Solution Manual For Derivatives Markets, 3rd Edition

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Chapter 1Introduction to DerivativesQuestion1.1This problem offers different scenarios in which some companies may have an interest to hedgetheir exposure to temperatures that are detrimental to their business. In answering the problem, itis 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 dislikesdaysat which it is abnormally coldbecause people are likely to drink less, and her businesssuffers. She will be interested in a cooling-degree-day futures contractbecause it will makepayments 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 herbusiness if it does not snowin the beginning of the seasonor if the snow is melting too fast atthe end of the season. She will be interested in a heating-degree-day futures contractbecauseit will make payments when her usual business suffers, thus compensating the losses.c)During the summer months, an electric utility company, such as one in the south of theUnited States, will sell a lot of energy during days of excessive heatbecause people will usetheir air conditioners, refrigerators,and fans more often, thus consuming a lot of energy andincreasing profits for the utility company. In this scenario, the utility company will have lessbusiness during relatively colder days, and the cooling-degree-day futures offers a possibilityto hedge such risk.Alternatively, we may think of a utility provider in the northeast during the winter months, aregion where people use many additional electric heaters. This utility provider will makemore money during unusually cold daysand may be interested in a heating-degree-daycontractbecause that contract pays off if the primary business suffers.d)An amusement park operatorfears bad weather and cold daysbecause people will abstainfrom going to the amusement park during cold days. She will buy a cooling-degree-dayfutureto offset her losses from ticket sales with gains from the futures contract.Question1.2A variety of counterparties are imaginable. For one, we could think about speculators who havedifferences in opinion and who do not believe that we will have excessive temperature variationsduring the life of the futures contracts. Thus, they are willing to take the opposing side, receivinga payoff if the weather is stable.Alternatively, there may be opposing hedging needs: Compare the ski-resort operator and thesoftdrink manufacturer. The cooling-degree-day futures contract will pay off if the weather isrelativelymild, and we saw that the resort operator will buy the futures contract. The buyer of

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2Chapter 1/Introduction to Derivativesthe cooling-degree-day futures will make a loss if the weather is cold(which means that theseller of the contract will make a gain). Since the soft drink manufacturer wants additionalmoney if it is cold, she may be interested in taking the opposite side of the cooling-degree-dayfutures.Question1.3a)Rememberthattheterminologybidandaskisformulatedfromthemarketmakersperspective. 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 = $4,125.00b)Similarly, you can sell at the market maker’s bid price. You will again have to pay acommission, and your broker will deduct the commission from the sales price of the shares.You receive:($40.95 × 100)− $20 = $4,075.00c)Your round-trip transaction costs amount to:$4,125.00 − $4,075.00 = $50Question1.4In this problem, the brokerage fee is variableand depends on the actual dollar amount of thesale/purchase of the shares. The conceptof the transaction cost remains the same: If you buy theshares, the commission is added to the amount you owe, and if you sell the shares, thecommission is deducted from the proceeds of the sale.a)($41.05 × 100)+($41.05 × 100)× 0.003= $4,117.315= $4,117.32b)($40.95 × 100)−($40.95 × 100)× 0.003= $4,082.715= $4,082.72c)$4,117.32 − $4,082.72= $34.6The variable(or proportional)brokerage fee is advantageous to us. Our round-trip transactionfees are reduced by $15.40.Question1.5In answering this question,it isimportant to remember that the market maker provides a serviceto the market. He stands ready to buy shares into his inventory and sell shares out of hisinventorythus providing immediacy to the market. He is remunerated for this service by earningthe bid-ask spread. The market maker buys the security at a price of $100,and he sells it at a

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Chapter 1/Introduction to Derivatives3price of $100.10. If he buys 100 shares of the security and immediately sells them to anotherparty, he is earns a spread of:100×($100.12− $100)= 100 × $0.12 = $12.00Question1.6A short sale of XYZ entails borrowing shares of XYZ and then selling themandreceiving cash.Therefore, initially, we will receive the proceeds from the sale of the assetless the proportionalcommission charge:300×($30.19)− 300 ×($30.19)× 0.005 = $9,057 × 0.995= $9,011.72When we close out the position, we will again incur the commission charge, which is added tothe purchasing cost:300×($29.87)+ 300 ×($29.87)× 0.005 = $8,961 × 1.005= $9,005.81Finally, we subtract the cost of covering the short position from our initial proceeds to receivetotal profits: $9,011.72 − $9,005.81 = $5.91. We can see thatpaying the commission chargetwice significantly reduces the profits we can make.Question1.7a)A short sale of JKI stock entails borrowing shares of JKI,then selling themandreceivingcash, and we learned that we sell assets at the bid price. Therefore, initially, we will receivethe 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 willalways buy at the ask. Therefore, we earn the following profit:400×($25.125)− 400 ×($23.0625)= $10,050 − $9,225.00= $825.00b)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])= $10,050 × 0.997 − $9,225 × 1.003= $10,019.85 − $9,252.675= $767.175.

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4Chapter 1/Introduction to Derivativesc)The proceeds from short salesminus the commission charge are $10,019.85(or $10,050 ifyou ignore the commission charge). Since thesix-month interest rate is givenand the periodof 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 $10,019.85:$10,019.85 ×(0.03)= $300.5955= $300.60orwithout taking into account the commission charge:$10,050.00 ×(0.03)= $301.50.Question1.8We learned from the main text that short selling is equivalent to borrowing moneyand that ashort 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 calledthe short rebate in the stock market. Usually, the short rebate is close to the prevailing marketinterest 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 canearn a nice profit in the form of the difference between the current market interest rate and theshort rebate.By signing an agreement as mentioned in the problem, you give your brokerage firm thepossibility to act as an equity lender, using the shares of your account. Brokers want you to signsuch an agreement because they can make additional profits.Question1.9If we borrow an asset from a lender, we have the obligation to make any payments to the lenderthat she is entitled to as a stockholder.Becausethe lender is entitled to the dividend on thedaythe stock goes ex dividendbut doesnot receive it from the companybecause we have sold herstock, 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 ofthe 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 theadditional required money.An unexpected increase of the dividendcould be a signthat the company is doing well.Empirically, we observe sharp priceincreases after such announcements. As we have a shortposition in the stock, we make money if the stock price falls. Therefore, an unexpected increasein the dividend is very bad for our position, and we should care!

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Chapter 1/Introduction to Derivatives5Question1.10The main textbook explains short selling as follows. “The sale of a stock you do not already ownis called a short-sale.”Short positions are those resulting from short sales. At NASDAQ forexample, each financial firm is required to report its overall short positions in all accounts inNASDAQ-listed securities twice a month. The short positions from these reports are aggregatedand the resulting number is called short interest in NASDAQ stocks. The short interest statisticsare widely observed,and some financial market participants believe that short interest is a goodmeasure of market sentiment.YoucaneasilyaccessstatisticsonshortinterestonindividualstocksonNasdaq(www.nasdaq.com). Once you are on the home page, go to the pull-down menu “Quotes andResearch,”then go to “Short Interest” in the folder “Company Profile.”Once you are on thatpage, 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 aparticularlyhigh short interest.It is theoretically possible to have short interest of more than 100 percent because some marketparticipants (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).Question1.11We are interested in borrowing the asset “money.” Therefore, we go to an owner(orif youprefer,toa collector)of the asset, called Bank. The Bank provides the $100 of the asset moneyin digital form by increasing our bank account. We sell the digital money by going to the ATMand withdrawing cash. After 90 days, we buy back the digital money for $102by depositing cashinto our bank account. The lender is repaid, and we have covered our short position.Question1.12We are interested in borrowing the asset “money” to buy a house. Therefore, we go to an ownerof the asset, called Bank. The Bank provides the dollar amount, say $250,000,in digital form inour mortgage account. As $250,000 is a large amount of money, the bank is subject to substantialcredit risk(e.g., we may lose our job)and demands acollateral.Although the money itself isnot subject to large variations in price(besides inflation risk, it is difficult to imagine a reason formoney to vary in value), the Bank knows that we want to buy a house, and real estate prices varysubstantially. Therefore, the Bank wants more collateral than the $250,000 they are lending.

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6Chapter 1/Introduction to DerivativesIn fact, as the Bank is only lending up to 80percentof the value of the house, we could get amortgage of $250,000 for a house that is worth $250,000 ÷ 0.8 = $312,500. We see that the bankfactored in ahaircutof $312,500 − $250,000 = $62,500 to protect itself from credit risk andadverse fluctuations in property prices.We buy back the asset money over a long horizon of time by reducing our mortgage throughannuity payments.Question1.13For example, go to the websiteof the CME group (www.cmegroup.com). Under the tab“Products & Trading,”you can choose the asset class you are interested in. If you click, forexample, on “Equity Index,”you will come to a product listing of all equity products CMEoffers. Once you click on any particular product, say E-mini S&P 500 futures, you get to aseparate page with the latestquotes of this product. If you click on the link “ContractSpecifications,”you will see all details regarding ticker symbol, contract size, tick size, tradedmonths, settlement, etc. of the product.CME group has a Daily Volume and Open Interest Report that summarizes exchange-widevolume, including futures and options volume. You can access it by clicking on the “MarketData” taband choosing the volume report.If the notional values were cut in half, one would expect trading volume and open interest todouble but the notional value traded to stay the same.Question 1.14We havethe followingfor thedifferent trades and different dealers:a)ABCTrade 1+5–5Trade 2+15–15Trade 3+10–10Trade 4–20+20Total0+5–5Therefore, trader A’s net position is zero, trader B is long five contracts, and trader C is shortfive contracts.

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Chapter 1/Introduction to Derivatives7b)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 = $5,000Notional value of open interest is equal to 5 contracts×$ 100/contract = $500.c)Now all traders have a net position of zero. Trading volume has increased by five contracts to55. The notional value of trading volume is equal to 55×$100 = $5,500. Open interest andthe notional value of open interest are equal to zero.d)Trading volume and open interest would beidentical; the notional values of trading volumeand open interest would be lower.

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Chapter 2An Introduction to Forwards and OptionsQuestion 2.1The payoff diagram of the stock is just a graph of the stock price as a function of thestock price:In order to obtain the profit diagram at expiration, we have to finance the initialinvestment. 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, ofthe bond to be repaid, and of the sum of the two positions, which is the profit graph. Thearrows show that at a stock price of $55,the profit at expiration is indeed zero.

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Chapter 2/An Introduction to Forwards and Options9Question 2.2Since we sold the stock initially, our payoff at expiration from being short the stock isnegative.

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10Part One/Insurance, Hedging, and Simple StrategiesIn order to obtain the profit diagram at expiration, we have to lend out the money wereceived from the short sale of the stock. We do so by buying a bond for $50. After oneyear,we receive from the investment in the bond: $50 × (1 + 0.1) = $55. The secondfigure shows the graph of the sold stock, of the money we receive from the investment inthe bond, and of the sum of the two positions, which is the profit graph. The arrows showthat at a stock price of $55,the profit at expiration is indeed zero.Question 2.3The position that is the opposite of a purchased call is a written call. A seller of a calloptionis said to be the option writeror to have a short position. The call option writer isthe counterparty to the option buyer, and his payoffs and profits are just the opposite ofthose of the call option buyer.Similarly, the position that is the opposite of a purchased put option is a written putoption. Again, the payoff and profit for a written put are just the opposite of those of thepurchased put.It is important to note that the opposite of a purchased call isNOTthe purchased put. Ifyou do not see why, please draw a payoff diagram with a purchased call and a purchasedput.

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Chapter 2/An Introduction to Forwards and Options11Question 2.4a)The payoff to a long forward at expiration is equal to:Payoff to long forward = Spot price at expiration–forward priceTherefore, we can construct the following table:Price of asset insixmonthsAgreed forward pricePayoff to the long forward4050−104550−55050055505605010b)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 insixmonthsStrike pricePayoff to the call option40500455005050055505605010c)If we compare the two contracts, we immediately see that the call option has aprotection for adverse movements in the price of the asset: If the spot price is below$50,the buyer of the call option can walk awayand need not incur a loss. The buyerof the long forward incurs a loss,buthe has the same payoff as the buyer of the calloption if the spot price is above $50. Therefore, the call option should be moreexpensive. It is this attractive option to walk away that we have to pay for.

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12Part One/Insurance, Hedging, and Simple StrategiesQuestion 2.5a)The payoff to a short forward at expiration is equal to:Payoff to short forward = forward price–spot price at expirationTherefore, we can construct the following table:Price of asset in six monthsAgreed forward pricePayoff to the short forward40501045505505005550−56050−10b)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 constructthe following table:Price of asset insixmonthsStrike pricePayoff to the call option40501045505505005550060500c)The same logic as in question 2.4 (c) applies. If we compare the two contracts, we seethat the put option has a protection for increases in the price of the asset: If the spotprice is above $50,the buyerof the put option can walk awayand need not incur aloss. 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 isbelow $50. Therefore, the put option should be more expensive. It is this attractiveoption to walk away if things are not as we want that we have to pay for.Question 2.6We need to solve the following equation to determine the effective annual interest rate:$91 ×(1+r) = $100. We obtainr= 0.0989,which means that the effective annualinterest rate is approximately 9.9percent.

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Chapter 2/An Introduction to Forwards and Options13Remember that when we drew profit diagrams for the forward or call option, we drew thepayoff on the vertical axis, and the index price at the expiration of the contract on thehorizontal axis. In this case, the particularity is that the default-free,zero-coupon bondwill pay exactly $100,no matter what the stock price is. Therefore, the payoff diagram isjust a horizontal line, intersecting they-axis at $100.The textbook provides the answer to the question concerning the profit diagram in thesection “Zero-Coupon Bonds in Payoff and Profit Diagrams.” When we were calculatingprofits, we saw that we had to find the future value of the initial investment. In this case,our initial investmentis $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 isdefault-freeand does not bear coupons, the effective annual interest rate is exactly the 9.9percentwe have calculated before. Therefore, the future value of $91 is $91 × (1 +0.0989) = $100,and our profit in six months is zero!Question 2.7a)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 2.1that in order to obtain the profit diagram at expiration ofa purchase of XYZ stock, we have to finance the initial investment. We did so byselling a bond for $50. After one year,we had to pay back: $50 × (1 + 0.1) = $55.Therefore, our total profit at expiration from the purchase of a stock that was financedby a loan was: $ST− $55,whereSTis 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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14Part One/Insurance, Hedging, and Simple Strategiesour long forward contract,and bothpositions do not require any initial cash—butthen, 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 forwardcontract is not entitled to receive the dividendbecause she only has a claim to buy thestock in the future for a given price, but she does not own it yet. Therefore, it doesmatternow whether we ownthe stock or the long forward contract. Becauseeverything 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 dividendif we own the stockat the ex-dividend date. This question hints at the very important fact that we have tobe careful to take into account all the benefits and costs of an asset when we try tocompare prices. We will encounter similar problems in later chapters.Question 2.8We saw in question 2.7(b) that there is no advantage in buying either the stock or theforward contract if we can borrow to buy a stock today (soboth strategies do not requireany initial cash) and if the profit from this strategy is the same as the profit of a longforward contract. The profit of a long forward contract with a price for delivery of $53 isequal to: $ST− $53,whereSTis the (unknown) value of one share of XYZ at expirationof the forward contract in one year. If we borrow $50 today to buy one share of XYZstock (that costs $50), we have to repay in one year:$50 × (1 +r). Our total profit in oneyear from borrowing to buy one share of XYZ is therefore:$ST− $50 × (1 +r). Now wecan equate the two profit equations and solve for the interest rater:$ST− $53= $ST− $50 × (1 +r)⇔$53= $50 × (1 +r)⇔$531$50=r⇔r= 0.06Therefore, theone-year effective interest rate that is consistent with no advantage toeither buying the stock or forward contract is 6 percent.Question 2.9a)If the forward price is $1,100,then the buyer of the one-year forward contractreceives at expiration after one year a profit of: $ST− $1,100,whereSTis the(unknown) value of the S&Rindex 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 theindex today, so that we do not need any initial cash. If we borrow $1,000 today to buythe 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 istherefore: $ST− $1,100. The profits from the two strategies are identical.

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Chapter 2/An Introduction to Forwards and Options15b)The forward price of $1,200 is worse for us if we want tobuy a forward contract. Tounderstand this, suppose the index after one year is $1,150. While we have alreadymade money in part a) with a forward price of $1,100,we are still losing $50 with thenew price of $1,200. As there was no advantage in buying eitherthestock or forwardat a price of $1,100,we now need to be “bribed” to enter into the forward contract.We somehow need to find an equation that makes the two strategies comparableagain. Suppose that we lend some money initially together with enteringinto theforward contract so that we will receive $100 after one year. Then, the payoff fromour modified forward strategy is: $ST− $1,200 + $100 = $ST− $1,100,which equalsthe payoff of the “borrow to buy index” strategy. We have found the future value ofthe premium somebody needs us to pay. We still need to find out what the premiumwe will receive in one year is worth today.We 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 noadvantage in buying either stock or forward at a price of $1,100,we now need to“bribe” someone to sell this advantageous forward contract to us. We somehow needto find an equation that makes the two strategies comparable again. Suppose that weborrow some money initially together with entering into the forward contract so thatwe will have to pay back $100 after one year. Then, the payoff from our modifiedforward strategy is: $ST− $1,000 − $100 = $ST− $1,100,which equals the payoff ofthe “borrow to buy index” strategy. We have found the future value of the premiumweneed to pay. We still need to find out what this premium we have to pay in oneyear is worth today. We simply need to discount it: $100/(1 + 0.10) = $90.91. Weshould be willing to pay $90.91 to enter into the one-year forward contract with aforward price of $1,000.Question 2.10a)Figure 2.6depicts the profit from a long call option on the S&R index withsixmonths to expiration and a strike price of $1,000 if the future price of the optionpremium is $95.68. The profit of the long call option is:max[0,ST− $1,000]− $95.68⇔max[−$95.68,ST− $1,095.68]whereSTis the (unknown) value of the S&R index at expiration of the call option insix months. In order to find the S&R index price at which the call option diagramintersects thex-axis, we have to setthe above equation equal to zero. We get:ST−$1,095.68 = 0⇔ST= $1,095.68. This is the only solution, as the other part of themaximum function, −$95.68,is always less than zero.b)The profit of thesixmonth forward contract with a forward price of$1,020 is: $ST−$1,020. In order to find the S&R index price at which the call option and the forwardcontract have the same profit, we need to set both parts of the maximum function ofthe profit of the call option equal to the profit of the forward contract and see whichpart permits a solution. First, we see immediately that $ST− $1,020 = $ST− $1,095.68

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16Part One/Insurance, Hedging, and Simple Strategiesdoes not have a solution. But we can solve the other leg:$ST− $1,020 = −$95.68⇔ST= $924.32,which is the value given in the exercise.Question 2.11a)Figure 2.8depicts the profit from a long put option on the S&R index withsixmonthsto 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 −ST]− $75.68⇔max[−$75.68,$924.32 −ST]whereSTis the (unknown) value of the S&R index at expiration of the put option insix months. In order to find the S&R index price at which the put option diagramintersects thex-axis, we have to set the above equation equal to zero. We get: $924.32−ST= 0⇔ST= $924.32. This is the only solution, as the other part of the maximumfunction, −$75.68,is always less than zero.b)The profit of the shortsix-month forward contract with a forward price of $1,020 is:$1,020 −$ST. In order to find the S&R index price at which the put option and thesold forward contract have the same profit, we need to set both parts of the maximumfunction of the profit of the put option equal to the profit of the forward contract andsee which part permits a solution. First, we see immediately that $1,020 − $ST=$924.32 − $STdoes not have a solution. But we can solve the other leg: $1,020 −ST=−$75.68⇔ST= $1,095.68,which is the value given in the exercise.Question 2.12a)Long ForwardThe maximum loss occurs if the stock price at expiration iszero (the stockpricecannot be less than zerobecause companies have limited liability). The forward thenpays 0–Forward price. The maximum gain is unlimited. The stock price at expirationcould theoretically grow to infinity;there is no bound. We make a lot of money if thestock price grows to infinity (or to a very large amount).b)Short ForwardThe profit for a short forward contract is forward price–stock price at expiration.The maximum loss occurs if the stock price rises sharply;there is no bound to it, so itcould grow to infinity. The maximum gain occurs if the stock price is zero.c)Long CallWe will not exercise the call option if the stock price at expiration is less than thestrike price. Consequently, the only thing we lose is the future value of the premiumwe paid initially to buy the option. As the stock price can grow very large (andwithout bound), and our payoff grows linearly in the terminal stock price once it ishigher than the strike, there is no limit to our gain.

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Chapter 2/An Introduction to Forwards and Options17d)Short CallWe have no control over the exercise decision when we write a call. The buyer of thecall option decides whether to exerciseitor not, and he will only exercisethe callifhe makes a profit. As we have the opposite side, we will never make any money atthe expiration of the call option. Our profit is restricted to the future value of thepremium, and we make this maximum profit whenever the stock price at expiration issmaller than the strike price. However, the stock price at expiration can be very largeand has no bound, and as our loss grows linearly in the terminal stock price, there isno limit to our loss.e)Long PutWe will not exercise the put option if the stock price at expiration is larger than thestrike price. Consequently, the only thing we lose whenever the terminal stock priceis larger than the strike is the future value of the premium we paid initially to buy theoption. We will profit from a decline in the stock prices. However, stock pricescannot be smaller than zero, so our maximum gain is restricted to strike price less thefuture value of the premium,and it occurs at a terminal stock price of zero.f)Short PutWe have no control over the exercise decision when we write a put. The buyer of theput option decides whether to exercise or not, and he will only exercise if he makes aprofit. As we have the opposite side, we will never make any money at the expirationof the put option. Our profit is restricted to the future value of the premium, and wemake this maximum profit whenever the stock price at expiration is greater than thestrike price. However, we lose money whenever the stock price is smaller than thestrike;hence,the largest loss occurs when the stock price attains its smallest possiblevalue, zero. We lose the strike price because somebody sells us an asset for the strikethat is worth nothing. We are only compensated by the future value of the premiumwe received.Question 2.13a)In order to be able to draw profit diagrams, we need to find the future values of thecall premia. They are:i)35-strike call: $9.12 × (1 + 0.08) = $9.8496ii)40-strike call: $6.22 × (1 + 0.08) = $6.7176iii)45-strike call: $4.08 × (1 + 0.08) = $4.4064

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18Part One/Insurance, Hedging, and Simple StrategiesWe can now graph the payoff and profit diagrams for the call options. The payoffdiagram 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 Options19b)Intuitively, whenever the 45-strike option pays off (i.e., has a payoff bigger thanzero), the40-strike and the 35-strike options pay off. However, there are someinstances in which the 40-strike option pays off and the 45-strike optiondoes not.Similarly, there are some instances in which the35-strike option pays offbutneitherthe 40-strike nor the 45-strike pay off. Therefore, the 35-strike offers more potentialthan the 40-and 45-strike, and the 40-strike offers more potential than the45-strike.We pay for these additional payoff possibilities by initially paying a higher premium.Question 2.14In order to be able to draw profit diagrams, we need to find the future values of the putpremia. They are:a)35-strike put: $1.53 × (1 + 0.08) = $1.6524b)40-strike put: $3.26 × (1 + 0.08) = $3.5208c)45-strike put: $5.75 × (1 + 0.08) = $6.21We get the following payoff diagrams:

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20Part One/Insurance, Hedging, and Simple StrategiesWe get the profit diagram by deducting the option premia from the payoff graphs. Theprofit diagram looks as follows:Intuitively, whenever the 35-strike put option pays off (i.e., has a payoff bigger thanzero), the40-strike and the 35-strike options also pay off. However, there are someinstances in which the40-strike option pays off and the 35-strike optiondoes not.Similarly, there are some instances in which the 45-strike option pays offbutneither the40-strike nor the 35-strike pay off. Therefore, the45-strike offers more potential than the40-and 35-strike, and the 40-strike offers more potential than the 35-strike. We pay forthese additional payoff possibilities by initially paying a higher premium. It makes sensethat the premium is increasing in the strike price.Question 2.15The nice thing that lead us to the notion of indifference between a forward contract and aloan-financed stock index purchase whenever the forward price equaled the future priceof the loan was that we could already tell today what we had to pay back in the future. Inother words, the return on the loan, the risk-free interest rater, was known today, and weremoved uncertainty about the payment to be made. If we were to finance the purchase ofthe index by short selling IBM stock, we would introduce additional uncertaintybecausethe future value of the IBM stock is unknown. Therefore, we could not calculate todaythe amount to be repaid, and it would be impossible to establish an equivalence betweenthe forward and loan-financed index purchase today. The calculation of a profit diagramwould only be possible if we assumed an arbitrary value for IBM at expiration of thefutures, and we would have to draw many profit diagrams with different values for IBMto get an idea of the many possible profits we could make.

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Chapter 2/An Introduction to Forwards and Options21Question 2.16The following is a copy of a spreadsheet that solves the problem:QuickTime™ and adecompressorare needed to see this picture.

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Chapter 3Insurance, Collars, and Other StrategiesQuestion3.1This question is a direct application of the Put-Call-Parity [equation (3.1)]of the textbook.Mimicking Table 3.1., we have:S&R IndexS&R PutLoanPayoff−(Cost + Interest)Profit900.00100.00−1,000.000.00−95.68−95.68950.0050.00−1,000.000.00−95.68−95.681,000.000.00−1,000.000.00−95.68−95.681,050.000.00−1,000.0050.00−95.68−45.681,100.000.00−1,000.00100.00−95.684.321,150.000.00−1,000.00150.00−95.6854.321,200.000.00−1,000.00200.00−95.68104.32The payoff diagram looks as follows:

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Chapter 3/Insurance, Collars, and Other Strategies23We can see from the table and from the payoff diagram that we have in fact reproduced a callwith the instruments given in the exercise. The profit diagrambelowconfirmsthis hypothesis.Question3.2This question constructs a position that is the opposite to the position of Table 3.1. Therefore, weshould get the exact opposite results from Table 3.1. and the associated figures. Mimicking Table3.1., we indeed have:S&R IndexS&R PutPayoff−(Cost + Interest)Profit−900.00−100.00−1,000.001,095.6895.68−950.00−50.00−1,000.001,095.6895.68−1,000.000.00−1,000.001,095.6895.68−1,050.000.00−1,050.001,095.6845.68−1,100.000.00−1,100.001,095.68−4.32−1,150.000.00−1,150.001,095.68−54.32−1,200.000.00−1,200.001,095.68−104.32On the next page, we see the corresponding payoff and profit diagrams. Please note that theymatch the combined payoff and profit diagrams of Figure 3.5. Only the axes have differentscales.

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24Part One/Insurance, Hedging, and Simple StrategiesPayoffdiagram:Profit diagram:

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Chapter 3/Insurance, Collars, and Other Strategies25Question3.3In 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) = $52.81,the call premium:$120.405 × (1 + 0.02) = $122.81,andthezero-coupon bond:$931.37 × (1 + 0.02) = $950.00Now, 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 looksexactly like a certain payoff of $950,plus a call with a strike price of 950. But this is thealternative given to us in the question. We have thus confirmed the equivalence of the twocombined positions for the payoff diagrams. The profit diagrams on the next page confirm theequivalence of the two positions (which is again an application of the Put-Call-Parity).

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26Part One/Insurance, Hedging, and Simple StrategiesProfitdiagram for a long 950-strike put and a long index combined:Question3.4This question is another application of Put-Call-Parity. Initially, we have the following cost toenter into the combined position: We receive $1,000 from the short sale of the index, and wehave to pay thecall premium. Therefore, the future value of our cost is: ($120.405 − $1,000) ×(1+ 0.02) =−$897.19. Note that a negative cost means that we initially have an inflow ofmoney.Now, we can directly proceed to draw the payoff diagram:

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Chapter 3/Insurance, Collars, and Other Strategies27We can clearly see from the figure that the payoff graph of the short index and the long calllooks like a fixed obligation of $950,which is alleviated by a long put position with a strike priceof 950. The following profit diagram, including the cost for the combined position, confirms this:Question3.5This question is another application of Put-Call-Parity. Initially, we have the following cost toenter into the combined position: We receive $1,000 from the short sale of the index, and wehave to pay thecall 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 ofmoney.

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28Part One/Insurance, Hedging, and Simple StrategiesNow, 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 exercise, weneed tofind 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 calllooks like a fixed obligation of $1,050,which is exactly the future value of the borrowed amount,and a long put position with a strike price of1,050. The following profit diagram, including thecost for the combined position we calculated above, confirms this. The profit diagram is thesame as the profit diagram for a loan and a long1,050-strike put with an initial premium of$101.214.

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Chapter 3/Insurance, Collars, and Other Strategies29Profitdiagram of going short the index and buying a1,050-strike call:Question3.6We now move from a graphical representation and verification ofthe Put-Call-Parity to amathematical representation. Let us first consider the payoff of (a). If we buy the index (let usname it S), we receive at the time of expiration T of the options simplyST.The payoffs of part (b) are a little bit more complicated. If we deal with options and themaximum function, it is convenient to split the future values of the index into two regions: onewhereST<Kand another one whereST≥K. We then look at each region separately, and hopeto be able to drawaconclusion when we look at the aggregate position.We have for the payoffs in (b):InstrumentST< K=950ST≥K=950Get repayment of loan$931.37×1.02=$950$931.37×1.02=$950Long call optionmax(ST−950,0)=0ST−950Short put option−max ($950−ST,0)0=ST−$950TotalSTST

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30Part One/Insurance, Hedging, and Simple StrategiesWe now see that the total aggregate position only gives usST, no matter what the final indexvalue is—but this is the same payoff as in part (a). Our proof for the payoff equivalence iscomplete.Now let us turn to the profits. If we buy the index today, we need to finance it. Therefore, weborrow$1,000and have to repay $1,020 after one year. The profit for part (a) is thus:ST−$1,020.The profits of the aggregate position in part (b) are the payoffs, less the future value of the callpremium plus the future value of the put premium (because we have sold the put), and less thefuture value of the loan we gave initially. Note that we already know that a riskless bond iscanceling out of the profit calculations. We can again tabulate:InstrumentST<KST≥KGet repayment of loan$931.37 × 1.02 = $950$931.37 × 1.02 = $950Future value of given loan−$950−$950Long call optionmax (ST− 950, 0) = 0ST− 950Future value call premium−$120.405 × 1.02 = −$122.81−$120.405 × 1.02 = −$122.81Short put option− max ($950 −ST,0)0=ST− $950Future value put premium$51.777 × 1.02 = $52.81$51.777 × 1.02 = $52.81TotalST− 1020ST− 1020Indeed, we see that the profits for parts(a) and (b) are identical as well.Question3.7Let us first consider the payoff of (a). If we short the index (let us name it S), we have to pay atthe timeof expiration T of the options: −ST.The payoffs of part (b) are more complicated. Let us look again at each region separately, andhope to be able to draw a conclusion when we look at the aggregate position.We have for the payoffs in (b):InstrumentST< KST≥KMake repayment of loan−$1,029.41×1.02= −$1,050−$1,029.41×1.02= −$1,050Short call option−max(ST−1,050,0)=0−max(ST−1,050,0)= 1,050 −STLong put optionmax ($1,050 −ST,0)0= $1,050 −STTotal−ST−ST

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