Economics 377, DINY Spring 2014
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Problem Set 2
Assigned problems due in class March 27, 2014
Put your answer on separate paper or in the space provided, whichever you prefer. Full
credit requires showing your work and giving a full explanation when required. Please staple
the assignment to the top of any extra sheets you turn in.
1. The file XOM-2014-03-19.docx shows the price of Exxon-Mobil XOM along with the
prices of put and call options written on XOM late-afternoon March 19, 2012 along with
some other information. The options expire on October 18, 2014. There are 213 calendar
days between the two dates, so take T = 213/365 = 0.5836, use r = 0.01 as the continuously
compounded interest rate, for simplicity ignore dividends, and treat the options as European
(though they are actually American). For option prices just use the mid-quote (average of
bid and ask prices).
A. Suppose a trader uses the stock, the Put-90 and Call-100 to form a stock and collar for
XOM. Draw the payoff and profit diagrams and label the key points using numerical values
where available.
Payoff (left side): Min Payoff = 90, Max Payoff = 100; the payoff is always positive with no
zero payoff. Kinks at 90 and 100.
Profit (right side): Cost equals erT × (3.50 + 93.73 − 1.70) = 96.09, Min profit = −6.09, Max
profit = 3.91, Zero profit (break even) at 96.09, Kinks at 90 and 100.
Figure below; see class notes for labels.
B. Suppose a trader uses the stock and the Put-90 to create for XOM a protective put.
Draw the payoff and profit diagrams and label the key points using numerical values where
available. What is the total cost of the protective put?
Buy stock and Put-90. Payoff (left side): Min Payoff = 90, no Zero Payoff, Max Payoff =
unlimited, Kink at 90.
Profit(right side): Cost equals erT × (93.73 + 3.50) = 97.80, Min profit = −7.80, Max profit
= unlimited, Zero Profit at 97.80, Kink at 90.
See figure below.
C. Verify (theoretically) that the payoff of a portfolio comprised of e−rT ×90 in cash (money
market) and the Call-90 exactly replicates the payoff of the protective put in B. Since the
Economics 377, DINY Spring 2014
Figure 1: Problem 1A
Figure 2: Problem 1B
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Economics 377, DINY Spring 2014
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payoff of this portfolio and the protective put are exactly the same, their costs must be the
same, which is put-call parity. Check how closely the parity is satisfied for these prices; it
will not be exact in actual data because of the inexact interest rate assumption and other
small deviations.
Payoff of protective put at T is ST + max(K − ST , 0) = max(K, ST ) = max(K, ST )
Payoff of call and an amount of cash equal to PV of strike is max(ST − K, 0) + erT × e−rT K =
max(ST − K, 0) + K = max(ST , K) = max(ST , 90)
The parity condition (in the absence of dividends) is p + S0 = c + P V (K).
For these prices p + S0 = 97.23 and c + e−0.01×0.5836 × 90 = 95.76. The agreement is close
but not great. This particular parity computation ignores the dividend paid by XOM, the
interest rate assumption could be wrong by a bit, plus there are the bid-ask spreads and
other transactions costs. So we shall say OK in this case. The apparent violation is still
quite small and would likely not be realizable in the real world.
2. The Striking Price column in Barron’s suggested a call tree strategy that entails buying
an in-the-money call, and writing both an at-the-money and out-of-the-money call. Use the
setup of Problem 1. to answer the questions below.
A. Suppose a trader uses the Call-85, Call-92.50, and Call-100 to create a call tree on XOM.
In the space below draw the payoff and profit diagrams for this call tree.
Payoff side: Zero Payoff for 0 ≤ S ≤ 85, and Negative Payoff greater than 104, MaxPayoff=7.50, Kinks at S=85,92.50, and 100.
Profit side: Cost equals erT (9.93 − 4.75 − 1.70) = 3.50, MaxProfit = 4.00, Zero profit
at 88.50 and at 104.00, minimum profit is unlimited negative, Kinks at 88, 92.50, 100. Figure
below; see your class notes for the key points.
B. How would you describe the call tree strategy, i.e., directional (bullish, bearish), nondirectional, or some combination? It is like a non-directional bet that XOM will stay
close to $93 near the maturity of the options, but to some extent bearish as well, because
the trader would have to believe XOM will not greatly rise in price and generate a huge
loss on the call tree. Actually, at these prices the call tree is a “dog” (unattractive) trading
strategy. The reverse call tree would look extremely attractive if the trades could actually
Economics 377, DINY Spring 2014
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Figure 3: Problem 2A
be implemented at the mid-quotes, which is very unlikely.
3. The spreadsheet 377 02.xlsx shows the value of the stock ST and the probability for
each of nine possible states in five months hence T = 5/12.
A. What
P9is the expected value of the payoff on the call-20? Remember the expected payoff
is just 1 πk Payoffk where Payoffk is the payoff of the call in state k (sometimes zero sometimes positive).
Call-20: Expected Payoff = 9.63
B. The fair value of the call-20 is the present value of the expected payoff. Do this computation:
Fair value (price) Call-20: = e−rT × Expected Payoff = 9.56
C. The same rules apply to all financial instruments. Compute the fair value (price) of
a. Put-30 = 4.96
b. Call-20, Call-35 bull spread = 9.56 − 1.61 = 7.94
c. Call-30, Put-30 Straddle Position 4.96 + 3.35 = 8.31
d. The stock itself e−rt × E(ST ) = 28.17
For the last item, note that the rule applies with equal force to the underlying stock, which
is a traded security.