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Sample Final Exam Finance 353 Derivatives Daytime MBA 1 Section I: Multiple Choice Questions. (5X10=50 points). Please read the instructions below carefully before answering questions in section I. Instructions: Please select one and only one answer for each of the following questions. No credit will be given if multiple selections are made. The information provided in each question may be more than enough for you to choose the right answer. Choose the correct answer for each question by writing down the letter A, B, C, or D in the space provided for each question. Each question is worth 5 points. You do not need to write down the steps for any of the questions in section I. Full credit will be given as long as the final answer is correct. However, a maximum of 3 points will be given for partial credit even if the final answer is incorrect. You will NOT earn full credit for the question if the steps you wrote are clearly wrong, even if the final answer is correct. Important: The following table contains the price and Delta of call options calculated using the Black-Scholes model. All quantities are computed assuming a volatility of 30% per annum, continuously compounded risk-free interest rate 5% per annum, maturity of 5 years, and dividend yield of 0. You may use the following information in answering any of the questions in this exam. Table 1: Option Price and Delta using Black-Scholes Model Stock Price Strike Price BS Call Price BS Call Delta 200 50 161.1965 0.997237 200 100 125.0315 0.959191 200 150 95.07198 0.872218 200 200 71.91561 0.760555 100 200 12.38903 0.372517 100 100 35.95781 0.760555 150 100 78.2095 0.9053 Question 1-3 are based on the following information. A financial institution has the following portfolio of OTC options on the S&P 500 index with a total value of $70000. All options have a maturity of 6 months. Table 2: Portfolio Positions and Portfolio Greeks used in Question 1-3 Type Position Delta of Option Vega of Option Call-1100 -1000 0.50 1.8 Call-1500 -500 0.80 0.2 Put-1200 -2000 -0.40 0.7 Call-1450 -500 0.70 1.4 2 The S&P 500 index is currently 1400. The expected return on S&P 500 index is 8% per annum, and the expected return on the above portfolio is 5% per annum. The risk-free interest rate is 3% per annum. All rates are continuously compounded. 1. A traded option call-1400 is available with a Delta of 0.6 and a Vega of 1.0. Which of the following positions in the S&P 500 index and the call-1400 will make the portfolio both Delta-neutral and Vega-neutral? A. A long position in 4000 call-1400 and a long position in 1950 shares of the stock. B. A long position in 4000 call-1400 and a short position in 1950 shares of the stock. C. A long position in 5000 call-1400 and a short position in 2550 shares of the stock. D. A long position in 450 shares of the stock. Answer: _____________________________ 2. Suppose the financial institution took the position in the S&P 500 index as described in answer D of question 1. Suppose also it will frequently rebalance its position in the S&P 500 index to maintain Delta neutrality of the portfolio at all times during the next 6 months. Assume frictionless market and no transaction cost. Which of the following best describes the expected return of the Delta-neutral portfolio over the next six months? A. 8% per annum. B. 3% per annum. C. Lower than 8% per annum but higher than 3% per annum. D. Higher than 8% per annum. Answer:______________________________ 3 3. Suppose you currently have a short position in a stock with market value $35,000,000. The stock has a beta of 1.2 with the S&P 500 index. Which of the following position in the S&P500 index options listed in table 2 Delta-hedges the market risk of your position in the stock? A. A short position in 60000 call-1100. B. A short position in 62500 put-1200. C. A long position in 62500 put-1200. D. A short position in 75000 put-1200. Answer:__________________________ 4. Consider a non-dividend-paying stock with current price S0 100 . Suppose the continuously compounded interest rate is r 5% per annum. Suppose the assumptions of the Black-Scholdes model are all satisfied. All of the options described below have a strike price of $100 and a maturity of 6 months. Which of the following scenario has the highest implied volatility according to the Black-Scholes model? A. An American call option on one share of the stock is traded at $11.385 when the stock price is S0 $99 . B. A European call option on one share of the stock is traded at $12.385 when the stock price is S0 100 . C. A European put option on one share of the stock is traded at $9.916 when the stock price is S0 100 . D. A European put option on one share of the stock is traded at $10.916 when the stock price is S0 99 . Answer: _________________________________ 4 5. The price of a 5-year forward contract on a dividend paying stock is 128.4031. What is the Black-Scholes price of a 5-year European call option on the stock with strike price $100. Assume the continuously compounded interest rate is 5% per annum, and the volatility of the stock return is 30% per annum. (a) $35.9578 (b) $78.2095 (c) $77.8801 (d) Not enough information to determine. Answer_________________________________ 6. Firm B’s current total asset value is 4 million and it has 1 million shares outstanding. Firm B is not expected to a pay dividend to equity holders in the next five years. The historical volatility of firm B’s total asset is 0.30 per annum. Assume the distribution of firm B’s total asset value satisfies the assumptions of the Black-Schole’s model. Firm B has issued 20,000 convertible bonds, each with a maturity of T 5 years. At maturity, each convertible bond can be redeemed for a face value of $100 or can be exchanged for 50 shares of firm B’s stock at the bond holder’s choice. If firm B’s assets are not enough to pay for all the convertible bonds, it will declare bankruptcy and the convertible bond holders will have claim to its assets. Assume that the continuously compounded interest rate is r 5% per annum. Which of the following is the best estimate of the no-arbitrage price of the convertible bond? A. $110.926 B. $189.074 C. $44.834 D. None of the above. Answer: _________________________________ 5 Question 7 and 8 are based on the following information: The current price of ABC stock is S0 $200 , the volatility of the stock is 0.30 . ABC stock is not expected to pay any dividend in the next five years. The continuously compounded risk-free interest rate is 5% per annum. CDE bank has issued a $1 million Equity-linked CD with a 5 year maturity. Each CD is structured to pay 100% principle protection, and pay participation of the ABC stock’s appreciation (This means the CD will pay a fraction of ABC stock’s return on top of the guaranteed principal payment if ABC appreciates in value at the maturity date, and pays only the principal amount if the ABC stock depreciates.). 7. What is the maximum amount of price participation that CDE bank could offer? A. 76.06% B. 61.52% C. 100% D. 50% Answer: _________________________________ 8. Which of the following is a perfect hedge of the risk associated with the issuance of the equity linked CD if your answer in 7 is indeed the price participation offered by the bank? A. A long position of 2339 shares of the ABC stock. B. A long position of 3075 shares of the ABC stock and a long position in 3075 at-the- money put options of ABC stock. C. A short position of 2339 shares of the ABC stock. D. A long position of 3075 at-the-money put option of the ABC stock. Answer: _________________________________ 6 9. The manager of a blue chip growth stock mutual fund is trying to hedge the market risk of $135 million portfolio position that has a beta of 1.37 and a correlation of 0.95 with the S&P 500 index. The current level of the S&P 500 Index is $1400, and the current LIBOR is 4.2% per annum. How many S&P 500 index futures contracts will you need to hedge the position (Hint: Each S&P 500 futures contract contains 250 S&P500 index)? (Assume you cannot take fractions of contracts in answering this question.) A. 528 B. 366 C. 551 D. 382 Answer: _________________________________ 10. Under which of the following circumstances is it NEVER optimal to exercise the American option before maturity? A. An exchange American call option whose strike asset is a dividend-paying stock and the underlying asset is a non-dividend paying stock. B. An American call option on a dividend paying stock. C. A dollar denominated American put option on Euro (That is, it is An American style currency put option on Euro, and the strike currency is US dollar.). D. An American put option on a non-dividend paying stock. Answer: _________________________________ 7 Please read the instructions below carefully before answering questions in section II and section III. Instructions: Please write your answers to section II and section III only on the separately provided Midterm Exam Book II. Any writing on this page will not be graded. Please show your steps in answering questions in section II and section III. Partial credits will be given even if the final answers are wrong. Section II: Hedging Interest Rate Risks (30) Firm ABC planed a project that would take nine months to complete, at a total cost of $100 million. Bank XYZ offered to provide a nine-month loan starting from Sep 20. The bank insisted that the interest payment be made quarterly and the loan rate be 200 basis points above the spot rate of 90-day LIBOR. Interest payment for each of the three quarters was due at the end of each quarter. Principal plus the interest payment for the third quarter was due at the end of the last quarter. The Eurodollar price indexes are given in the table 3. 1) What is the annualized three-month spot rate of LIBOR on Sep 20 (for a three-month loan initiated on Sep 20)? What is the annualized implied three-month forward rate of LIBOR on Sep 20, for a loan initiated on Dec 20? What is the annualized implied three-month forward rate of LIBOR on Sep20, for a loan initiated on Mar 20 of the next year? (2+2+2=6 points in total) Maturity of the Contract Eurodollar Price Index September 93 December 92.7 March 92.2 Table 3: Eurodollar Price Indexes 2) Assume no arbitrage and a frictionless market. What is the annualized 6-month spot rate of LIBOR on Sep 20 (for a three-month loan initiated on Sep 20)? What is the annualized 9-month spot rate of LIBOR on Sep 20 (for a three-month loan initiated on Sep 20)? (2+2=4 points in total) 8 3) Suppose firm ABC decided to use Eurodollar futures contracts to hedge the interest rate risk. Which Eurodollar futures contracts should the firm use to hedge the interest rate risk? How many contracts should the firm use? Should the firm take long positions or short positions in the Eurodollar futures contracts? (2+2+1=5 points in total) 4) Suppose a swap dealer offers an interest rate swap contract to firm ABC, which allows firm ABC to pay a fixed swap rate, and receive floating interest payment on the 100-million principle. Terms of the swap contract is detailed in table 4. What is the annualized no-arbitrage swap rate? (5 points) Interest Rate Swap Term Sheet Floating Rate Payment Fixed Rate Payment Payer Swap Dealer Payer Firm ABC Floating Rate Three-month LIBOR Fixed Rate ??? Notional $ 100 million Notional $ 100 million Date Count 30/360 Date Count 30/360 Interval 3 month Interval 3 month Maturity 9 months Maturity 9 months Table 4: The Interest Rate Swap Term Sheet 9 Section III. An American Call Option in a Binomial Model (20 points). Consider a stock with current price $80 per share, and a continuously compounded dividend yield of 5% per annum. The continuously compounded interest rate is 3% per annum. The stock return is lognormally distributed with volatility 0.40 per annum. Consider an American call option on the stock with strike price $75 and maturity of 12 months. 1) Calculate the u and d using McDonald’s recipe and set up a two-period binomial model of the stock price movement. (You need to draw the binomial tree, and write down the stock price on the binomial tree.) 2) Calculate the risk-neutral probability, and describe the optimal exercise rule of the American call option by circling the nodes at which it is optimal to exercise the option on the binomial tree you drew in part 1). Calculate the value of the American call at date 0. 3) Calculate the Delta of the American call option at date 0. Calculate the Delta of an otherwise identical European call option. Which option has a higher Delta? Explain your result. 10

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