Problem 1: Investing in Land for a Wind Farm
A group of investors is considering the purchase of a parcel of land in order to construct a wind farm for producing electricity. There are two periods, period 0 and period 1, and the investors must decide whether or not to purchase the land by paying a price L in period 0.If the investors purchase the land, they must also decide whether or not to set up the wind farm in period 0 or in period 1 or not to set up the wind farm at all.
If the wind farm is set up in period 0, then the investors must pay the construction cost C0 = 150 in period 0. The investors also obtains a profit in period 0 with a present value V0 = 50 in addition to the expected present value of the profit produced by the wind farm in period 1.
Due to uncertainty about the price in period 1 that can be charged for electricity from the wind farm, from the perspective of period 0 the present value of the profit from the wind farm in period 1 is uncertain. With probability 0.6, the present value (in period 0) of the profit from the wind farm in period 1 takes the high value VH = 240. With probability 0.4, the present value of the profit from the wind farm in period 1 takes the low value VL = 40. (Note that the present values, V0, VL, and VH, do not include the cost of constructing the wind farm.)
The investors have three choices in period 0. First, the investors can choose not to purchase the land. In this case, they pay nothing and receive nothing, so the net present value associated with this choice is 0. Second, the investors can choose to purchase the land and set up the wind farm in period 0.
Finally, the investors can choose to purchase the land in period 0 by paying L but defer setting up the wind farm until period 1. The investors must pay an additional deferral cost D in period 0 in order to have the option to set up the wind farm in period 1.
By waiting until period 1, the investors learn whether the present value of the profit which they obtain from the wind farm will be VH or VL in period 1 before having to decide whether to set up the wind farm. Of course, ff the investors decide to set up the wind farm in period 1, they must also pay the cost of constructing the wind farm. For simplicity, assume that the present value of the construction cost is the same value C0 = 150 whether the wind farm is set up in period 0 or period 1.
Of course, the investors do not have to pay the construction cost C0 if they chooses not to set up the wind farm. However, since the deferral cost D is paid in period 0, that cost must be paid even if the investors decides not to set up the wind farm in period 1. By choosing to wait, the investors also lose the present value of the profit from the wind farm in period 0, V0.
Assume initially that the land has no resale value if the investors choose not to set up the wind farm. Also assume that the investors are risk neutral and wish to maximize the expected net present value produced by the opportunity to purchase the land and set up a wind farm. All the present values are specified in multiples of 100,000 pounds.
Q1. Draw a decision tree representing the decision problem facing the investors.
Q2. Suppose that L= 12 and D = 6. What is the optimal policy for the investors to follow? What is the expected net present value obtained by the investors in this case? Justify your answer.
Q3. Continue to suppose that D = 6. What is the maximum amount L that the investors should be willing to pay for the land to set up the wind farm? Explain your reasoning.
Q4. Suppose that L = 12. What is the maximum amount D that the investors should be willing to pay for the option to defer setting up the wind farm in period 1? Explain your reasoning.
Q5. Suppose now that the investors have an additional choice in period 0. By paying the construction cost C0 = 150 and an additional price P in period 0, the investors can set up the wind farm in period 0 with an option to abandon the project in period 1 after they find out whether the present value of the profit from the wind farm in period 1 will be VH or VL. By abandoning the project in period 1 if the present value of the profit from the wind farm is low, the investors obtain a "scrap value" in period 1 with a present value of S instead of VL. Similarly, the investors obtain S instead of VH by abandoning the wind farm in period 1 after learning that the profit in period 1 will be high.
i. Explain why the option to abandon the project in period 1 resembles a put option for a share of stock.
ii. Suppose that the option to defer setting up the project in period 1 is not available and that S = 80. What is the maximum amount P that the investors should be willing to pay for the option to abandon the project in period 1? Explain your answer. Continue to suppose that L = 12.
How much would your answer change if L were equal to 6?
Problem 2: Valuation on a Multiplicative Binomial Lattice
This problem reviews some of the main ideas of valuation on a binomial lattice and the properties of put and call options. You may wish to review the relevant lecture material and readings.
Suppose that the price of a share of QQF stock is S(0) = £120 in period 0. At the beginning of period 1, the price of a share can either move upward to S(1) = u S(0) or downward to S(1) = d S(0). Suppose that u = 5/4 = 1.25 and d = 4/5 = .80, so that S(1) = u S(0) = £150 after an up move and S(1) = d S(0) = £96 after a down move. Suppose that the probability of an up move is p = 0.6.
Similarly, suppose that, at the beginning of period 2, the share price either moves up or down by the same multiplicative factors and with the same probability (0.6) of an up move. (If the probability of an up move in a period is 0.6, then the probability of a down move in a period is 1 - 0.6 = 0.4.) Hence, if the share price in period 1 is S(1), then the share price at the beginning of period 2 is either S(2) = u S(1) = 5/4 S(1) or S(2) = d S(1) = 4/5 S(1).
For simplicity, suppose that a period is a year, and let the riskless interest rate be r = .04, that is, 4 % per period.
Q1. Draw a multiplicative binomial lattice that represents the possible paths of the share price from period 0 (when S(0) = 120) to period 2.
Q2.
(i) Briefly explain what is meant by an Arrow-Debreu 1-period-ahead up security and the 1-period-ahead up state price.
(ii) Write down two equations that describe a replicating portfolio of QQF shares and riskless bonds that has a payoff of 1 in period 1 if the price of a QQF share moves upward at the beginning of period 1 and has a payoff of 0 otherwise. Briefly explain your reasoning.
Calculate the number of shares of QQFF stock and the amount of riskless bonds in the replicating portfolio.
(Recall that buying riskless bonds is equivalent to lending money at the riskless rate of interest and selling riskless bonds short is equivalent to borrowing money at the riskless rate. A negative number of shares in a portfolio corresponds to selling those shares short.)
Q3.
(i) Using the replicating portfolio you determined in question 3, calculate the numerical value of πu, the 1-period-ahead up state price for the state where the price of a share of QQF stock makes an up move from period 0 to period 1. Also calculate the numerical value of πd, the 1-period-ahead down state price for the state where the price of a share of QQF stock makes a down move from period 0 to period 1. Explain your reasoning.
(ii) Replicating portfolios, risk neutral probabilities, and state prices can each be used to calculate the market value in period 0 of any asset or portfolio of assets with payoffs in period 1 that are determined by the movement in the price of a share of QQF stock. Write down the formulas that connect the relevant risk neutral probabilities for period 1, pu and pd, and the 1-period- ahead state prices, πu and πd. What are the numerical values of the risk neutral probabilities, pu and pd?
Q4. Consider a portfolio that consists of the following two assets: one European call option for one share of QQF stock that expires in period 1 and has an exercise price of K = 110, and one European put option for one share of QQF stock that expires in period 1 and has an exercise price of
K = 130.
What are the possible payoffs of this portfolio in period 1? Explain your reasoning.
Using the 1-period-ahead up and down state prices, πu and πd, write down a formula for the market value in period 0 of this portfolio. Also, write down a formula for the market value in period 0 of the portfolio in terms of the risk neutral probabilities, pu and pd. Using your formulas, calculate a numerical value for the market value of the portfolio in period 0. Briefly explain your answers.
Q5. There are three possible share prices for QQF stock in period 2: S(2) = 187.5, S(2) = 120, and S(2) = 76.8.
(i) How many price paths on your multiplicative binomial lattice lead to each of these prices in period 2?
(ii) What are the numerical values of the risk neutral probabilities associated with each of the possible values for the share price, S(2), in period 2? What are the numerical values of the Arrow-Debreu state prices associated with each possible value of S(2) in period 2? Briefly explain your answers.
(iii) What is the price in period 0 of a security that pays 90 pounds in period 2 if the share price in period 2 is either S(2) = 187.5 or S(2) = 76.8, and otherwise pays nothing? Explain your reasoning.
Q6. Consider the following project. The project pays an amount 1300 in period 2 if the price of a share of QQF stock in period 2 is 187.5, that is, if the share price has moved upward in both period 1 and period 2. The project also pays an amount 600 in period 2 if the price of a share of QQF stock in period 2 is 120, and the project pays an amount 180 in period 2 if the price of a share of QQFF stock in period 2 is 76.8. For simplicity, suppose that there are no other costs or payoffs associated with the project.
(i) Explain how to estimate the market value in period 0 of this project. Calculate a numerical value for the market value of the project.
(ii) Calculate the expected present value of this project in period 0.
(ii) Does your estimate of the market value in period 0 suggest that investors are risk neutral? If so, briefly explain why. If not, calculate the risk premium associated with this project and briefly explain your reasoning.
Q7. This question asks about general properties of put and call options not necessarily only options on QQF stock.
Consider a portfolio that consists of 1 American put option to sell 1 share of some stock and 1 American call option to buy 1 share of the same stock.
Suppose that the exercise price of both the call and put options is given by the same value K and that the two options have the same .expiration date.
(i) Draw a payoff diagram showing the payoff of this portfolio at the expiration date as a function of the price of the stock at the expiration date. (Unless otherwise stated, the price of the stock in such a payoff diagram is always the price at the expiration date of the option or options.) Make sure to carefully label your diagram.
Also, briefly explain how you determined the payoff of the portfolio for various values of the share price at the expiration date.
(ii) To what extent would you agree or disagree with each of the following two propositions? In each case, explain your answer.
(a) An increase in the exercise price of the call option is more likely to decrease rather than increase the value at the expiration date of the portfolio of put and call options. (Assume that the exercise price of the put option is not changed from its original value.)
(b) Due to discounting, an increase in the time to expiration of each of the two options is likely to decrease the value of the portfolio.
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