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1 Name: Thursday, February 28 th M375T=M396C Introduction to Actuarial Financial Mathematics Spring 2013, The University of Texas at Austin In-Term Exam I Instructor: Milica Čudina Notes: This is a closed book and closed notes exam. The maximal score on this exam is 100 points. Time: 75 minutes MULTIPLE CHOICE 1 (5) a b c d e TRUE/FALSE 1 (2) TRUE FALSE 2 (2) TRUE FALSE 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 FOR GRADER S USE ONLY: DEF N T/F M.C. Σ

2 2 Standing assumptions: No-arbitrage All options are European in style DEFINITIONS 1. (5 points) Write the definition of an arbitrage opportunity. Solution: See your notes.

3 TRUE/FALSE QUESTIONS Please, circle the correct answer on the front page of this exam. 1. (2 pts) Source: Sample FM(DM) Problem #27. The position consisting of one long homeowners insurance contract benefits from falling prices in the underlying asset. Solution: TRUE Recall our comparison of the homeowner s insurance policy to the put option. The payoff of the put option is decreasing in the price of the underlying asset. 2. (2 pts) Consider a portfolio consisting of the following four European options with the same expiration date T on the underlying asset S: long one call with strike 40, long two calls with strike 50, short one call with strike 65. Let S(T ) = 69. Then, the payoff from the above position at time T is less than 60. Solution: FALSE The payoff is (69 40) + 2(69 50) (69 65) = 63. 3

4 4 FREE-RESPONSE PROBLEMS 1. (20 points) Consider a European call option and a European put option on a nondividend-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, V C (0) V P (0) = 0.15, T = 4, K = 70. We employed a no-arbitrage argument to get the put-call parity: V C (0) V P (0) = S(0) K rt r = 1 ( ) T ln K. S(0) V C (0) + V P (0) Using in the data provided, we get r = 0.06.

5 2. (20 points) Let the initial price of a non-dividend-paying stock be $20 and the risk-free continuously compounded interest rate be r = 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! Solution: One equality which is always true in arbitrage-free market-models is the put-call parity. Let us examine if it holds for the above data. In our usual notation: V C (0) V P (0) = = 1, S(0) Ke rt = 20(1 e 0.05 ) = So, the put-call parity is violated. This observation helps us construct the arbitrage portfolio. Noticing that the portfolio consisting of the outright purchase of the asset and borrowing K rt can be considered to be relatively cheap as compared to the portfolio consisting of the long call and the short put, we decide to do the following at time 0: (1) buy one share of stock, (2) borrow K rt at the risk free rate to be repaid at time T, (3) write a European call option on this asset with strike K and exercise date T, and (4) buy a European put option on this asset with strike K and exercise date T. The initial cost of this portfolio is: S(0) Ke rt (V C (0) V P (0)) < 0. The negative initial cost means that we initially receive money. This money can be invested at the risk-free rate (thus creating a fully-leveraged portfolio) or just kept (at the zero interest rate). At time T, the payoff/worth of our portfolio is always: S(T ) K (S(T ) K) + + (K S (T )) + = 0. Since we started with an inflow of money and broke-even at time T regardless of the final stock price, the above portoflio is an arbitrage portfolio. 5

6 6 3. (20 points) Which of the positions listed will benefit from the underlying asset s price decline? Draw the payoff curves for each position and justify your answer. (i) Short put (ii) Long put (iii) Short call (iv) Short stock (v) Short forward contract Solution: Only the short put is long in the underlying asset.

7 4. (8 points) A stock currently sells for $ A 6 month call option with strike $35.00 has a premium of $2.27. Assuming a 4% continuous dividend yield, what is the price of the associated put option as dictated by put-call parity? Solution: We have: V P (0, 35, 0.5) = V C (0, 35, 0.5) e δt S 0 + e rt 35 V P (0, 35, 0.5) = $2.27 e e = $

8 8 MULTIPLE CHOICE QUESTIONS Please, circle the correct answer on the front page of this exam. 1. The initial price of the market index is $900. After 3 months the market index is priced at $920. The nominal rate of interest convertible monthly is 4.8%. The premium on the long call, with a strike price of $930, is $2.00. What is the profit or loss at expiration for this long call? (a) $2.00 loss (b) $2.02 loss (c) $2.02 gain (d) $2.00 gain (e) None of the above. Solution: (b) In our usual notation, the profit is (S(T ) K) + C (1 + j) 3 with C denoting the price of the call and j the effective monthly interest rate. We get ( ) The premium on a 2-month call option on the market index with an exercise price of 1050 is $9.30 when originally purchased. After 2 months the position is closed and the index spot price is If interest rates are 0.5% effective per month, what is the call s profit? (a) $9.30 (b) $9.39 (c) $12.61 (d) $22.00 (e) None of the above. Solution: (c) The value at expiration of the cost of the call is The payoff of the call is = 22. So the profit is = Jafee Corp. common stock is priced at $36.50 per share. The company just paid its $0.50 quarterly dividend. Interest rates are 6.0%. A $35.00 strike European call, maturing in 6 months, sells for $3.20. What is the price P of a 6-month, $35.00 strike put option? (a) 0 P < $1.25 (b) $1.25 P < $1.45 (c) $1.45 P < $1.55 (d) $1.55 P < $1.66 (e) $1.66 P

9 9 Solution: (d) V P (0) = V C (0) + Ke rt F P 0,T (S) = V C (0) + Ke rt S(0) + De rt 1 + De rt 2 = e 0.06/ e 0.06/ e 0.06/

10 10 4. (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) $ (b) $ (c) $ (d) $950 (e) None of the above. Solution: (c) Per futures contract, the initial deposit into the margin account is = 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 150e 0.05 (10/365) So, the price of an index futures contract should drop by = to cause a margin call. In other words, the index futures price needs to be = 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/loss is made by shorting the corresponding European put whose premium is $2.50? (a) $0.15 loss (b) $0.15 gain (c) $0.36 loss (d) $0.36 gain (e) None of the above. Solution: (b) or (e) We can obtain the no-arbitrage premium of the corresponding put as dictated by put-call parity, as follows: V P (0, K = 70, T = 0.5) = V C (K = 70, T = 0.5) + e rt K S(0) + P V 0,T (Dividends) = e e e = = = Since we have decided to short the put at a premium higher by $0.12, the answer is 0.015e

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