0.6282×4+ 0.3718×20)=9.4636
Value at node A = e–-0.05×1(0.6282×1.4147 +0.3718×12)=5.0894
Delta

Delta (D) is an important parameter in
the pricing and hedging of option.

The delta
(D) inofthestock
the change
price option
of a stock option
=
the change in the price of the underlying stock
Delta

(Figure 11.7) Delta At the end of the
first time step is
1.4147  9.4636
= 0.4024
60  40

At the 0end
time
 4 of the second 4
 20 step is
either72  48 = 0.1667 or 48  32 = 1

The two-step examples show that
Matching Volatility With u and d
In practice, when constructing a binomial tree
to represent the movements in a stock price.
We choose the parameters u and d to match
the volatility of the stock
s price.
Dt
u=e
d = 1 u = e s
s = volatility
Dt = the length of the time step
Dt
Options On Other Assets
Option on stocks paying a continuous
dividend yield
 Dividend yield at rate = q
 Total return from dividends and capital gains in a
risk-neutral world = r.
=> Capital gains return = r-q
e( r  q ) Dt  d
p=
 The stock expected value after one
step of
u time
d
length Dt is S0e(r-q) Dt
s Dt
u
=
e
; d =e
1 /(r-q)
u Dt =>
pS0u+(1-p)S0d=S
0
Options On Other Assets
ad
p=
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 rf is the foreign
risk - free rate
a = 1 for a futures contract
Options On Other Assets
ad
p =Dt
Option on stock indices ( a= e(r-q)
ud

)
European 6-month call option on an index level
when index level is 810,K=800, rf=5%,
σ=20%,q=2%
Node time
:
Dt = 0.25,
s Dt
u=e
=e
= 1.1052,
d = 1 / u = 0.9048,
0.2 0.25
0
( 0.05 0.02)0.25
a=e
= 1.0075
(1.0075  0.9048)
p=
= 0.5126
(1.1052  0.9048)
810
53.39
0.25
895.19
100.66
732.92
5.06
0.5
989.34
189.34
810.00
10
663.17
0.00
Options On Other Assets
ad
p
=
Option on currencies ( a= e(r-rf) Dt u  d

)
Three-step tree:American 3-month call.when the
value of the currency is 0.61,K=0.6,rf=5%,
σ=20%,foreign rf=7%
0
0.0833
0.1667
0.25
Dt = 0.0833,
u = e 0.2
0.0833
= 1.0594,
d = 1 / u = 0.9439,
a = e ( 0.050.07)0.0833 = 0.9983
(0.9983  0.9439)
p=
= 0.471
(1.0594  0.9439)
0.61
0.019
0.632
0.033
0.589
0.007
0.654
0.054
0.61
0.015
0.569
0.00
0.677
0.077
0.632
0.032
0.589
0.00
0.550
0.00
Options On Other Assets
ad
p
=
Option on futures ( a= 1 u  d

)
Three-step tree: American 9-month put. when the
futures price is 31,K=30,rf=5%, σ=30%
0
0.25
Dt = 0.25,
u = e 0.3
0.25
= 1.1618,
d = 1 / u = 0.8607,
a =1
(1  0.8607)
p=
= 0.4626
(1.1618  0.8607)
31
2.84
36.02
0.93
26.68
4.54
0.5
41.85
0
31
1.76
22.97
7.03
0.75
48.62
0
36.02
0
26.68
3.32
19.77
10.23