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Solution to Sample Midterm Finance 353 Derivatives Daytime MBA Fall 2009 Section I: Multiple Choice Questions. (4X15=60 points). 1. Assuming frictionless market, which of the following cannot be the profit diagram of any derivative contract? Answer: C Shorting the derivative contract creates an arbitrage opportunity. 2. Suppose you enter into a short futures contract to sell July Silver for $5.20 per ounce on the New York Commodity Exchange. The size of the contract is 5,000 ounces. The initial margin is $4,000, and the maintenance margin is $3,000. Which of the following prices will lead to a margin call? A. $5.45 per ounce. B. $ 5.30 per ounce. C. $4.89 per ounce. D. $5.10 per ounce. Answer: A. In order to receive a margin call, you need to lose $1,000. Since one contract is 5,000 ounces, and you are shorting the contract, you will lose $1,000 if silver price increase by more than 1000/5000=$0.20 Additional question: which of the above price would allow you to withdraw $1,000 from the margins account? Answer: C 3. Assuming the risk-free interest is 9% per annum with continuous compounding and that the dividend yield on a stock index varies throughout the year. In February, May, August, and November, dividends are paid at a rate of 5% per annum. In other months, dividends are paid at a rate of 2% per annum. Suppose the value of the index on July 31, 2006 is 300. What is the futures price (per share) for a contract deliverable on December 31, 2006? A. $265.41 B. $307.34. C. $305.04 D. $317.91 Hint: Use the average dividend yield. Answer: B Aug Sep Oct Nov Dec 5% 2% 2% 5% 2% Hence the average dividend yield is (5%+2%+2%+5%+2%)/5=3.2%. Price of the futures 5 ( 0.090.032)* contract is thus: F0,T S 0 e ( r )T 300 * e 12 307.3383 Question 4, 5, and 6 are based on the following information: A stock is expected to pay a dividend of $1 per share in 2 months and in 5 months. The current stock price is $50, and the continuous compounding risk free interest rate is 8% per annum. An investor has just taken a short position in a 6-month forward contract on the stock. 4. What is the arbitrage free price of the forward contract? A. $52.04 B. $50.01 C. $47.96 D. $48.05 Answer: B. The arbitrage free forward price is 6 4 1 0.08* 0.08* 0.08* F0,T S0erT FV ( D) 50* e 12 1* e 12 1* e 12 50.007 5. Suppose your answer in question 4 is in fact the price specified in the investor’s forward contract. What is the value of the forward contract? A. $0 B. $50.01 C. $47.96 D. $2.00 Answer: A The date 0 value of a forward contract is always 0. 6. Suppose your answer in question 4 is in fact the price specified in the investor’s forward contract. Three months later, the price of the stock is $48 and the risk free rate of interest is still 8% per annum. What is the value of the short position of the forward contract? A. $0 B. $2.004 C. $-2.00 D. $2.00 Answer: B The arbitrage free forward price is 3 1 0.08* 0.08* F0,T S0erT FV ( D) 48* e 1* e12 47.963 12 However, the forward contract allows you to sell the same stock three months later at $50, thereby making a profit of 50.007-47.963 =$2.044. Therefore the 3 0.08 present value of the contract is 2.044 e 12 $2.004 7. Suppose the continuously compounded Euro-denominated Libor is 4.5% per annum, the continuously compounded dollar denominated Libor is 3% per annum, and the current exchange rate is 0.84 $/€. Which of the following is mostly likely to be the 6- month forward rate of dollar denominated in Euro traded in Frankfurt? A. 0.8337 €/$. B. 1.1994 €/$. C. 1.2085 €/$. D. 1.1728 €/$ Answer: B Use the formula for forward price of currencies: 0.030.045 6 F0,T x0 e r r*T 0.84 e 0.8337 $/€ Therefore the Euro-denominated 12 dollar price is 1/0.8337=1.1994 €/$ 8. Suppose the S&P 500 currently has a level of 900. The continuously compounded return on a one-year T-bill is 5% per annum. You wish to hedge a $135,000 thousand portfolio that has a beta of 0.75 (relative to the S&P 500 index). Suppose you can trade fractions of contracts. How many S&P 500 E-mini futures contracts should you short to hedge your portfolio? A. 3.0 B. 2.25 C. 2.14 D. 2.85 Hint: (One S&P 500 E-mini futures contract is of size $50XS&P500 index.) I 135000 Answer: B Use the formula 0.75 2.25 S0* 50 900 Question 9 and 10 are based on the following information: The contract size of Gold futures traded on CBOT is 100 fine troy ounces (We will use oz hereafter) per contract. The spot price and futures price on Feb 2, 2007 is as follows: Price Spot Price 710.00 $/oz April Futures 720.00 $/oz June Futures 735.00 $/oz A gold producer is expecting to sell 100,000 oz of gold in April and in June, respectively. Assume interest rate is zero in answering question 9 and question 10. 9. Suppose the gold producer wants to use a stack hedge strategy to hedge the risk of gold price fluctuations. Which of the following is a valid stack hedge strategy? A. Purchase 1000 April futures on Feb 2, Accept delivery on the 1000 April futures on April 2. Purchase 1000 June futures on April 2, and accept delivery on the 1000 June futures on June 2. B. Sell 1000 April futures on Feb 2, and deliver the 1000 April futures on April 2. Sell 1000 June futures on April 2, and deliver the 1000 June futures on June 2. C. Sell 2000 April futures on Feb 2. Deliver 1000 April futures contracts on April 2 and settle the rest of the 1000 April futures contracts by cash. Sell 1000 June futures on April 2, and deliver the 1000 June futures on June 2. D. Sell 2000 June futures on Feb 2. Sell 100,000 oz gold on the spot market in April. Deliver the 1000 June futures on June 2, and cash settle the rest of the 1000 June futures on June 2. Answer: C is stack hedge. 10. Under which of the following circumstances does the stack hedge strategy perform as well as a strip hedge strategy? A. On April 2, the spread between spot price and June futures price is -15 $/oz (meaning Spot price - June futures price =-$15). B. On April 2, the spread between spot price and June futures price is -10 $/oz. C. The spot price on June 2 is $15 higher than the spot price on April 2. D. The spot price on April 2 is $10 higher than the spot price on Feb 2. Answer: A. The total gain on the stack hedge strategy is: F0,1 F0,1 P F1,2 . The total gain on the 1 strip hedge strategy will be F0,1 F0,2 . So the difference between the two is F 0,1 F0,2 P F1,2 . Therefore if the spread between spot price on April 2 and the 1 June futures price is -15 $/oz ( F0,1 F0,2 720 735 15 ), stack hedge perform as well as strip hedge. 11. Suppose in March, you entered into 10 long positions of the September Eurodollar futures contract when the price index is 92. Suppose also, on the last trading day of the Sep Eurodollar futures contract1, the continuously compounded 3-month Libor is 6% per annum. Which of the following is the gain on your position? (Please ignore marking to market in answering this question.) A. -$48869.35 B. $49607.92 C. $48869.35 D. $488.6935 1 0.06 Answer: The 3-month effective interest in Sep is e 4 1 0.015113 . Therefore the price index of the Eurodollar futures contract must be: 100 1.5113 4 93.9548 . The gain on the Eurodollar futures in therefore: 93.9548 92 2500 4886.935 . Since you have 10 contracts, you should choose C. 1 This is usually the second London bank business day immediately preceding the third Wednesday of the contract month (September). 12. A stock that pays 4% continuous dividend yield currently sells for $86.00. A 10- month European call option with a strike of $95.00 has a premium of $5.7405, and a European put option with the same strike price and maturity date is traded at $14.6371. What is the continuously compounded risk free rate that excludes arbitrage? A. 3.5% per annum. B. 3.13% per annum. C. 3.75% per annum. D. 2.95% per annum. Answer: C Using the put-call parity: c p S 0 e T Ke rT , we have 5.7405 14.6371 10 10 10 10 0.04 r r 0.04 86 e 12 95e 12 . That is, 95 e 5.7405 14 .6371 86 e 12 12 92 .0772 . ln 92.0772 / 95 Therefore we have: r 0.0375 3.75% 10 / 12 13. Which of the following actions can never be optimal? A. Exercising before maturity an American put option on a stock that does not pay any dividend. B. Exercising before maturity an American call option on a stock that pays continuous dividend. C. Exercising an American call option on a stock at maturity just after a large dividend is paid. D. Exercising an American put option on a stock at maturity just after a large dividend is paid. Answer: C You should exercise just before the dividend is paid. Since the option is about to expire, the time value and option value of the option are both zero. Therefore it is optimal to exercise the call and get the dividend. 14. A stock price is currently $40. It is known that at the end of one month it will be either $42 or $38, each with 50% probability. The stock does not pay any dividend during the next month. The risk-free interest rate is 8% per annum with continuous compounding. What is the value of a 1-month European call option with a strike price of $39? A. $1.689 B. $1.701 C. $ 1.133 D. $1.126 Answer: A We can use the risk-neutral probability method here. R e 0.08 / 12 1.00669 , 38 1.00669 * 40 0.5669 . Therefore the value of the call is c 1 0.5669 3 0 42 38 R 40 40 0.08 / 12 e 0.5669 3 1.6893 15. The risk free interest rate is 10% per annum, the current price of the stock index ABC is $50, and volatility of the stock is 30% per annum. The Black-Sholes price of a 3-month at-the-money European put option on the stock is _____. A. $3.61 B. $2.37 C. $0.047 D. $0.072 Answer: r 0.1, S 0 50 , 0.30 , T 3 / 12 0.25, K 50 . Using BS formula, S0 ln rT d1 Ke 1 p Ke rT N d 2 S 0 N d1 , where T 0.2417 and T 2 S0 ln rT d2 Ke 1 T 0.0917 . We have p 2.37 T 2 ( N 0.0917 and N 0.2417 can be obtained from the normal distribution table. For example, N 0.0917 N 0.09 0.17 N 0.10 N 0.09 0.5359 0.17 0.5398 0.5359 0.5366 , similarly, N 0.2417 N 0.24 0.17 N 0.25 N 0.24 0.5948 0.17 * 0.5987 0.5948 0.5955 . Therefore N 0.0917 1 N 0.0917 0.4634 , N 0.2417 1 N 0.2417 0.4045 ) Section II. Constructing Arbitrage Portfolios (10X2=20 points) The following scenario creates arbitrage opportunities for investor. Construct an arbitrage portfolio and complete the Cash Flow Table of the portfolio constructed to verify it is indeed an arbitrage opportunity. 1. The current exchange rate between Euro and Yen is 0.02 €/¥. The continuously compounded interest rate of Euro is 4% per annum, and the continuously compounded interest rate of Yen is 1% per annum. The price of one futures contract with 6-month maturity is 254,000 (One Yen futures contract contains ¥ 12,500,000). Portfolio Cash Flow at Date 0 Cash Flow at Maturity Strategy In Yen In euro In Yen In Euro Borrow € 0.01 6 0.04 6 12 .5M e 12 0.02 0.24875 e 12 0.24875 M 0.253778 M Buy ¥ 0.24875 / 0.02 0.24875M 12.4375M Lend ¥ 12.4375M 6 0.01 12.43766 M e 12 12.5M Short 1 12.5M 0.254M futures Total €221.7 The no arbitrage price of a futures contract of Yen is F0,T x0 erEUROrYEN T 6 0.03 12 ,500 ,000 0.02 e 253778 .3 . Therefore the futures contract is over-priced. 12 We sell the futures, and buy Yen today. 2. The current price of the stock is $100 per share. The continuously compounded interest rate is 5% per annum. The stock is expected to pay quarterly dividend in the amount of $2. The price of an at-the-money call option with 6 months to maturity is $3, and the price of a put with the same strike price and maturity is $2.25. (Assume both the call and the put expires immediately after the second dividend payment.) Portfolio Strategy Cash Flow at Date 0 Cash Flow at Maturity ST >=K ST <K Long 1 put -2.25 0 100- ST Long 1 share of stock -100 ST ST 3.926 (PV(D)) Short 1 call 3 100-ST 0 Short Ke rT bond 97.53 -100 -100 Total 2.206 0 0 The put-call parity implies: c Ke rT p S 0 PV ( D ) . Using the prices given above: 6 0.05 rT c Ke 3 100 * e 12 100 .531 , and the right hand side of the parity is: 3 6 0.05 0.05 p S 0 PV D 2.25 100 2 e 12 2e 12 98.32 . Therefore the put is rT T under-priced: c Ke S 0 e p , we’d like to buy low and sell high, that is long put, long stock, short call, and short bond. Section III: Binomial Option Pricing Models. (20 points) Show your steps. Partial credits will be given even if the final answer is wrong. Consider a non-dividend paying stock XYZ. Suppose the historical standard deviation of the log annual return of the stock is 0.30 . Suppose the continuous compounded interest rate per annum is 8%, and the current price of the stock is S 0 $100 . 1) Construct the following two-step binomial tree of the stock price movement over a period of 6 months. S uu uuS0 ( Node uu) S u uS0 ( Node u) S du duS0 S 0 100 ( Node du) S d dS 0 ( Node d ) S dd ddS 0 ( Node dd ) Compute the appropriate parameters of the model (that is, u , d , and the effective one- period interest rate R ) from the historical moments of the stock return data. (Hint: Use the formula: u e r h h , d e r h h , R erh ) u d R 1.1853 0.8781 1.020 2) Compute the delta of an at-the-money call option at node u, and d. Which is greater? Node 0 u d Δ 0.6074 1 0.1513 3) Compute the price of the call option. * Rd ud 0.4626 , c 2 * Cuu 2 * 1 *Cud 10.275 R 1 2 4) Compute the price of a put option with the same strike price and maturity. Using put- 6 0.08 call parity, c p S 0 Ke rT 100 100 e 12 3.921 . Therefore p c 3.921 6.354