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

Real OptionsModeling Options

Prof. Luiz Brandão

[email protected]

2009

Discrete ModelsModeling Simple Options

IAG PUC – Rio Brandão

3

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 () 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.


4

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?


5

Binomial Model

Vu3

Vu2

Vud

Vud2

Vu2dVu

Vd2

Vd3

Vd

V

p

p

1-p

1-p

p

1-p

Model Parameters:

t

t

u e

d e

Vu

V

p

1 - pVd

(1 )k dp

u d


6

Modeling the Underlying Asset

10

10u

10d

10ud

10u2

10d2

The parameters for the binomial approximation are:

With these parameters we can model the evolution of the value of the project in time.

1.284

1 0.779

tu e

d u


7



The last column shows the value of the project in each state with the expansion.

With Expansion

10


12,84

7,79

10,0

6,07

16,49

12,0

6,11

21,73

1.284

1 0.779

tu e

d u


8



The last column shows the value of the project in each state with the expansion.

With Expansion


12,84

7,79

10,0

6,07

16,49

12,0

6,11

21,73

1.284

1 0.779

tu e

d u


9

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

(1 )rf dp

u d

p

p

1-p

1-p


10

Solving by Risk Neutral Probability

12,33

16,43

8,86

p

1-p

12,0

6,11

21,73

(1 )0.5566

rf dp

u d

p

p

1-p

1-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.

The value of the project with the option to expand increases to $ 12.33 million


11

Solving with DPL Notation:

Decision nodes, uncertainty and value

Inserting nodes in a tree

“Detach Tree” command

“Perform Subtree” command

Objective Function

Cash Flows (Get/Pay)

Expected Valueof this node

Probability


12

12,84

7,79

21,17

4,72

MMZ: Three Periods In the same example as before,

additional periods can be modeled as follows:

10

12,84

7,79

10,0

6,07

16,49

With Expansion

15,26

7,69

28,55

3,08

Aplication


14

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

Steps to solve Real Option Problems


15

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

European Option (with DPL)


16

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:

0.25 1.28

1 0.78

tu e e

d u

(1 ) 1.05 0.780.537

1.28 0.78

r dp

u d


17

Step 2: Binomial Lattice

Weston Inc.

up

Y2/(1+r)^2 down

Y2/(1+r)^2

up

down

Y2Y1

44.863 54%

[44.863] up

27.211 46%

[27.211] down

Y2

54%

[36.686] up

27.211 54%

[27.211] up

16.504 46%

[16.504] down

Y2

46%

[22.251] down

Y1 [30.000]


18

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

Yes

(1.30*Y2-5)/(1+r)^2

No

Y2/(1+r)^2

up

down

Expand?

up

down

Y2Y1

53.787

[53.787] Yes

44.863

[44.863] No

Expand?

54%

[53.787] up

30.839

[30.839] Yes

27.211

[27.211] No

Expand?

46%

[30.839] down

Y2

54%

[43.157] up

30.839

[30.839] Yes

27.211

[27.211] No

Expand?

54%

[30.839] up

16.920

[16.920] Yes

16.504

[16.504] No

Expand?

46%

[16.920] down

Y2

46%

[24.392] down

Y1 [34.465]


19

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


20

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


21

Solution 761.90

[761.90] Yes

725.62

[725.62] Yes

1652.72

[1652.72] No

Abandon2

51%

[1652.72] up

725.62

[725.62] Yes

907.03

[907.03] No

Abandon2

49%

[907.03] down

Y2 [1285.58] No

Abandon1

51%

[1285.58] up

761.90

[761.90] Yes

725.62

[725.62] Yes

907.03

[907.03] No

Abandon2

51%

[907.03] up

725.62

[725.62] Yes

497.79

[497.79] No

Abandon2

49%

[725.62] down

Y2 [817.71] No

Abandon1

49%

[817.71] down

Y1 [1055.23]


22

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.


23

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


24

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 OptionsModeling Options

Prof. Luiz Brandão

[email protected]

2009

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

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Transcript of Real Options Modeling Options Prof. Luiz Brandão [email protected] 2009.