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Solution Manual For Derivatives Markets, 3rd Edition

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    Chapter 1 Introduction to Derivatives Question 1.1 This problem offers different scenarios in which some companies may have an interest to hedge their exposure to temperatures that are detrimental to their business. In answering the problem, it is useful to ask the question: Which scenario hurts the company, and how can it protect itself? a ) A soft drink manufacturer probably sells more drinks when it is abnormally hot. She dislikes days at which it is abnormally cold because people are likely to drink less, and her business suffers. She will be interested in a cooling - degree - day futures contract because it will make payments when her usual business is slow. She hedges her business risk. b ) A ski - resort operator may fear large losses if it is warmer than usual. It is detrimental to her business if it does not snow in the beginning of the season or if the snow is melting too fast at the end of the season. She will be interested in a heating - degree - day futures contract because it will make payments when her usual business suffers, thus compensating the losses. c ) During the summer month s , an electric utili ty company, such as one in the s outh of the United States, will sell a lot of energ y during days of excessive heat because people will use their air conditioners, refrigerators , and fans more often, thus consuming a lot of energy and increasing profits for the utility company. In this scenario, the utility company will have less business during relatively colder days, and the cooling - degree - day futures offers a possibility to hedge such risk. Alternatively, we may thin k of a utility provider in the northe ast during the winter months, a region where people use many additional electric heaters. This utility provider will make more money during unusually col d days and may be interested in a heating - degree - day contract because that contract pays off if the primary business suffers. d ) An amusement park operator fears bad weather and cold days because people will abstain from going to the amusement park during cold days. She will buy a cooling - degree - day future to offset her losses from ticket sales with gains from the futures contract. Question 1.2 A variety of counter parties are imaginable. For one, we could think about speculators who have differences in opinion and who do not believe that we will have excessive temperature variations during the life of the futures contracts. Thus, they are willing to take the opposing side, receiving a payoff if the weather is stable. Alternatively, there may be opposing hedging needs: Compare the sk i - resort operator and the soft drink manufacturer. The cooling - degree - day futures contract will pay off if the weather is relatively mild, and we saw that the resort operator will buy the futures contract. The buyer of

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    2 Chapter 1/Introduction to Derivatives the cooling - degree - day futures will make a loss if the weather is cold ( which means that the seller of the contract will make a gain ) . Since the soft drink manufacturer wants additional money if it is cold, she may be interested in taking the opposite side of the cooling - degree - day futures. Question 1.3 a ) Remember that the terminology bid and ask is formulated from the market makers perspec tive. Therefore, the price at which you can buy is called the ask price. Furthermore, you will have to pay the commission to your broker for the transaction. You pay: ( $41 . 05 × 100 ) + $20 = $ 4, 125 . 00 b ) Similarly, you can sell at the market maker’s bid price. You w ill again have to pay a commis sion, and your broker will deduct the commission from the sales price of the shares. You receive: ( $40 . 95 × 100 ) − $20 = $ 4, 075 . 00 c ) Your round - trip transaction costs amount to: $ 4, 125 . 00 − $ 4, 075 . 00 = $50 Question 1.4 In this problem , the brokerage fee is variable and depends on the actual dollar amount of the sale/purchase of the shares. The concept of the transaction cost remains the same: If you buy the shares, the commission is added to the amount you owe, and if you sell the shares, the commission is deducted from the proceeds of the sale. a ) ( $41 . 05 × 100 ) + ( $41 . 05 × 100 ) × 0 . 003 = $ 4, 117 . 315 = $ 4, 117 . 32 b ) ( $40 . 95 × 100 ) − ( $40 . 95 × 100 ) × 0 . 003 = $ 4, 082 . 715 = $ 4, 082 . 72 c ) $ 4, 117 . 32 − $ 4, 082 . 72 = $34 . 6 The variable ( or proportional ) brokerage fee is advantageous to us. Our round - trip transaction fees are reduced by $15.40. Question 1.5 In answering this question , it is important to remember that the market maker provides a service to the market. He stands ready to buy shares into his inventory and s ell shares out of his inventory thus providing immediacy to the market. He is remunerated for this service by earning the bid - ask spread. The market maker buys the security at a price of $10 0, and he sells it at a

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    Chapter 1/Introduction to Derivatives 3 price of $100.10. If he buys 100 shares of the security and immediately sells them to another party, he is earns a spread of: 100 × ( $100 . 12 − $100 ) = 100 × $0 . 12 = $12 . 00 Question 1.6 A short sale of XYZ entails borrowing shares of XYZ and then selling them and receiving cash. Therefore, initially, we will receive the proc eeds from the sale of the asset less the proportional commission charge: 300 × ( $30 . 19 ) − 300 × ( $30 . 19 ) × 0 . 005 = $ 9, 057 × 0 . 995 = $ 9, 011 . 72 When we close out the position, we will again incur the commission charge, which is added to the purchasing cost: 300 × ( $29 . 87 ) + 300 × ( $29 . 87 ) × 0 . 005 = $ 8, 961 × 1 . 005 = $ 9, 005 . 81 F inally, we subtract the cost of covering the short position from our initial proceeds to receive total profits: $ 9, 011 . 72 − $ 9, 005 . 81 = $5 . 91. We can see that paying the commission charge twice significantly reduces the profits we can make. Question 1.7 a ) A short sale of JKI stock entails borrowing shares of JKI , then selling them and receiving cash, and we learned that we sell assets at the bid price. Therefore, initially, we will receive the proceeds from the sale of the asset at the bid ( ignoring the commissions and interest ) . After 180 days, we cover the short position by buying the JKI stock, and we saw that we will always buy at the ask. Therefore, we earn the following profit: 400 × ( $25 . 125 ) − 400 × ( $23 . 0625 ) = $1 0, 050 − $ 9, 225 . 00 = $825 . 00 b ) We have to pay the commission twice. The commission will reduce our profit: 400 × ( $25 . 125 ) − 400 × ( $25 . 125 ) × 0 . 003 − ( 400 × [ $23 . 0625 ] + 400 × [ $23 . 0625 ] ) = $1 0, 050 × 0 . 997 − $ 9, 225 × 1 . 003 = $1 0, 01 9 . 85 − $9 , 252 . 675 = $767 . 175 .

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    4 Chapter 1/Introduction to Derivatives c ) The proceeds from short sales minus the commission charge are $1 0, 019.85 ( or $1 0, 050 if you ignore the commission charge ) . Since the six - month interest rate is given and the period of our short sale is exactly half a year, we can directly calculate the interest we could earn ( and that we now lose ) on a deposit of $1 0, 019.85: $1 0, 019 . 85 × ( 0 . 03 ) = $300 . 5955 = $300 . 60 or without taking into account the commission charge: $1 0, 050 . 00 × ( 0 . 03 ) = $301 . 50 . Question 1.8 We learned from the main text that short selling i s equivalent to borrowing money and that a short seller will often have to deposit the proceeds of the short sale with the lender as collateral. A short seller is entitled to earn interest on his collateral, and the interest rate he earns is called the short rebate in the stock market. Usually, the short rebate is close to the prevailing market interest rate. Sometimes, though, a particular stock is scarce and difficult to borrow. In this case, the short rebate is substantially less than the current market interest rate, and an equity lender can earn a nice profit in the form of the difference between the current market interest rate and the short rebate. By signing an agreement as mentioned in the problem, you give your brokerage firm the possibility to act as an equity lender, using the shares of your account. Brokers want you to sign such an agreement because they can make additional profits. Question 1.9 If we borrow an asset from a lender, we have the obligation to make any payments to the lender that she is entitled to as a stockholder. Because the lender is entitled to the dividend on the day the stock goes ex dividend but does not receive it from the company because we have sold her stock, we must provide the dividend. This payment is tax - deductible for us. In a perfect capital market, we would expect that the stock price falls exactly by the amount of the dividend on the ex - date. Therefore, we should not care. However, two complications may arise. First, we may have borrowed a large amount of shares, and the increased dividend forces us to pay more to the lender, and we may not have the additional required money. A n unexpected increase of the dividend could be a sign that the company is doing well . E mpirically, we observe sharp price increases after such announce ments. As we have a short position in the stock, we make money if the stock price falls. Therefore, an unexpected increase in the dividend is very bad for our position, and we should care!

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    Chapter 1/Introduction to Derivatives 5 Question 1.10 The main textbook explains short selling as follows. “The sale of a stock you do not already own is called a short - sale.”Short positions are those resulting from short sales. At NASDAQ for example, each financial firm is required to report its overall short positions in all accounts in NASDAQ - listed securities twice a month. The short positions from these reports are aggregated and the resulting number is called short interest in NASDAQ stocks. The short interest statistics are widely observed , and some financial market participants believe that short interest is a good measure of market sentiment. You can easily access statistics on short interest on individual stocks on Nasdaq (www.nasdaq.com). Once you are on the home page, go to the pull - down menu “Quotes and Research ,” then go to “Short Interest” in the folder “Company Profile .” Once you are on that page, you can enter up to 25 tickers and retrieve the short interest in the stocks of your choice. In general, stocks that lend themselves to some speculation and stocks around corporate events (mergers and acquisition, dividend dates, etc.) with uncertain outcomes will have a particularly high short interest. It is theoretically possible to have short interest of more than 100 percent because some market participants (e.g., market makers) have the ability to short sell a stock without having a locate (i.e., having someone who actually owns the stock and has agreed to lend it). Question 1.11 We are interested in borrowing the asset “money.” Therefore, we go to an owner ( or if you prefer, to a collecto r ) of the asset, called Bank. The Bank provides the $100 of the asset money in digital form by increasing our bank account. We sell the digital money by going to the ATM and withdrawing cash. After 90 days, we buy back the digital money for $10 2 by depositing cash into our bank account. The lender is repaid, and we have covered our short position. Question 1.12 We are interested in borrowing the asset “money” to buy a house. Therefore, we go to an owner of the asset, called Bank. The Bank provides the dollar amount, say $25 0, 00 0, in digital form in our mortgage account. As $25 0, 000 is a large amount of money, the bank is subject to substantial credit risk ( e.g., we may lose our job ) and demands a collateral . Although the money itself is not subject to large variations in price ( besides inflation risk, it is difficult to imagine a reason for money to vary in value ) , the Bank knows that we want to buy a house, and real estate prices vary substantially. Therefore, the Bank wants more collateral than the $25 0, 000 they are lending.

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    6 Chapter 1/Introduction to Derivatives In fact, as the Bank is only lending up to 80 percent of the value of the house, we could get a mortgage of $25 0, 000 for a house that is worth $25 0, 000 ÷ 0 . 8 = $31 2, 500. We see that the bank factor ed in a haircut of $31 2, 500 − $25 0, 000 = $6 2, 500 to protect itself from credit risk and adverse fluctuations in property prices. We buy back the asset money over a long horizon of time by reducing our mortgage through annuity payments. Question 1.13 For ex ample, go to the web site of the CME group (www.cmegroup.com). Under the tab “Products & Trading ,” you can choose the asset class you are interested in. If you click, for example, on “Equity Index ,” you will come to a product listing of all equity products CME offers. Once you click on any particular product, say E - mini S&P 500 futures, you get to a separate page with the latest quotes of this product. If you click on the link “Contract Specifications ,” you will see all details regarding ticker symbol, contract size, tick size, traded months, settlement, etc. of the product. CME group has a Daily Volume and Open Interest Report that summarizes exchange - wide volume, including futures and options volume. You can access it by cl icking on the “Market Data” tab and choosing the volume report. If the notional values were cut in half, one would expect trading volume and open interest to double but the notional value traded to stay the same. Question 1.14 We have the following for the diffe rent tra des and different dealers : a) A B C Trade 1 +5 – 5 Trade 2 +15 – 15 Trade 3 +10 – 10 Trade 4 – 20 +20 Total 0 +5 – 5 Therefore, trader A’s net position is zero, trader B is long five contracts, and trader C is short five contracts.

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    Chapter 1/Introduction to Derivatives 7 b) Trading volume is equal to 5 + 15 + 10 + 20 = 50. Open interest is equal to 5. Notional value of trading volume is equal to 50 contracts × $100/ contract = $5,000 Notional value of open interest is equal to 5 contracts × $ 100 /c ontract = $500. c) Now all traders have a net position of zero. Trading volume has increased by five contracts to 55. The notional value of trading volume is equal to 55 × $100 = $5,500. Open interest and the notional value of open interest are equal to zero. d) Trading volume and open interest would be identical; the notional values of trading volume and open interest would be lower.

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    Chapter 2 An Introduction to Forwards and Options Question 2.1 The payoff diagram of the stock is just a graph of the stock price as a function of the stock price: In order to obtain the profit diagram at expiration, we have to finance the initial investment. We do so by selling a bond for $50. After one year , we have to pay back: $50 × (1 + 0.1) = $55. The second figure (on the next page) shows the graph of the stock, of the bond to be repaid, and of the sum of the two positions, which is the profit graph. The arrows show that at a stock price of $5 5, the profit at expiration is indeed zero.

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    Chapter 2/An Introduction to Forwards and Options 9 Question 2.2 Since we sold the stock initially, our payoff at expiration from being short the stock is negative.

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    10 Part One/Insurance, Hedging, and Simple Strategies In order to obtain the profit diagram at expiration, we have to lend out the money we received from the short sale of the stock. We do so by buying a bond for $50. After one year , we receive from the investment in the bond: $50 × (1 + 0.1) = $55. The second figure shows the graph of the sold stock, of the money we receive from the investment in the bond, and of the sum of the two positions, which is the profit graph. The arrows show that at a stock price of $5 5, the profit at expiration is indeed zero. Question 2.3 The position that is the opposite of a purchased call is a written call. A seller of a call option is said to be the option writer or to have a short position. The call option writer is the counterparty to the option buyer, and his payoffs and profits are just the opposite of those of the call option buyer. Similarly, the position that is the opposite of a purchased put option is a written put option. Again, the payoff and profit for a written put are just the opposite of those of the purchased put. It is important to note that the opposite of a purchased call is NOT the purchased put. If you do not see why, please draw a payoff diagram with a purchased call and a purchased put.

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    Chapter 2/An Introduction to Forwards and Options 11 Question 2.4 a) The payoff to a long forward at expiration is equal to: Payoff to long forward = Spot price at expirat ion – forward price Therefore, we can construct the following table: Price of asset in six months Agreed forward price Payoff to the long forward 40 50 −10 45 50 −5 50 50 0 55 50 5 60 50 10 b) The payoff to a purchased call option at expiration is: Payoff to call option = max[ 0, spot price at expiration – strike price] The strike is given: It is $50. Therefore, we can construct the following table: Price of asset in six months Strike price Payoff to the call option 40 50 0 45 50 0 50 50 0 55 50 5 60 50 10 c) If we compare the two contracts, we immediately see that the call option has a protection for adverse movements in the price of the asset: If the spot price is below $5 0, the buyer o f the call option can walk away and need not incur a loss. The buyer of the long forward incurs a loss, but he has the same payoff as the buyer of the call option if the spot price is above $50. Therefore, the call option should be more expensive. It is this attractive option to walk away that we have to pay for.

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    12 Part One/Insurance, Hedging, and Simple Strategies Question 2.5 a) The payoff to a short forward at expiration is equal to: Payoff to short forward = forward price – spot price at expiration Therefore, we can construct the following table: Price of asset in six months Agreed forward price Payoff to th e short forward 40 50 10 45 50 5 50 50 0 55 50 −5 60 50 −10 b) The payoff to a purchased put option at expiration is: Payoff to put option = max[ 0, strike price – spot price at expiration] The strike is given: It is $50. Therefore, we can construct the following table: Price of asset in six months Strike price Payoff to the call option 40 50 10 45 50 5 50 50 0 55 50 0 60 50 0 c) Th e same logic as in question 2.4 ( c) applies. If we compare the two contracts, we see that the put option has a protection for increases in the price of the asset: If the spot price is above $5 0, the buyer of the put option can walk away and need not incur a loss. The buyer of the short forward incurs a loss and must meet her obligations. However, she has the same payoff as the buyer of the put option if the spot price is below $50. Therefore, the put option should be more expensive. It is this attractive option to walk away if things are not as we want that we have to pay for. Question 2.6 We need to solve the following equation to determine the effective annual interest rate: $91 × (1 + r ) = $100. We obtain r = 0.098 9, which means that the effective annual interest rate is approximately 9.9 percent .

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    Chapter 2/An Introduction to Forwards and Options 13 Remember that when we drew profit diagrams for the forward or call option, we drew the payoff on the vertical axis, and the index price at the expiration of the contract on the horizontal axis. In this case, the particularity is that the default - free , zero - coupon bond will pay exactly $10 0, no matter what the stock price is. Therefore, the payoff diagram is just a horizontal line, intersecting the y - axis at $100. The textbook provides the answer to the question concerning the profit diagram in the section “Zero - Coupon Bonds in Payoff and Profit Diagrams.” When we were calculating profits, we saw that we had to find the future value of the initial investment. In this case, our initial investment is $91. How do we find the future value? We use the current risk - free interest rate and multiply the initial investment by it. However, as our bond is default - free and does not bear coupons, the effective annual interest rate is exactly the 9.9 percent we have calculated before. Therefore, the future value of $91 is $91 × (1 + 0.0989) = $10 0, and our profit in six months is zero! Question 2.7 a) It does not cost anything to enter into a forward contract — we do not pay a premium. Therefore, the payoff diagram of a forward contract coincides with the profit diagram. The graphs have the following shape: b) We have seen in question 2.1 that in order to obtain the profit diagram at expiration of a purchase of XYZ stock, we have to finance the initial investment. We did so by selling a bond for $50. After one year , we had to pay back: $50 × (1 + 0.1) = $55. Therefore, our total profit at expirat ion from the purchase of a stock that was financed by a loan was: $ S T − $5 5, where S T is the value of one share of XYZ at expiration. But this profit from buying the stock , and financing it is the same as the profit from

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    14 Part One/Insurance, Hedging, and Simple Strategies our long forward contract , and both positions do not require any initial cash — but then, there is no advantage in investing in either instrument. c) The dividend is only paid to the owner of the stock. The owner of the long forward contract is not e ntitled to receive the dividend because she only has a claim to buy the stock in the future for a given price, but she does not own it yet. Therefore, it does matter now whether we own the stock or the long forward contract. Because everything else is the same as in part a) and b), it is now beneficial to own the share: We can receive an additional payment in the form of the dividend if we own the stock at the ex - dividend date. This question hints at the very important fact that we have to be careful to take into account all the benefits and costs of an asset when we try to compare prices. We will encounter similar problems in later chapters. Question 2.8 We saw in question 2.7 ( b) that there is no advantage in buying either the stock or the forward contract if we can borrow to buy a stock today (so both strategies do not require any initial cash) and if the profit from this strategy is the same as the profit of a long forward contract. The profit of a long forward contract with a price for delivery of $53 is equal to: $ S T − $5 3, where S T is the (unkn own) value of one share of XYZ at expiration of the forward contract in one year. If we borrow $50 today to buy one share of XYZ stock (that costs $50), we have to repay in one year: $50 × (1 + r ). Our total profit in one year from borrowing to buy one share of XYZ is therefore: $ S T − $50 × (1 + r ). Now we can equate the two profit equations and solve for the interest rate r : $ S T − $53 = $ S T − $50 × (1 + r ) ⇔ $53 = $50 × (1 + r ) ⇔ $53 1 $50  = r ⇔ r = 0.06 Therefore, the one - year effective interest rate that is c onsistent with no advantage to either buying the stock or forward contract is 6 percent. Question 2.9 a) If the forward price is $ 1, 10 0, then the buyer of the one - year forward contract receives at expiration after one year a profit of: $ S T − $ 1, 10 0, where S T is the (unknown) value of the S&R index at expiration of the forward contract in one year. Remember that it costs nothing to enter the forward contract. Let us again follow our strategy of borrowing money to finance the purchase of the index today, so that we do not need any initial cash. If we borrow $ 1, 000 today to buy the S&R index (that costs $ 1, 000), we have to repay in one year: $ 1, 000 × (1 + 0.10) = $ 1, 100. Our total profit in one year from borrowing to buy the S&R index is therefore: $ S T − $ 1, 100. The profits from the two strategies are identical.

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    Chapter 2/An Introduction to Forwards and Options 15 b) The forward price of $ 1, 200 is worse for us if we want to buy a forward contract. To un derstand this, suppose the index after one year is $ 1, 150. While we have already made money in part a) wi th a forward price of $ 1, 10 0, we are still losing $50 with the new price of $ 1, 200. As there was no advantage in buying either the stock or forward at a price of $ 1, 10 0, we now need to be “bribed” to enter into the forward contract. We somehow need to find an equation that makes the two strategies comparable again. Suppose that we lend some money initially together with entering into the forward contract so that we will receive $100 after one year. Then, the payoff from our modified forward strategy is: $ S T − $ 1, 200 + $100 = $ S T − $ 1, 10 0, which equals the payoff of the “borrow to buy index” strategy. We have found the future value of the premium somebody needs us to pay. We still need to find out what the premium we will receive in one year is worth today. W e need to discount it: $100/ (1 + 0.10) = $90.91. c) Similarly, the forward price of $ 1, 000 is advantageous for us. As there was no advantage in buying either stock or forward at a price of $ 1, 10 0, we now need to “bribe” someone to sell this advantageous forward contract to us. We somehow need to find an equation that makes the two strategies comparable again. Suppose that we borrow some money initially together with entering into the forward contract so that we will have to pay back $100 after one year. Th en, the payoff from our modified forward strategy is: $ S T − $ 1, 000 − $100 = $ S T − $ 1, 10 0, which equals the payoff of the “borrow to buy index” strategy. We have found the future value of the premium we need to pay. We still need to find out what this premi um we have to pay in one year is worth today. We simply need to discount it: $100/(1 + 0.10) = $90.91. We should be willing to pay $90.91 to enter into the one - year forward contract with a forward price of $ 1, 000. Question 2.10 a) Figure 2. 6 depicts the profit from a long call option on the S&R index with six months to expiration and a strike price of $ 1, 000 if the future price of the option premium is $95.68. The profit of the long call option is: max[ 0, S T − $ 1, 000] − $95.68 ⇔ max[−$95.6 8, S T − $ 1, 095.68] where S T is the (unknown) value of the S&R index at expiration of the call option in six months. In order to find the S&R index price at which the call option diagram intersects the x - axis, we have to set the above equation equal to zero. We get: S T − $ 1, 095.68 = 0 ⇔ S T = $ 1, 095.68. This is the only solution, as the other part of the maximum function, −$95.6 8, is always less than zero. b) The profit of the six month forward contract with a forward price of $ 1, 020 is: $ S T − $ 1, 020. In order to find the S&R index price at which the call option and the forward contract have the same profit, we need to set both parts of the maximum function of the profit of the call option equal to the profit of the forward cont ract and see which part permits a solution. First, we see immediately that $ S T − $ 1, 020 = $ S T − $ 1, 095.68

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    16 Part One/Insurance, Hedging, and Simple Strategies does not have a solution. But we can solve the other leg: $ S T − $ 1, 020 = −$95.68 ⇔ S T = $924.3 2, which is the value given in the exercise. Question 2. 11 a) Figure 2. 8 depicts the profit from a long put option on the S&R index with six months to expiration and a strike price of $ 1, 000 if the future value of the put premium is $75.68. The profit of the long put option is: max[ 0, $ 1, 000 − S T ] − $75.68 ⇔ max[−$75.6 8, $924.32 − S T ] where S T is the (unknown) value of the S&R index at expiration of the put option in six months. In order to find the S&R index price at which the put option diagram intersects the x - axis, we have to set the above equation equal to zero. We get: $924.32 − S T = 0 ⇔ S T = $924.32. This is the only solution, as the other part of the maximum function, −$75.6 8, is always less than zero. b) The profit of the short six - month forward contract with a forward price of $ 1, 020 is: $ 1, 020 − $ S T . In order to find the S&R index price at which the put option and the sold forward contract have the same profit, we need to set both parts of the maximum function of the profit of the put option equal to the profit of the forward contract and see which part permits a solution. First, we see immediately that $ 1, 020 − $ S T = $924.32 − $ S T does not have a solution. But we can solve the other leg: $ 1, 020 − S T = −$75.68 ⇔ S T = $ 1, 095.6 8, which is the value given in the exercise. Question 2.12 a) Long Forward The maximum loss occurs if the stock price at expiration is zero (the stock price cannot be less than zero because companies have limited liability). The forward then pays 0 – Forward price. The maximum gain is unlimited. The stock price at expiration could theoretically grow to infinity ; there is no bound. We make a lot of money if the stock price grows to infinity (or to a very large amount). b) Short Forward The profit for a short forward contract is forward price – stock price at expiration. The maximum l oss occurs if the stock price r ises sharply ; there is no bound to it, so it could grow to infinity. The maximum gain occurs if the stock price is zero. c) Long Call We will not exercise the call option if the stock price at expiration is less than the strike price. Consequently, the only thing we lose is the future value of the premium we paid initially to buy the option. As the stock price can grow very large (and without bound), and our payoff grows linearly in the terminal stock price once it is higher than the strike, there is no limit to our gain.

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    Chapter 2/An Introduction to Forwards and Options 17 d) Short Call We have no control over the exercise decision when we write a call. The buyer of the call option decides whether to exercise it or not, and he will only exercise the call if he makes a profit. As we have the opposite side, we will never make any money at the expiration of the call option. Our profit is restricted to the future value of the premium, and we make this maximum profit whenever the stock price at expiration is smaller than the strike price. However, the stock price at expiration can be very large and has no bound, and as our loss grows linearly in the terminal stock price, there is no limit to our loss. e) Long Put We will not exercise the put option if the stock price at expiration is larger than the strike price. Consequently, the only thing we lose whenever the terminal stock price is larger than the strike is the future value of the premium we paid initially to buy the option. We will profit from a decline in the stock prices. However, stock prices cannot be smaller than zero, so our ma ximum gain is restricted to strike price less the future value of the premium , and it occurs at a terminal stock price of zero. f) Short Put We have no control over the exercise decision when we write a put. The buyer of the put option decides whether to exercise or not, and he will only exercise if he makes a profit. As we have the opposite side, we will never make any money at the expiration of the put option. Our profit is restricted to the future value of the premium, and we make this maximum profit whenever the stock price at expiration is greater than the strike price. However, we lose money whenever the stock price is smaller than the strike ; hence , the largest loss occurs when the stock price attains its smallest possible value, zero. We lose the strike price because somebody sells us an asset for the strike that is worth nothing. We are only compensated by the future value of the premium we received. Question 2.13 a) In order to be able to draw profit diagrams, we need to find the future values of the call premia. They are: i) 35 - strike call: $9.12 × (1 + 0.08) = $9.8496 ii) 40 - strike call: $6.22 × (1 + 0.08) = $6.7176 iii) 45 - strike call: $4.08 × (1 + 0.08) = $4.4064

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    18 Part One/Insurance, Hedging, and Simple Strategies We can now graph the payoff and profit diagrams for the call options. The payoff diagram looks as follows: We get the profit diagram by deducting the option premia from the payoff graphs. The profit diagram looks as follows:

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    Chapter 2/An Introduction to Forwards and Options 19 b) Intuitively, whenever the 45 - strike option pays off (i.e., has a payoff bigger than zero), the 40 - strike and the 35 - strike options pay off. However, there are some instances in which the 40 - strike option pa ys off and the 45 - strike option does not. Similarly, there are some instances in which the 35 - strike option pays off but neither the 40 - strike nor the 45 - strike pay off. Therefore, the 35 - strike offers more potential than the 40 - and 45 - strike, and the 40 - strike offers more potential than the 45 - strike. We pay for these additional payoff possibilities by initially paying a higher premium. Question 2.14 In order to be able to draw profit diagrams, we need to find the future values of the put premia. They are: a) 35 - strike put: $1.53 × (1 + 0.08) = $1.6524 b) 40 - strike put: $3.26 × (1 + 0.08) = $3.5208 c) 45 - strike put: $5.75 × (1 + 0.08) = $6.21 We get the following payoff diagrams:

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    20 Part One/Insurance, Hedging, and Simple Strategies We get the profit diagram by deducting the option premia from the payoff graphs. The profit diagram looks as follows: Intuitively, whenever the 35 - strike put option pays off (i.e., has a payoff bigger than zero), the 40 - strike and the 35 - strike options also pay off. However, there are some instances in which the 40 - strike option pa ys off and the 35 - strike option does not. Similarly, there are some instances in whic h the 45 - strike option pays off but neither the 40 - strike nor the 35 - strike pay off. Therefore, the 45 - strike offers more potential than the 40 - and 35 - strike, and the 40 - strike offers more potential than the 35 - strike. We pay for these additional payoff possibilities by initially paying a higher premium. It makes sense that the premium is increasing in the strike price. Question 2.15 The nice thing that lead us to the notion of indifference between a forward contract and a loan - financed stock index purchase whenever the forward price equaled the future price of the loan was that we could already tell today what we had to pay back in the future. In other words, the return on the loan, the risk - free interest rate r , was known today, and we removed uncertainty about the payment to be made. If we were to finance the purchase of the index by short selling IBM stock, we would in troduce additional uncertainty because the future value of the IBM stock is unknown. Therefore, we could not calculate today the amount to be repaid, and it would be impossible to establish an equivalence between the forward and loan - financed index purchase today. The calculation of a profit diagram would only be possible if we assumed an arbitrary value for IBM at expiration of the futures, and we would have to draw many profit diagrams with different values for IBM to get an idea of the many possible profits we could make.

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    Chapter 2/An Introduction to Forwards and Options 21 Question 2.16 The following is a copy of a spreadsheet that solves the problem: QuickTime™ and a decompressor are needed to see this picture.

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    Chapter 3 Insurance, Collars, and Other Strategies Question 3.1 This question is a direct appl ication of the Put - Call - Parity [ equati on (3.1)] of the textbook. Mim icking Table 3.1., we have: S&R Index S&R Put Loan Payoff −(Cost + Interest) Profit 900.00 10 0.00 −1 , 000.00 0.00 −95.68 −95.68 950.00 50.00 −1 , 000.00 0.00 −95.68 −95.68 1 , 000.00 0.00 −1 , 000.00 0.00 −95.68 −95.68 1 , 050.00 0.00 −1 , 000.00 50.00 −95.68 −45.68 1 , 100.00 0.00 −1 , 000.00 100.00 −95.68 4.32 1 , 150.00 0.00 −1 , 000.00 150.00 −95.68 54.32 1 , 200.00 0.00 −1 , 000.00 200.00 −95.68 104.32 The payoff diagram looks as follows:

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    Chapter 3/Insurance, Collars, and Other Strategies 23 We can see from the table and from the payoff diagram that we have in fact reproduced a call with the instruments given in the exercise. The profit diagram below confirms this hypothesis. Question 3.2 This question constructs a position that is the opposite to the position of Table 3.1. Therefore, we should get the exact opposite results from Table 3.1. and the associated figures. Mimicking Table 3.1., we indeed have: S&R Index S&R Put Payoff −(Cost + Interest) Profit −900.00 −100.00 −1 , 000.00 1 , 095.68 95.68 −950.00 −50.00 −1 , 000.00 1 , 095.68 95.68 −1 , 000.00 0.00 −1 , 000.00 1 , 095.68 95.68 −1 , 050.00 0.00 −1 , 050.00 1 , 095.68 45.68 −1 , 100.00 0.00 −1 , 100.00 1 , 095.68 −4.32 −1 , 150.00 0.00 −1 , 150.0 0 1 , 095.68 −54.32 −1 , 200.00 0.00 −1 , 200.00 1 , 095.68 −104.32 On the next page, we see the corresponding payoff and profit diagrams. Please note that they match the combined payoff and profit diagrams of Figure 3.5. Only the axes have different scales.

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    24 Part One /Insurance , Hedging, and Simple Strategies Pa yoff diagram: Profit diagram:

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    Chapter 3/Insurance, Collars, and Other Strategies 25 Question 3.3 In order to be able to draw profit diagrams, we need to find the future value of the put premium, the call premium , and the investment in zero - coupon bonds. We have for: the put premium: $51.777 × (1 + 0.02) = $52.8 1, the call premium: $120.405 × (1 + 0.02) = $122.81 , an d the zero - coupon bond: $931.37 × (1 + 0.02) = $950.00 Now, we can construct the payoff and profit diagrams of the aggregate position: Payoff diagram: From this figure, we can already see that the combination of a long put and the long index looks exactly like a certain payoff of $95 0, plus a call with a strike price of 950. But this is the alternative given to us in the question. We have thus confirmed the equivalence of the two combined positions for the payoff diagrams. The profit diagrams on the next page confirm the equivalence of the two positions (which is again an application of the Put - Call - Parity).

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    26 Part One /Insurance , Hedging, and Simple Strategies Profit d iagram for a long 950 - strike put and a long index combined: Question 3.4 This question is another application of Put - Call - Parity. Initially, we have the following cost to enter into the combined position: We receive $ 1, 000 from the short sale of the index, and we have to pa y the call premium. Therefore, the future value of our cos t is: ($120.405 − $ 1, 000) × (1 + 0.02) = −$897.19. Note that a negative cost means that we initially have an inflow of money. Now, we can directly proceed to draw the payoff diagram:

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    Chapter 3/Insurance, Collars, and Other Strategies 27 We can clearly see from the figure that the payoff graph of the short i ndex and the long call looks like a fixed obligation of $95 0, which is alleviated by a long put position with a strike price of 950. The following profit diagram, including the cost for the combined position, confirms this: Question 3.5 This question is another application of Put - Call - Parity. Initially, we have the following cost to enter into the combined position: We receive $ 1, 000 from the short sale of the index, and we have to pa y the call premium. Therefore, the future value of our cost is: ($71.802 − $ 1, 000) × (1 + 0.02) = −$946.76. Note that a negative cost means that we initially have an inflow of money.

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    28 Part One /Insurance , Hedging, and Simple Strategies Now, we can directly proceed to draw the payoff diagram: In order to be able to compare this position to the other suggested position of the ex ercise, we nee d to find the future value of the borrowed $ 1, 029.41. We have: $ 1, 029.41 × (1 + 0.02) = $ 1, 050. We can now see from the figure that the payoff graph of the short index and the long call looks like a fixed obligation of $ 1, 05 0, which is exactly the future value of the borrowed amount, and a long put position with a strike price of 1, 050. The following profit diagram, including the cost for the combined position we calculated above, confirms this. The profit diagram is the same as the profit diagram for a loan and a long 1, 050 - strike put with an initial premium of $101.214.

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    Chapter 3/Insurance, Collars, and Other Strategies 29 Profit d iagram of going short the index and buying a 1, 050 - strike call: Question 3.6 We now move from a graphical representation and verification of the Put - Call - Parity to a mathe matical representation. Let us first consider the payoff of (a). If we buy the index (let us name it S), we receive at the time of expiration T of the options simply S T . The payoffs of part (b) are a little bit more complicated. If we deal with optio ns and the maximum function, it is convenient to split the future values of the index into two regions: one where S T < K and another one where S T ≥ K . We then look at each region separately, and hope to be able to draw a conclusion when we look at the agg regate position. We have for the payoffs in (b): Instrument S T < K = 950 S T ≥ K = 950 Get repayment of loan $931 . 37 × 1 . 02 = $950 $931 . 37 × 1 . 02 = $950 Long call o ption max (S T − 950 , 0 ) = 0 S T − 950 Short put o ption − max ($950 − S T , 0) 0 = S T − $950 Total S T S T

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    30 Part One /Insurance , Hedging, and Simple Strategies We now see that the total aggregate position only gives us S T , no matter what the final index value is — but this is the same payoff as in part (a). Our proof for the payoff equivalence is complete. Now let us turn to the profits. If we buy the index today, we need to finance it. Therefore, we borrow $ 1, 00 0 and have to repay $ 1, 020 after one year. The profit for part (a) is thus: S T − $ 1, 020. The profits of the aggregate position in part (b) are the payoffs, less the future value of the call premium plus the future value of the put premium (because we have sold the put), and less the future value of the loan we gave initially. Note t h at we already know that a risk less bond is canceling out of the profit calculations. We can again tabulate: Instrument S T < K S T ≥ K Get repayment of loan $931.37 × 1.02 = $950 $931.37 × 1.02 = $950 Future value of given loan −$950 −$950 Long call o ptio n max ( S T − 950, 0) = 0 S T − 950 Future value call premium −$120.405 × 1.02 = −$122.81 −$120.405 × 1.02 = −$122.81 Short put o ption − max ($950 − S T , 0) 0 = S T − $950 Future value put premium $51.777 × 1.02 = $52.81 $51.777 × 1.02 = $52.81 Total S T − 1020 S T − 1020 Indeed, we see that the profits for part s (a) and (b) are identical as well. Question 3.7 Let us first consider the payoff of (a). If we short the index (let us name it S), we have to pay at th e time of expiration T of the options: − S T . The payoffs of part (b) are more complicated. Let us look again at each region separately, and hope to be able to draw a conclusion when we look at the aggregate position. We have for the payoffs in (b): Instrument S T < K S T ≥ K Make repayment of loan − $1 , 029 . 41 × 1 . 02 = − $1 , 050 − $1 , 029 . 41 × 1 . 02 = − $1 , 050 Short call o ption − max ( S T − 1 , 050 , 0 ) = 0 − max ( S T − 1 , 050 , 0 ) = 1 , 050 − S T Long put o ption max ($1 , 050 − S T , 0) 0 = $1 , 050 − S T Total − S T − S T

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