Download Option Concepts Homework II

Option Concepts II Homework Solutions
ISyE 4803C (Fall 06)
Problem 1. a.
0
100
1
130
76.92
Stock Price
2
169
100
59.17
Risk-neutral probability p of an up move = (106 – 76.92) / (130 – 76.92) = 0.548
0
7.42
1
0
17.41
2
0
0
40.83
1
0
Max{17.41, 100 – 76.92} = 23.08
2
0
0
40.83
European Put Price
Note: 7.42 = 40.83(0.452)2 / (1.06)2
b.
0
9.84
American Put Price
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.
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 the expansion is undertaken the new project value will
be 2(3000) – 800 = 5200, which is greater than the original 3000, so the company should expand IF the
market goes up. (The additional expenditure of 800 only occurs if the expansion option is exercised, 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, the
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 4.
a. The portfolio –2S1 + 1S2 costs nothing initially and yields a payoff vector of (20,10). Alternatively, the
portfolio –3S1 + 1S2 costs –5 initially (i.e. you receive 5 at time 0) and yields a payoff vector of (0,0).
b. S3 behaves like the money market security. One may use S1 and S3 to construct the replicating portfolio
9.7S1 – 450.476S3 that will match the payoffs of S2. This portfolio costs 131.524 to acquire, which does
not equal the current price, 300, of S2. An arbitrage opportunity exists. Note that the risk-neutral
probability associated with the market spanned by S1 and S3 is q = 0.46. One may also use S2 and S3 to
construct the replicating portfolio (1/9.7)S2 + 46.439S3 that will match the payoffs of S1. This portfolio
costs 77.637 to acquire, which does equal the current price, 60, of S1. An arbitrage opportunity exists.
Note that the risk-neutral probability associated with the market spanned by S2 and S3 is q = 0.8427.
Remark: In a complete market, which we have here, all securities can be valued via discounted expectation
with the risk-neutral probability. The risk-neutral probability must be unique. Why? Consider the security
ei that pays off 1 in state i and 0 otherwise. Let Θ be a replicating portfolio (i.e. DΘ = e i). The cost of this
portfolio is pTΘ. We know from our theory that we can alternatively determine the cost of this portfolio as
(1/R)Eq[ei] = (1/R) qi, where q is a risk-neutral probability vector. Hence, the cost of the portfolio Θ is
(1/R) αi . All replicating portfolios must have the same cost; otherwise there would be arbitrage. Thus, for
any two risk-neutral probability vectors, α and β, it must be the case that (1/R) αi = (1/R) βi or αi = βi. We
conclude that α = β, and so the risk-neutral probability vector is unique, as claimed. Since the respective
risk-neutral probabilities, 0.46 and 0.8427, are not equal the market (p, D) in (b) must exhibit arbitrage.
Problem 5.
220
8.94
0.36*
0.6
0.493*
150
5.85
0.0447S – 0.855M
0.8
0.4
115
5.88
-0.00883S + 6.90M
0.64*
120
4.47
0.34*
0.2
0.7
0.506*
90
6.38
-0.082S + 13.76M
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 the
option cheap, so buy it. How? SELL the replicating portfolio of {-0.00883S + 6.90M} and collect 5.88,
i.e., acquire the portfolio {0.00883S - 6.90M} 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 + 0.855M} 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 – 13.76M} 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 have 1.15(1.04) 2 = 1.24, risk-free!