10-0
Finance 457
Introduction to Binomial Trees
10
Chapter Ten
McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-1
Finance 457
Chapter Outline
10.1 A one-step binomial model
10.2 Risk Neutral Valuation
10.3 Two-step binomial trees
10.4 A put example
10.5 American Options
10.6 Delta
10.7 Matching volatilities with u and d
10.8 Binomial Trees in Practice
McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-2
Finance 457
Prospectus:
• The last chapter
concerned itself
with the value of an
option at expiry.
McGraw-Hill/Irwin
• This section
considers the value
of an option prior to
the expiration date.
• A much more
interesting question.
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-3
Finance 457
An Option-Pricing Formula
• We will start with a
binomial option
pricing formula to
build our intuition.
McGraw-Hill/Irwin
• Then we will
graduate to the
normal
approximation to the
binomial for some
real-world option
valuation.
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-4
Finance 457
Binomial Option Pricing Model
Suppose a stock is worth $25 today and in one period
will either be worth $28.75 or $21.25.
The risk-free rate is 5%. What is the value of an at-themoney call option?
S0
S1
$28.75
$25
$21.25
McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-5
Finance 457
Binomial Option Pricing Model
1. A call option on this stock with exercise price of
$25 will have the following payoffs.
2. We can replicate the payoffs of the call option.
With a levered position in the stock.
S0
S1
c1
$28.75
$3.75
$21.25
$0
$25
McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-6
Finance 457
Binomial Option Pricing Model
Borrow the present value of $21.25 today and buy 1 share.
The net payoff for this levered equity portfolio in one period is
either $7.50 or $0.
The levered equity portfolio has twice the option’s payoff so the
portfolio is worth twice the call option value.
S0
( S1 - debt ) = portfolio c1
$28.75 - $21.25 = $7.50
$3.75
$25
$21.25 - $21.25 =
McGraw-Hill/Irwin
$0
$0
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-7
Finance 457
Binomial Option Pricing Model
The levered equity portfolio value today is
today’s value of one share less the present value
of a $21.25 debt:
$25  $21.25e
S0
 rf
( S1 - debt ) = portfolio c1
$28.75 - $21.25 = $7.50
$3.75
$25
$21.25 - $21.25 =
McGraw-Hill/Irwin
$0
$0
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-8
Finance 457
Binomial Option Pricing Model
We can value the option today as
half of the value of the levered
1
equity portfolio: C 0  $25  $21.25e  r f
2

S0
( S1 - debt ) = portfolio c1
$28.75 - $21.25 = $7.50
$3.75
$25
$21.25 - $21.25 =
McGraw-Hill/Irwin
$0
$0
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.

10-9
Finance 457
The Binomial Option Pricing Model
If the interest rate is 5%, the call is worth:


1
rf
C 0  $25  $21.25e
 $2.39
2
c0
S0
( S1 - debt ) = portfolio c1
$28.75 - $21.25 = $7.50
$3.75
$2.39 $25
$21.25 - $21.25 =
McGraw-Hill/Irwin
$0
$0
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-10
Finance 457
Binomial Option Pricing Model
The most important lesson (so far) from the binomial
option pricing model is:
the replicating portfolio intuition.
Many derivative securities can be valued by
valuing portfolios of primitive securities
when those portfolios have the same
payoffs as the derivative securities.
McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-11
Finance 457
Delta and the Hedge Ratio
• In the example just previous, we replicated the
payoffs of the call option with a levered equity
portfolio. This has everything to do with anything
for the rest of the semester, so let’s take a minute to
wrap our brains around it now rather than later.
• The delta of a stock option is the ratio of change in
the price of the option to the change in the price of
the underlying asset:
fu  fd

S 0u  S 0 d
• The delta is the number of units of stock we should
hold for each option shorted in order to create a
riskless hedge.
McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-12
Finance 457
Delta and the Hedge Ratio
• This practice of the construction of a riskless hedge
is called delta hedging.
• The delta of a call option is positive.
– Recall from the example:
fu  fd
$3.75  0
$3.75 1




S 0u  S 0 d
$28.75  $21.25 $7.5 2
• The delta of a put option is negative.
• Deltas change through time.
-This is a feature of options that we will return to in chapter 14
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10-13
Finance 457
The Risk-Neutral Approach to Valuation
S0u
p
fu
1- p
S0d
S0
f
fd
We could value f as the value of the replicating portfolio. An
equivalent method is risk-neutral valuation
f  e  rT [ p  f u  (1  p)  f d ]
McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-14
Finance 457
The Risk-Neutral Approach to Valuation
S0u
p
S0
p is the risk-neutral
probability of an “up” move.
f
S0 is the value of the
underlying asset today.
fu
1- p
S0d
fd
S0u and S0d are the values of the asset in the next period
following an up move and a down move, respectively.
fu and fd are the values of the derivative asset in the next
period following an up move and a down move, respectively.
McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-15
Finance 457
The Risk-Neutral Approach to Valuation
S0u
p
fu
S0
f
f  e  rT [ p  f u  (1  p)  f d ]
1- p
S0d
fd
• The key to finding p is to note that it is already impounded
into an observable security price: the value of S0:
S0  e
rf T
[ p  S 0 u  (1  p)  S 0 d ]
A minor bit of algebra yields:
McGraw-Hill/Irwin
d
p
ud
e
rf T
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-16
Example of the Risk-Neutral Valuation of a Call:
Finance 457
Suppose a stock is worth $25 today and in one period will
either be worth 15% more or 15% less. (u = 1.15; d = 0.85)
The risk-free rate is 5%. What is the value of an at-the-money
call option?
The binomial tree would look like this:
$28.75
p
$25.00
Cu
c0
1- p
$21.25
Cd
McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-17
Example of the Risk-Neutral Valuation of a Call:
Finance 457
The next step would be to compute the risk neutral
probabilities
rf T
e
d
p
ud
2
e .05  0.85

p
3
1.15  0.85
$25.00
c0
2
3
1
3
$28.75
Cu
$21.25
Cd
McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-18
Example of the Risk-Neutral Valuation of a Call:
Finance 457
After that, find the value of the call in the up state
and down state.
c0  e
rf T
1
2

   $3.75   $0 
3
3

$25.00
c0  $c20.39
2
3
1
3
$28.75
$3.75
$21.25
$0
McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-19
Risk-Neutral Valuation and the Replicating Portfolio
Finance 457
This risk-neutral result is consistent with valuing the
call using a replicating portfolio.
c0  e
rf T
1
2

   $3.75   $0   $2.39
3
3



1
rf
c0  $25  $21.25e
 $2.39
2
McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-20
Finance 457
More on the Binomial Model
• The binomial option pricing model is an alternative
to the Black-Scholes option pricing model—
especially given the computational efficiency of
spreadsheets such as Excel.
• In some situations, it is a superior alternative.
• For example if you have path dependency in your
option payoff, you must use the binomial option
pricing model.
– Path dependency occurs when how you arrive at a price (the path you
follow) for the underlying asset is important.
– One example of a path dependent security is a “no regret” call option
where the exercise price is the lowest price of the stock during the
option life.
McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-21
Finance 457
3 Period Binomial Option Pricing Example
• There is no reason to stop with just two
periods.
• Find the value of a three-period at-the-money
call option written on a $25 stock that can go
up or down 15 percent each period when the
risk-free rate is 5 percent.
McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-22
Finance 457
Three Period Binomial Process: Stock
Prices
$25.00  (1.15)
3
$25.00  (1.15)
$25.00  (1.15)
38.02
2
2/3
33.06
$25.00  (1.15) 2 (1  .15)
2/3
1/3
28.75
$25.00  (1.15)(1  .15)
2/3
28.10
2/3
1/3
$25
24.44
$25.00  (1.15)  (1  .15) 2
2/3
1/3
1/3
21.25
$25.00  (1  .15)
$25.00  (1.  15) 2
1/3
18.06
20.77
2/3
$25.00  (1  .15)3
1/3
15.35
McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-23
Finance 457
Three Period Binomial
Process: Call Option Prices
CU ,U ,U  max[$ 38.02  $25,0]
CU ,U 
1
2

e .05   $13.02   $3.10
3
3

33.06
CU 
e
.05
2
1
[  $9.25   $1.97]
3
3
28.75
2/3
$25
6.54
CD 
e
4.57
1/3
.05
2/3
CU , D  C D ,U 
e
.05
C0  e
McGraw-Hill/Irwin
C D ,U ,U  CU , D ,U  CU ,U , D 
max[$ 28.10  $25,0]
1/3
2
1
[  $3.10  2/3
$0]
3
3
24.44
2
1
[  $1.97   $0]
3
3 2/3
C D,D 
21.25
3.10
max[$ 20.77  $25,0]
1/3
2
1
e .05 [  0   0] 2/3
3
3
18.06
2
1
[  $6.50   $1.25]
3
3
28.10
CU , D , D  C D ,U , D  C D , D ,U 
1.98
1/3
.05
2/3
9.28
1/3
1.26
38.02
13.02
20.77
0
C D,D,D 
max[$ 15.35  $25,0]
0
1/3
15.35
Copyright © 2002 by The McGraw-Hill Companies, Inc.
0 All rights reserved.
10-24
Finance 457
Valuation of a Lookback Option
• When the stock price falls due to the stock market
as a whole falling, the board of directors tends to
reset the exercise price of executive stock options.
• To see how this reset provision adds value, let’s
price that same three-period call option (exercise
price initially $25) with a reset provision.
• Notice that the exercise price of the call will be the
smallest value of the stock price depending upon the
path followed by the stock price to get there.
McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-25
Finance 457
Three Period Binomial Process:
Lookback Call Option Prices
38.02
33.06
28.10
28.75
28.10
24.44
20.77
$25
28.10
24.44
20.77
21.25
20.77
18.06
15.35
McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-26
Three Period Binomial Process:
C
 max[$ 38.02  $25,0]
Lookback Call Option Prices
$38.02
U ,U ,U
Finance 457
33.06
$28.10
max[$ 28.10  $25,0]  3.10 $3.10
CU ,U , D 
CU , D,U  max[$ 28.10  $24.44,0]  3.66
28.75
24.44
CU , D, D  max[$ 20.77  $24.44,0]  0
$25
C D ,U ,U  max[$ 28.10  $21.25,0]  6.85
24.44
21.25
C D ,U , D  max[$ 20.77  $21.25,0]  0
C D , D ,U
max[$ 20.77  $18.06,0]  2.71
18.06
C D, D, D  max[$ 15.36  18.06,0]
McGraw-Hill/Irwin
$13.02
28.10
$3.66
$20.77
$0
$28.10
$6.85
$20.77
$0
$20.77
$2.71
$15.35
$0
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-27
Finance 457
Three Period Binomial Process:
Lookback Call Option Prices
CU ,U  e
.05
1
2


$
13
.
02


$
3
.
10
3

3


$38.02
$13.02
33.06
9.25
$28.10
$3.10
28.75
28.10
$3.66
CU , D  e
.05
C D ,U  e
.05
$25
1
2


$
3
.
66


$
0
3

3


3
2


$
6
.
85


$
0
3

3


24.44
2.33
$20.77
$0
$28.10
$6.85
24.44
4.35
$20.77
$0
21.25
$20.77
$2.71
C D,D  e
McGraw-Hill/Irwin
.05
1
2


$
2
.
71


$
0
3

3


18.06
1.72
$15.35
$0
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-28
Finance 457
Three Period Binomial Process:
Lookback Call Option Prices
$38.02
$13.02
33.06
1
2

CU  e .05   $9.25   $2.33
3
3

9.25
$28.10
$3.10
28.75
$28.10
6.61
$3.66
24.44
2.33
$20.77
$0
$25
5.25
1
2

C D  e .05   $4.35   $1.72
3
3

$28.10
$6.85
24.44
$20.77
4.35
21.25
$20.77
3.31
C0  e
McGraw-Hill/Irwin
.05
$0
1
2


$
6
.
61


$
3
.
31
3

3


$2.71
18.06
1.72
$15.35
$0
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-29
Finance 457
10.4 A put example
38.02
At the money. Before we start, we
expect value less than $5.25
2/3
33.06
2/3
1/3
28.75
28.10
2/3
2/3
1/3
$25
24.44
2/3
1/3
1/3
21.25
20.77
2/3
1/3
18.06
1/3
15.35
McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
10-30
Finance 457
10.4 A put example
pU ,U ,U  $0
pU ,U 
1
2

e .05   $0   $0
3
3

pU 
2
1
e .05 [  $0   $1.32]
3
3
2/3
$25
pD 
e
1.09
1/3
.05
2/3
pU , D  p D ,U 
e
.05
1/3
2
1
[  $0   $4.2/3
23]
3
3
24.44
2
1
[  $1.32   $5.72]
3
3 2/3
1.32
p D,D 
21.25
1/3
0
20.77
4.23
p D,D,D 
18.06
2
1
p0  e .05 [  $.42   $2.63]
3
3
28.10
pU , D , D  p D ,U , D  p D , D ,U 
2
1
e .05 [  $4.23   $9.652/3
]
3
3
1/3
McGraw-Hill/Irwin
p D,U ,U  pU , D,U  pU ,U , D 
0
1/3
2.63
2/3
33.06
28.75
0.43
38.02
0
5.72
1/3
15.35
Copyright © 2002 by The McGraw-Hill Companies, Inc.
All rights reserved.
9.65
10-31
Finance 457
10.4 A put example
• We can check our work with put-call parity:
c0  e
 rT
K  p0  S 0
$4.57  e .053 $25  $1.09  $25
McGraw-Hill/Irwin
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10-32
Finance 457
10.5 American Options
• At each node prior to expiry, compare immediate
exercise with the option’s value.
• If the proceeds of immediate exercise are higher
than the value of the option, exercise.
• Use the exercise value at that node to work
backward through the tree to find the value of an
American option at time 0.
McGraw-Hill/Irwin
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10-33
Finance 457
Optimal Early Exercise: American Put
38.02
0
2
1
$3.02  e .05 [  $1.32   $6.945]
3
3
2/3
33.06
2/3
0
1/3
28.75
2/3
0.43
2/3
1/3
$25
1.09
1.21
2/3
1/3
1.32
1/3
2.63
2/3
1/3
3.75
p0  e
.05
2
1
[  $.43   $3.75]
3
3
McGraw-Hill/Irwin
0
24.44
21.25
3.02
28.10
20.77
4.23
18.06
5.72
6.94
1/3
15.35
Copyright © 2002 by The McGraw-Hill Companies, Inc.9.65
All rights reserved.
10-34
Finance 457
Optimal Exercise of American Calls
• There are two cases to consider:
– A stock paying a known dividend yield
– The dollar amount of the dividend is known.
McGraw-Hill/Irwin
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10-35
Finance 457
Known Dividend Yield
S0 u3(1-d)
S0 u2
S0 u2(1-d)
S0 u
S0 u(1-d)
S0
S0
S0 (1-d)
S0 d
1
d
u
S0 d(1-d)
S0 d2
S0 d2(1-d)
Ex-dividend date
S0 d3(1-d)
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10-36
Finance 457
Known Dollar Dividend
(S0 u2– D) u
S0 u2
S0 u2– D
S0 u
(S0 u2– D) d
(S0 – D) u
S0
S0
S0 – D
S0 d
1
d
u
(S0 – D) d
(S0 d2 – D)u
S0 d2
S0 d2 – D
Ex-dividend date
(S0 d2 – D)d
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10-37
Finance 457
10.7 Matching Volatility with u and d
• In practice, we choose the parameters u and d to
match the volatility of the stock price.
ue
 δt
d e
McGraw-Hill/Irwin
 δt
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10-38
Finance 457
10.8 Binomial Trees in Practice
The BOPM is easily incorporated into Excel spreadsheets
After 30 or so steps, the results are excellent.
14% s
1 Maturity
1n
1 Dt
$ 25.00 S 0
$ 25.00 X
Stock Price
5% r f
Exercise Price
1.1500 u
Ordinary Call
0.8500 d
1.0500 a
66.67% Risk Neutral Prob
33.33% 1- R.N. Prob
McGraw-Hill/Irwin
$
$
$
28.75
25.00
3.75
$
$
$
21.25
25.00
-
q
$ 25.00
$ 25.00
$ 2.38
1- q
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