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M339D=M389D Introduction to Actuarial Financial Mathematics University of Texas at Austin Sample In-Term Exam II - Solutions Instructor: Milica Cudinaˇ
Notes: This is a closed book and closed notes exam. The maximal score on this exam is 100 points. Time: 50 minutes
TRUE/FALSE
1 (2) TRUE FALSE
2 (2) TRUE FALSE
3 (2) TRUE FALSE
4 (2) TRUE FALSE
5 (2) TRUE FALSE
MULTIPLE CHOICE
1 (5) a b c d e
2 (5) a b c d e
3 (5) a b c d e
4 (5) a b c d e
5 (5) a b c d e
6 (5) a b c d e
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8 (5) a b c d e
DEFINITIONS
Problem 2.1. (5 points) Write the definition of an arbitrage portfolio.
FREE-RESPONSE PROBLEMS
Problem 2.2. (20 points) A certain common stock is priced at $42.00 per share. Assume that the continuously compounded interest rate is r = 10.00% per annum. Consider a $50−strike European call, maturing in 3 years which currently sells for $10. 80. What is the price of the corresponding 3−year, $50−strike European put option?
Solution: Due to put-call parity, we must have
VP (0) = VC (0) + e−rT^ K − S(0) = 10.80 + e−^0.^30 · 50 − 42. 00 ≈ 5. 84.
Problem 2.3. (20 points) Consider a European call option and a European put option on a non-dividend-paying stock. Assume:
(1) The current price of the stock is $55. (2) The call option currently sells for $0.15 more than the put option. (3) Both options expire in 4 years. (4) Both options have a strike price of $70.
Calculate the continuously compounded risk-free interest rate r.
Solution: In our usual notation,
S(0) = 55, VC (0) − VP (0) = 0. 15 , T = 4, K = 70.
We employed a no-arbitrage argument to get the put-call parity:
VC (0) − VP (0) = S(0) − K−rT^ ⇒ r =
T
ln
K
S(0) − VC (0) + VP (0)
Using the data provided, we get r = 0.06.
Problem 2.4. (30 points) Let the initial price of a non-dividend-paying stock be $20 and the risk-free continuously compounded interest rate be r = 0.05. Assume that the current premium for an at-the-money European put on this asset with expiration date in one year equals $0.50. The premium for the European call with the same strike and expiration date and on the same asset is $1.50. Is there an arbitrage opportunity? If your answer is affirmative, provide an arbitrage portfolio and show that it is an arbitrage portfolio. If your answer is negative, justify it!
MULTIPLE CHOICE QUESTIONS
Please, circle the correct answer on the front page of this exam.
Problem 2.5. (5 points) Source: Problem #2 from the Sample FM(DM) questions. You are given the following information:
(1) The current price to buy one share of XYZ stock is 500. (2) The stock does not pay dividends. (3) The risk-free interest rate, compounded continuously, is 6%. (4) A European call option on one share of XYZ stock with a strike price of K that expires in one year costs $66. 59. (5) A European put option on one share of XYZ stock with a strike price of K that expires in one year costs $18. 64.
Determine the strike price K.
(a) $ (b) $ (c) $ (d) $ (e) None of the above.
Solution: (c) This problem is a simple application of put-call parity. In our usual notation,
VC (0) − VP (0) = S(0) − e−rT^ K ⇒ K = erT^ (S(0) − VC (0) + VP (0)) = e^0.^06 ·^1 (500 − 66 .59 + 18.64) = 480.
Problem 2.6. (5 points) A stock is currently priced at $118 per share. It is scheduled to pays a continuous dividend in the amout proportional to its price with the yield of 2.0% per annum. A nine-month 120-strike European call and put options on this stock have equal prices. Let the continuously-compounded annual risk-free rate of interest be denoted by r. Then,
(a) r ≤ 0. 04 (b) 0. 04 < r ≤ 0. 045 (c) 0. 045 < r ≤ 0. 05 (d) 0. 05 < r ≤ 0. 055 (e) None of the above.
Solution: (b) By the put-call parity, we have
0 = VC (0) − VP (0) = S(0)e−δT^ − Ke−rT^ ⇒ − rT = ln
S(0)
K
− δT
⇒ r =
T
ln
K
S(0)
Problem 2.7. (5 points) Consider an investment in S&P 500 Index futures contracts at a price of $1000. The initial margin requirement is 15.0% of the notional value. The maintenance margin is $100. If the continuously compounded interest rate is 5.0% what will the futures price need to be for a margin call to occur 10 days from now? Assume no settlement within the 10 days (i.e., the futures price does not change within the 10 days).
(a) $939. (b) $940. (c) $949. (d) $ (e) None of the above.
Solution: (c) Per futures contract, the initial deposit into the margin account is
1000 · 0 .15 = 150.
Over the course of the next 10 days, interest is accrued and the balance in the account at the end of the 10 days is
150 e^0.^05 ·(10/365)^ ≈ 150. 21.
So, the price of an index futures contract should drop by 150. 21 −100 = 50.21 to cause a margin call. In other words, the index futures price needs to be 1000 − 50 .21 = 949. 79.
Problem 2.8. (5 points) A certain common stock is priced at $74.20 per share. The company just paid its $1.10 quarterly dividend. Assume that the interest rate is r = 6.0%. Consider a $70 strike European call, maturing in 6 months which currently sells for $6. 50. How much (arbitrage) profit is made by shorting the corresponding European put whose premium is $2.50?
(a) $0.124 gain (b) $0.15 gain (c) $0.36 loss (d) $0.36 gain (e) None of the above.
Solution: (a) We can obtain the no-arbitrage premium of the corresponding put as dictated by put-call parity, as follows:
VP (0, K = 70, T = 0.5) = VC (K = 70, T = 0.5) + e−rT^ K − S(0) + P V 0 ,T (Dividends) = 6.50 + e−^0.^03 · 70 − 74 .20 + e−^0.^06 ·^0.^25 · 1 .10 + e−^0.^06 ·^0.^5 · 1. 10 = − 67 .70 + 0. 97 · 70 + 0. 98 · 1 .10 + 0. 97 · 1. 10 = − 67 .70 + 68.97 + 1.08 = 2. 38.
Since we have decided to short the put at a premium higher by $0.12, the answer is 0. 12 e^0.^03 ≈ 0 .124.
Problem 2.11. (5 points) You are given that the level swap prices for a one-year swap and a two-year swap are $2.25 and $2.60, respectively. Assume that the zero-coupon bond yield rates for a one-year and a two-year bond are 4% and 5%, respectively. What is the forward price for a two-year forward contract on the same asset?
(a) About $1. (b) About $2. (c) About $3. (d) About $3. (e) None of the above.
Solution: (b) Let x be the forward price for a two-year forward contract. Then, it needs to satisfy the following equation
- 25
- 04
x
- 052
Solving for x above, we get x ≈ 2. 97.
Problem 2.12. (5 points) The S&P 500 Index is priced at $950.46. The annualized dividend yield on the index is 1.40%. The continuously compounded annual interest rate is 8.40%. What is the forward price for a forward contract with delivery date 9 months from today?
(a) $937. (b) $942. (c) $984. (d) $1001. (e) None of the above.
Solution: (d) In our usual notation,
F 0 ,T (S) = S(0)e(r−δ)T^ = 950. 46 e(0.^084 −^0 .014)0.^75 ≈ 1001. 69.
Problem 2.13. (5 points) The forward prices on a barrel of crude oil are $43 and $45 for delivery in one year and two years, respectively. The effective annual yield rates on zero- coupon government bonds are 4.0% and 4.5% for one-year and two-year bonds, respectively. What is the two year (level) swap price on a barrel of crude oil?
(a) About $43. (b) About $43. (c) About $44. (d) About $45. (e) None of the above.
Solution: (b)
x =
43
- 04 +^
45
- 0452 1
- 04 +^
1
- 0452
