Agenda
An Investment Criterion
Incorporating Real Options
1.
2.
3.
4.
Objectives
Option Pricing
Real Options
New Criterion
James Alleman, Hirofumi Suto &
Paul Rappoport
Experts Dialogue: Managing Risk in the Competitive Environment
International Telecommunication Union
Geneva, Switzerland
28-29 October 2004
Copyright © 2003 & 2004 James Alleman, All rights reserved
Alleman, Suto, & Rappoport
Agenda
1.
2.
3.
4.
5.
2
1. Objectives
Objectives
Option Pricing
Real Options
New Criterion
Application
• Develop an Investment Criterion
Incorporating Real Options
or
• A Simple Decision-Making
Criterion Under Uncertainty
Alleman, Suto, & Rappoport
3
Alleman, Suto, & Rappoport
Agenda
1.
2.
3.
4.
5.
Agenda
Objectives
Option Pricing
Real Options
New Criterion
Application
Alleman, Suto, & Rappoport
4
1. Objectives
2. Option Pricing
2.1 Call Option
2.2 Binomial Lattice Model
2.3 Continuous Additive Model
3. Real Options
4. New Criterion
5. Application
5
Alleman, Suto, & Rappoport
6
1
Agenda
2.1 Call Option
1. Objectives
2. Option Pricing
• The right to buy a stock, not obligation
2.1 Call Option
2.2 Binomial Lattice Model
2.3 Continuous Additive Model
– At certain price = K: exercise price
– At certain time = T: time to expiration
3. Real Options
4. New Criterion
5. Application
Alleman, Suto, & Rappoport
7
Alleman, Suto, & Rappoport
2.1 Call Option
Agenda
1. Objectives
2. Option Pricing
• Example: IBM Stock Call Option
2.1 Call Option
2.2 Binomial Lattice Model
2.3 Continuous Additive Model
– Exercise Price K: $23
– Time to Expire T: 1 year
3. Real Options
4. New Criterion
5. Application
• How much would you pay for this option
if the stock is traded at $20 now?
Alleman, Suto, & Rappoport
9
2.2 Binomial Lattice Model
10
• 2.2.2 Payoff of Call Option (K= $23)
$34
Stock
Call Option
p = .5
$20
1-p = .5
Alleman, Suto, & Rappoport
2.2 Binomial Lattice Model
• 2.2.1 Stock behavior
Stock
8
p = .5
$20
$C?
$13
1-p = .5
$34
max($34-$23,0)=$11
Exercise
$13
max($13-$23,0)=$0
Not Exercise
t=0
Alleman, Suto, & Rappoport
t=1
11
t=0
Alleman, Suto, & Rappoport
t=1
12
2
2.2 Binomial Lattice Model
2.2 Binomial Lattice Model
• 2.2.3 One Price Principle
• 2.2.2 Payoff of Call Option (K= $23)
Call Option
Payoffs are same
$11
p = .5
$C?
Prices should be same
1-p = .5
t=0
$0
t=1
Alleman, Suto, & Rappoport
13
One price principle,
No arbitrage principle
Alleman, Suto, & Rappoport
2.2 Binomial Lattice Model
2.2 Binomial Lattice Model
• 2.2.4 Replicating Portfolio
1.7x
$x of Stock
$b of Bond
$34
+
x
$20
• 2.2.4 Replicating Portfolio
1.08b
b
$x of Stock
$13
Alleman, Suto, & Rappoport
$b of Bond
1.7x
+
Alleman, Suto, & Rappoport
Replicating Portfolio
1.08b
$x of Stock
+
1.08b
1.08b
b
1.08b
0.65x
Replicating Portfolio
1.7x+1.08b
$b of Bond
1.7x
x
x+b
1.7x+1.08b = 11
x+b
0.65x+1.08b
Alleman, Suto, & Rappoport
16
• 2.2.4 Replicating Portfolio
b
0.65x
1.08b
2.2 Binomial Lattice Model
• 2.2.4 Replicating Portfolio
x
1.08b
b
0.65x
2.2 Binomial Lattice Model
$x of Stock
+
x
15
$b of Bond
1.7x
1.08b
0.65x
14
0.65x+1.08b = 0
17
Alleman, Suto, & Rappoport
18
3
2.2 Binomial Lattice Model
2.2 Binomial Lattice Model
• 2.2.4 Replicating Portfolio
• 2.2.4 Replicating Portfolio
1.7x+1.08b = 11
0.65x+1.08b = 0
1.7x+1.08b = 11
0.65x+1.08b = 0
Alleman, Suto, & Rappoport
19
2.2 Binomial Lattice Model
11 − 0
= 10.5
1.7 − .65
1
1.7 × 0 − .65 × 11
b=
×
= −6.3
1.08
1.7 − .65
Price of Portfolio= x + b = 10.5 − 6.3 = 4.2
1.7x+1.08b = 11
0.65x+1.08b = 0
11 − 0
= 10.5
1.7 − .65
1
1.7 × 0 − .65 × 11
= −6.3
b=
×
1.08
1.7 − .65
x=
Price of Portfolio= x + b = 10.5 − 6.3 = 4.2
21
2.2 Binomial Lattice Model
Price of Call Option, C = 4.2
Alleman, Suto, & Rappoport
22
2.2 Binomial Lattice Model
• 2.2.5 Risk-neutral Probability
C = x +b=
20
• 2.2.4 Replicating Portfolio
x=
Alleman, Suto, & Rappoport
Alleman, Suto, & Rappoport
2.2 Binomial Lattice Model
• 2.2.4 Replicating Portfolio
1.7x+1.08b = 11
0.65x+1.08b = 0
11 − 0
= 10.5
1.7 − .65
1
1.7 × 0 − .65 × 11
= −6.3
b=
×
1.08
1.7 − .65
x=
• 2.2.5 Risk-Neutral Probability
1.08 − .65 11 1.7 − 1.08
0
×
+
×
1.7 − .65 1.08 1.7 − .65 1.08
C = x +b=
1.08 − .65 11 1.7 − 1.08
0
×
+
×
1.7 − .65 1.08 1.7 − .65 1.08
q = .4
1-q = .6
Risk-Neutral Probability
Alleman, Suto, & Rappoport
23
Alleman, Suto, & Rappoport
24
4
2.2 Binomial Lattice Model
2.2 Binomial Lattice Model
• 2.2.6 General Option Pricing Formula
• 2.2.6 General Option Pricing Formula
C = x +b=
1.08 − .65 11 1.7 − 1.08
0
×
+
×
1.7 − .65 1.08 1.7 − .65 1.08
C =q×
Cu
C
+ (1 − q ) × d
R
R
Alleman, Suto, & Rappoport
C = x +b=
1.08 − .65 11 1.7 − 1.08
0
×
+
×
1.7 − .65 1.08 1.7 − .65 1.08
C =q×
25
2.2 Binomial Lattice Model
Cu
C
+ (1 − q ) × d
R
R
P
Alleman, Suto, & Rappoport
26
2.2 Binomial Lattice Model
• 2.2.7 Risk-neutral Probability
• 2.2.6 General Option Pricing Formula
Stock
C = Eˆ [P ]
$34
q = .4
$20
Eˆ [• ] :Expectation with q
Cu
C
P = PV[payoff]
or d
R
R
1-q = .6
t=0
= Expectation of P with Risk-neutral Probability
Alleman, Suto, & Rappoport
27
Alleman, Suto, & Rappoport
1. Objectives
2. Option Pricing
$1.08
2.1 Call Option
2.2 Binomial Lattice Model
2.3 Continuous Additive Model
q = .4
$1
t=0
1 = .4 ×
Alleman, Suto, & Rappoport
28
Agenda
• 2.2.7 Risk-neutral Probability
1-q = .6
t=1
34
13
20 = .4 ×
+ .6 ×
1.08
1.08
2.2 Binomial Lattice Model
Bond
$13
3. Real Options
4. New Criterion
5. Application
$1.08
t=1
1.08
1.08
+ .6 ×
1.08
1.08
29
Alleman, Suto, & Rappoport
30
5
2.3 Continuous Additive Model
2.3 Continuous Additive Model
– 2.3.1 Build Model that satisfies risk-neutral
S0
S1
t=0
– 2.3.1 Build Model that satisfies risk-neutral
RS0
S0
t=1
S1
t=0
RS0
t=1
S  1
E  1  = RS0 = S0
R R
Alleman, Suto, & Rappoport
31
Alleman, Suto, & Rappoport
2.3 Continuous Additive Model
2.3 Continuous Additive Model
– 2.3.2 Payoff of Call Option
– 2.3.1 Build Model that satisfies risk-neutral
t=0
RS0
t=1
σ
S1
≡V
R
V
S0 K
Alleman, Suto, & Rappoport
33
f(V)
S1
V
Probability
S0
32
Alleman, Suto, & Rappoport
34
2.3 Continuous Additive Model
2.3 Continuous Additive Model
– 2.3.2 Payoff of Call Option
– 2.3.3 Call Option Price
C = Eˆ [P (V )] =
+∞
∫ P (V )f (V )dV
V
Alleman, Suto, & Rappoport
f(V)
V-K
S0 K
35
Probability
Payoff : P(V)
f(V)
V-K
S0 K
Alleman, Suto, & Rappoport
Probability
Payoff : P(V)
−∞
V
36
6
2.3 Continuous Additive Model
Agenda
– 2.3.3 Call Option Price
+∞
+∞
−∞
K
V-K
S0 K
Alleman, Suto, & Rappoport
V
37
Agenda
Alleman, Suto, & Rappoport
– Value Operation Flexibility
– Applying Option Pricing Theory
3.1 What’s Real Options
3.2 Defer Option
3.3 An Example Project
3.4 ExNPV
– Types of Real Options
•
•
•
•
39
Defer
Expand
Switch
Abandon etc.
Alleman, Suto, & Rappoport
Agenda
40
3.2 Defer Option
1. Objectives
2. Option Pricing
3. Real Options
– Real Option:
Right to wait to invest
until the market is good
3.1 What’s Real Options
3.2 Defer Option
3.3 An Example Project
3.4 ExNPV
4. New Criterion
5. Application to DSL
Alleman, Suto, & Rappoport
38
3.1 What ‘s Real Options?
1. Objectives
2. Option Pricing
3. Real Options
4. New Criterion
5. Application
Alleman, Suto, & Rappoport
Objectives
Option Pricing
Real Options
New Criterion
Application
f(V)
Probability
∫ P (V )f (V )dV = ∫ (V − K )f (V )dV
Payoff : P(V)
C = Eˆ [P (V )] =
1.
2.
3.
4.
5.
– Call Option Analogy:
Right to buy the stock if the stock price
becomes high
41
Alleman, Suto, & Rappoport
42
7
3.2 Defer Option
Defer Option
Agenda
1. Objectives
2. Option Pricing
3. Real Options
Call Option
Present value of a project’s
future cash flow
S
Stock price
Investment to
acquire the project assets
K
Exercise price
Length of time the decision
may be deferred
T
Time to expiration
Time value of money
rf
Risk-free rate of return
Riskiness of the project assets
σ2
Variance of returns
on stock
Alleman, Suto, & Rappoport
43
3.1 What’s Real Options
3.2 Defer Option
3.3 An Example Project
3.4 ExNPV
4. New Criterion
5. Application to DSL
Alleman, Suto, & Rappoport
3.3 Example: A Project
44
3.3 Example: A Project
• NPV
Defer Option
Variable
Present value of operating future cash flow
S
$100 million
Investment to Equipment at time T=1
KT
$110 million
Length of time the decision may be deferred
T
1 year
Risk-free rate
rf
6%
Riskiness of the project
σ
$30 million
Alleman, Suto, & Rappoport
45
– PV[Cash out] = PV[KT]
= 110/1.06
= 103.8
– PV[Cash in] = S
= E[ PV[S1]]
= 100
– NPV = PV[cash in] – PV[cash out]
= 100 – 103.8
= - 3.8
Alleman, Suto, & Rappoport
3.3 Example: A Project
46
3.4 ExNPV
• ROV (Real Option Value)
• Expanded NPV
+∞
C=
∫ (V − K )f (V )dV = 10.2
10.2 M
6.4 M
K
σ =30
V-K
- 3.8 M
NPV
V
K=110/1.06=103.8
Alleman, Suto, & Rappoport
S=100
47
+
=Conventional
Value of Project
Alleman, Suto, & Rappoport
ROV
=
=Flexibility value
to defer
ExNPV
48
8
Agenda
1.
2.
3.
4.
5.
Agenda
Objectives
Option Pricing
Real Options
New Criterion
Application
1.
2.
3.
4.
Objectives
Option Pricing
Real Options
New Criterion
4.1 Basic Ideas
4.2 Case: NPV < 0
4.3 What do d and D* mean?
5. Application
Alleman, Suto, & Rappoport
49
Alleman, Suto, & Rappoport
Agenda
1.
2.
3.
4.
50
4.1 Basic Ideas
Objectives
Option Pricing
Real Options
New Criterion
•
When is ExNPV = 0, when NPV < 0 ?
4.1 Basic Ideas
4.2 Case: NPV < 0
4.3 What do d and D* mean?
5. Application
Alleman, Suto, & Rappoport
51
Alleman, Suto, & Rappoport
4.1 Basic Ideas
Agenda
When is ExNPV = 0, when NPV < 0 ?
ExNPV > 0
1.
2.
3.
4.
ExNPV < 0
Wait and watch
the market!
52
Objectives
Option Pricing
Real Options
New Criterion
4.1 Basic Ideas
4.2 Case: NPV < 0
4.3 What do d and D* mean?
Do not invest
5. Application to DSL
Alleman, Suto, & Rappoport
ExNPV = 0
53
Alleman, Suto, & Rappoport
54
9
4.2 Case: NPV < 0
4.2 Case: NPV < 0
If NPV < 0, when is ExNPV = 0? i.e.
• ExNPV = ROV + NPV = 0
When is ExNPV = 0, give NPV < 0?
• ExNPV = ROV + NPV = 0
+∞
∫ (V − K )f (V )dV + (S − K ) = 0
K
V-K
σ
V
Alleman, Suto, & Rappoport
55
4.2 Case: NPV < 0
V −S
σ
• To solve the equation
V −S
v=
: Normal distribution (0,1): fN(v)
σ
D≡
Alleman, Suto, & Rappoport
57
|S −K |
σ
| NPV |
riskiness
=
58
4.2 Case: NPV < 0
• Solve the equation
• Solve the equation
+∞
+∞
K
K
∫ (V − K )f (V )dV + (S − K ) = 0
∫ (V − K )f (V )dV + (S − K ) = 0
+∞
N
: Normal distribution (0,1): fN(v)
Alleman, Suto, & Rappoport
4.2 Case: NPV < 0
∫ (v − D )f
+∞
∫ (v − D )f
(v )dv − D = 0
N
D
(v )dv − D = 0
D
1
1
 1 
exp − D 2  − D
2π
2π
 2 
Alleman, Suto, & Rappoport
56
4.2 Case: NPV < 0
• To solve the equation
v=
S K
Alleman, Suto, & Rappoport
59
−D
−∞
LHS ( left hand side )
Alleman, Suto, & Rappoport
 1
∫ exp − 2 v
2

dv − D = 0

60
10
4.2 Case: NPV < 0
4.2 Case: NPV < 0
LHS (left hand side)
• Solve the equation
Criterion for Case:
ExNPV > 0
1
ExNPV < 0
0
-1
D* = 0.276
-2
-3
0
0.5
1
1.5
2
2.5
3
D
Alleman, Suto, & Rappoport
61
Alleman, Suto, & Rappoport
4.2 Case: NPV < 0
4.2 Case: NPV < 0
Criterion for Case:
Criterion for Case:
D < D*
D > D*
D < D*
Wait and watch
the market!
where
D≡
|S −K |
σ
=
| NPV |
riskiness
where
63
Do not invest
D≡
|S −K |
σ
=
| NPV |
riskiness
Alleman, Suto, & Rappoport
Agenda
64
Agenda
Objectives
Option Pricing
Real Options
New Criterion
1.
2.
3.
4.
4.1 Basic Ideas
4.2 Case: NPV < 0
4.3 What do d and D* mean?
Objectives
Option Pricing
Real Options
New Criterion
4.1 Basic Ideas
4.2 Case: NPV < 0
4.3 What do d and D* mean?
5. Application to DSL
Alleman, Suto, & Rappoport
D > D*
Wait and watch
the market!
Alleman, Suto, & Rappoport
1.
2.
3.
4.
62
5. Application to DSL
65
Alleman, Suto, & Rappoport
66
11
4.2 & 4.3 Combined Criterion
d≡
when
S−K
• Summary of Criterion
σ
-D*
-0.276
Not Invest
Combined Criterion
0
Invest
carefully
Wait and
watch
the market
NPV < 0
|NPV| > ROV |NPV| < ROV
NPV
ROV
d
D*
0.276
NPV > 0
|NPV| < ROV |NPV| > ROV
d
d < - D*
- D* < d < 0
0 < d < D*
D* < d
Decision
Not Invest
Wait and watch
Invest carefully
Invest
Invest
ROV
ROV
ROV
ROV
E[NPV]=m'-I
Alleman, Suto, & Rappoport
67
V
E[NPV]=m'-I
E[NPV]=m'-I
V
V
E[NPV]=m'-I
Alleman, Suto, & Rappoport
4.4 Meaning of d and D*?
V
68
4.4 Meaning of d and D*?
• d = NPV/riskiness
• d = NPV/riskiness
– Uncertainty adjusted NPV
– Risk normalized NPV
– Uncertainty adjusted NPV
– Risk normalized NPV
• d = D*
– The point of ExNPV = 0
– Break-even point of NPV plus ROV
Alleman, Suto, & Rappoport
69
Alleman, Suto, & Rappoport
σ
= −0.276
V-K
σ
S K
Alleman, Suto, & Rappoport
d=
S −K
σ
V
= −0.276
V-K
σ
S K
71
Alleman, Suto, & Rappoport
Probability
S −K
Probability
d=
f(V)
If d = - D*, what is the probability the
project payoff > 0 ?
f(V)
If d = - D*, what is the probability the
project payoff > 0 ?
Payoff : P(V)
4.4 Meaning of d and D*?
Payoff : P(V)
4.4 Meaning of d and D*?
70
V
Probability = 39%
72
12
If d = D*, what is the probability the
project payoff > 0 ?
If d = D*, what is the probability the
project payoff > 0 ?
Loss Function L(V)
4.4 Meaning of d and D*?
Loss Function L(V)
= 0.276
K-V
d=
σ
= 0.276
K-V
V
K S
S −K
Probability
σ
f(V)
S −K
Probability
d=
V
K S
Alleman, Suto, & Rappoport
73
Probability = 61%
Alleman, Suto, & Rappoport
4.4 Meaning of d and D*?
Tradeoffs of Losses
1
0.4
0.8
Expected
Loss
Loss
0.6
Probability
0.8
Expected
Loss
Loss
1
Probability
Payoff > 0
0.6
0.4
0.2
0.2
0
-D * 0 D *
1
0
2
-2
-1
-D * 0 D *
d
75
Alleman, Suto, & Rappoport
4.4 Meaning of d and D*?
2
76
Agenda
Tradeoffs of Losses
1.
2.
3.
4.
5.
1
Probability
Payoff > 0
0.8
Expected
Opportunity
Loss
0.6
0.4
Probability
Loss
1
d
Alleman, Suto, & Rappoport
Expected
Loss
Probability
Probability
Payoff > 0
-1
74
4.4 Meaning of d and D*?
Tradeoffs of Losses
-2
f(V)
4.4 Meaning of d and D*?
Objectives
Option Pricing
Real Options
New Criterion
Application
0.2
0
-2
-1
-D * 0 D *
1
2
d
Alleman, Suto, & Rappoport
77
Alleman, Suto, & Rappoport
78
13
5.1 Simple Case
5.1 Simple Case
Six Independent DSL Projects
Variable
A
B
C
D
E
F
S
$100.00 $100.00 $100.00 $100.00 $100.00 $100.00
KT
$90.00 $90.00 $110.00 $110.00 $110.00 $110.00
T
0.0
2.0
0.0
0.5
1.0
2.0
30%
30%
30%
20%
30%
40%
σ
Six Independent DSL Projects
Variable
A
B
C
D
E
F
S
$100.00 $100.00 $100.00 $100.00 $100.00 $100.00
KT
$90.00 $90.00 $110.00 $110.00 $110.00 $110.00
T
0.0
2.0
0.0
0.5
1.0
2.0
30%
30%
30%
20%
30%
40%
σ
rf
6%
6%
6%
6%
6%
6%
S : Current value of future CF
KT : Investment at time T
T : Time to expiration
σ : Volatility
rf : Risk-free rate of return
rf
6%
79
$42.43
$19.90
0.469
$0.00
-$10.00
-infinite
$14.14
-$6.84
-0.484
invest
invest
do not
invest
do not
invest
6%
6%
PV ( KT ) =
d=
KT
(1 + rf )T
NPV
S − PV (KT )
=
Riskiness
Sσ T 80
5.1 Simple Case
Six Independent DSL Projects
Variable
A
B
C
D
E
F
S
$100.00 $100.00 $100.00 $100.00 $100.00 $100.00
$90.00 $90.00 $110.00 $110.00 $110.00 $110.00
KT
T
0.0
2.0
0.0
0.5
1.0
2.0
σ
30%
30%
30%
20%
30%
40%
rf
6%
6%
6%
6%
6%
6%
$0.00
$10.00
+infinite
6%
Riskiness = Sσ T
Alleman, Suto, & Rappoport
5.1 Simple Case
Exercise
decision
6%
S : Current value of future CF
KT : Investment at time T
T : Time to expiration
σ : Volatility
rf : Risk-free rate of return
Alleman, Suto, & Rappoport
Riskiness
NPV
d
6%
$30.00
-$3.77
-0.126
$56.57
$2.10
0.037
C D
E
-D*
-0.276
Not Invest
wait &
invest
watch carefully
Alleman, Suto, & Rappoport
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F
0
Wait and
watch
the market
B
d
D*
0.276
Invest
carefully
A
Invest
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Conclusion
An Investment Criterion
Incorporating Real Options
• Decision index d = NPV/Riskiness
gives uncertainty adjusted NPV
• d = D* = 0.276 gives the break-even
point of NPV plus ROV
James Alleman, Hirofumi Suto &
Paul Rappoport
• Make a decision by observing d
Experts Dialogue: Managing Risk in the Competitive Environment
International Telecommunication Union
Geneva, Switzerland
28-29 October 2004
Copyright © 2003 & 2004 James Alleman, All rights reserved
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