Introduction to Financial Engineering
ISyE 6227
Additional Real Options Homework Solutions
Problem 1. a.
0 / 1 / 2100 / 130 / 169
76.92 / 100
Stock Price / 59.17
Risk-neutral probability p of an up move = (106 – 76.92) / (130 – 76.92) = 0.548
0 / 1 / 27.42 / 0 / 0
17.41 / 0
European Put Price / 40.83
Note: 7.42 = 40.83(0.452)2 / (1.06)2
b.
0 / 1 / 29.84 / 0 / 0
Max{17.41, 100 – 76.92} = 23.08 / 0
American Put Price / 40.83
Problem 2.
Use Put-Call Parity: S + P - C = K(1 + r)-3 = 74(1.0496)-3 = 64, which implies P = 87.57 – 80 = 7.57.
Problem 3.
220/ 8.94
0.36*
0.6
150
/ 0.493* / 5.85
0.0447S – 0.855B
0.8 / 0.4
115 / 0.64* / 120
5.88 / 4.47
-0.00883S + 6.90B / 0.34*
0.2
0.7
0.506* / 90
/ 6.38
-0.082S + 13.76B
0.3
0.66* / 80
7.75
a. Objective and risk-neutral (marked with an asterisk) probabilities are shown along the lattice.
c. { [(0.8)(0.6)](8.94) + [(0.8)(0.4) + (0.2)(0.7)](4.47) + [(0.2)(0.3)(7.75] } / (1.20)2 = 4.73.
d. Replicating portfolio shown on lattice. Must use risk-neutral probabilities and risk-free rate.
e. Decision-Tree value = 4.73, which is less than the CORRECT value of 5.88. Market is pricing option cheap, so buy it. How? SELL the replicating portfolio of { -0.00883S + 6.90B } and collect 5.88, i.e., acquire the portfolio { 0.00883S - 6.90B } which entails a liability of 5.88, use 4.73 to buy the option, and place the difference 1.15 in bank to collect interest at 4% per year. As the stock price path unfolds REBALANCE portfolio as prescribed in each node. You will always have EXACTLY how much money you need to rebalance. For example, if the stock price = 150 at time t =1, the portfolio you acquired at time t = 0 is now worth –5.85, namely, you owe 5.85. You can pay off this obligation by acquiring the (new) replicating portfolio { -0.0447S + 0.855B } in which you would collect 5.85. Similarly, if the stock price = 90 at time t = 1, the portfolio you acquire at time t = 0 is now worth –6.38, namely, you owe 6.38. You can pay off this obligation by acquiring the (new) replicating portfolio { 0.082S – 13.76B } in which you would collect 6.38. At time t = 2 the payoffs of your rebalanced portfolio at time t =1 will either be –8.94, -4.47 or –7.75, namely, you owe these amounts. Fortunately, the option you purchased at time t = 0 will exactly match these obligations at time t = 2 (by construction). At the end you will have 1.15(1.04)2 = 1.24, risk-free!
Problem 4.
a. PV = 0.20(3000) + 0.80(500) / (1.25) = 800.
b. NPV = 800 – 1200 = -400 < 0.
c. Since NPV is negative the company should not invest in this project.
d. Consider time t = 1. If the market goes up, and we undertake the expansion the new project value will be 2(3000) – 800 = 5200, which is greater than the original 3000, so we would expand IF the market goes up. (The additional expenditure of 800 only occurs if we choose to expand, and this decision would only be made after the market outlook has revealed itself at time t = 1.) If, on the other hand, the market goes down, it clearly pays to liquidate and obtain the value of 0.75(1200) = 900. Thus, our final project payoffs are either 5200 or 900. Now what is this worth at time t = 0? Consider the original project as the traded underlying security. It’s value today is 800, and its value next period is either 3000 or 500. With a risk-free rate of 10%, the risk-neutral probability of going up is 0.152 = (3000 – 880) / (3000 – 500). Thus, the discounted expected value of the project with the options (using the risk-neutral probability and the risk-free rate) is { (0.152)(5200) + (0.848)(900) } / (1.1) = 1412.36. The new NPV is therefore 1412.36 – 1200 = 212.36, so the project with the flexibility should be undertaken.
Problem 5.
0 / 1 / 2 / 3 / 4 / 512.50 / 18.75 / 28.13 / 42.19 / 63.28 / 94.92
8.33 / 12.50 / 18.75 / 28.13 / 42.19
5.56 / 8.33 / 12.50 / 18.75
3.70 / 5.56 / 8.33
Project Value Event Tree Without Delay Option / 2.47 / 3.70
1.65
Capital Investment
10.00 / 11.00 / 12.10 / 13.31 / 14.64 / 16.11
Maintenance Cost
0.50 / 0.50 / 0.50 / 0.50 / 0.50
0 / 1 / 2 / 3 / 4 / 5
4.12 / 8.41 / 16.03 / 28.88 / 48.64 / 78.81
1.47 / 3.06 / 6.43 / 13.49 / 26.08
1.20 / 1.20 / 1.27 / 2.64
1.20 / 1.20 / 1.20
Project Value Event Tree With Delay Option / 1.20 / 1.20
1.20
The numbers in bold signify the development project should be undertaken. Since the developer could begin development today for an NPV of 2.5 million, and since the value of the development when the delay option is exists is 4.12 million, the developer would be willing to pay up to 1.62 million to acquire this delay option.
Problem 6.
0 / 1 / 2100.00 / 149.18 / 222.55
67.03 / 100.00
Firm Value Event Tree / 44.93
Let F denote the face value of the debt to be paid to the bondholders at time t = 2. The bondholders pay 50.00 today to acquire the payoff of Max{F, V} at time t = 2. Given F we would value the debt today at time t = 0 by taking the discounted expected payoff using the risk-neutral probabilities. Since the debt value today is given at 50.00 we seek the value for F for which E[ Max{F, V}] = 50.00. Begin by assuming that F < 100.00. The risk-neutral probability of an up-move is 0.464, and so the risk-neutral probability of a down move is 0.536. It is reasonable to assume that F < 100.00; otherwise, the yield on the debt would exceed 35%. When F < 100.00 the bondholders will receive F unless there are 2 consecutive down moves, which occurs with probability (0.536)2 = 0.287. Thus, the expected final payoff for the bondholders under the risk-neutral probabilities is
F – (0.287)[F – 44.93] = 0.713F + 12.91,
which must equal the future value at time t = 2 of 50.00 = (50.00) e0.10 = 55.26. Thus, F = 59.40. The yield on this debt is ½ ln (59.40/50.00) or 8.61%. With a continuously compounded market expected rate of return of 15% the objective probability of a down move is 0.402. Accordingly, the market’s expected value of the final payoff for the bondholders is
F – (0.402)2[F – 44.93] = 0.838F + 7.26 = 57.04,
which implies a market expected rate of return on the debt of ½ ln (57.04/50.00) or 6.59%.