Real Options
Discrete Pricing Methods
Prof. Luiz Brandão
brandao@iag.puc-rio.br
2009
Discrete Pricing Methods

Continuous Time Methods

Black and Scholes Equation

Simulation Methods

Discrete Pricing Methods

Replicating Portfolio

Risk Free Portfolio

Risk Neutral Probabilities
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2
Risk Free Portfolio
A Simple Project

A project will have the value of $160M or 62.5M within a year,
depending on the state of the economy, with a probability of 0.50
Projeto
0.50
$160M
$100M
0.50

$62.5M
We assume that the risk adjusted discount rate is 11.25%, which
provides a project value of $100, and that the risk free rate is
5%.
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4
Option to Expand

Suppose now, that the project has an option to expand capacity
by 50% within year at a cost of $50M.

With this option, the result of the project becomes:
Project with
Option
0.50
0.50

max [160,1.5(160)-50]= $190M
max [ 62.5, 1.5(62.5)-50] = $62.5M
Does this option add value to the project?
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5
Risk Free Portfolio

We can isolate the option to expand from the project, as show
below. In this case, the option to expand adds $30 in one year if
the economy improves, and zero otherwise.
Project
0.50
Option
$160M
0.50
$30
$100M
0.50

$62.5M
0.50
$0
We create a portfolio composed of the project and n Call
options:  = 100 + nC.
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6
Risk Free Portfolio

The value in a year will be:
Risk Free
Portfolio
0.50
160 + 30n
100 + n C
0.50
62.5

In order for this portfolio to be risk free, it is necessary that the
returns be identical, regardless of the state of the economy.

In this case we must then have 160 + 30 n = 62.5, which results in
n=-3.25
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7
Risk Free Portfolio


The PV of the portfolio is therefore
= 100 - 3.25C

Since the portfolio is risk free, we must discount its cash flows at
the risk free rate.

We then find:
100 - 3.25 C = 62.5/(1+0.05)
C = $12.45

The total value of the project will be $112.45
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8
Risk Neutral Probabilities
Risk Neutral Probabilities

Consider the following risky project, where the possible cash flows
in one year will be 70 or 40:
High
0.50
Low
0.50
High
0.4167
Low
0.5833
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
70
PV 
40
0.5(70)  0.5(40)
 50
1  0.10

We can also adjust for risk by using
the risk free rate of 5% and adjusting
the probability to 0.4167.

The value of the project is:
70
40
We can adjust for risk by using the
risk adjusted discount rate of 10%.
The value of the project is:
PV 
0.4167(70)  0.5833(40)
 50
1.05
10
Risk Neutral Probability

The Risk Neutral Probability (p) is the probability that
makes us obtain the same previous PV when we discount
the cash flows using the risk free discount rate.

That probability can be determined from the existing
relationship between, the discount rate, the objective
probabilities, the cash flows of the project, and the Present
Value.

The risk neutral probability method p is equivalent to the
replicating portfolio method and produces the same results.

Note that these probabilities are not “true” probabilities in
the sense that they don’t reflect the real chances that any
particular cash flow will occur. They are simply another
way of determining the project’s market value.
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11
Risk Neutral Probability


This method can also be used to
determine the value of a project
with options.
Projeto
p
Define a probability p such that:
p
V0 (1  r )  Vd
Vu  Vd
V0 =100
1-p
100(1  0.05)  62.5
 0.4359
160  62.5
pCu  (1  p)Cd
C
1 r
p

We are able to determine the
value of the call through:
C
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30 p  0(1  p )
 12.45
1.05
Brandão
Vu =160
Vd =62.5
Opçâo
p
Cu =30
C0
1-p
Cd = 0
12
Value of the Project with Expansion

Similar to what we did with the
option, we are able to find the
value of the project with an
option to expand through:
pVu  (1  p)Vd
V
1 r
V

Projeto
p
Vu =190
V0 =?
1-p
Vd =62.5
190 p  (1  p )62.5
 112.45
1.05
We are able to observe that the risk neutral probability method
gives the same results as the replicating portfolio and the risk
neutral portfolio methods.
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13
Project’s Value with Abandonment

Using risk neutral
probabilities we can also
calculate the value of the
project with an option to
abandon:
V
pVu  (1  p)Vd
1 r
V
160 p  (1  p )92.5
 116.12
1.05
Projeto
p
Vu =160
V0 =?
1-p
Vd =92.5

This method provides the same results in a simpler way.

One of the advantages of this method is that the risk neutral
probability is constant for the entire project.
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14
Risk Neutral Probability

The four variables below should be consistent among
themselves.
Cash
Flow
Objective
Probability
PV
WACC

Given three we can determine the fourth.
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15
Risk Neutral Probability

We can obtain the same PV discounted using the risk
free discount rate if we use the risk neutral probability.
Cash
Flow
Risk Neutral
Probability
PV
Risk Free
Discount Rate


Given the cash flow, risk free discount rate and the
value of the project, we can determine the risk neutral
probability.
This permits us to determine the value of a project
without having to create a replicating portfolio.
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16
Example
Example

Initech obtained a concession that allows it to invest in a project
in two years.

Data:(Values in $1,000)

The value of the project today is $1,000

In one year, the value will be $1,350 or $741, depending on the
market conditions.

In two years, the value will be $1,821, $1,000 or $549.

WACC is 15%

The risk free discount rate is 7%

Initech can opt to extend the project by 30% at a cost of $250 it
in two years if it decides to build it.

What is the value of this option?
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18
Solution by Risk Neutral Probability

Model the underlying asset
1822,1
p
1349,9
1-p
p
1000
1000,0
p
1-p
740,8
1-p
548,8
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19
Solution by Risk Neutral Probability
p
V0 (1  r )  Vd
 0.54045
Vu  Vd
2118,8

Model the underlying asset

Model the exercise of the
options

Determine the risk neutral
probability p that incorporates
the project’s risk in each node.

Solve the binomial tree by
rolling back the project payoffs
discounted at the risk free rate.

One advantage of this method
is that in the majority of cases
the probability p is the same
for all the nodes in the project.
p
1521,2
1-p
p
1097,4
1050,0
p
1-p
766,1
1-p
548,8
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Advantages

Both the binomial tree and the analytic model of Black and
Scholes can be utilized to value options.

The B&S equation, however, allows for valuing only a limited
combination of problems.

The binomial model is more flexible and allows you to model and
resolve a much greater and more complex range of practical
problems, especially in the case of American options.

The solution by risk neutral probability also provides significant
advantages in relation to the method of replicating portfolio
method.

The evaluation through replicating portfolio is tedious and can be
impractical to solve complex problems.
The use of risk neutral probabilities in binomial trees is a practical
alternative to the resolution of the problem of valuating real
21
options projects.
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
Comparative Table
Black and Scholes
Binomial Tree
European Option
European and American Options
Only one source of uncertainty
Multiple source of uncertainty
Only one option
Multiple Options
No Dividends
Dividends
Simple Options
Composed Options
Call or Put
Call, Put, Call + Put
Underlying Asset follows an
GBM
Underlying Asset follows an GBM
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22
Real Options
Discrete Pricing Methods
Prof. Luiz Brandão
brandao@iag.puc-rio.br
2009