Livre corrigé 2-3-5 PDF

1/OneDrive/Bureau/M1%20Finance/IMarché%20dérivé/Livre%20Corrigé.pdf 15 Demander I'IA Bing sur 257 0) | CID CHAPTER 1 Introduction Problem 1.1 Whar is the difference benween a long forward position and a short forward position? When a trader enters into a long forward contract, she is agreeing t...

1/OneDrive/Bureau/M1%20Finance/IMarché%20dérivé/Livre%20Corrigé.pdf 15 Demander I'IA Bing sur 257 0) | CID CHAPTER 1 Introduction Problem 1.1 Whar is the difference benween a long forward position and a short forward position? When a trader enters into a long forward contract, she is agreeing to buy the underlying asset for a certain price at a certain time in the future. When a trader enters into a short forwatd contract, she is agreeing to self the underlying asset for a certain price at a certain time in the futuro. Problem 1.2. Explain carefully the difference between hedging, speculation, and arbitrage. A trader is hedeing when she has an exposure to the price of an asset and takes a position in a derivative to offset the exposure. In a speculation the trader has no exposure to offsct. She is betting on the future movements in the price of the asset. Arbitrage involves taking a position in two or more different markets to lock in a profit. Problem 1.3. What is the difference berween entering into a long forward contract when the forward price is 850 and taking a long position in a call option with a strike price of $502 In the first case the trader is obligated to buy the asset for $50. (The trader does not have a choice.) In the second case the trader has an option to buy the asset for $50. (The trader does not have to exercise the option.) Problem 1.4. Explain carefully the difference between selling a call option and burying a put option. Selling a call option involves giving someone else the right to buy an asset from you. It gives vou a payoft of - max(S, - K ,0) - min(K - S, ,0) Buving a put option involves buying an option from someone else. It gives a payoff of max(K - S,,0) In both cases the potential payoff is K - S.. When you write a call option, the payoff is negative or zero. (This is because the counterparty chooses whether to exercise.) When you buy a put option, the payoff is zero or positive. (This is because you choose whether to exercise. Problem 1.5. An investor enters into a short forward contract to sell 100,000 British pounds for US dollars at an exchange rate of 1.4000 US dollars per pound. How much does the investor gain or lose if the exchange rate at the end of the contract is (a) 1.3900 and (6) 1.4200? (a) The investor is obligated to sell pounds for 1,4000 when they are worth 1.3900. The gain is (1.4000-1.3900) ×100,000 = $1,000. C2012 Pearson Education e (b) The investor is obligated to sell pounds for 1.4000 when they are worth 1.4200. The loss is (1.4200-1.4000]× 100,000 = 52,000 Problem 1.6. A mader enters into a short cotton fitures contract when the futures price is 30 cents per pound. The contract is for the delivery of 50,000 pounds. How much does the trader gain or lase if the cotton price at the end of the contract is (a) 48.20 cents per pound; (b) 51.30 cents per pound! (a) The trader sells for 50 cents per pound something that is worth 48.20 cents per pound. Gain = (50.5000 - $0.4820) × 50, 000 - 5900. (b) The trader sells for 50 cents per pound something that is worth 51.30 cents per pound. Loss - (50.5130 - 50.5000) × 50,000 - $650. Problem 1.7. Suppose that you write a put contract with a strike price of $40 and an expiration date in three months. The current stock price is 541 and the contract is on 100 shares. What have you committed vourself to? How much could your gain or lose? You have sold a put option. You have agreed to buy 100 shares for $40 per share if the party on the other side of the contract chooses to exercise the right to sell for this price. The option will be exercised only when the price of stock is below $40. Suppose, for example, that the option is exercised when the price is $30. You have to buy at $40 shares that are worth $30; you lose $10 per sharc, or $1,000 in total. If the option is exercised when the price is $20, you lose $20 per share, or $2,000 in total. The worst that can happen is that the price of the stock declines to almost zero during the three-month period. This highly unlikely event would cost you $4,000. In return for the possible futurc losses. you receive the price of the option from the purchaser. Problem 1.8. What is the difference berween the over-the-counter market and the exchange-traded market? What are the bid and offer quotes of a market maker in the over-the-counter market? The over-the-counter market is a telephone- and computer-linked network of financial institutions, fund managers, and corporate treasurers where two participants can enter into any mutually acceptable contract. An exchange-traded market is a market organized by an exchange where traders cither meet physically or communicate clectronically and the contracts that can be traded have been defined by the exchange. When a market maker quotes a bid and an otter, the bid is the price at which the market maker is prepared to buy and the offer is the price at which the market maker is prepared to sell. Problem 1.9. You would like to speculate on a rise in the price of a certain stock. The current stock price is 529, and a three-month call with a strike of $30 costs $2.90. You have $5.800 to invest. Identify two alternative strategies, one involving an investment in the stock and the other involving investment in the option. What are the potential gains and losses from each? One strategy would be to buy 200 shares. Another would be to buy 2,000 options. If the share price does well the second strategy will give rise to greater gains. For example, if the share 2 C2012 Pearson Education price goes up to 540 you gain [2,000 × ($40 - $3071-$5,800 - $14,200 from the second strategy and only 200× ($40-529) - $2,200 from the first strategy. However, if the share price does badly, the second strategy gives greater losses. For example, if the share price goes down to $25, the first strategy leads to a loss of 200× ($29- $25) = $800, whereas the second strategy leads to a loss of the whole $5,800 investment. This example shows that options contain built in leverage. Problem 1.10. Suppose you own 5,000 shares that are worth 325 each. How can put options be used to provide you with insurance against a decline in the value of your holding over the next four monthy? You could buy 50 put option contracts (cach on 100 shares) with a strike price of $25 and an expiration date in four months. If at the end of four months the stock price proves to be less than $25, you can exercise the options and sell the shares for $25 cach Problem 1.11. When first issued, a stock provides funds for a company. Is the same true of an exchange- traded stock option? Discuss. An exchange-traded stock option provides no funds for the company. It is a security sold by one investor to another. The company is not involved. By contrast, a stock when it is first issued is sold by the company to investors and does provide funds for the company. Problem 1.12. Explain why a futures contract can be used for ether speculation or hedging. If an investor has an exposure to the price of an asset, he or she can hedge with futures contracts. If the investor will gain when the price decreases and lose when the price increases, a long futures position will hedge the risk. If the investor will lose when the price decreases and gain when the price increases, a short futures position will hedge the risk. Thus either a long or a short futures position can be entered into for hedging purposes. If the investor has no exposure to the price of the underlying asset, entering into a futures contract is speculation. If the investor takes a long position, he or she gains when the asset's pricc increases and loses when it decreases. If the investor takes a short position, he or she loses when the asset's price increases and gains when it decreases Problem 1.13. Suppose that a March call option to buy a share for $50 costs $2.50 and is held until March. Under what circumstances will the holder of the option make a prof? Under what circunstances will the option he exercised? Draw a diagram showing how the profit on a long position in the option depends on the stock price at the maturity of the option. The holder of the option will gain if the price of the stock is above $52.50 in March. (This ignores the time value of money.) The option will be exercised if the price of the stock is above $50.00 in March. The profit as a function of the stock price is shown in Figure SI.. 3 92012 Pearson Education Stock Price 50 55 60 Figure SI.I Profit from long position in Problem 1.13 Problem 1.14. Suppose that a June put option to sell a share for $60 costs $4 and is held until June. Under what circumstances will the seller of the option (le., the party with a short position) make a profit? Under what circumstances will the option he exercised? Draw a diagram showing how the profit from a short position in the option depends on the stock price at the maturity of the option. The seller of the option will lose money if the price of the stock is below $56.00 in June. (This ignores the time value of money.) The option will be exercised if the price of the stock is below $60.00 in June. The profit as a function of the stock price is shown in Figure S1.2. Stock Price 60 65 70 Figure SI.2 Profit from short position in Problem 1.14 Problem 1.15. It is May and a trader writes a Septemher call option with a strike price of $20. The stack 22012 Pearson Education Itromenaton is May and a trader writes a September call option with a strike price of $20. The stock 02012 Pearson Edacation price is 318, and the option price is 52. Describe the investor's cash flows if the option is held until September and the stock price is $25 at this time The trader has an inflow of S2 in May and an outflow of $5 in September. The $2 is the cash received from the sale of the option. The $5 is the result of the option being exercised. The investor has to buy the stock for $25 in September and sell it to the purchaser of the option for $20. Problem 1.16. A trader writes a December put option with a strike price of 830. The price of the option is 54. Under what circamstances does the trader make a gain? The trader makes a gain if the price of the stock is above $26 at the time of exercise. (This ignores the time value of money.) Problem 1.17. A company knows that it is die to receive a certain amount of a foreign currency in four months. What type of option contract is appropriate for hedging A long position in a four-month put option can provide insurance against the exchange rate falling below the strike price. It ensures that the foreign eurrency can be sold for at least the strike pricc. Problem 1.18. A US company expects to have to paul million Canadian dollars in six months. Explain how the exchange rate risk can be hedged using (a) a forwurd contract; (b) an option The company could enter into a long forward contract to buy 1 million Canadian dollars in six months. This would have the effect of locking in an exchange rate equal to the current forward exchange rate. Alteratively the company could buy a call option giving it the right (but not the obligation) to purchase 1 million Canadian dollars at a certain exchange rate in six months. This would provide insurance against a strong Canadian dollar in six months while still allowing the company to benefit from a weak Canadian dollar at that time. Problem 1.19. 1 mader enters into a short forward contraes on 100 million yen. The forward exchange rate is 50.0080 per ven. How much does the trader gain or lose ifthe exchange rate at the end of the contract is (a) $0.0074 per yen; (b) 50.0091 per yen? a) The trader sells 100 million yen for $0.0080 per yen when the exchange rate is $0.0074 per yen. The gain is 100 x 0.0006 millions of dollars or $60,000. b) The trader sells 100 million yen for $0.0080 per yen when the exchange rate is $0.0091 per wen. The loss is 100 x 0.001| millions of dollars or $1 10,000 Problem 1.20. The Chicago Board of Trade offers a futures contract on long-term Treasury bonds. Characterize the mestors likely to use this contract. Most investors will use the contract because they want to do one of the following: a) Hedge an exposure to long-term interest rates 5 02012 Pearson Education b) Speculate on the future direction of long-term interest rates. c) Arbitrage between the spot and futures markets for Treasury bonds. This contract is discussed in Chapter 6. Problem 1.21. "Options and futures are zero-sum games. " What do you think is meant by this statement? The statement means that the gain (loss) to the party with the short position is equal to the loss (gain) to the party with the long position. In aggregate, the net gain to all parties is zero. Problem 1.22. Describe the profit from the following portfolio: a long forward contract on an asset and a long European put option on the asset with the same maturity as the forward contract and a strike price that is equal to the forward price of the asset at the time the portfolio is set up. The terminal value of the long forward contract is: S, - F where S, is the price of the asset at maturity and F, is the delivery price, which is the same as the forward price of the asset at the time the portfolio is set up). The terminal value of the put option is: max (F. - S,.0) The terminal value of the portfolio is therefore S, - Fo + max (F - 5, .0) - max (0,5, - F.I This is the same as the terminal value of a European call option with the same maturity as the forward contract and a strike price equal to F. This result is illustrated in the Figure SI.3. The profit equals the terminal value of the call option less the amount paid for the put option. (It does not cost anything to enter into the forward contract. 20 Profit 15 10 Asset Price 30 40 ----- Forward - - - Put Total 15 -20 Figure SI.3 Profit from portfolio in Problem 1.22 Problem 1.23. In the 1980s, Bankers Trust developed index currency option notes (ICONs). These are bonds 6 C2012 Pearson Education in which the amount received by the holder at maturity varies with a foreign exchange rate. One example was ifs trade with the Long Term Credir Bank of Japan. The ICON specified that if the yen U.S. dollar exchange rate, S,, is greater than 169 yen per dollar at maturity (in 1995). the holder of the bond receives $1,000. If it is less than 169 yen per dollar, the amown received by the holder of the bond is 1,00 - пах 0.1,00 159 When the exchange rate is below 84.5, nothing is received by the holder at maturity. Show that this ICON is a combination of a regular bond and two options. Suppose that the yen exchange rate (yen per dollar) at maturity of the ICON isS,. The payoff from the ICON is 1,000 S, > 169 if 1,0-1,0 152-1) I° 84558, 5169 When 84.55 S, s169 the payoff can be written if S, <84.5 169,000 2.000 The payoff from an ICON is the payoff from: (a) A regular bond (b) A short position in call options to buy 169.000 yen with an exercise price of 1/169 (c) A long position in call options to buy 169,000 yen with an exercise price of 1/84.5 This is demonstrated by the following table, which shows the terminal value of the various components of the position Bond Short Calls Long Calls 84.5 5 5, 5169 1000 1000 - 169,000(4 - #) S, < 84.5 1000 - 169,000(+ - ta) 169,000(4 - #s 5, > 169 0 0 Whole position 1000 2000 - 169.cus 0 Problem 1.24. On July 1, 2011, a compury enters into a forward contract to buy 10 million Japanese ven an January 1, 2012. On September 1, 2011, it enters into a forward contract to sell 10 million Japanese yen on Januars J, 2012. Describe the payoff from this strategy. Suppose that the forward price for the contract entered into on July 1, 2011 is F and that the forward price far the contract entered into on September 1, 2011 is F, with both F, and f being measured as dollars per yen. If the value of one Japanese yen (measured in US dollars) 7 92012 Pearson Education Cane January 1-2012, then the value of the first contend while tie volue 1008, - 71 ans of dollars) at that wen sold) at that time is: The total neyoff Cram the two contracts(per is therefore 1475, - 5,1 Tris if die forward price far delivery on January 1, 2012 increased between July 1, 2011 and 1045, F0 + 1005, -5,1 - 1005, - F0 September 1011 the company will make a profit. (Note that the yen/USD exchange rate is SMALL eXT ressed as the number of yen per USD not as the number of USD per yen) Problem 1.25. Suppose that USD-sterling spot and forward exchange rates are as follows: Spor 190-day forward 1180-day forward 1.4580 1.4556 1.4518 What opportunities are open to an arbiirager in the following situations? (6) A 180-day European call prion to buy El for $1.42 ensts 2 cents. A 90-day European put option to sell El for $1.49 costs 2 cents. (a) The arbitrageur buys a 180-day call option and takes a short position in a 180-day forward contract. IC S, is the terminal spot rate, the profit from the call option is max(3, - 1.42,0) - 0.02 The profit from the short forward contract is 1.4518 - 5, The profit from the strategy is therefore тах(5, - 1.42,0) - 0.02 + 1.4518 - 5, or mux 5, - 1.42,0) + 1.4318 - 5, This is 1.4318-5, when S, 1.42 0,1 [B when Sy =1.42 This shows that the profit is always positive. The time value of money has been ignored in these calculations. However, when it is taken into account the strategy is still likely to be profitable in all circumstances. (We would require an extremely high interest rate for S0.0118 interest to be required on an outlay of S0.02 over a 180-day period.) (b) The trader buys 90-day put options and takes a long position in a 90 day forward contract. If S, is the terminal spot rate, the profit from the put option is max41.49 - 5, _07 - 0.02 The profit from the long forward contract is Sr 1.4556 The profit from this strategy is therefore (ax(1.49 - 5.03 - 0.02 + S, - 1,4556 таĸс1.49 - S+/01 - Sy - 1.4756 This is Sp=1.4756 when Sr51 49 0.0144 when Sr1.49 The profit is therefore always positive. Again, the time value of money has been ignored but is unlikcly to atfect the overall profitability of the strategy. (We would require interest rates to be extremely high for 50.0144 interest to be required on an outlay of 50.02 over a 190-day period.) C2012 Pearson Education CHAPTER 2 Mechanics of Futures Markets Problem 2.1. Distinguish berween the terms open interest and trading volume. The open interest of a futures contract at a particular time is the total number of long positions outstanding. (Equivalently, it is the total number of short positions outstanding.) The trading volume during a certain period of time is the number of contracts traded during this period. Problem 2.2. What is the difference hetween a local and a futures commission merchant? A futures commission merchant trades on behalfof a client and charges a commission. A local trades on his or her own behalf. Problem 2.3. Suppose that sou enter into a short futures contract to sell July silver for 517.20 per ounce. The size of the contract is 5,000 ounces. The initial margin is $4.000, and the maintenance margin is $3,000. What change in the futures price will lead to a margin call? What happens if vow do not meet the margin call? There will be a margin call when $1,000 has been lost from the margin account. This will occur when the price of silver increases by 1,000/5.000 = $0.20. The price of silver must therefore rise to S17.40 per ounce for there to be a margin call. If the margin call is not met. your broker closes out your position. Problem 2.4. Suppose that in September 2012 a company takes a long pasition in a contract on May 2013 crude oil futures. It closes out is position in March 20/3. The futures price (per barrel) is 568.30 when it enters into the contract, $70.50 when it closes out its position, and $69.10 at the end of December 2012. One contract is for the delivery of 1,000 barrels. What is the company's total profit? When is it realized? How is it taxed if it is (aj a hedger and (bI a speculator? Assume that the compary has a December 31 vear-end. The total profit is ($70.50 - $68,30) × 1,000 = $2,200. Ofthis ($69.10 - 568.30)× 1,000 or 5800 is realized on a day-by-day basis between September 2012 and December 31, 2012. A further (S70.50 - $69.10) × 1,000 - 51,400 is realized on a day-by-day basis between January 1,2013, and March 2013. A hedger would be taxed on the whole profit of $2.200 in 2013. A speculator would be taxed on $800 in 2012 and $1,400 in 2013. Problem 2.5. What does a stop order to sell at 52 mean? When might it he used? What does a limit order to sell at 52 mean? When might it be used? A stop order to sell at $2 is an order to sell at the best available price once a price of $2 or 10 22012 Pourson Education less is reached. It could be used to limit the losses from an xisting long pusition. A Vimit order to sell at 52 is an order to sell at a price of $2 or more. It could be used to instruct a broker that a short position should be taken, providing it can be done at a price more favorable than $2 Problem 2.6. What is the difference berween the operation of the margin accounts administered by a clearing house and shose administered by a broker? The margin account administered by the clearing house is marked to market daily, and the clearing house member is required to bring the account back up to the prescribed level daily. The margin account administered by the broker is also marked to market daily. However, the account does not have to be brought up to the initial margin level on a daily basis. It has to be brought up to the initial margin level when the balance in the account falls below the maintenance margin level. The maintenance margin is usually about 75% of the initial margin. Problem 2.7. What differences exist in the way prices are quoted in the foreign exchange futures market, the foreign exchange spot market, and the foreign exchange forward marker? In futures markets, prices are quoted as the number of US dollars per unit of foreign currency. Spot and forward rates are quoted in this way for the British pound, euro, Australian dollar, and New Zealand dollar. For other major currencies, spot and forward rates are quoted as the number of units of foreign currency per US dollar Problem 2.8. The party with a short position in a futures contract sometimes has options as to the precise asset that will he delivered, where delivery will take place, when delivery will take place, and so on. Do these options increase or decrease the fixtures price? Explain your reasoning. These options make the contract less attractive to the party with the long position and more attractive to the party with the short position. They therefore tend to reduce the futures price. Problem 2.9. What are the most impartant aspects of the desige of a new futures contract? The most important aspects of the design of a new futures contract are the specification of the underlying asset, the size of the contract, the delivery arrangements, and the delivery months. Problem 2.10. Explain how margins protect investors against the possibility of defaul. A margin is a sum of money deposited by an investor with his or her broker. It acts as a guarantee that the investor can cover any losses on the futures contract. The balance in the margin account is adjusted daily to reflect gains and losses on the futures contract. If losses are above a certain level, the investor is required to deposit a further margin. This system makes it unlikely that the investor will default. A similar system of margins makes it unlikely that the investor's broker will default on the contract it has with the clearing house member and unlikely that the clearing house member will default with the clearing house. 11 52012 Person Education Problem 2.11. 4 trader buys two July futures contracts on frozen orange juice. Each contract is for the delivery of 15.000 pounds. The current futures price is 160 cents per pound, the initial margin is $6.000 per contract, and the maintenance margin is $4,500 per contract. What price change would lead to a margin call? Under what circumstances could $2,000 be withdrawn from the margin account? There is a margin call if more than $1,500 is lost on one contract. This happens if the futures price of frozen orange juice falls by more than 10 cents to below 150 cents per pound. $2,000 can be withdrawn from the margin account if there is a gain on one contract of $1,000. This will happen if the futures price rises by 6.67 cents to 166.67 cents per pound. Problem 2.12. Show that, if the futures price of a commodity is greater than the spot price during the deliery period, then there is an arbitrage opportunity. Does an arbitrage opportunity exist if the fitures price is less than the spot price? Explain your answer. If the futures price is greater than the spot price during the delivery period, an arbitrageur buys the asset, shorts a futures contract, and makes delivery for an immediate profit. If the futures price is less than the spot price during the delivery period, there is no similar perfect arbitrage strategy. An arbitrageur can take a long futures position but cannot force immediate delivery of the asset. The decision on when delivery will be made is made by the party with the short position. Nevertheless companies interested in acquiring the asset may find it altractive to enter into a long futures contract and wait for delivery to be made. Problem 2.13. Explain the difference berween a marker-iFtouched order and a stop order. A market-if-touched order is executed at the best available price after a trade occurs at a specified price or at a price more favorable than the specifled price. A stop order is executed at the best available price after there is a bid or offcr at the specified price or at a price less favorable than the specified price. Problem 2.14. Explain what a stop-limit order to sell at 20.30 with a limit of 20.10 means. A stop-limit order to sell at 20.30 with a limit of 20.10 means that as soon as there is a bid at 20.30 the contract should be sold providing this can be done at 20.10 or a higher price. Problem 2.15. Ar the end of one day a clearing house member is long 100 contracts, and the settlement price is $50,000 per contract. The original margin is $2,000 per contract. On the following duy the member becomes responsible for clearing an additional 20 long contracts, entered into at a price of 551,000 per contract. The settlement price at the end of this day is $50.200. How much does the member have to add to its margin account with the exchange clearing horse? The clearing house member is required to provide 20 x $2,000 = $40,000 as initial margin for the new contracts. There is a gain of (50,200 - 50,000)× 100 - $20,000 on the existing contracts. There is also a loss of (51.000 - 50,200) × 20 - $16,000 on the new contracts. The 12 12012 Pearson Education member must therefore add 40,000 - 20,000 + 16,000 - 536,000 to the margin account. Problem 2.16. Or July 1, 2012, a Japanese company enters into a forward contract to buy SI million with pen on January 1, 2013. On Septemher 1, 2012. it enters into a forward contract la sell SI willow on Janary 1, 20/3. Describe the profit or loss the compuny will make in dollars as a finction of the forward exchange rates on July 1, 2012 and September 1, 2012. Suppose F and F are the forward exchange rates for the contracts entered into July 1, 2012 and September 1, 2012, and S is the spot rate on January 1, 2013. (All exchange rates are measured as yen per dollar). The payoff from the first contract is (S - F) million yen and the payoff from the second contract is (F, - 5) million yen. The total payoff is therefore (S -F) + (P, - 5) = (F. - F) million yen. Problem 2.17. The forward price on the Swiss franc for delivery in 45 days is quoted as 1.1000. The futures price for a contract that will be delivered in 45 davs is 0.9000. Explain these two quotes. Which is more favorable for an investor wanting to sell Swiss francs? The 1.1000 forward quote is the number of Swiss franes per dollar. The 0.9000 futures quote is the number of dollars per Swiss franc. When quoted in the same way as the futures price the forward price is 1/1.1600 - 0.9091. The Swiss franc is therefore more valuable in the forward market than in the futures market. The forward market is therefore more attractive for an investor wanting to sell Swiss francs. Problem 2.18. Suppose you call vour broker and issue instructions to sell one July hogs contract. Describe what happens. Live hog futures are traded on the Chicago Mercantile Exchange. The broker will request some initial margin. The order will be relayed by telephone to your broker's trading desk on the floor of the change (or to the trading desk of another broker). It will then be sent by messenger to a commission broker who will execute the trade according to your instructions. Confirmation of the trade eventually reaches you. If there are adverse movements in the futures price your broker may contact you to request additional margin. Problem 2.19. "Speculation in futures markets is pure gambling. It is not in the public interest to allow speculators to trade on a futures exchange." Discuss this viewpoint Speculators are important market participants because they add liquidity to the market. However, contracts must be useful for hedging as well as speculation. This is because regulators gencrally only approve contracts when they are likely to be of interest to hedgers as well as speculators. Problem 2.20. Live cattle futures trade with June, August, October, December, Februan, and April 13 02012 Pearson Education maturities. Why do you think that the open interest for the June contract is less than that for the August contract in Table 2.2? Normally, the shorter the maturity of a contract is, the higher the open interest. However, traders tend to close out their positions in the month immediately before the maturity month This means that the open interest for the closest maturity month can be less than that for the next closest maturity month Problem 2.21. What do you think would happen if an exchange started trading a contract in which the quality ofthe underiving asset was incompletely specified? The contract would not be a success. Parties with short positions would hold their contracts until delivery and then deliver the cheapest form of the asset. This might well be viewed by the party with the long position as garbage! Once news of the quality problem became widely known no one would be prepared to buy the contract. This shows that futures contracts are feasible only when there are rigorous standards within an industry for defining the quality of the asset. Many futures contracts have in practice failed because of the problem of defining quality. Problem 2.22. *When a futures contraet is traded on the floor of the exchange, it may be the case that the open interest increases by one, stays the same, or decreases by one. " Explain this statement. If both sides of the transaction are entering into a new contract, the open interest increases by one. If both sides of the transaction are closing out existing positions, the open interest decreases by one. If one party is entering into a new contract while the other party is closing out an existing position, the open interest stays the same. Problem 2.23. Suppose that on October 24, 2072, a compury sells one April 2013 live-cattle futures contracts. It closes out its position on January 21, 2013. The futures price (per pound) is 91.20 cents when it enters into the contract, 88.30 cent when it closes out its position, and 88.80 cents at the end of December 2012. One contract is for the delivery of 40,000 pounds of cattle. What is the total profit? How is it raxed if the company is (a) a hedger and (b) a speculator? Assume that the company has a December 31 rear end. The total profit is 40,000 × (0.9120 - 0.8830) - 51,160 If the company is a hedger this is all taxed in 2013. If it is a speculator 40,000× (0.9120 - 0.8880) = 5960 is taxed in 2012 and 40,000 × (0.8880 - 0.8830) = $200 is taxed in 2013 Problem 2.24. A catile farmer expects to have 120,000 pounds of live cattle to sell in three months. The livecartle futures contract traded by the CME Group is for the delivery of 40,000 pounds of cutile. How can the farmer use the contract for hedging? From the farmer's viewpoint, what are the pros and cons of hedging? 14 02012 Pearson Education The farmer can short 3 contracts that have 3 months to maturity. If the price of cattle falls, the gain on the futures contract will offsct the loss on the sale of the cattle. If the price of cattle rises, the gain on the sale of the cattle will be offset by the loss on the futures contract. Using futures contracts to hedge has the advantage that it can at no cost reduce risk to almost zero. Its disadvantage is that the farmer no longer gains from favorable movements in cattle prices. Problem 2.25. It is July 2011. A mining company has just discovered a small deposit of gold. It will take six months to construct the mine. The gold will then he extracted on a more or less continuous basis for one year. Futures contracts on gold are available with delivery months every tho months from August 2011 to December 2012. Each conract is for the delivery of 100 ounces. Discuss how the mining company might use futures markets for hedging. The mining company can estimate its production on a month by month basis. It can then short futures contracts to lock in the price received for the gold. For example, if a total of 3,000 ounces are expected to be produced in September 2011 and October 2011, the price received for this production can be hedged by shorting 30 October 2011 contracts. 15 C2012 Pearson Education CHAPTER 5 Determination of Forward and Futures Prices Problem 5.1. Explain what happens when an investor shorts a certain share. The investor's broker borrows the shares from another client's account and sells them in the usual way. To close out the position, the investor must purchase the shares. The broker then replaces them in the account of the client from whom they were borrowed. The party with the short position must remit to the broker dividends and other income paid on the shares. The broker transfers these funds to the account of the client from whom the shares were borrowed. Occasionally the broker runs out of places from which to borrow the shares. The investor is then short squeezed and has to close out the position immediately. Problem 5.2. What is the difference between the forward price and the value of a forward contract? The forward price of an asset today is the price at which you would agree to buy or sell the asset at a future time. The value of a forward contract is zero when you first enter into it. As time passes the underlying asset price changes and the value of the contract may become positive or negative. Problem 5.3. Suppose that you enter into a six-month forward contract on a non-dividend-paying stock when the stock price is $30 and the risk-free interest rate (with continuous compounding) is 12% per anram. What is the forward price? The forward price is 13061205 = $31.86 Problem 5.4. A stock inder currently stands at 350. The risk-free interest rate is 8% per annum (with continuous compounding) and the dividend yield on the index is 4% per annum. What should the futures price for a four-month contract be? The futures price is 13506998-454958 - S354.7 Problem 5.5. Explain carefully why the futures price of gold can be calculated from its spot price and other observable variables whereas the futures price of copper cannot. Gold is an investment asset. If the futures price is too high, investors will find it profitable to increase their holdings of gold and short futures contracts. If the futures price is too low, they will find it profitable to decrease their holdings of gold and go long in the futures market. Copper is a consumption asset. If the futures price is too high, a strategy of buy copper and short futures works. However, because investors do not in general hold the asset, the strategy of sell copper and buy futures is not available to them. There is therefore an upper bound, but 30 Education C2012 Pearson no lower bound, to the futures price. Problem 5.6. Explain carefully the meaning of the terms convenience yield and cost of carry. What is the relationship berween funes price, spot price, convenience yield, and cost of carry? Corvenience yield measures the extent to which there are benefits obtained from ownership of the physical asset that are not obtained by owners of long futures contracts. The cost of carry is the interest cost plus storage cost less the income cared. The futares price, F., and spot price, S, are related by where c is the cost of carry, y is the convenience yield, and 7 is the time to maturity of the futures contract. Problem 5.7. Explain why a foreign currency can be treated as an asset providing a known yield. A foreign currency provides a known interest rate, but the interest is received in the foreign currency. The value in the domestie currency of the income provided by the foreign currency is therefore known as a percentage of the value of the forcign currency. This means that the income has the properties of a known yield Problem 5.8. Is the futures price of a stock index greater than or less than the expected future value of the index? Explain your answer. The futures price of a stock index is always less than the expected future value of the index. This follows from Section 5.14 and the fact that the index has positive systematic risk. For an alternative argument, let g be the expected return required by investors on the index so that E(S,) - S,e'*-*7. Because y 3 r and 7, - Ser-wt, it follows that E(S,) > F. Problem 5.9. A one-year long forward contract on a non-dividend-paying stock is entered into when the stuck price is $40 and the risk-free rate of interest is 1096 per annum with continuous compounding. a) What are the fonward price and the inirial value of the forward contract? b) Six munths luter. the price of the stock is $45 and the risk-free interest rate is still 109% Whar are the forward price and the value of the forward contract? (a) The forward price. Fo, is given by equation (5. 1) as: 40€011contract = 44.21 is zero. or $44.21. The initial value of F, the= forward b) The delivery price & in the contract is $44.21. The value of the contract. f, after six months is given by equation (5.5) as: 5 = 45 - 44.21e 9191 = 2.95 31 C2012 Pearson Education 36 sur 257 \ 5) | iD i.e. it is 52.95. The forward price is: or $47.31. 4569785 - 47.31 Problem 5.10. The risk-free rate of interest is 7% per annum with continous compounding, and the dividend vield on a stock index is 3.296 per annum. The current value of the index is 150. What is the six-month futures price? Using equation (5.3) the six month futures price is or $152.88. =152.88 Problem 5.11. Asstme that the risk-frec interest rate is 996 per annum with continuous compounding and that the dividend yield on a stock index varies throughout the vear. In February. Mery. August, and November, dividends are paid at a rate of 5% per arum. In other months. dividends are paid at a rate of 2%6 per annum. Suppose that the value of the index on July 31 is 1,300. What is the futures price for a contract deliverable on December 31 of the same gear? The futures contract lasts for five months. The dividend wield is 2% for three of the months and 5% for two of the months. The average dividend yield is therefore The futures price is therefore or 51331.80 (3x2+2×5) =3.2% 1300@*9-697220419 - 1.331.80 Problem 5.12. Suppose that the risk-free interest rate is 10% per annum with continuous compounding und that the dividend yield on a stock index is 4% per amm. The index is standing at 400, and the futures price for a contract deliverable in four months is 405. What arbitrage opportunities does this create? The theoretical futures price is 400e 1-ADthat 412the .= 408.08 The actual futures price is only 405. This shows index futures price is too low relative to the index. The correct arbitrage strategy is (a) Buy futures contracts (b) Short the shares underlying the index. Problem 5.13. Estimate the difference between short-term interest rates in Mexico and the United States on May 26, 2010 from the information in Table 5.4. The settlement prices for the futures contracts are to Sept: 0.76375 Dec:0.75625 32 02012 Pearson Education 32 C20[2 Pearson Education The December price is about 0.98% below the September price. This suggests that the shortterm interest rate in the Mexico exceeded short-term interest rale in the United States by about 0.98% per threc months or about 3.92% per year. Problem 5.14. The rwo-mouth interest rates in Switzerland and the United States are 296 and 5% per annum, respectively, with continucus compounding. The spot price of the Swiss franc is SO.8000. The futures price for a contract deliverable in tuo mandas is 50.8100. What arbitrage opportinities does this create? The theorctical futures price is (0,8000e 414-046 n2. 17 - 0.8040 The actual futures price is too high. This suggests that an arbitrageur should buy Swiss francs and short Swiss francs futures, Problem 5.15. The spot price of sihver is $15 per ounce. The storage costs are $0.24 per ounce per year parable quarterly in advance. Assuing that interest rates are 103 per annum for all maturities, calculate the futures price of silver for delivery in nine months. The present value of the storage costs for nine months are 0.06 ÷ 0.060 2. 0-0.44 4 0.062 *185 - 0.176 or SO.176. The futures price is from caution (5.11) given by F where F, = (15.000 + 0.176)e91079.= 16.36 i.e.. it is $16.36 per ounce. Problem 5.16. Suppose that F, and F, are rwo futures contracts on the same commodiry with times to maturity, 4, and ty, where 4, >4, Prove that Is Fert where r is the interest rate (assumed constants and there are no storage costs. For the purposes of this problem, assure that a futures contract is the same as a forward contract. an investor could make a riskless profit by (a) Taking a long position in a futures contract which matures at time 4, (b) Taking a short position in a futures contract which matures at time r, When the first futures contract matures, the asset is purchased for F, using funds borrowed al rate r. It is then held until time 4, at which point it is exchanged for F; under the second contract. The costs of the funds borrowed and accumulated interest at time 1, is Fen positive profit of is then realized at lime 4. This type of arbitrage opportunity cannot exist for long. Hence: 33 02012 Pearson Education 1/OneDrive/Bureau/M1%20Finance/IMarché%20dérive/Livre%20Corrigé.pdf 38 Demander l'IA Bing sur 257 0) | iD Problem 5.17. When a known fiture cash oufflow in a foreign currency is hedged by a company using a forward contract, there is no foreign exchange risk. When it is hedged ting futures contracts, the daily settlement process does leave the compary exposed to some risk. Explain the nature of this risk. In particular, consider whether the company is better off using a futures contract or a forward contraer when a) The value of the foreign currency falls rapidly during the life of the contract b) The value of the foreign currency rises rapidly during the life of the contract c) The value of the foreign currency first rises and then falls hack to its intrial value d) The vale of the foreign currency first falls and then rises back to its initial value Assume that the forward price equals the futures price. In total the gain or loss under a futures contract is equal to the gain or loss under the corresponding forward contract. However the timing of the cash flows is different. When the time value of money is taken into account a futures contract may prove to be more valuable or less valuable than a forward contract. Of course the company does not know in advance which will work out better. The long forward contract provides a perfect hedge. The long futures contract provides a slightly imperfect hedge. a) In this case the forward contract would lead to a slightly better outcome. The company will make a loss on its hedge. If the hedge is with a forward contract the whole of the loss will be realized at the end. If it is with a futures contract the loss will be realized day by day throughout the contract. On a present value hasis the former is preferable. b) In this case the futures contract would lead to a slightly better outcome. The company will make a gain on the hedge. If the hodge is with a forward contract the gain will be realized at the end. fit is with a futures contract the gain will be realized day by day throughout the life of the contract. On a present value basis the latter is preferable. c) In this case the futures contract would lead to a slightly better outcome. This is because it would involve positive cash flows early and negative cash flows later. d) In this case the forward contract would lead to a slightly better outcome. This is because, in the case of the futures contract, the early cash flows would be negative and the later cash flow would be positive. Problem 5.18. It is sometimes argued that a forward exchange rate is an unbiased predictor of future exchange rates. Under what circumstances is this so? From the discussion in Scetion 5.14 of the text, the forward exchange rate is an unbiased predictor of the füture exchange rate when the exchange rate has no systematic risk. To have no syslematic risk the exchange rate must be uncorrelated with the return on the market. Problem 5.19. Show that the growth rate in an index futures price equals the excess return of the portfolio underlying the index over the risk-free rate. Assume that the risk-free interest rate and the dividend vield are constant. Suppose that F, is the futures price at time zero for a contract maturing at time 7 and F is the futures price for the same contract at time 1. It follows that F, = 5,a dir 34 C2012 Pearson Education )neDrive/Bureau/M1%20Finance/IMarché%20dérivé/Livre%20Corrigé.pdf Demander lA Bing 39 sur 257 0) | ED where S, and S are the spot price at times zero and 4, r is the risk-free rate, and o is the dividend yield. These equations imply that = S, Define the excess return of the portfolio underlying the index over the risk-free rate as z. The total retum is r + r and the return realized in the form of capital gains is r + x-q. It follows that S, = S,of-w"s and the equation for F/F, roduccs to E=e which is the required result. Problem 5.20. Show that equation (5.3) is true by considering an investment in the asset combined with a shor position in a futures contract. Assume that all incore from the asser is reinvested in the asset. Use an argument similar to that in footnoses 2 and 4 and explain in detail what an arbitrageur would do if equation (5.3) did not hold. Suppose we buy M units of the asset and invest the income from the asset in the asset. The income from the asset causes our holding in the asset to grow at a continuously compounded rate g. By time 7 our holding has grown to Ne* units of the asset. Analogously to footnotes 2 and 4 of Chapter 5, we therefore buy N units of the asset at time zero at a cost of S. per unit and enter into a forward contract to sell Net unit for F, per unit at time 7 - This generates the following cash flows: Time 0: - NS, Time I: NA Because there is no uncertainty about these cash flows, the present value of the time T inflow must equal the time zero outflow when we discount at the risk-free rate. This means that F. - Sure This is equation (5.3). , an arbitrageur should borrow money at rate r and buy N units of the assel. At the same time the arbitrageur should enter into a forward contract to sell NeT units of the asset at time 7. As income is received, it is reinvested in the asset. Al time T the loan is repaid and the arbitrageur makes a profit of N (FeaT - S,e') at time 7 . an arbitrageur should short N units of the asset investing the proceeds at rate r. At the same time the arbitrageur should enter into a forward contract to buy Net units of the asset at time r. When income is paid on the asset, the arbitrageur owes money on the short position. The investor meets this obligation from the cash proceeds of shorting further units. The result is that the number of units shorted grows at rate , to Net. The cumulative short position is closed out at time T and the arbitrageur makes a profit of N(S,e" - F,e*). 35 2012 Pearson Education Problem 5.21. Explain careful what is meant by the expected price of a commodity on a particular future dale. Suppose that the futures price of crude oil declines with the maturity of the contract at the rate of 29 per year. Assume that speculators tend to he short crude oil futures and hedgers tended to be long crude oil fisures. What does the Keynes and llicks argument imply about the expected future price of oil? To understand the caning of the expected future price of a commodity, suppose that there are N different possible prices at a particular future time: Pi. P. _ P.. Define q, as the (subjective) probability the price being P, (with g, + q, + _ + 9, = 1). The expected future price is ÇuP Different people may have different expected future prices for the commodity. The expected future price in the market can be thought of as an average of the opinions of different market participants. Of course, in practice the actual price of the commodity at the future time may prove to be higher or lower than the expected price. Keynes and Hicks argue that speculators on average make money from commodity futures trading and hedgers on average lose money from commodity futures trading. If speculators lend to have short positions in crude oil futures, the Keynes and Hicks argument implies that futures prices overstate expected future spot prices. If crude oil futures prices decline at 2% per year the Keynes and Hicks argument therefore implies an even faster decline for the expected price of crude oil in this case. Problem 5.22. The Value Line Index is designed to reflect changes in the vale of a portfolio of over 1,600 equally weighted stocks. Prior to March 9, 1988, the change in the index from one day to the next was calculated as the geometric average ofthe changes in the prices of the stocks underiving the index. In these circumstances, does equation (5.8) correctly relate the futures price of the index to its cash price? If not, does the equation overstate or understate the futures price? When the geometric average of the price relatives is used, the changes in the value of the index do not correspond to changes in the value of a portfolio that is traded. Equation (5.8) is therefore no longer correct. The changes in the value of the portfolio are monitored by an index calculated from the arithmetic average of the prices of the stocks in the portfolio. Since the geometric average of a set of numbers is always less than the arithmetic average, equation (5.8) overstates the futures price. It is rumored that at one time (prior to 1988), equation (5.8) did hold for the Value Line Index. A major Wall Street firm was the first to recognize that this represented a trading opportunity. It made a financial killing by buying the stocks underlying the index and shorting the futures. Problem 5.23. A U.S, company is interested in using the fittures contructs traded by the CME Group to hedge its Australian dollar exposure. Define r as the interest rule (all maturities) on the U.S. dollar and r as the interest rate (all maturities) on the Australian dollar. Assume that r and rare constant and that the company uses a contract expiring at time T to hedge an 36 Education 02012 Pearson