Answer each of the following questions with True or False. You do not need to provide any explanation. Each question is worth 2 points. (a) The most common form of swaps is a plain vanilla fixed-for-floating interest rate swap where the notional values are exchanged at the beginning and at the end of the contract. (b) Options and Futures are designed to enable the transfer of market risk, while swaps are designed to enable the transfer of credit risk. (c) A margin call will be issued only if the investor’s margin account balance becomes negative. (d) Options market makers who are delta-hedging a long call position would most likely sell the underlying when markets are rising. (e) A rise in the convenience yield decreases the futures price while a rise in storage costs increases the futures price. 1 2. Futures Market Arbitrage (18 points) You are the analyst for Safeway Food Stores (SWY). The company is currently quoted at $21.19. The stock is a high dividend paying stock and is expected to pay a dividend of $0.75 at the end of the year (exactly 3 months from now), a $1 dividend in June 2011 (9 months from now), and a $1.25 dividend in December 2011 (15 months from now). The 12 month futures prices for SWY is $20.12, the continuously compounded risk-free rate is 2%, and it is possible to borrow and lend at this rate. (a) (2 points) Notice that the 12 months futures price is below the spot price of SWY. Does this imply that an arbitrage opportunity exists? Explain! (b) (2 points) Is this market said to be in contango or in backwardation? Explain! (c) (4 points) What is the fair price of a 12 month SWY futures? (d) (8 points) Is there an arbitrage opportunity? If yes, explain how you could exploit it, and illustrate all transactions! (e) (2 points) Also, if an arbitrage opportunity exists, you can synthetically earn interest on an investment at a rate different from the risk-free rate, find that rate and explain, if applicable. 3. Decomposing Option Value (20 points) Using common industry lingo, a typical trader often decomposes the value of an option (OP, option price) into its “time value” and its “intrinsic value”. The intrinsic value (IV) of an option is the value of the option if you exercise it now. The time value (TV) of an option is the price of the option (OP) minus its intrinsic value (IV). For this question, assume that there are no dividend payments and ignore insurance value. (a) (2 points) Given a call option on a stock trading at S with a strike of K, write an expression for the intrinsic value (IV) of the stock trading at S. (b) (14 points) Given an in-the-money American call option, fill in how the value changes with increase, decrease, no change, or it depends, if the following characteristics experience a small increase. If your answer is “it depends,” please provide a quick clarification. Characteristic intrinsic value (IV) time value (TV) Strike Volatility (Underlying) Option Price Price (Underlying) Time to Maturity (c) (2 points) Show mathematically that the time value (TV) of an option is strictly less than or equal to the option price (OP). 2 (d) (2 points) If an American call option were out-of-the-money, what would this imply for its time value (TV), intrinsic value (IV) and option price (OP)? 4. Option Payoff Diagrams (22 points) Use the information in Table 1 to draw the NET and GROSS payoff diagrams for the following strategies, given that the stock price of Microsoft (MSFT) is 49.06 on April 29, 2015. Use the bid and ask prices, when you are short and long respectively, in your calculations. (a) (4 points) Buy a share of MSFT stock and buy an October put at 47. (b) (4 points) Invest into a zero-coupon bond worth 47 USD and buy an October call at 47. (c) (2 points) Compare the last two option payoff diagrams. What do you observe? (d) (4 points) Buy a 47-52.50 October bullish call spread (i.e. buy the lower strike call and sell the higher strike call). (e) (4 points) Buy a 47-52.50 October bearish put spread (i.e. buy the higher strike put and sell the lower strike put). (f) (4 points) Buy an October butterfly spread (i.e. buy a call at 47, buy a call at 52.50 and write two calls at 50) CALLS PUTS Last Bid Ask Strike Last Bid Ask 10.15 10.00 10.40 40.00 2.50 2.44 2.74 8.30 7.90 8.30 43.00 3.55 3.35 3.75 6.68 6.70 7.10 45.00 4.30 4.15 7.00 5.75 5.60 6.00 47.00 5.23 5.10 8.50 4.40 4.25 4.60 50.00 6.90 6.75 7.20 3.44 3.30 3.65 52.50 8.30 8.25 8.75 2.72 2.58 2.85 55.00 10.40 9.95 13.00 2.01 1.97 2.20 57.5 12.00 11.80 15.00 1.59 1.49 5.00 60.00 13.95 13.80 14.35 Table 1: OCTOBER 2015 OPTION PRICES OF MICROSOFT (MSFT), April 29, 2015 5. Black-Scholes-Merton and the Merton Model (30 points). Company XYZ has issued equity S and 1 year zero-coupon debt D with a face value of 90. Assume that the value of the assets of the firm V are initially at 100. According to the Merton model (Merton JF 1974), debt and equity are contingent claims on the assets of the firm. (a) (5 points) Draw the payoff diagram for both the equity and debt of the firm as a function of the firm’s value one year from now. Draw the individual payoffs of debt and equity, without the financing costs. (b) (5 points) What is the value of Company XYZ’s equity at t = 0. Assume an annual continuously compounding interest rate of 0% (zero), a time horizon of T = 1 year and asset volatility of σV = 20%? 3 (c) (5 points) What is the value of Company XYZ’s debt at t = 0, assuming an annual continuously compounded interest rate of 0%, a time horizon of T = 1 year and asset volatility of σV = 20%? Indicate two ways how you can calculate the value of debt! (d) (4 points) What is the risk-neutral probability of default? (e) (5 points) What is the delta, gamma and vega of XYZ’s equity? (f) (6 points) You, as a shareholder, would like to hedge yourself against market risk by making your holdings delta-, gamma- and vega-neutral. How can you do this by using only the company’s stock and the two options indicated in the Table 2? Hint: Assume that each call (put) option contract gives the right to buy (sell) 1 unit of the underlying asset. Table 2: Options and Greeks ∆ Γ V ega Option 1 0.6 0.5 2.0 Option 2 0.5 0.8 1.2 4
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