1. Real Options Discrete Pricing Methods Prof. Luiz Brandão brandao@iag.puc-rio.br 2009

2. IAG PUC – Rio Brandão 2 Discrete Pricing Methods  Continuous Time Methods Black and Scholes Equation  Simulation Methods  Discrete Pricing Methods Replicating Portfolio Risk Free Portfolio Risk Neutral Probabilities

3. Risk Free Portfolio

4. IAG PUC – Rio Brandão 4 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%.

5. IAG PUC – Rio Brandão 5 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?

6. IAG PUC – Rio Brandão 6 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 Option $30 0.50

7. IAG PUC – Rio Brandão 7 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

8. IAG PUC – Rio Brandão 8 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

9. Risk Neutral Probabilities

10. IAG PUC – Rio Brandão 10 Risk Neutral Probabilities  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:  Consider the following risky project, where the possible cash flows in one year will be 70 or 40:

11. IAG PUC – Rio Brandão 11 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.

12. IAG PUC – Rio Brandão 12 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: C d = 0 Opçâo C u =30 p 1-p C 0

13. IAG PUC – Rio Brandão 13 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:  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.

14. IAG PUC – Rio Brandão 14 Project’s Value with Abandonment  Using risk neutral probabilities we can also calculate the value of the project with an option to abandon:  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.

15. IAG PUC – Rio Brandão 15 Risk Neutral Probability  The four variables below should be consistent among themselves.  Given three we can determine the fourth. Cash Flow WACC Objective Probability PV

16. IAG PUC – Rio Brandão 16 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 Free Discount Rate Risk Neutral Probability PV

17. Example

18. IAG PUC – Rio Brandão 18 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?

19. IAG PUC – Rio Brandão 19 1000 1349,9 740,8 p 1-p 1000,0 548,8 1822,1 p p 1-p  Model the underlying asset Solution by Risk Neutral Probability

20. IAG PUC – Rio Brandão 20 Solution by Risk Neutral Probability 1097,4 1521,2 766,1 p 1-p 1050,0 548,8 2118,8 p p 1-p  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.

21. IAG PUC – Rio Brandão 21 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.

22. IAG PUC – Rio Brandão 22 Comparative Table Black and ScholesBinomial Tree European OptionEuropean and American Options Only one source of uncertaintyMultiple source of uncertainty Only one optionMultiple Options No DividendsDividends Simple OptionsComposed Options Call or PutCall, Put, Call + Put Underlying Asset follows an GBM

23. Real Options Discrete Pricing Methods Prof. Luiz Brandão brandao@iag.puc-rio.br 2009