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Real Options. Modeling Options Prof. Luiz Brandão [email protected] 2009. Discrete Models. Modeling Simple Options. Modeling Simple Options.

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### Real Options

Modeling Options

Prof. Luiz Brandão

[email protected]

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.
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?
Binomial Model
• Model Parameters:

10

Modeling the Underlying Asset
• The parameters for the binomial approximation are:
• With these parameters we can model the evolution of the value of the project in time.

10u2

10u

10ud

10d

10d2

With Expansion

10

Modeling the Underlying Asset
• The parameters for the binomial approximation are:
• With these parameters we can model the evolution of the value of the project in time.
• The last column shows the value of the project in each state with the expansion.

16,49

21,73

12,84

10,0

12,0

7,79

6,07

6,11

With Expansion

Modeling the Underlying Asset
• The parameters for the binomial approximation are:
• With these parameters we can model the evolution of the value of the project in time.
• The last column shows the value of the project in each state with the expansion.

16,49

21,73

12,84

10,0

12,0

7,79

6,07

6,11

Solving by Risk Neutral Probability
• 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

p

p

1-p

1-p

Solving by Risk Neutral Probability
• 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

21,73

p

16,43

1-p

p

12,33

12,0

p

1-p

8,86

1-p

6,11

Solving with DPL
• Notation:
• Decision nodes, uncertainty and value
• Inserting nodes in a tree
• “Detach Tree” command
• “Perform Subtree” command

Expected Value

Objective Function

of this node

Probability

Cash Flows (Get/Pay)

With Expansion

28,55

21,17

16,49

15,26

12,84

12,84

10,0

10

7,69

7,79

7,79

6,07

4,72

3,08

MMZ: Three Periods
• In the same example as before, additional periods can be modeled as follows:

### 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
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
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:
Weston Inc.
• Step 2: Binomial Lattice
Weston Inc.
• Step 3: Model the options
• Step 4: Solve the decision tree.
• The project value is \$ 34.465 and the NPV is \$ 9.465
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:
• Should Betalog invest in this project?
• What is the optimal investment strategy?

4.2

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

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.
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

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 value

### Real Options

Modeling Options

Prof. Luiz Brandão

[email protected]

2009