6. Use the same set of information given in the problem above. (a) Use S&P 500 future prices to calculate the implied dividend yield on S&P 500. For simplicity, assume you can borrow or deposit money at the rates implied by Gold's futures prices. (b) Now suppose you believe that we are headed for a slow-down in economic activity and that the dividend yield will be lower than the value implied in part (a). What June-2008 contracts you would buy or sell to make money, assuming your view is correct? Again, assume you can borrow or deposit money at the rates implied by Gold's futures prices. 7. The Wall Street Journal gives the following futures prices for crude oil on September 6, 2006: Maturity Oct Dec Jun 07 Dec 07 Futures price ($/barrel) 67.50 69.60 72.66 73.49 and the spot price of oil is $67.50/barrel. Use the interest rates you found in the previous problem. (a) Compute the net convenience yield (in eective annual rate) for these maturities. (You can use the market information provided in the above problem.) (b) Briey discuss the convenience yield you obtained. 8. The data is the same as in the two problems above. You are running a renery and need 10 million barrels of oil in three months. (a) How do you use oil futures to hedge the oil price risk? The contract size is 1,000 barrels for futures. 5 c 2008, Andrew W. Lo and Jiang Wang 1.4 Forward and Futures 1 QUESTIONS (b) Suppose that you can also rent a storing facility for 10 million barrels of oil for three months at an annualized cost of 5% (in terms of the value of oil stored). Describe how you can utilize it to lock into a xed oil price for your future demand. (c) Which of these two strategies is better? Explain why. 9. The data is the same above. Now suppose instead that you are not in the oil business but can also rent the storage facility at the same cost. Can you take advantage of the current market conditions and the rental opportunity? If yes, please explain how (i.e., describe the actions you need to take). If not, briey explain why. 10. The current price of silver is $13:50 per ounce. The storage costs are $0:10 per ounce per year payable quarterly at the beginning of each quarter and the interest rate is 5% APR compounded quarterly (1:25% per quarter). (a) Calculate the future price of silver for delivery in nine months. Assume that silver is held for investment only and that the con- venience yield of holding silver is zero. (b) Suppose the actual price of the futures contract traded in the market is below the price you calculated in part (a). How would you construct a risk-free trading strategy to make money? What if the actual price is higher? To get full credit, say precisely what you will buy or sell, and how much money you will borrow or deposit into a bank account and for how long. 11. A pension plan currently has $50M in S&P 500 index and $50M in one-year zero-coupon bonds. Assume that the one-year interest rate is 6%. Assume that the current quote on the S&P 500 index is 1; 350, each futures contract is written on 250 units of the index and the dividend yield on the index is approximately 3% per year, i.e., $1; 000 invested in the index yields $30 in dividends at the end of the year. (a) Suppose you invest $1; 350  250 in one-year zero-coupon bonds and at the same time enter into a single futures contract on S&P 500 index with one year to maturity. Assume that in one year the index nishes at 1; 200. What is the total value of your position? How does this compare with buying 250 units of the index and holding them for a year? Assume that in one year the index nishes at 1; 400. Repeat the analysis. (b) If this plan decides to switch to a 70=30 stock/bond mix for a period of one year, how would you implement this strategy using S&P 500 futures? How many contracts with one year to maturity 6 c 2008, Andrew W. Lo and Jiang Wang 1.4 Forward and Futures 1 QUESTIONS would you need? Assume that the index nishes the year at 1; 400, describe the plan's portfolio in one year and one day from now (right after the futures expire). What is the stock/bond mix? 12. Spot price for soybean meal is $152.70 per ton and the 12-month soybean-meal futures is traded at $148.00. The 1-year interest rate is 3%. (a) What is the net convenience yield on soybean meal for the 12 month period? (b) You need 1,000 tons of soybean meal in 12 months. How would you lock into a price today using the futures contracts? (The size for each soybean-meal futures contract is 100 tons.) 13. The spot price for smoked salmon is $5,000 per ton and its six-month futures price is $4,800. The monthly interest rate is .0025 (.25%). (a) What is the average monthly net convenience yield on smoked salmon for the next six months? (b) If you are a manager of Bread&Circus and need 10 tons of smoked salmon in six months. How can you avoid the risk in the price of smoked salmon over the next six months using futures? (c) Suppose that your net convenience yield for smoked salmon is 1.2%. How does this change your hedging strategy? 14. A wine wholesaler needs 100,000 gallons of Cheap Chardonnay for de- livery in Boston in June 2007. A producer oers to deliver the wine at that time for $500; 000 paid now, in December 2006. The wholesaler can also buy Cheap Chardonnay futures contracts for June 2007. The current futures price is $51; 000 for each 10,000 gallon futures contract. The wholesaler is determined to lock in the cost of the 100,000 gallons needed in June. (a) The wholesaler considers the futures contract, but worries that the contract will not lock in her cost, because futures prices may uctuate widely between now and June. Is her concern justied? Why or why not? (b) Do you recommend that the wholesaler pay the producer now or take a long position in Chardonnay futures? (Additional assump- tions may be needed to answer. Make sure they are reasonable.) Explain briey. 7 c 2008, Andrew W. Lo and Jiang Wang 1.4 Forward and Futures 1 QUESTIONS 15. You are a distributor of canola seed and need to make deliveries of 10,000 bushels one month from now. You currently have no canola seed in inventory. The current spot price of canola seed is $7.45 per bushel and the futures price for delivery in one month is $7.65. You would like to hedge the uncertainty about the spot price one month from now. (a) If your storage cost is $.15 per bushel (paid at the end of month), what would you do? (b) Suppose that in the short run, your storage cost increases to $.25 per bushel. What would you do? 16. Assume perfect markets: no transaction costs and no constraints. In addition assume that the one-month risk-free interest rate will remain constant over a three-month period. Two futures contracts with two and three months maturity are traded on a nancial asset without any intermediate payout. The price for these contracts are F2 = $100 and F3 = $101, respectively. (a) What is the spot price of the underlying asset today? (b) Suppose that a one-month futures contract is trading at price F1 = $98. Does this imply an arbitrage opportunity? How would you take advantage of this opportunity? To get full credit, be precise on what you would buy or sell, and how much money you would deposit into a bank account and/or borrow. 17. Assume perfect markets: no transaction costs and no constraints. The one-month risk-free interest rate will remain constant over a six-month period. Two futures contracts are traded on a nancial asset without payouts: a three-month (futures price F(t; t + 3)) and a six-month (futures price F(t; t + 6)) contract. You can observe that F(t; t + 3) = $120 and F(t; t + 6) = $122. (a) What is the spot price of the underlying asset at time t? (b) Suppose that a three-month futures contract is trading at price F(t; t + 3) = $119:5. Does this imply an arbitrage opportunity? How would you take advantage of this opportunity? 8 c 2008, Andrew W. Lo and Jiang Wang 1.5 Options 1 QUESTIONS 1.5 Options 1. A stock price is currently $50. It is known that at the end of two months it will be either $53 or $48. The risk-free interest rate is 10% per annum with continuous compounding. What is the value of a two- month European call option with a strike price of $49? 2. A stock price is currently $80. It is known that at the end of four months it will be either $75 or $85. The risk-free interest rate is 5% per annum with continuous compounding. What is the value of a four- month European put option with a strike price of $80? 3. Today's price of three traded call options on BackBay.com, all expiring in one month, are as follows: Strike Price Option Price $50 $71 2 60 $3 70 $11 2 You are considering buying a buttery spread" consisting of the fol- lowing positions:  Buy 1 call at strike price of $50  Sell (write) 2 calls at strike price of $60  Buy 1 call at strike price of $70. (a) Plot the payo of your total position for dierent values of the stock price on the maturity date. (b) What is the dollar investment required to establish the spread? (c) For what stock prices on the maturity date will you be making an overall prot? 4. You are given the following prices: Security Maturity (years) Strike Price ($) JEK stock - - 94 Put on JEK stock 1 50 3 Put on JEK stock 1 60 5 Call on JEK 1 50 ? Call on JEK 1 60 ? Tbill (FV=100) 1 - 91 What is the price of the two call options? 9 c 2008, Andrew W. Lo and Jiang Wang 1.5 Options 1 QUESTIONS 5. (a) Here are the payo diagrams of some popular trading strategies using just put and call options with same maturities. How would you replicate them? Identifygthe number and strikes of call or put options that have to be bought or sold in order to generate these payos. (All angles are 45 degrees!) (a) Payoff a b Stock Price Payoff Stock Price (b) a (c) a b c Stock Price Figure 1: Payo Diagrams (b) Now suppose for institutional reasons, you are short on volatility in this market, i.e. you will lose money if the market becomes volatile. For example you can imagine that your are an investment bank working on a few major M&A deals which may fall apart if the market goes down too much or goes up too much. In either case, you will lose money if the market becomes volatile. Explain which if any of the above three payos would work well to hedge your exposure. What is the cost of this hedge to you? 6. A buttery spread is a combination of option positions that involve three strike prices. To create a buttery spread, a trader purchases an option with a low strike price and an option with a high strike price and sells two options with an intermediate strike price. For this problem, 10 c 2008, Andrew W. Lo and Jiang Wang 1.5 Options 1 QUESTIONS assume that the intermediate strike price is halfway between the low and the high strike prices and that the options are European. Denote the intermediate strike price by X, the low strike price by X  a, and the high strike price by X + a, a > 0. (a) Graph the payo diagrams at maturity of the buttery spread in which the underlying options are call options. Holding the intermediate strike price xed, what happens to the payos as the low and high strike price converge to the intermediate price? (b) Suppose that a trader purchases a buttery spread (using call options) for which the intermediate strike price equals to today's stock price. Based only on this trade, what is the trader's view of the future direction of the market? 7. IBM shares are now traded at $80.00. The term structure of interest rates is at at 5%. (a) Plot the terminal payo from a European put option on 1 share of IBM with a exercise price of $85 and a maturity of 3 months (not including the price of the option). (b) Suppose that you purchase the put option at $4.00 from the mar- ket. Specify the ranges of IBM share price at the options maturity date for which you will be making a net prot. 8. Suppose that after a recent news about the economy, IBM share price remains the same, but the prices of its options shot up. How is this possible? 9. You are given the following information. Use this information to de- termine the unknown prices. Table 2: Stock, Options, and T-bill prices Secutiry Maturity (years) Strike Price($) 401 Stock - - $100 Put on 401 Stock 1 $50 $3 Put on 401 Stock 1 $60 $5 Calle on 401 Stock 1 $50 $57.50 Calle on 401 Stock 1 $60 ? Tbill(FV=100) 1 - ? 10. Joseph Jones, a manager at Computer Science, Inc. (CSI), received 1,000 shares of company stock as part of his compensation package. The stock currently sells at $40 at share. Joseph would like to defer selling the stock until the next tax year. In January, however, he will 11 c 2008, Andrew W. Lo and Jiang Wang 1.5 Options 1 QUESTIONS need to sell all his holdings to provide for a down payment on his new house. Joseph is worried about the price risk involved in keeping his shares. At current prices, he would receive $40,000 for the stock. If the value of his stock holdings falls below $35,000, his ability to come up with the necessary down payment would be jeopardized. On the other hand, if the stock value rises to $45,000, he would be able to maintain a small cash reserve even after making the down payment. Joseph considers three investment strategies: (a) Strategy A is to write January call options on the CSI shares with strike price $45. There calls are currently selling for $3 each. (b) Strategy B is to buy January put options on CSI with strike price $35. These options also sell for $3 each. (c) Strategy C is to establish a zero-cost collar by writing the January calls and buying the January puts. Evaluation each of these strategies with respect to Joseph's investment goals. What are the advantages and disadvantages of each? Which would you recommend? 11. You write a call option with strike $50 and buy a call with strike $60. The options are on the same stock and have the same maturity date. One of the calls sells for $3; the other sells for $9. (Assume zero interest rate.) (a) Draw the payo graph for this strategy at the option maturity date. (b) Draw the prot graph for this strategy. (c) What is the break-even point for this stratgy? Are you bullish or bearish on the stock? 12. Consider an increasingly popular deposit contract with payos linked to the performance on the S&P 500 Index on the U.S. stock market. For every dollar invested in the contract, the rate of return in one year is equal to 60% of the realized rate of return of the S&P 500 Index during this year if this rate of return is positive; otherwise, you get your money back. In essence, you are protected from the downside risk of the S&P 500 Index, while you are still able to participate in the upside potential of the stock market. The one year riskless interest rate is 10%. For simplicity, assume that the stocks in the index do not pay dividends. (a) Draw a graph for the payo one year from now for a one dollar investment in the contract with the horizontal axis being the re- 12 c 2008, Andrew W. Lo and Jiang Wang 1.5 Options 1 QUESTIONS alized rate of return on the S&P 500 Index. Also write down the payo symbolically. (b) Show that the payo one year from now for a one dollar investment in this contract is the payo to a portfolio of a default-free bond and a European call option on the S&P 500 Index. (c) Suppose that the rates of return on the S&P 500 Index can take two possible values one year from now, 20% and -20% with prob- abilities 60% and 40%, respectively.

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