Options, Futures and Other Derivatives 7th solution manual.pdf

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Options, Futures and Other Derivatives 7th solution manual.pdf
In both cases the potential payoff is K- Sr. 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 beca use you choose whether to exercise Problem 1.5 An investor enters into a short forward contract to sell 100, 000 British pounds for U.S dollars at an exchange rate of 1. 9000 U.S. dollars per pound. How much does the investor gain or lose if the exchange rate at the end of the contract is(a)1.8900 and(b)1.9200? (a) The investor is obligated to sell pounds for 1.9000 when they are worth 1.8900. The gain Is(19000-18900)×100,000=$1,000 (b)The investor is obligated to sell pounds for 1. 9000 when they are worth 1.9200. The loss is(1.9200-1.9000)×100,000=$2,000 Problem 1.6 a trader enters into a short cotton futures contract when the futures price is 50 cents per pound. The contract is for the delivery of 50, 000 pounds. How much does the trader gain or lose if the cotton price at the end of the contract is(a)48.20 cents per pound;(b 51.30 cells per pound? (a) The trader sells for 50 cenis per pound sonething that is worth 48.20 cents per pound Gain=($05000-80.4820)×50,000=8900. (b) The trader sells for 50 cents per pound something that is worth 51. 30 cents per pound Loss=($0.5130-$0.5000)×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 $41 and the contract is What have you committed yourself to? How much could you gain or lose? on 100 shares. You have sold a put option. You have agreed to buy 100 shares for $40 per share if the party on the other sidc of thc cont ract 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 exanple, that the option is exercised when the price is $30. You have to buy at $40 shares thal are worth $30; you lose $10 per share, or $1,000 in total. If the option is exercised when the price is $ 20, you lose $20 per share, or 82, 000 in total. The worst, that can happen is that the price of the stock declines to almost zero during the thrco-month period. This highly unlikely event would cost you $4, 000. In rcturn for the possible future losses, you receive the price of the option from the purchaser. Problem 1. 8 What is the difference between the over-the-counter market and the exchange-traded tarket? 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 oy an exchange where traders either meet physically or comMunicate electronically and the tracts that can be traded have been defined by the exchange. When a market maker quotes a bid and an offer, 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 $29, 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 invelving investment in the optie WAThat-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 price goes up to $40 you gain [2, 000 x($40-$30)-$5, 800=$14, 200 from the second strategy and only 200 X($40-$29)=$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 $25, the first strategy leads to a loss of 200 X($S29-$25)=$800, whereas the second strategy leads to a loss of the whole $5, 800 investmcnt. This example shows that options contain built in leverage Problem 1.10 Suppose that you own 5,000 shares worth $25 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 months? You could buy 5,000 put options (or 50 contracts) with a strike price of $25 and an expiration date in 4 months. This provides a type of insurance. If at the end of 4 months the stock price proves to be less than 25 you can exercise the options and sell the shares for $25 each. The cost of this strategy is the price you pay for the put options. Problem 1.11 When first issued, a stock provides funds for a company. Is the same true of a stock option? Discuss A slock option provides no funds for the company. It is a security sold by one trader to another. The company is not involved. By contrast, a stock when it is first issued is a clain sold by the company to investors and does provide funds for the company Problem 1.1.2 Explain why a forward contract can be used for either speculation or hedging If a trader has an exposure to the price of an asset, she can hedge with a forward contract. If the exposure is such that the trader will gain when the price decreases and option is in these circumstances less than the price received for the option. The option will be exercised if the stock price at maturity is less than $60.00. Note that if the stock price is betwcen $56.00 and S60.00 the seller of the option makes a profit even though the option is exercised. The profit from the short position is as shown in Figure S1.2 6420 Stock Price 250 70 Figure S1. 2 Profit from short position In ProbleM 1.14 Problem 1.15 It is May and a trader writes a September call option with a strike price of $20. The stock price is $18, and the option price is $2. Describe the trader's cash flows if the option is held until Septernber and the stock price is $25 at that time The trader receives an infow of $2 in May. Since the option is exercised, the trader also has an outflow of $5 in September. The $2 is the cash received from the sale of the option. The $5 is the result of buying the stock for $25 in September and selling it to the purchaser of the option for $20. One contract consists of 100 options and so the cash Hows for a contract are multiplied by 100 Problem 1.16 a trader writes a December put option with a strike price of $30. The price of the option is s4. Under what circumstances does the trader make a gain? The trader makes a gain if the price of the stock is above $26 in December.(This ignores the time value of money. Problem 1.17. a company knows that it is due to receive a certain amount of a foreign currency in four months. What type of option contract is appropriate for hedging? 6 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 currency can be sold for at least the strike price Problem 1.18 A United States company expects to have to pay 1 million Canadian dollars in six months. Explain how the exchange rate risk can be hedged using(a)a forward 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 urrent forward exchange rate. Alternatively the company could buy a call option giving it the right(bul not the obligaTion)lo purchase 1 million Canadian dollar 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 fron a weak Canadian dollar at that time Problem 1.19 A trader enters into a short forward contract on 100 million yen. The for ward exchange rate is $0.0080 per yen. How much does the trader gain or lose if the exchange rate at the end of the contract is(a)$0.0074 per yen;(b)$0.0091 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 yen. The loss is 100 x 0.0011 millions of dollars or $110,000 Problem 1. 20 The Chicago Board of Trade offers a futures contract on long-term Treasury bonds Characterize the traders likely to use this contract Most traders who use the contract will wish to do one of the following (a) Hedge their exposure to long-term interest rates (b) Speculate on the future direction of long-term interest rates (c arbitrage between cash and futures markels 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 a short position in an option is always equal to the loss(gain) to the party with the long position. The sum of uhe gains 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 where Sr is the price of the asset at maturity and Fo is the forward price of the asset at the time the portfolio is set up. (The delivery price in the forward contract is Fo The terminal value of the put option is max(Fo The terminal value of the portfolio is therefore T-Fo+ max(Fc max(0, ST- FC This is the same as the terminal value of a European call option with the same maturity as the forward contract and an exercise price equal to Fo. This result is illustrated in the Figure S1.3. 'The profit equals the terminal value less the amount paid for the option Problen 1.23 In the 1980s, Bankers Trust developed indec currency option notes(ICONs) .These are bonds in which the amount received by the holder at maturity varies with a foreign exchange rate. One example was its trade with the Long Term Credit Bank of Japan. The ICon specified that if the yen-US dollar exchange rate, ST, 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 amount received by the holder of the bond is 169 1,000-max|0,1,000 T When the exchange rate is below 84.5, nothing is received by the holder at maturity, Show that this ICon is is a combination of a regular bond and two options Suppose that the yen exchange rate (yen per dollar)at maturity of the ICon is ST The payoff from the icon is 1,000 169 0001.00(3-1)i845≤Sr≤169 if ST <84.5 Profit Asset Price Figure S1.3 Profit from portfolio in Problcm 1.22 When 84.5< S]< 169 the payoff can be written 2,000169,000 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 Terminal Terminal Terminal Terminal Valuc o Value of Value of Regular bond Short calls Long calls Whole position >169 1,000 1000 84.5<Sr<1691000 169,00( 169 2000-36900 Sr<845 1000 Problem 1.24. On July 1, 2008, a company enters into a forward contract to buy 10 milion Japanese yen On January 1, 2009. On September 1, 2008, it enters into a forward contract to sell 10 million Japanese yen on January 1, 2009. Describe the payoff from this strategy. Suppose that the forward price for the contract entered into on July 1, 2008 is Fi and that the forward price for the contract entered into on September 1, 2008 is F2 with both F1 and E2 being measured as dollars per yen. If the value of one Japanese yen(mcasured in U.S. dollars)is ST on January 1, 2009, then the value of the first contract (in millions of dollars)at that time is 0(S7-F1) while the valuc of the second contract (per yen sold) at that time is T The total payoff from the two contracts is therefore 10(S7-1)+10(F2-Sr)=10(72-F1) Thus if the forward price for delivery on January 1, 2009 increases between July 1, 2008 nd September 1, 2008 the company will make a profit Problem 1.25 Suppose that USD-sterling spot and forward exchange rates are as follows Spot 2.0080 90-day forward 2.0056 180-day forward 20018 What opportunities are open to an arbitrageur in thc following situations? a. A 180-day European call option to buy t] for $1. 97 costs 2 cents b. A 90-day European put option to sell f1 for $2.04 costs 2 cents (a)The trader buys a 180-day call option and takes a short position in a 180-day forward contract. If S is the terminal spot rate the profit from the call option is max(Sr-197,0)-0.02 The profit from the short forward contract is 2.0018-Sr The profit from the strategy is therefore mnax(ST-1.97,0)-002+20018-S7 Or max(Sr-1.97:0)+1.9818-7 This is 1.9818- ST when ST<1.97 00118 whlen ST>1.97 This shows that the profit is always positive. The tirne value of Inoney has been ignored in these calculations. However, when it is taken into account the strategy is still likely to be profitable in all circunstances.(We would require all extremely high interest rate for 90.0118 interest to be required on an outlay of $0.02 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 ST is the terminal spot rate, the profit from the put option is max(204-S7,0)-0020 The profit from the long forward contract is Sr-2.0056 The profit from this strategy is therefore max(204-Sr,0)-0.020+Sr-20056 max(204-Sr,0)+Sr-2.0256 This is Sr-2.0256 when ST>2.04 0.0144 hen ST<2.04 The profit is therefore always positive. Again, the time value of money has been ignored but is unlikely to affect the overall profitability of the stratcgy. (We would rcquirc interest rates to be extremely high for $0.0144 interest to be required on an outlay of $0.02 over a 90-day period. ASSIGNMENT QUESTIONS Problem 1.26 The price of gold is currently $600 per ounce. The forward price for delivery in one year is $800. An arbitrageur can borrow money at 10% per annum. What should the arbitrageur do? Assume that the cost of storing gold is zero and that gold provides 10 nicene The arbitrageur could borrow money to buy 100 ounces of gold today and short futures colltracls on 100 ounces of gold for delivery in one year. This means that gold is purchased for $600 per ounce and sold for $800 per ounce. The return (33. 3% per annum)is far greater than the 10%o cost of the borrowed funds. This is such a profitable opportunity that the arbitrageur should buy as many ounces of gold as possible and short futures contracts on

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