SAMPLE SOLUTIONS FOR DERIVATIVES MARKETS
Question #1
Answer is D
If the call is at-the-money, the put option with the same cost will have a higher strike price.
A purchad collar requires that the put have a lower strike price. (Page 76)
Question #2
Answer is C
66.59 – 18.64 = 500 – K exp(–0.06) for K = 480 (Page 69)
Question #3
Answer is D
The accumulated cost of the hedge is (84.30-74.80)exp(.06) = 10.09.
Let x be the market price.
If x < 0.12 the put is in the money and the payoff is 10,000(0.12 – x) = 1,200 – 10,000x. The sale of the jalapenos has a payoff of 10,000x – 1,000 for a profit of 1,200 – 10,000x + 10,000x – 1,000 – 10.09 = 190.
From 0.12 to 0.14 neither option has a payoff and the profit is 10,000x – 1,000 – 10.09 = 10,000x – 1,010.
If x >0.14 the call is in the money and the payoff is –10,000(x – 0.14) = 1,400 – 10,000x. The profit is 1,400 – 10,000x + 10,000x – 1,000 – 10.09 = 390.
The range is 190 to 390. (Pages 33-41)
Question #4
Answer is B
The prent value of the forward prices is 10,000(3.89)/1.06 + 15,000(4.11)/1.0652 +
20,000(4.16)/1.073 = 158,968. Any quence of payments with that prent value is acceptable. All but B have that value. (Page 248)
Question #5
Answer is E
If the index exceeds 1,025, you will receive x – 1,025. After buying the index for x you will have spent 1,025. If the index is below 1,025, you will pay 1,025 – x and after buying the index for x you will have spent 1,025. One way to get the cost is to note that the forward price is 1,000(1.05) = 1,050. You want to pay 25 less and so must spend 25/1.05 = 23.81 today. (Page 112)
Question #6
Answer is E
In general, an investor should be compensated for time and risk. A forward contract has no investment, so the extra 5 reprents the risk premium. Tho who buy the stock expect to earn both the risk premium and the time value of their purcha and thus the expected stock value is greater than 100 + 5 = 105. (Page 140)
Question #7
Answer is C
All four of answers A-D are methods of acquiring the stock. The prepaid forward has the payment at time 0 and the delivery at time T. (Pages 128-129)
Question #8
Answer is B
Only straddles u at-the-money options and buying is correct for this speculation. (Page 78)
Question #9
Answer is D
This is bad on Exerci 3.18 on Page 89. To e that D does not produce the desired outcome, begin with the ca where the stock price is S and is below 90. The payoff is S + 0 + (110 – S) – 2(100 – S) = 2S – 90 which is not constant and so cannot produce the given diagram. On the other
hand, for example, answer E has a payoff of S + (90 – S) + 0 – 2(0) = 90. The cost is 100 + 0.24 + 2.17 – 2(6.80) = 88.81. With interest it is 93.36. The profit is 90 – 93.36 = –3.36 which matches the diagram.
Question #10
Answer is D
[rationale-a] True, since forward contracts have no initial premium
[rationale-b] True, both payoffs and profits of long forwards are opposite to short forwards.
[rationale-c] True, to invest in the stock, one must borrow 100 at t=0, and then pay back 110 = 100*(1+.1) at t=1, which is like buying a forward at t=1 for 110. [rationale-d] Fal, repeating the calculation shown above in part c), but with 10% as a continuously compounded rate, the stock investor must now pay back
100*e.1 = 110.52 at t=1; this is more expensive than buying a forward at t=1
for 110.00.
[rationale-e] True, the calculation would be the same as shown above in part c), but now the stock investor gets an additional dividend of 3.00 at t=.5, which the
forward investor does not receive (due to not owning the stock until t=1). [This is bad on Exerci 2-7 on p.54-55 of McDonald]
[McDonald, Chapter 2, p.21-28]
Question #11
Answer is C
Solution: The 35-strike call has future cost (at t=1) of 9.12*(1+.08) = 9.85
The 40-strike call has future cost (at t=1) of 6.22*(1+.08) = 6.72
The 45-strike call has future cost (at t=1) of 4.08*(1+.08) = 4.41
If S1<35, the profits of the 3 calls, respectively, are -9.85, -6.72, and -4.41.
If 35<S1<40, the profits of the 3 calls, respectively, are S1-44.85, -6.72, and -4.41.
If 40<S1<45, the profits of the 3 calls, respectively, are S1-44.85, S1-46.72, and -4.41.
If S1>45, the profits of the 3 calls, respectively, are S1-44.85, S1-46.72, and S1-49.41.
The cutoff points for when the relative profit ranking of the 3 calls change are:
S1-44.85=-6.72, S1-44.85=-4.41, and S1-46.72=-4.41, yielding cutoffs of 38.13, 40.44, and 42.31.
If S1<38.13, the 45-strike call has the highest profit, and the 35-strike call the lowest.
If 38.13<S1<40.44, the 45-strike call has the highest profit, and the 40-strike call the lowest.
If 40.44<S1<42.31, the 35-strike call has the highest profit, and the 40-strike call the lowest.
If S1<42.31, the 35-strike call has the highest profit, and the 45-strike call the lowest.
We are looking for the ca where the 35-strike call has the highest profit, and the 40-strike call has the lowest profit, which occurs when S1 is between 40.44 and 42.31.
[This is bad on Exerci 2-13 on p.55-56 of McDonald]
[McDonald, Chapter 2, p.33-37]
Question #12
Answer is B
Solution: The put premium has future value (at t=.5) of 74.20 * (1+(.04/2)) = 75.68 Then, the 6-month profit on a long put position is: max(1,000-S.5,0)-75.68. Correspondingly, the 6-month profit on a short put position is 75.68-max(1,000-S.5,0). The two profits are opposites (naturally, since long and short positions have opposite payoff and profit). Thus, they can only be equal if producing 0 profit. 0 profit is only obtained if 75.68 = max(1,000-S.5,0), or 1,000-S.5 = 75.68, or S.5 = 924.32. [McDonald, Chapter 2, p.39-42]
Question #13
Answer is D
Solution: Buying a call, in conjunction with a short position in a stock index, is a form of insurance called a cap. Answers (A) and (B) are incorrect becau they relate to a floor, which is the purcha of a put to insure against a long position in a stock index. Answer (E) is incorrect becau it relates to writing a covered call, which is the sale of a call along with a long position in the stock index, so th
at the investor is lling rather than buying insurance. Note that a cap can also be thought of as ‘buying’ a covered call. Now, let’s calculate the profit:
2-year profit = payoff at time 2 – the future value of the initial cost to establish the position = (-75 + max(75-60,0)) – (-50 + 10)*(1+.03)2 = -75+15+40*(1.03)2 = 42.44-60 = -17.56. Thus, we’ve lost more from holding the short position in the index (since the index went up) than we’ve gained from owning the long call option.
[McDonald, Chapter 3, p.59-65]
Question #14
Answer is A
Solution: This consists of standard applications of the put-call parity equation on p.69. Let C be the price for the 40-strike call option. Then, C + 3.35 is the price for the 35-strike call option. Similarly, let P be the price for the 40-strike put option. Then, P – x is the price for the 35-strike put option, where x is what we’re trying to find. Using put-call parity, we have:
(C + 3.35) + 35*e-.02 - 40 = P – x (this is for the 35-strike options)
C + 40*e-.02 – 40 = P (this is for the 40-strike options)
Subtracting the first equation from the cond, 5*e-.02 – 3.35 = x = 1.55.
[McDonald, Chapter 3, p.68-69]
Question #15
Answer is C
Solution: The initial cost to establish this position is 5*2.78 – 3*6.13 = -4.49. Thus, you are receiving 4.49 up front. This grows to 4.49*e .08*.25 = 4.58 after 3 months. Then, the following payoff/profit table can be constructed at T=.25 years:
S T : 5*max(S T – 40, 0) – 3*max(S T – 35, 0) + 4.58 = Profit
S T <
35 0 - 0 + 4.58 = 4.58 35 <= S T <= 40 0 - 3*(S T – 35) + 4.58 = 109.58-3S T S T > 40 5*(S T -40) - 3*(S T – 35) + 4.58 = 2S T -90.42
Thus, the maximum profit is unlimited (as S T increas beyond 40, so does the profit) Also, the maximum loss is 10.42 (occurs at S T = 40, where profit = 109.58-120 = -10.42)
[Notes] The above problem is an example of a ratio spread.
[McDonald, Chapter 3, p.73]
Question #16
Answer is D
Solution: The ‘straddle’ consists of buying a 40-strike call and buying a 40-strike put. This costs 2.78 + 1.99 = 4.77 at t=0, and grows to 4.77*e .02 = 4.87 at t=.25. The ‘strangle’ consists of buying a 35-strike put and a 45-strike call. This costs 0.44 + 0.97 = 1.41 at t=0, and grows to 1.41*e .02 = 1.44 at t=.25. For S T <40, the ‘straddle’ has a profit of 40-S T -4.87 = 35.13, and for S T >=40, the ‘straddle’ has a profit of S T -40-4.87 = 44.87. For S T <35, the ‘strangle’ has a profit of 35-S T -1.44 = 33.56, and for S T >45, the ‘strangle’ has a profit of S T -45-1.44 = 46.44. However, for 35<=S T <=45, the ‘strangle’ has a profit of -1.44 (since both options would not be exercid). Comparing the payoff structures between the ‘straddle’ and ‘strangle,’ we e that if S T <35 or if S T >45, the ‘strad
dle’ would outperform the ‘strangle’ (since 35.13 > 33.56, and since -44.87 > -46.44). However, if 35<=S T <=45, we can solve for the two cutoff points for S T , where the ‘strangle’ would outperform the ‘straddle’ as follows:
-1.44 > 35.13 – S T, and -1.44 > S T - 44.87. The first inequality gives S T > 36.57, and the cond inequality gives S T < 43.43. Thus, 36.57 < S T < 43.43.
[McDonald, Chapter 3, p.78-80]
