Define a forward contract. Explain at what time are cash flows generated for this contract. How is settlement determined?
6. Explain who bears default risk in a forward contract.
7. What risks are being managed by trading derivatives on exchanges?
8. Explain the difference between a forward contract and an option.
9. What is the difference between value and payoff in the context of derivative securities.
10. What is a short position in a forward contract? Draw the payoff diagram for a short position at a forward price of $103, if the possible range of the underlying stock price is $50-150.
11. Forward prices may be derived using the notion of absence of arbitrage, and market efficiency is not necessary. What is the difference between these two concepts?
12. Suppose you are holding a stock position, and wish to hedge it. What forward contract would you use, a long or a short? What option contract might you use? Compare the forward versus the option on the following three criteria: (a) uncertainty of hedged position cash-flow, (b) Up-front cash-flow and (c) maturity-time regret.
13. What derivatives strategy might you implement if you expected a bullish trend in stock prices? Would your strategy be different if you also forecast that the volatility of stock prices will drop?
14. What are the underlyings in the following derivative contracts? (a) A life insurance contract. (b) A home mortgage. (c) Employee stock options. (d) A rate lock in a home loan
15. Assume you have a portfolio that contains stocks that track the market index. You now want to change this portfolio to be 20% in commodities and only 80% in the market index. How would you use derivatives to implement your strategy?
16. In the previous question, how do you implement the same trading idea without using futures contracts?
17. You buy a futures contract on the S&P 500. Is the correlation with the S&P 500 index positive or negative? If the nominal value of the contract is $100,000 and you are required to post $10,000 as margin, how much leverage do you have?
16. What is the “closing out” of a position in futures markets? Why is closing out of contracts permitted in futures markets? Why is unilateral transfer or sale of the contract typically not allowed in forward markets?
17. An investor enters into a long position in 10 silver futures contracts at a futures price of $4.52/oz and closes out the position at a price of $4.46/oz. If one silver futures contract is for 5,000 ounces, what are the investor’s gains or losses?
18. What is the settlement price? The opening and closing price?
19. An investor enters into a short futures position in 10 contracts in gold at a futures price of $276.50 per oz. The size of one futures contract is 100 oz. The initial margin per contract is $1,500, and the maintenance margin is $1,100.
20. The current price of gold is $642 per troy ounce. Assume that you initiate a long position in 10 COMEX gold futures contracts at this price on 7-July-2006. The initial margin is5% of the initial price of the futures, and the maintenance margin is 3% of the initial price. Assume the following evolution of gold prices over the next five days, and compute the margin account assuming that you meet all margin calls.
Date Price per Ounce 7-Jul-06 642 8-Jul-06 640 9-Jul-06 635 10-Jul-06 632 11-Jul-06 620 12-Jul-06 625
21. When is a futures market in “backwardation”? When is it in “contango”?
22. Suppose there are three deliverable bonds in a Treasury Bond futures contract whose current cash prices (for a face value of $100,000) and conversion factors are as follows: (a) Bond 1: Price $98,750. Conversion factor 0.9814.
(b) Bond 2: Price $102,575. Conversion factor 1.018. (c) Bond 3: Price $101,150. Conversion factor 1.004. The futures price is $100,625. Which bond is currently the cheapest-to-deliver?
v23. You enter into a short crude oil futures contract at $43 per barrel. The initial margin is $3,375 and the maintenence margin is $2,500. One contract is for 1,000 barrels of oil. By how much do oil prices have to change before you receive a margin call?
24. You take a long futures contract on the S&P 500 when the futures price is 1,107.40, and close it out three days later at a futures price of 1,131.75. One futures contract is for 250× the index. Ignoring interest, what are your losses/gains?
25. An investor enters into 10 short futures contract on the Dow Jones Index at a futures price of 10,106. Each contract is for 10× the index. The investor closes out five contracts when the futures price is 10,201, and the remaining five when it is 10,074. Ignoring interest on the margin account, what are the investor’s net profits or losses?
26. A bakery enters into 50 long wheat futures contracts on the CBoT at a futures price of $3.52/bushel. It closes out the contracts at maturity. The spot price at this time is $3.59/ bushel. Ignoring interest, what are the bakery’s gains or losses from its futures position?
27. An oil refining company enters into 1,000 long one-month crude oil futures contracts on NYMEX at a futures price of $43 per barrel. At maturity of the contract, the company rolls half of its position forward into new one-month futures and closes the remaining half. At this point, the spot price of oil is $44 per barrel, and the new one-month futures price is $43.50 per barrel. At maturity of this second contract, the company closes out its remaining position. Assume the spot price at this point is $46 per barrel. Ignoring interest, what are the company’s gains or losses from its futures positions?
28. Define the following terms in the context of futures markets: market orders, limit orders, spread orders, one-cancels-the-other orders.
29. Distinguish between market-if-touched orders and stop orders.
30. You have a commitment to supply 10,000 oz of gold to a customer in three months’ time at some specified price and are considering hedging the price risk that you face. In each of the following scenarios, describe the kind of order (market, limit, etc.) that you would use will be sustained. Thus, unless there is high volatility and a reversal of direction, this approach may not be profitable and might turn out to be loss-making.
32. The spread between May and September wheat futures is currently $0.06 per bushel. You expect this spread to widen to at least $0.10 per bushel. How would you use a spread order to bet on your view?
33. The spread between one-month and three-month crude oil futures is $3 per barrel. You expect this spread to narrow sharply. Explain how you would use a spread order given this outlook.
34. Suppose you anticipate a need for corn in three months’ time and are using corn futures to hedge the price risk that you face. How is the value of your position affected by a strengthening of the basis at maturity?
35. A short hedger is one who is short futures in order to hedge a spot cash flow risk. A long hedger is similarly one who goes long futures to hedge an existing risk. How does a weakening of the basis affect the positions of short and long hedgers?
5. A security is currently trading at $97. It will pay a coupon of $5 in two months. No other payouts are expected in the next six months. (a) If the term structure is flat at 12%, what should the be forward price on the security for delivery in six months? (b) If the actual forward price is $92, explain how an arbitrage may be created
6. Suppose that the current price of gold is $365 per oz and that gold may be stored costlessly. Suppose also that the term structure is flat with a continuously compounded rate of interest of 6% for all maturities. (a) Calculate the forward price of gold for delivery in three months. (b) Now suppose it costs $1 per oz per month to store gold (payable monthly in advance). What is the new forward price? (c) Assume storage costs are as in part (b). If the forward price is given to be $385 per oz, explain whether there is an arbitrage opportunity and how to exploit it.
7. A stock will pay a dividend of $1 in one month and $2 in four months. The risk-free rate of interest for all maturities is 12%. The current price of the stock is $90.
(a) Calculate the arbitrage-free price of (i) a three-month forward contract on the stock and (ii) a six-month forward contract on the stock. (b) Suppose the six-month forward contract is quoted at 100. Identify the arbitrage opportunities, if any, that exist, and explain how to exploit them.
8. A bond will pay a coupon of $4 in two months’ time. The bond’s current price is $99.75. The two-month interest rate is 5% and the three-month interest rate is 6%, both in continuously compounded terms.
(a) What is the arbitrage-free three-month forward price for the bond?
(b) Suppose the forward price is given to be $97. Identify if there is an arbitrage opportunity and, if so, how to exploit it.
9. Suppose that the three-month interest rates in Norway and the US are, respectively, 8% and 4%. Suppose that the spot price of the Norwegian Kroner is $0.155.
10. Consider a three-month forward contract on pound sterling. Suppose the spot exchange rate is $1.40/£, the three-month interest rate on the dollar is 5%, and the three-month interest rate on the pound is 5.5%. If the forward price is given to be $1.41/£, identify whether there are any arbitrage opportunities and how you would take advantage of them
11. Three months ago, an investor entered into a six-month forward contract to sell a stock. The delivery price agreed to was $55. Today, the stock is trading at $45. Suppose the three-month interest rate is 4.80% in continuously compounded terms. (a) Assuming the stock is not expected to pay any dividends over the next three months, what is the current forward price of the stock? (b) What is the value of the contract held by the investor? (c) Suppose the stock is expected to pay a dividend of $2 in one month, and the one-month rate of interest is 4.70%. What are the current forward price and the value of the contract held by the investor?
12. An investor enters into a forward contract to sell a bond in three months’ time at $100. After one month, the bond price is $101.50. Suppose the term-structure of interest rates is flat at 3% for all maturities. (a) Assuming no coupons are due on the bond over the next two months, what is the forward price on the bond? (b) What is the marked-to-market value of the investor’s short position? (c) How would your answers change if the bond will pay a coupon of $3 in one month’s time?
13. A stock is trading at $24.50. The market consensus expectation is that it will pay a dividend of $0.50 in two months’ time. No other payouts are expected on the stock over the next three months. Assume interest rates are constant at 6% for all maturities. You enter into a long position to buy 10,000 shares of stock in three months’ time. (a) What is the arbitrage-free price of the three-month forward contract? (b) After one month, the stock is trading at $23.50. What is the marked-to-market value of your contract? (c) Now suppose that at this point, the company unexpectedly announces that dividends will be $1.00 per share due to larger-than-expected earnings. Buoyed by the good news, the share price jumps up to $24.50. What is now the marked-to-market value of your position?
15. This question generalizes the previous one from two deliveries to many. Consider a contract that requires the short position to make deliveries of one unit of an underlying at time points t1, t2, . . . , tN . The common delivery price for all deliveries is F. Assume the interest rates for these horizons are, respectively, r1, r2, . . . , rN in continuouslycompounded annualized terms. What is the arbitrage-free value of F given a spot price of S?
16. In the absence of interest-rate uncertainty and delivery options, futures and forward prices must be the same. Does this mean the two contracts have identical cash-flow implications? (Hint: Suppose you expected a steady increase in prices. Would you prefer a futures contract with its daily mark-to-market or a forward with its single mark-tomarket at maturity of the contract? What if you expected a steady decrease in prices?)
17. Consider a forward contract on a non-dividend-paying stock. If the term-structure of interest rates is flat (that is, interest rates for all maturities are the same), then the arbitrage-free forward price is obviously increasing in the maturity of the forward contract (i.e., a longer-dated forward contract will have a higher forward price than a shorterdated one). Is this statement true even if the term-structure is not flat?
18. The spot price of copper is $1.47 per lb, and the forward price for delivery in three months is $1.51 per lb. Suppose you can borrow and lend for three months at an interest rate of 6% (in annualized and continuously-compounded terms)
(a) First, suppose there are no holding costs (i.e., no storage costs, no holding benefits). Is there an arbitrage opportunity for you given these prices? If so, provide details of the cash flows. If not, explain why not. (b) Suppose now that the cost of storing copper for three months is $0.03 per lb, payable in advance. How would your answer to Part (a) change? (Note that storage costs are asymmetric: you have to pay storage costs if you are long copper, but you do not receive the storage costs if you short copper.)
19. The SPX index is currently trading at a value of $1,265, and the FESX index (the Dow Jones EuroSTOXX Index of 50 stocks, referred to from here on as “STOXX”) is trading at e3,671. The dollar interest rate is 3% , and the euro interest rate is 5%. The exchange rate is $1.28/euro. The six-month futures on the STOXX is quoted at e3,782. All interest rates are continuously compounded. There are no borrowing costs for securities. (a) Compute the correct six-month forward futures prices of the SPX, STOXX, and the currency exchange rate between the dollar and the euro. (b) Is the futures on the STOXX correctly priced? If not, show how to undertake an arbitrage strategy assuming you are not allowed to undertake borrowing or lending transactions in either currency.
20. The current level of a stock index is 450. The dividend yield on the index is 4% (in continuously compounded terms), and the risk-free rate of interest is 8% for six-month
1. What is meant by the term “convenience yield”? How does it affect futures prices?
2. True or false: An arbitrage-free forward market can be in backwardation only if the benefits of carrying spot (dividends, convenience yields, etc.) exceed the costs (storage, insurance, etc.).
3. Suppose an active lease market exists for a commodity with a lease rate ` expressed in annualized continuously-compounded terms. Short-sellers can borrow the asset at this rate and investors who are long the asset can lend it out at this rate. Assume the commodity has no other cost of carry. Modify the arguments in the appendix to the chapter to show that the theoretical futures price is F = e (r−`)T S
10. Copper is currently trading at $1.28/lb. Suppose three-month interest rates are 4% and the convenience yield on copper is c = 3%
14. A three-month forward contract on a non-dividend-paying asset is trading at 90, while the spot price is 84. (a) Calculate the implied repo rate. (b) Suppose it is possible for you to borrow at 8% for three months. Does this give rise to any arbitrage opportunities? Why or why not?
17. A three month-forward contract on an index is trading at 756 while the index itself is at 750. The three-month interest rate is 6%. (a) What is the implied dividend yield on the index? (b) You estimate the dividend yield to be 1% over the next three months. Is there an arbitrage opportunity from your perspective?
18. The spot US dollar-euro exchange rate is $1.10/euro. The one-year forward exchange rate is $1.0782/euro. If the one-year dollar interest rate is 3%, then what must be the one-year rate on the euro?
19. You are given information that the spot price of an asset is trading at a bid-ask quote of 80 − 80.5, and the six-month interest rate is 6%. What is the bid-ask quote for the six-month forward on the asset if there are no dividends?
20. Redo the previous question if the interest rate for borrowing and lending are not equal, i.e., there is a bid-ask spread for the interest rate, which is 6.00–6.25%.
21. In the previous question, what is the maximum bid-ask spread in the interest rate market that is permissible to give acceptable forward prices?
22. Stock ABC is trading spot at a price of 40. The one-year forward quote for the stock is also 40. If the one-year interest rate is 4% and the borrowing cost for the stock is 2%, show how to construct a risk-less arbitrage in this stock
23. You are given two stocks, A and B. Stock A has a beta of 1.5, and stock B has a beta of −0.25. The one-year risk-free rate is 2%. Both stocks currently trade at $10. Assume a CAPM model where the expected return on the stock market portfolio is 10%. Stock A has an annual dividend yield of 1% and stock B does not pay a dividend. (a) What is the expected return on both stocks? (b) What is the one-year forward price for the two stocks? (c) Is there an arbitrage? Explain
5. In the presence of basis risk, is a one-for-one hedge, i.e., a hedge ratio of 1, always better than not hedging at all?
6. If the correlation between spot and futures price changes is ρ = 0.8, what fraction of cash-flow uncertainty is removed by minimum-variance hedging?
7. The correlation between changes in the price of the underlying and a futures contract is +80%. The same underlying is correlated with another futures contract with a (negative) correlation of −85%. Which of the two contracts would you prefer for the minimumvariance hedge?
8. Given the following information on the statistical properties of the spot and futures, compute the minimum-variance hedge ratio: σS = 0.2, σF = 0.25, ρ = 0.96.
9. Assume that the spot position comprises 1,000,000 units in the stock index. If the hedge ratio is 1.09, how many units of the futures contract are required to hedge this position?
10. You have a position in 200 shares of a technology stock with an annualized standard deviation of changes in the price of the stock being 30. Say that you want to hedge this position over a one-year horizon with a technology stock index. Suppose that the index value has an annual standard deviation of 20. The correlation between the two annual changes is 0.8. How many units of the index should you hold to have the best hedge?
11. You are a portfolio manager looking to hedge a portfolio daily over a 30-day horizon. Here are the values of the spot portfolio and a hedging futures for 30 days.
Carry out the following analyses using Excel: (a) Compute σ(∆S), σ(∆F ), and ρ. (b) Using the results from (a), compute the hedge ratio you would use. (c) Using this hedge ratio, calculate the daily change in value of the hedged portfolio. (d) What is the standard deviation of changes in value of the hedged portfolio? How does this compare to the standard deviation of changes in the unhedged spot position?
12. Use the same data as presented above to compute the hedge ratio using regression analysis, again using Excel. Explain why the values are different from what you obtained above
(a) Which currency should the company use for hedging purposes? (b) What is the minimum-variance hedge position? Indicate if this is to be a long or short position.
14. You use silver wire in manufacturing. You are looking to buy 100,000 oz of silver in three months’ time and need to hedge silver price changes over these three months. One COMEX silver futures contract is for 5,000 oz. You run a regression of daily silver spot price changes on silver futures price changes and find that δs = 0.03 + 0.89δF + What should be the size (number of contracts) of your optimal futures position. Should this be long or short?
15. Suppose you have the following information: ρ = 0.95, σS = 24, σF = 26, K = 90, R = 1.00018. What is the minimum-variance tailed hedge?
23. You are trying to hedge the sale of a forward contract on a security A. Suggest a framework you might use for making a choice between the following two hedging schemes: (a) Buy a futures contract B that is highly correlated with security A but trades very infrequently. Hence, the hedge may not be immediately available. (b) Buy a futures contract C that is poorly correlated with A but trades more frequently.
24. Download data from the Web as instructed below and answer the questions below:
(a) Extract one year’s data on the S&P 500 index from finance.yahoo.com. Also download corresponding period data for the S&P 100 index. (b) Download, for the same period, data on the three-month Treasury Bill rate (constant maturity) from the Federal Reserve’s Web page on historical data: www.federalreserve.gov/releases/h15/data.htm. (c) Create a data series of three-month forwards on the S&P 500 index using the index data and the interest rates you have already extracted. Call this synthetic forward data series F. (d) How would you use this synthetic forwards data to determine the tracking error of a hedge of three-month maturity positions in the S&P 100 index? You need to think (a) about how to set up the time lags of the data and (b) how to represent tracking error
25. Explain the relationship between regression R2 and tracking error of a hedge. Use the data collected in the previous question to obtain a best tracking error hedge using regression
1. Explain the difference between the following terms: (a) Payoff to an FRA. (b) Price of an FRA. (c) Value of an FRA.
2. What characteristic of the eurodollar futures contract enabled it to overcome the settlement obstacles with its predecessors?
3. How are eurodollar futures quoted?
4. It is currently May. What is the relation between the observed eurodollar futures price of 96.32 for the November maturity and the rate of interest that is locked-in using the contract? Over what period does this rate apply?
5. What is the price tick in the eurodollar futures contract? To what price move does this correspond?
6. What are the gains or losses to a short position in a eurodollar futures contract from a 0.01 increase in the futures price?
7. You enter into a long eurodollar futures contract at a price of 94.59 and exit the contract a week later at a price of 94.23. What is your dollar gain or loss on this position?
8. What is the cheapest to deliver in a Treasury bond futures contract? Are there other delivery options in this contract?
18. If you expect interest rates to rise over the next three months and then fall over the three months succeeding that, what positions in FRAs would be appropriate to take? Would your answer change depending on the current shape of the forward curve?
19. A firm plans to borrow money over the next two half-year periods, and is able to obtain a fixed-rate loan at 6% per annum. It can also borrow money at the floating rate of Libor + 0.5%. Libor is currently at 4%. If the 6 × 12 FRA is at a rate of 6%, find the cheapest financing cost for the firm.
20. You enter into an FRA of notional 6 million to borrow on the three-month underlying Libor rate six months from now and lock in the rate of 6%. At the end of six months, if the underlying three-month rate is 6.6% over an actual period of 91 days, what is your payoff given that the payment is made right away? Recall that the Actual/360 convention applies.
23. You anticipate a need to borrow USD 10 million in six-months’ time for a period of three months. You decide to hedge the risk of interest-rate changes using eurodollar futures contracts. Describe the hedging strategy you would follow. What if you decided to use an FRA instead?
24. In Question 23, suppose that the underlying three-month Libor rate after six months (as implied by the price of the eurodollar futures contract expiring in 6 months) is currently at 4%. Assume that the three-month period has 90 days in it. Using the same numbers from Question 23 and adjusting for tailing the hedge, how many futures contracts are needed? Assume fractional contracts are permitted
25. Using the same numbers as in the previous two questions, compute the payoff after six months (i.e., at maturity) under (a) an FRA and (b) a tailed eurodollar futures contract if the Libor rate at maturity is 5%, and the locked-in rate in both cases is 4%. Also compute the payoffs if the Libor rate ends up at 3%. Comment on the difference in payoffs of the FRA versus the eurodollar futures.
26. The “standard bond” in the Treasury bond futures contract has a coupon of 6%. If, instead, delivery is made of a 5% bond of maturity 18 years, what is the conversion factor for settlement of the contract? Assume that the last coupon on the bond was just paid.
27. Suppose we have a flat yield curve of 3%. What is the price of a Treasury bond of remaining maturity seven years that pays a coupon of 4%? (Coupons are paid semiannually.) What is the price of a six-month Treasury bond futures contract? Make any assumption you require concerning the maturity of the delivered bond to find this price.