Real Options Discrete Pricing Methods Prof. Luiz Brandão [email protected] 2009.

Real OptionsDiscrete Pricing Methods

Prof. Luiz Brandão

[email protected]

2009

IAG PUC – Rio Brandão

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Discrete Pricing Methods Continuous Time Methods

Black and Scholes Equation

Simulation Methods

Discrete Pricing Methods Replicating Portfolio

Risk Free Portfolio

Risk Neutral Probabilities

Risk Free Portfolio


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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

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%.

$62.5M

Projeto 0.50

0.50

$160M

$100M


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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:

Does this option add value to the project?

max [ 62.5, 1.5(62.5)-50] = $62.5M

Project withOption max [160,1.5(160)-50]= $190M

0.50

0.50


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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.

We create a portfolio composed of the project and n Call options: = 100 + nC.

$62.5M

Project

$160M

$100M

0.50

0.50$0

Option

$300.50

0.50


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Risk Free Portfolio The value in a year will be:

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

62.5

Risk FreePortfolio 160 + 30n

100 + n C

0.50

0.50


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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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High 70

Low 40

0.50

0.50

We can adjust for risk by using the risk adjusted discount rate of 10%. The value of the project is:

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:

High 70

Low 40

0.4167

0.5833

0.4167(70) 0.5833(40)50

1.05PV

0.5(70) 0.5(40)50

1 0.10PV

Consider the following risky project, where the possible cash flows in one year will be 70 or 40:


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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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Risk Neutral Probability This method can also be used to

determine the value of a project with options.

Define a probability p such that:

We are able to determine the value of the call through:

Vd =62.5

Projeto

Vu =160p

1-p

V0 =1000 (1 ) d

u d

V r Vp

V V

100(1 0.05) 62.5

0.4359160 62.5

p

(1 )

1u dpC p C

Cr

30 0(1 )12.45

1.05

p pC

Cd = 0

Opçâo

Cu =30p

1-p

C0


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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:

Vd =62.5

Projeto

Vu =190p

1-p

V0 =?(1 )

1u dpV p V

Vr

190 (1 )62.5112.45

1.05

p pV

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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Project’s Value with Abandonment Using risk neutral

probabilities we can also calculate the value of the project with an option to abandon:

Vd =92.5

Projeto

Vu =160p

1-p

V0 =?(1 )

1u dpV p V

Vr

160 (1 )92.5116.12

1.05

p pV

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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Risk Neutral Probability The four variables below should be consistent among

themselves.

Given three we can determine the fourth.

CashFlow

WACC

ObjectiveProbability

PV


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Risk Neutral Probability We can obtain the same PV discounted using the risk

free discount rate if we use the risk neutral probability.

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.

Cash Flow

Risk FreeDiscount Rate

Risk NeutralProbability

PV

Example


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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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1000

1349,9

740,8

p

1-p

1000,0

548,8

1822,1

p

p

1-p

1-p

Model the underlying asset

Solution by Risk Neutral Probability


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Solution by Risk Neutral Probability

1097,4

1521,2

766,1

p

1-p

1050,0

548,8

2118,8

p

p

1-p

1-p

0 (1 )0.54045d

u d

V r Vp

V V

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.


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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 options projects.


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Comparative TableBlack 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

Real OptionsDiscrete Pricing Methods

Prof. Luiz Brandão

[email protected]

2009

Real Options Discrete Pricing Methods Prof. Luiz Brandão [email protected] 2009.

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