Real Options
Modeling Options
Prof. Luiz Brandão
brandao@iag.puc-rio.br
2009
Discrete Models
Modeling Simple Options
Modeling Simple Options

A stochastic process allows us to adopt a probability distribution
for forecasted prices, instead of estimating each individual
forecasted price directly on the decision tree.

For assets that follow a GBM, we can use the CCR model.

To do this we need only the initial value (S), an estimate of the
volatility (s) and the risk free rate (r).

After the binomial model is completed, we apply the
corresponding option exercises and roll back the tree using the
risk neutral probabilities and discounting at the risk free rate.
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3
Example

MMZ Corp. is analyzing the feasibility of a MiniMall construction
project in a suburban area. Preliminary studies and required
licenses will take two years.

Project Information

If operating today, the value of the project would be $10 million.

Volatility is estimated to be 25%

The risk free rate is 6%.

The investment cost is $11 million.

If the market picks up in the next two years, MMZ has an option
to expand the mall to a neighboring area at an additional cost of $
3 million. This expansion would increase the value of the mall by
50% two years from now.

What is your recommendation to the firm?
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4
Binomial Model

Model Parameters:
Vu3
Vu2
Vu
p
Vu
p
p
Vud
V
V
Vu2d
1-p
p
1-p
Vud2
Vd
1-p
Vd
1-p
Vd2
Vd3
u  es
d  e s
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t
t
(1  k )  d
p
ud
5
Modeling the Underlying Asset

The parameters for the
binomial approximation
are:
10u2
u  es t  1.284
d  1 u  0.779
10u
10ud
10

With these parameters we
can model the evolution of
the value of the project in
time.
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10d
10d2
6
Modeling the Underlying Asset

The parameters for the
binomial approximation
are:
With Expansion
u  es t  1.284
d  1 u  0.779

21,73
10,0
12,0
6,07
6,11
12,84
10

16,49
With these parameters we
can model the evolution of
the value of the project in
time.
7,79
The last column shows the
value of the project in each
state with the expansion.
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Modeling the Underlying Asset

The parameters for the
binomial approximation
are:
u  es t  1.284
d  1 u  0.779


With these parameters we
can model the evolution of
the value of the project in
time.
With Expansion
16,49
21,73
10,0
12,0
6,07
6,11
12,84
7,79
The last column shows the
value of the project in each
state with the expansion.
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8
Solving by Risk Neutral Probability
p
(1  rf )  d
ud
p
1-p
p

Risk neutral probabilities allow
is to determine the correct
project value when options are
present.

In this case, we determine a
probability p that incorporates
the project risk in each node.

This allows us to discount the
cash flows/values at the risk
free rate.

One advantage of this method
is that in most cases, the
probability p is the same for
the whole tree.
p
1-p
1-p
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Solving by Risk Neutral Probability
p
(1  rf )  d
 0.5566
ud
21,73

Risk neutral probabilities allow is
to determine the correct project
value when options are present.

In this case, we determine a
probability p that incorporates
the project risk in each node.

This allows us to discount the
cash flows/values at the risk free
rate.

One advantage of this method is
that in most cases, the
probability p is the same for the
whole tree.

The value of the project with the
option to expand increases to $
12.33 million
10
p
16,43
1-p
p
12,33
12,0
p
1-p
8,86
1-p
6,11
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Solving with DPL

Notation:
Expected Value
of this node
Objective Function
Probability
Cash Flows (Get/Pay)

Decision nodes, uncertainty and value

Inserting nodes in a tree

“Detach Tree” command

“Perform Subtree” command
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MMZ: Three Periods

With Expansion
In the same example as before,
additional periods can be
modeled as follows:
21,17
28,55
12,84
15,26
7,79
7,69
4,72
3,08
16,49
12,84
10,0
10
7,79
6,07
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12
Aplication
Steps to solve Real Option Problems



Step 1: Determine the Parameters for the underlying asset

The underlying asset is the project without options

We use the binomial lattice to model the stochastic process of a GBM
Step 2: Build the binomial model

We use DPL to model the undelying asset with risk neutral
probabilities.

The resulting lattice shows the evolution of project value with time
Step 3: Model the project options.


Options are modeled by inserting decision nodes in the binomial
lattice, transforming it in a decision tree.
Step 4: Solve the decision tree
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European Option (with DPL)

Parameters:

Underlying Asset Price: $100

Volatility: 30%

Distribution: GBM (Lognormal)

Time to Expiration: 3 anos

Exercise Price: $140

r = 5%

Solution with DPL

Solution with B&S

Solution with Simulation
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15
Weston Inc. (Option to Expand)

Weston Inc. is analyzing the purchase of a concession project
valued at $30M which it plans to sell in two years. The volatility
of the project value is assumed to be 25% and the corporate
cost of capital is 15%.

The initial investment required for the concession is 25M, and
at the end two years there is an option to expand the project
by 30% at a cost of $5M. The risk free rate is 5%.

What is the NPV of this project?

Step 1: Parameters u, d, and risk neutral probability p:
u  es t  e0.25  1.28
d  1 u  0.78
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p
(1  r )  d 1.05  0.78

 0.537
ud
1.28  0.78
16
Weston Inc.
Y1

Y2
Step 2: Binomial Lattice
up
down
up
Y1
[30.000]
54%
down
46%
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Y2
[36.686]
Brandão
Y2
[22.251]
up
Y2/(1+r)^2
down
Y2/(1+r)^2
up
[44.863]
54%
down
44.863
[27.211]
46%
up
27.211
[27.211]
54%
down
27.211
[16.504]
46%
16.504
17
Weston Inc.
Expand?



Step 3: Model the
options
Y2
Y1
up
up
down
down
Yes
(1.30*Y2-5)/(1+r)^2
Step 4: Solve the
decision tree.
The project value is
$ 34.465 and theup
NPV is $ 9.465 54%
No
Y2/(1+r)^2
Y2
[43.157]
down
46%
Y1
[34.465]
down
46%
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up
54%
Y2
[24.392]
up
54%
down
46%
Expand?
[53.787]
Yes
No
Expand?
[30.839]
Yes
No
Expand?
[30.839]
Yes
No
Expand?
[16.920]
Yes
No
[53.787]
53.787
[44.863]
44.863
[30.839]
30.839
[27.211]
27.211
[30.839]
30.839
[27.211]
27.211
[16.920]
16.920
[16.504]
16.504
18
BetaLog (Option to Abandon)


Data:

A project has a life of three years and a value of $100M.

The initial investment will be $110M and the volatility is 30%.

The firm can abandon the project at any moment in exchange
for a residual value of $90M.

The WACC is 15% and the risk free rate is 7%.
Questions:
4.2

Should Betalog invest in this project?

What is the optimal investment strategy?
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Rio Verde Mining (Option to
Abandon)

Rio Verde Mining (RVM) has been invited to participate in a
project. The expected value of its share in this project is valued at
$1.000M.

Given large uncertainties about this project, RVM is negotiating to
include an exit clause in its contract, which would allow it to
abandon the project by selling it to its partners for $800M at any
time over the next two years.

All project cash flows will be reinvested in the project, which will
be sold at market price in two years.

It is estimated that the volatility of the project is 30%, and that
the risk free rate is 6%.

The partners of RVM agree to include this clause in the contract in
exchange for the payment of $50 million. What is your
recommendation for the firm?
5.6
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Solution
Yes
[761.90]
761.90
up
Abandon1
[1285.58]
up
No
51%
Y2
[1285.58]
51%
down
Y1
[1055.23]
Abandon2
[1652.72]
Abandon2
[907.03]
49%
Yes
[725.62]
No
725.62
[1652.72]
Yes
1652.72
[725.62]
No
725.62
[907.03]
907.03
Yes
[761.90]
761.90
down
49%
Abandon1
[817.71]
up
No
Y2
[817.71]
Abandon2
[907.03]
51%
down
49%
Abandon2
[725.62]
Yes
[725.62]
No
725.62
[907.03]
Yes
907.03
[725.62]
No
725.62
[497.79]
497.79
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21
Rio Verde Mining: Additional Options


Option to Expand

Suppose the investors now decided to extend the life of this project
to three years, after which there will be an opportunity to expand
the project by 25% at a cost of $200M.

What is the impact of this opportunity on the value of the project?
Option to Contract

In case the market price of the project turns out to be less than
expected, there is an option to contract operations of the mine by
50% at any time through the sale of assets worth $470M. In this
case, the amount received by RVM in case of abandon is also
halved.

Assume that once the project has been contracted, this decision is
irreversible and can be exercised only once. On the other hand, the
project may still be expanded after being contracted.
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MegaCorp Inc.

MegaCorp is analyzing a new project which will be sold at the end
of three years. The expected value of this project is $300.000, its
volatility is 35%, and the risk free rate is 5% per year.

The firm has contracted an insurance that allows it to abandon the
project at the end of the three years and receive $250.000.

MegaCorp can also expand this project by 40% at a cost of
$100.000 at the end of the second year.

Consider that the project does not generate cash flows for
MegaCorp during this period.

What is the value of the project with these options?
5.6
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Ecotech Ltd

A Ecotech Ltd. has an opportunity to invest in a project of
sustainable exploration of forestry products in the Amazon rain
forest. The project has na expected value of $30 milion, a volatilty
of 25%, a WACC of 15% and a risk free rate of 6% a.a.

If the investment is sucessfully undertaken, by the end of the third
year there is the probability of adding a processing plant which will
double the sales of the project. This plant will have a cost of $20
million.

Assume the project will be sold to international investors at its
market price at the end of five years. If the additional investment is
made, the project may also be abandoned for $50 million

Questions:

If the required initial investment is $35 milhões, should the project be
undertaken?

What is the probability that the processing plant will be implemented?

Perform a sensitivity analysis on the cost of volatility and the abandon
24
value
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Real Options
Modeling Options
Prof. Luiz Brandão
brandao@iag.puc-rio.br
2009