Options and Futures
Options and Futures FIN 444
Cal State Fullerton
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CHAPTER 11 FORWARD AND FUTURES HEDGIN G SPREAD AND TARGET STRATEGIES ENDOFCHAPTER QUESTIONS AND PROBLEMS 1 Short Hedge and Long Hedge The terms short and long refer to the position taken in the futures contract A short long hedge means that you are short long futures Since a hedge implies opposite positions in the spot and futures markets a short long hedge means that you are long short in the spot market 2 The Basis a The basis is defined as the difference between the spot price and the futures price b At expiration the spot price must equal the futures price give or take a small differential for transaction costs Therefore over the life of the contract the spot and futures prices will converge and the basis will go to zero at expiration c The basis is the difference between the spot price and the futures price If the basis is positive and strengthens the spot price increases more or decreases less than the futures price or the spot price goes up and the futures price goes down Since a short hedge is long the spot and short the futures this is beneficial Since the long hedge is long the futures and short the spot this hurts the long hedge 3 The Basis a The dealer is long sugar in the spot market and should sell sugar futures to set up a hedge S0 00479 f0 00550 b0 S0 7 f0 00479 7 00550 700071 75 ST SO G T fO We are not given ST but it will not matter since ST and fT will cancel So make up a value of ST say 00465 7 00465 7 00479 00465 7 00550 00071 In terms of the basis 7 7b0 bt 7 700071 0 00071 In dollars 7 11200000071 79520 Thus the profit on the hedge is 71 times the original basis times the number of pounds b b St7f 00574700590700016 7t St 7 S0 7 ft 7 f0 00574 7 00479 7 00590 7 00550 00055 In terms of the basis 7 7b0 b 7 700071 40016 00055 The basis went from 700071 to 410016 a profit of 00055 In dollars 7 11200000055 616 Chapter 11 81 EndofChapter Solutions Thus the basis strengthened so the hedger gained though not as much as if the hedge had been held to expiration 4 Contract Choice The most important factor is to have a strong correlation between the spot and futures prices It is also important that the futures contract have sufficient liquidity If the contract is not very liquid then the hedger may be unable to close the position at the appropriate time without making a significant price concession This weakens the effectiveness of the hedge by making the futures price less dependent on the spot market and the normal costofcarry relationship between the two markets In addition the contract should be correctly priced or at least priced in favor of the hedger For example a short long hedger would not want to sell buy a futures contract that was underpriced overpriced as this would reduce the hedging effectiveness 5 Contract Choice The rule of thumb is that the contract chosen should expire as soon as possible after the hedge termination date but not during the month of the hedge termination date This is because there is sometimes unusual price behavior in the expiration month resulting from a possible shortage of the deliverable good If the contract expires before the hedge is terminated the hedger will have to roll the contract into the next expiration This would incur additional transaction costs The appropriate expirations are a September b March of the next year c March of the current year d September 6 Why Hedge One reason firms hedge is because they can do it more effectively than their shareholders They are better able to assess the risks and they have lower transaction costs Of course this does not address the question of why the shareholders would want to hedge in the first place but this may be because they want to find a more acceptable combination of risk and return Firms also hedge for tax advantages to reduce the probability of bankruptcy which has many costs associated with it and also because the managers are hedging to protect their own wealth which is tied so closely to that of the firm 7 Minimum Variance Hedge Ratio a The minimum variance hedge ratio is defined by specifying the equation for the profit from a hedge consisting of one unit of the spot commodity and N futures contracts N is the number of futures contracts that minimizes the variance of the profit on the hedge The measure of hedging effectiveness is the amount of risk reduced divided by the original risk This measures the percentage of the risk in the spot position that is eliminated by the hedge b Price Sensitivity Hedge Ratio The price sensitivity formula gives a value of N that assures that the value of the overall position does not change as interest rates change The price sensitivity formula and the minimum variance hedge ratio are both risk minimizing hedge ratios The latter incorporates past information on the covariance between the spot and futures price changes while the former utilizes more current information on the sensitivity of the spot and futures prices to changes in interest rates If the past relationship between spot and futures prices holds in the future the two formulas would produce identical hedge results 8 Contract Choice The decision of whether to buy or sell futures when hedging is extremely important There are three easy approaches The first is to identify the worst outcome for an unhedged position and to assume that it will occur Then select a futures transaction that will profit if this worst outcome does happen The second approach is to identify if the current spot position is short or long Then take the opposite position in futures Chapter ll 82 EndofChapter Solutions The third approach is to identify the spot transaction that you will undertake at the end of the hedge The futures transaction at the end of the hedge will then be the opposite of this spot transaction Given this futures transaction do the opposite futures transaction today 9 Short Hedge and Long Hedge a The firm is exposed to the risk of a falling stock market Thus to profit it would need to execute a short hedge b The investor is exposed to the risk of the bond price rising Thus to profit he would need to execute a long hedge c The firm is long the currency and is exposed to the risk of a fall in the value of the currency Thus to profit it would need to execute a short hedge 10 Foreign Currency Hedges The exposure is to 100000SF225 SF22500000 The dealer would like to lock in the cost in dollars Thus he could buy 225 million Swiss francs in a forward contract at the rate of 03881 with an expiration of August 16 When the contract expires it does not matter what the spot rate is as the 225 million Swiss francs are purchased at the contract rate of 03881 Thus the total cost is 2250000003881 8732250 11 Foreign Currency Hedges The bank currently holds a long position in Canadian dollars worth 500000007564 3782000 and needs to protect against a decline in the value of the Canadian dollar Therefore it needs to sell Canadian dollar futures which are priced at 10000007541 75410 It needs 5000000100000 50 contracts Januag 2 The bank sells 50 contracts Februag 28 The 5000000 Canadian dollars are converted at a rate of 07207 for 500000007207 3603500 a decrease in value of 178500 The futures contracts are bought for 07220 or 10000007220 72200 This produces a profit on the futures of 5075410 7 72200 160500 This reduces the loss on the currency conversion to only 18000 The hedge eliminated 90 percent of the loss 12 Intermediate and LongTerm Interest Rate Hedges a The spot bonds are worth 0788755000000 3943750 Chapter 11 83 EndofChapter Solutions The futures price is 7125 The number of futures contracts is Nf 7 781832394375071250111411232 7 515 So buy 52 contracts at a price of 71250 each b The spot bonds are worth 082755000000 4137500 This is an increase of 193750 The futures price is 764375 The profit from the futures transaction is 527643750 7 71250 269750 The net profit on the hedge is 269750 7 193750 76000 13 Intermediate and LongTerm Interest Rate Hedges a The manger is long the bonds and is exposed to a fall in the price of the bonds so the appropriate transaction is to sell 13 contracts b The spot bonds are worth 1013751000000 1013750 The profit is 1013750 7 1074375 760625 The futures price is 7715625 The profit from the futures transaction is 7137715625 7 7746875 406250 The overall loss is 60625 7 406250 5656250 Note that the hedge reduced only a small portion of the loss This suggests that the relationship between the spot and futures prices obtained from the price sensitivity hedge ratio did not hold perfectly 14 Stock Market Hedges First find the portfolio beta on October 1 Stock Shares Price Value Weight Donnelly 10000 19625 19625000 00697 Goodrich 6200 31375 19452500 00691 Raytheon 15800 49375 78012500 02772 Maytag 8900 55375 49283750 01751 Kroger 11000 42125 46337500 01647 Comdisco 14500 19375 28093750 00998 Chapter 11 84 EndofChapter Solutions Chapter 11 85 Cessna 9900 2975 Foxboro 4500 2475 29452500 01047 11137500 00396 281395000 B 100006971050069111502772 09001751 08501647 14500998 120o1o47 09500396 7 1067 The futures price is 37620500 188100 N 28139501881001067 71596 Sell 16 contracts On December 31 the market value of the portfolio is 1000027375 620032875 1580053625 890077875 1100047875 1450028625 990030 125 450026 337486250 The gain on the portfolio is 337486250 7 2813950 56091250 The futures price is 42490500 212450 The profit on the futures transaction is 716212450 7 188100 7389600 The overall gain is 56091250 7 389600 17131250 The portfolio gained in value but some of the gain was offset by the loss on the futures After the fact the firm should not have hedged but of course it did not know that the market would have increased Stock Market Hedges The stock is worth 2000032875 657500 The futures price is 37530500 187650 The number of futures contracts required is N 657500187650110 385 EndofChapter Solutions Chapter 11 86 So buy 4 contracts On June 1 the stock is bought for 2000038625 772500 The increased cost of the stock is 772500 7 657500 115000 The futures price is 38730500 193650 The profit on the futures transaction is 4193650 7 187650 24000 The net additional cost of the stock is 115000 7 24000 91000 The hedge eliminated about 21 percent of the increase in the cost of the stock Intramarket Spreads Because you are bearish you sell the more volatile contract7the nearby7Which is the December and buy the March contract The outcome is July 15 Sell December at 76932 7628125 Buy March at 75932 7528125 Net price 1 November 15 Buy December at 791332 7940625 Sell March at 78932 7828125 Net price 71125 The investor sold the spread at 1 and bought it back at 1125 The result is a loss of 0 125 per 100 of face value or 125 per 100000 of face value Alpha Capture S 10000027625 2762500 f 500393 196500 N 276250019650095 1336 So sell 13 contracts On December 31 the stock is worth 10000028875 2887500 EndofChapter Solutions Chapter 11 87 The futures price is 50043230 216150 The profit on the futures contract is 713216150 7 196500 7255450 The net value of the stock on December 31 is 2887500 7 255450 2632050 Consequently the result is about a 47 percent loss This example illustrates the speculative nature of this type of transaction The market increased by about 10 percent39 however the stock increased by only about 45 percent Thus its beta was really only about 045 instead of 095 Because the beta was not accurately predicted the transaction was unable to do what it was supposed to do7remove the systematic risk leaving the alpha However even if the beta were accurately measured the actual alpha might not be the expected 10 percent Target Duration with Bond Futures a To lower duration you must sell futures 5833 9448456 112 Nf 5165 843 72094 11225 or 52 contracts b Profit on spot 8952597 7 9448456 7495859 Profit on futures 75268500 7 72094 186 888 7308971 Target Beta with Stock Index Futures a To lower the beta you must sell futures Nf l3 T B sXSO 1 7 1151050000042575500 7740 so sell 7 contracts b Profit on spot 7 9870000 7 10500000 7 7630000 Profit on futures 7 7740235 7 42575500 7 81 900 7548100 The portfolio return was 754810010500000 410522 The market fell 40235 7 4257542575 410550 This was an effective beta of095 Tactical Asset Allocation a To synthetically sell 5 million of domestic stock with a beta of 110 would require ND futures as follows 5000000 N 07110 7722 W E 250000 Endof Chapter Solutions In other words sell 22 contracts to reduce the beta on 5 million to zero To synthetically buy 5 million of foreign stock with a beta of 105 would require NF futures as follows N Ff 1 5000000 150000 10057 0 35 In other words buy 35 contracts to increase the beta on 5 million to 105 b The domestic stock futures price goes to 2500001018 254500 The profit is 722254500 7 250000 799000 The foreign stock futures price goes to l50000l014 152100 The profit is 35152100 7 150000 73500 The value of the domestic stock goes up to 20000000102 20400000 The total value of the portfolio is 20400000 7 99000 73500 20400000 21 Stock Market Hedges The transaction costs to sell each group of shares are as follows Northrup Grumman 20 H I He1nz 20 Washington Post 20 Disney 20 Wang Labs 20 Wisconsin Energy 20 General Motors 20 Union Pacific 20 Royal Dutch Shell 20 Illinois Power 20 14870003 7 46610 8755003 7 28265 1245003 7 5735 8750003 7 28250 33995003 7 103985 12480003 7 39440 14750003 7 46250 12900003 7 40700 7500003 7 24500 3550003 7 12650 3763 85 To determine the number of futures needed we need the beta of the portfolio The market values of the stocks and their weights are as follows Northrup Grumman H I Heinz Wisconsin Energy General Motors Union Pacific Royal Dutch Shell Illinois Power The beta is Chapter 11 14870 18125 8755 36125 1245 264 8750 1345 33995 425 12480 29 14750 4875 12900 715 7500 7875 3550 155 Value Weight 26951875 0055 31627438 0065 32868000 0067 117687500 0241 14447875 0030 36192000 0074 71906250 0147 92235000 0189 59062500 0121 55 02500 0011 488480938 EndofChapter Solutions Chapter 11 89 B 11000551050065105006D 12502411200030 065074 7 09501471200189 0750121 7 0600011 1048 The number of futures is N 104848848093836945500 277 or 28 Cost of trading futures 282750 770 The stocks would cost 376385 plus 25 for each tbill the funds were placed in So the minimum would be 378885 The futures cost 770 or about 15 the cost of selling the stocks Hedging Strategies Current spot position is 1000950 950000 in Swiss francs worth SF95000007254SF 689130 You will need to buy Swiss franc futures because you will be hurt on the spot purchase of the stock if the franc rises You will require 950000125000 76 contracts So buy 8 at 07250 At expiration your futures profit is 807295 7 07250125000 4500 The stock will cost 10009265007301 676438 which is 12692 less Thus your overall gain is 12692 4500 17192 The Stocks performance was a loss of 1000950 7 9265007254 17047 The currency strengthened for a gain of 10009265007301 7 07254 4355 The futures hedge generated a gain of 4500 To the hedger the loss on the stock is a gain because the stock will be cheaper to buy The gain on the currency is a loss to the hedger because more dollars will be required to acquire the necessary units of currency The reason the stock price of 92650 rather than 950 is used is because that is the number of Swiss francs that will have to be acquired per share Thus the hedge39s performance consists of 17047 from the stock 41355 from the currency 4 500 from the futures 1 7 l 92 Endof Chapter Solutions CHAPTER 8 THE STRUCTURE OF FORWARD AND FUTURES MARIGETS ENDOFCHAPTER QUESTIONS AND PROBLEMS 1 Chapter 8 60 Introduction A forward contract obligates the holder of the long position to purchase the commodity at a future date A call option grants the holder of the call the right but not the obligation to purchase the commodity at a future date A put option grants the holder of the put the right but not the obligation to sell the commodity at a future date A call is more like a forward contract than a put because long positions in the two contracts are bullish The holder of the forward contract however is obligated to buy the good at the future date The holder of the call can simply let the option expire if the market price of the commodity is less than the exercise price A call holder pays a premium for the right to not exercise The holder of a long forward contract does not pay a premium and gives up the right to not exercise Organized Futures Trading While both forward and futures contracts are agreements to purchase a good at a future date a futures contract provides liquidity by having a central marketplace and standardized contract terms This allows holders of futures contracts to sell them in the market at any time prior to expiration Futures trading is governed by the formal regulations of the futures exchange Most important the losses incurred by futures traders are guaranteed by the clearinghouse which requires the daily settlement of gains and losses That is the holders of profitable contracts do not have to worry about whether their gains will be paid by the holders of losing contracts Forward contracts however are subject to default risk Forward contracts can be tailored to the unique needs of firms For example a firm may need to execute a hedge in which the expiration is a specific date Futures contracts expire only on certain dates which may not fit the needs of the firm Development of Options on Futures Markets In an option on a futures contract the underlying is a futures contract Thus if the holder exercises a call option on a futures it creates a long position in an futures contract and if the holder exercises a put option on a futures it creates a short position in a futures contract As in options on assets an option on a futures requires that the buyer pay the premium up front There are two expirations the expiration of the option and the expiration of the underlying futures though for some contracts these expirations are the same In that case exercise of the option on the futures creates a futures contract that immediately expires thereby turning into the spot asset and making the option on the futures the same as an option on the asset Daily Settlement A 1000 OL 4200 S 75200 A OL S Change in Open Interest a 500 4700 75200 none b 1700 3500 75200 none c 1200 4200 75400 increase by 200 d 200 4200 4400 decrease by 800 If A trades with OL one or the other is merely offsetting and thus open interest does not change If A trades with the shorts both are reducing or increasing their positions so open interest changes In other words if traders trade with others who hold the same positions open interest will not change If they trade with those holding opposite positions open interest will change Organized Futures Trading a A centralized trading facility The exchange is a formal market place for trading the contracts b Standardized terms This establishes that certain contracts are identical and thus are perfect substitutes for each other EndofChapter Solutions Chapter 8 61 c Rules The exchange establishes rules and regulations that permit trading to transpire in an orderly manner d Clearinghouse The clearinghouse associated with the exchange provides a guarantee that each party to the contract will perform as expected The clearinghouse also provides the bookkeeping system that keeps tract of the transactions and the margin deposits e Contract development The exchange continuously monitors economic conditions and develops new contracts designed to meet the changing needs of hedgers and speculators General Classes of Futures Traders Locals are in business for themselves They attempt to profit by buying at low prices and selling at high prices In so doing they provide liquidity to the public Commission brokers simply execute transactions for other parties who do not have access to the trading floor They make their income by the commissions they receive on each transaction A futures commission merchant is a firm that solicits public orders It may have a commission broker on the floor of the exchange or it may engage an independent broker to execute its trades Mechanics of Futures Trading An open outcry system involves traders on the floor of an exchange who call out bids and offers On an electronic system traders are off the floor of the exchange and communicate their bids and offers by a computer link In addition some electronic systems actually have the computer match bids with offers Classi cation by Trading Strategy If a position in the futures market is accompanied by an opposite position in the spot market the transaction is a hedge The hedger does not necessarily have to have a long or short position in the spot market A hedge can be established if the hedger is reasonably certain of taking a future position in the spot market The hedge protects against price changes in the interim period until the spot transaction is made A speculative strategy is not normally accompanied by a transaction or contemplated transaction in the spot market Classi cation by Trading Strategy A spread strategy is a long position in one futures contract and a short position in anot er futures contract The prices of the two contracts are normally highly correlated so that the gains on one contract are at least partially offset by the losses on the other The objective is to take a small amount of risk in the hope of a small profit An arbitrage strategy involves a near riskless transaction in one or more futures contracts and possibly a spot transaction Arbitrage trading is usually triggered by a deviation from the theoretical relationship between the prices of two instruments Both transactions can be viewed as hedges A hedge is a position in the spot market and an opposite position in the futures market Thus it is similar to a spread in that the gain on one position is at least partially offset by the loss on the other Arbitrage is like hedging in that it is designed to have low risk and it often involves a position in the spot market and an opposite position in the futures market General Classes of Futures Traders These three types of futures traders differ primarily in the length of time they hold their positions Scalpers attempt to profit from small changes in the price of the contract They hold their positions for very short time intervals sometimes less than a minute Day traders usually hold their positions for less than a day Near the end of the trading day they close out their positions so that they have no open positions overnight Position traders hold their transactions open for different lengths of time This could be several days or weeks They attempt to profit by capitalizing on trends that typically last longer than a day Costs and Pro tability of Exchange Membership An individual can buy a full membership which provides the right to go onto the trading floor and engage in futures transactions Some exchanges also offer limited memberships which permit trading in certain contracts only Alternatively an individual can lease a seat from another individual already owning a seat EndofChapter Solutions Chapter 8 62 Contract Terms and Conditions Daily price limits determine the maximum and minimum price at which a contract can trade during a day At the end of a given day the settlement rice plus or minus the daily price limit establishes the maximum and minimum prices for trades the following day There are rules however that relax the limits under certain conditions The purpose of daily price limits is to prevent the margin accounts from being depleted so quickly that losses cannot be covered Daily Price Limits and Trading Halts Circuit breakers are rules that restrict trading after prices have moved by a specified amount They were instituted after the crash of 1987 They are designed to permit markets to quotcool offquot and in some cases additional margin to be collected Generally the idea is that by prohibiting panic trading investors are encouraged to absorb and analyze information before trading something they are supposed to do anyway Circuit breakers have the disadvantage however of simply disguising the true equilibrium price and can actually induce more panic If a building is on fire locking the doors might reduce injuries due to trampling but would hardly put out the fire Role of the Clearinghouse The clearinghouse intervenes in each contract guaranteeing to the buyer that the seller39s losses will be covered and guaranteeing to the seller that the buyer39s losses will be covered This allows a trader to enter into a transaction without having to check the creditworthiness of the other party The clearinghouse requires that each trader maintain a margin account to cover losses The clearinghouse also maintains a cash reserve to cover losses in the event of a failure to cover a loss by a trader or firm As a last resort the clearinghouse can assess the member firms a charge to make up any losses not already covered Daily Settlement An offsetting trade means to simply take an opposite position in the same contract For example a trader who buys a gold futures contract can offset the trade by selling a gold futures contract with the same expiration month This establishes a long and short position in the same contract which is equivalent to not having a position at all A cash settlement is permitted at expiration on certain contracts The settlement price on the last day of trading is automatically equal to the spot price The account is marked tomarket on the last day and all open positions are automatically closed If the contract provides for delivery the holder of the short position must deliver the commodity to the holder of the long position who pays the futures price on that day subject to some adjustments provided in certain contracts Forward contracts are designed to be held to expiration The terms of the contract are written so as to accommodate delivery if that is the intention of the party Many forward contracts however are cash settled at expiration If the holder of a forward contract decides to terminate the position early he would simply reenter the forward market and request a new offsetting contract While this is similar to offsetting a futures contract the forward market may not necessarily have the same liquidity as it did when the contract was opened While the contract can generally be offset it may end up being very costly to offset In addition since both contracts still exist credit risk remains EndofChapter Solutions 16 Daily Settlement Date Settlement Price Settlement Price MarktoMarket Other Entries Account Balance 71 45395 226975 850 9000 9850 72 45450 227250 275 10125 73 45200 226000 71250 8875 77 44355 221775 74225 4650 78 44165 220825 7950 4350 8050 79 44285 221425 600 8650 710 44415 222075 650 9300 711 44225 221125 7950 8350 714 43830 219150 71975 6375 715 43505 217525 71625 4750 716 43550 217750 225 4250 9225 Chapter 8 63 Explanation of Other Entries Initial margin deposit of 9000 78 Balance on 77 was 4650 which is below 6000 maintenance margin Required to deposit 4350 to bring balance up to initial margin of 9000 716 Balance on 715 was 4750 which is below 6000 maintenance margin Required to deposit 4250 to bring balance up to initial margin of 9000 Daily Settlement You start off with 3375 in your account It can drop to 2500 a difference of 875 before you get a margin call The price changes in increments of 001 Since you have a contract on 1000 barrels each move of 001 is a change in the margin account balance of 1000001 10 To loss 875 there must be 87510 875 moves of 001 or a decrease in the contract price of 0875 Since each move is a minimum of 001 it must fall by 088 That would take the price from 2742 to 2742 7 088 2654 Hedge FundsManaged Funds In many respects a futures fund and a hedge fund are quite similar A hedge fund is a fund that uses futures and other derivatives to invest typically in highly risky positions A futures fund however normally does not use overthecounter derivatives whereas a hedge fund often does Hedge funds tend to be fairly secretive often registered offshore and appeal primarily to wealthy investors The hedge fund industry is much larger than the futures fund industry Regulation of Futures and Forward Markets The objective of federal regulation of the futures markets is to authorize futures exchanges to approve new contracts and modifications of existing contracts to ensure that price information is available to the public to authorize individuals to provide services related to futures trading and to oversee the markets to prevent manipulation EndofChapter Solutions Regulation of Futures and Forward Markets An industry selfregulatory organization in the futures markets exists to provide a means in which the futures industry regulates itself This takes some of the burden of regulation off of the federal government The industry provides a means of licensing participants policing its members and adjudicating disputes Transaction Costs in Forward and Futures Trading There are three types of futures trading costs commissions bidasked spreads and delivery costs All three costs in uence the profitability of various futures trading strategies Commissions paid by the public to brokers are assessed on the basis of a dollar charge per contract The commission is paid at the order s initiation and includes both the opening and closing commissions that is a roundtrip commission is charged regardless of Whether the trader ultimately closes out the contract makes or takes delivery or makes a cash settlement There are also exchange fees and NFA fees Many floor traders quote prices at which they are Willing to simultaneously buy at the bid price and sell at the ask price The bidask spread is the cost to the public of liquidityithe ability to buy and sell quickly Without a large price concession A futures trader who holds a position to delivery faces the potential for incurring a substantial delivery cost In the case of most financial instruments this cost is rather small For commodities however it is necessary to arrange for the commodity s physical transportation delivery and storage Delivery and Cash Settlement All contracts eventually expire Physical settlement requires the seller to make physical delivery of the appropriate underlying instrument Most futures contracts allow for more than one deliverable instrument The contract usually specifies that the price paid by the long to the short be adjusted to re ect a difference in the quality of the deliverable good On cashsettled contracts such as stock index futures the settlement price on the last trading day is fixed at the closing spot price of the underlying instrument such as the stock index All contracts are marked to market on that day and the positions are deemed to be closed One exception to this procedure is the Chicago Mercantile Exchange s SampP 500 futures contract which closes trading on the Thursday before the third Friday of the expiration month but bases the final settlement price on the opening stock price on Friday morning This procedure was installed to avoid some problems created when a contract settles at the closing prices APPENDIX 8B SOLUTIONS 1 2 Chapter 8 64 First year Price at year end 42240500 211200 Taxable gain 211200 7 205150 6050 Tax 605006020 605004031 147620 Second year Price when sold 42730500 213650 Taxable gain 213650 7 211200 2450 Tax 245006020 245004031 59780 Total tax on the transaction 47620 59780 2074 First year EndofChapter Solutions Taxable gain 10500 7 10000 500 Tax 50006020 50004031 122 Second year It is assumed that you bought the commodity at the price at 11200 You would have to pay tax on the accrued profit since the end of the year Taxable gain 11200 7 10500 700 Tax 70006020 70004031 17080 3 You would not pay tax on the commodity until it is sold Chapter 8 65 Endof Chapter Solutions ENDOF 1 Chapter 6 CHAPTER 6 BASIC OPTION STRATEGIES CHAPTER QUESTIONS AND PROBLEMS Different Holding Periods When a call is purchased the buyer pays for both the time value and the intrinsic value of the option As the call gets closer and closer to expiration it will lose its time value At expiration of the call the holder collects only the intrinsic value By selling the call prior to expiration the holder is able to recover some of the time value previously purchased For a given stock price this increases the profit or decreases the loss however the shorter the holding period the less time the stock price has to move upward The tradeoff in deciding whether to sell an option early is between cutting the loss of time value and giving the stock enough time to make a substantial move Buy a Call The call with a higher exercise price will be far more speculative because the stock price must go higher in order to break even39 however the premium on such a call will be lower A call with a lower exercise price will have a greater chance of expiring inthemoney39 however the premium will be higher The tradeoff is between taking a gamble on the call with a higher exercise price at the cost of a small premium or buying the safer call with a lower exercise price at the cost of a larger premium Puts and Stock The Protective Put A protective put establishes a minimum price at which a stock can be sold In a bear market the stock will lose value that can be recovered by exercising the put This makes the put like an insurance policy that pays off in the event of a loss The premium on the put is like the premium on the insurance policy If the price of the stock goes up the insurance is not needed so the put is allowed to expire Puts and Stock The Protective Put The higher the exercise price the higher the price at which the stock can be sold This reduces the overall loss in a bear market but of course will require a higher premium It is therefore like taking a lower deductible in an insurance policy By forcing the insurer the put writer to assume more of the potential loss the cost of the insurance the put premium rises Call Put Option Transactions Both strategies are indeed bullish Buying a call gives you the option to buy the stock at a favorable price Writing a put on the other hand gives you the obligation to buy the stock at what might be an unfavorable price Both the call buyer and put writer will profit in a bull market but the call buyer will do better in a strong bull market because the profit will be higher the higher the stock price The put writer39s profit is limited to the original premium received In a bear market the call buyer will have limited losses while the put writer will lose more the lower the stock price All of this makes it sound as though the call buyer does better than the put writer but this is not necessarily so The call buyer is paying a premium while the put writer is receiving a premium The put writer earns interest on the premium while the call buyer forgoes interest Calls and Stock The Covered Call The covered call cuts losses on the downside and gains on the upside In a bull market the call is likely to be exercised and the stock called away This limits the gain in a bull market to the amount of the premium plus the difference between the exercise price and the original stock price A protective put cuts losses in a bear market but does allow gains in a bull market which are higher the higher the stock price Since a protective put is a synthetic call the comparison is more appropriately seen as that of writing a covered call versus buying a call Should we own stock and protect it with a short call or own simply the call Both strategies are bullish but the call or protective put would appeal more in situations where the investor is more concerned about taking advantage of a bull market than providing protection in a bear market Calls and Stock The Covered Call You could concentrate on writing outofthemoney calls however this would not guarantee that the stock would not be called away The position must be carefully monitored If the call moves inthemoney it could be repurchased Then you should write another outofthemoney call If the stock price continues to move upward you continue to roll out of one exercise price into a higher 35 EndofChapter Solutions Chapter 6 exercise price Disadvantages of this strategy are the potential for generating high transaction costs and the trouble of monitoring the portfolio as well as the fact that writing outofthe money calls produces relatively small premiums Synthetic Puts and Calls If P S0 7 C lt X equot then the synthetic call put plus stock is underpriced or the actual call is overpriced So buy the synthetic call and sell the actual call The payoffs from this portfolio at expiration are 13X Sr gtX Short call 0 7ST 7 X Long stock ST ST Long put X 7 S I 0 X X The cost of this portfolio is P S0 7 C which we said was less than the present value of the exercise price So we are essentially buying a portfolio that pays off X dollars for certain and paying a price that is less than the present value of X Calls and Stock The Covered Call N5 71 NC 1 1r Max0 ST 7X7C7ST7S0 lfSTgX1r7C7STS0 lfSTgtX1rST7X7C7STS0 S07X7C The breakeven is at a value of ST X Setting this profit equation to zero 7 C 7 ST S0 0 and solving for ST ST S0 7 C Maximum profit occurs if ST 0 1r 7 C S0 lLinimum profit occurs if ST gt X 1 S0 7X 7 C which is negative since C must be greater than S0 7 X Puts and Stock The Protective Put N5 7l 1 7Max0 X 7 ST P 7 ST 7 S0 lfST ltX1r7XSTP7STS0 PS0 7X lfST 2X1IP7STS0 EndofChapter Solutions Chapter 6 37 The breakeven is at a value of ST gt X Setting the profit equation to zero P7 ST So 0 and solving for ST ST P So Maximum profit occurs at ST ltX 1E P So 7X lLinimum profit occurs if ST oo n7w Calls and Stock The Covered Call Time value is directly related to time to expiration If the covered writer closes out the position prior to expiration the call will be more expensive to repurchase This reduces the profit for a given stock price The shorter the holding period however the less time the stock price has to move downward Writers of options benefit from short holding periods by giving the stock less time to move but they have to buy back more of the original time value they received Calls and Stock The Covered Call The lower the exercise price the more protection the option is providing This is because the call with the lower exercise price commands a higher premium and the premium cushions the loss on the downside The disadvantage of a lower exercise price is that the option is more likely to be exercised and the upside gain is lower Buy a Call C 525 1 100Max0ST 71657 525 Option Value at ST Expiration Profit 150 0 7525 155 0 7525 160 0 7525 165 0 7525 1 70 5 725 1 75 1 0 475 1 80 15 975 Note If you use Stratlyz7exls to generate the graphs in this chapter the horizontal axis will be labeled as the Stock Price at Expiration Here it has been changed to Stock Price at End of Holding Period EndofChapter Solutions 1500 1000 500 Pro t 500 1000 150 155 160 165 170 175 180 Stock Price at End of Holding Period Breakeven X C 165 525 17025 Maximum loss 525 the premium 14 Buy a Call X 165 rc 00535 0 021 Tit 20365 00548 based on 20 days between 81 and 821 Plugging into the BlackScholes Merton model we obtain the option values on August 1 for stock prices of 150 155 180 1r 100Option Value on 81 7 525 The results using a spreadsheet are S Option Value on 821 Profit 150 00901 751599 155 04295 748205 160 14203 738297 165 34782 717718 170 67247 14747 175 108992 56492 180 155951 103451 Note If you use Stratlyz7exls to generate the graphs in this chapter the horizontal axis will be labeled as the Stock Price at Expiration Here it has been changed to Stock Price at End of Holding Period 1500 1000 W 500 W 0 1 1 l l l 500 7 Profit 1000 150 155 160 165 170 175 180 Stock Price at End of Holding Period Chapter 6 38 Endof Chapter Solutions Chapter 6 Approximate breakeven stock price 168 Maximum loss 525 the premium Buy 21 Put P 675 1r 100Max0165 7 ST 7 675 Option Value at ST Expiration 150 15 155 10 160 5 165 0 170 0 175 0 180 0 Note If you use Stratlyz7exls to generate the graphs in this chapter the horizontal axis will be labeled as the Stock Price at Expiration Here it has been changed to Stock Price at End of Holding Period 1000 500 0 i i Profit 500 1000 150 1 55 160 1 65 Stock Price at End of Holding Period 170 175 180 Breakeven X 7 P 165 7 675 15825 ximum loss 6 Maximum gain if ST 0 15825 Calls and Stock The Covered Call C 6 1r 100sT 716513 7Maxo sT 7170 6 Option Value Profit S T at Expiration from Option 150 0 600 155 0 600 160 0 600 165 0 600 170 0 600 175 5 100 180 10 7400 39 Profit from Stock 71 5 1 3 Ov erall Profit EndofChapter Solutions Chapter 6 Note If you use Stratlyz7exls to generate the graphs in this chapter the horizontal axis will be labeled as the Stock Price at Expiration Here it has been changed to Stock Price at End of Holding Period 1500 1000 a 500 E 0 1 1 1 1 1 E n 500 1000 1500 150 155 160 165 170 175 180 Stock Price at End of Holding Period Breakeven So 7 C 16513 7 6 15913 Maximum profit 1087 Maximum loss if ST 0 Profit from call 600 Profit from stock 716513 Overall profit 715913 maximum loss 15913 Calls and Stock The Covered Call X 170 o 021 rc 00571 T7t 45365 01233 based on 45 days from 91 to 1016 Plugging into the BlackScholes Merton model we obtain the option values on August 1 for stock prices of 150 155 180 Profit 100St 7 16513 7 Option Value on 91 7 6 The results using a spreadsheet are Option Value Profit Profit Overall St on 91 from Option from Stock Pro t 150 02714 57286 71513 794014 155 07297 52703 71013 718597 160 16540 43460 7513 77840 165 32381 27619 713 26319 170 56014 3986 487 52686 175 87436 727436 987 71264 180 125540 765540 1487 83160 Note If you use Stratlyz7exls to generate the graphs in this chapter the horizontal axis will be labeled as the Stock Price at Expiration Here it has been changed to Stock Price at End of Holding Period 40 EndofChapter Solutions 18 19 Chapter 6 Profit 1000 500 500 7 1000 1500 155 160 165 170 175 Stock Price at End of Holding Period 180 Approximate breakev en stock price 162 Puts and Stock The Protective Put P 475 1r 100sT 716513 Max0165 7 ST 7 475 Option Value at Expiration 5 ST 150 155 160 165 170 175 180 10 5 0 0 0 0 Profit from Put Profit from Stock 987 1487 Note If you use Stratlyz7exls to generate the graphs in this chapter the horizontal axis will be labeled as the Stock Price at Expiration Here it has been changed to Stock Price at End of Holding Period Profit 1500 1000 500 0 500 1000 150 155 160 165 170 175 Stock Price at End of Holding Period 180 Breakeven P So 475 16513 16988 Maximum profit 00 Maximum loss 48750 Buy a Call C 00385 X 100 Contract size is 100000 EndofChapter Solutions Premium is 10000000385 3850 ST 090 095 100 105 110 Option Value at Expiration 000 000 005 010 Profit 3850 73850 73850 1150 6150 Note If you use Stratlyz7exls to generate the graphs in this chapter the horizontal axis Will be labeled as the Stock Price at Expiration Here it has been changed to Spot Exchange Rate at End of Holding Period Profit 090 095 100 105 110 Spot Exchange Rate at End of Holding Period Breakeven exchange rate X C 100 00385 10385 20 Buy 21 Put P 7 00435 x 7 100 Premium 10000000435 4350 Chapter 6 ST 090 095 100 105 110 Option Value at Expiration 005 000 000 000 Profit 5650 650 7 4350 7 4350 7 4350 Note If you use Stratlyz7exls to generate the graphs in this chapter the horizontal axis will be labeled as the Stock Price at Expiration Here it has been changed to Stock Price at End of Holding Period Endof Chapter Solutions Profit 3 8 1000 r 3000 7 090 095 100 105 110 Spot Exchange Rate at End of Holding Period Breakeven exchange rate X 7 P 100 7 00435 09565 21 Calls and Stock The Covered Call C 00385 X 100 So 09825 Contract size is 100000 Premium is 10000000385 3850 The currency costs 10000009825 98250 Net cost 98250 7 3850 94400 Option Value ST at Expiration Profit 090 000 74400 095 000 600 100 000 5600 105 005 5600 110 010 5600 Note If you use Stratyz7exls to generate the graphs in this chapter the horizontal axis will be labeled as the Stock Price at Expiration Here it has been changed to Stock Price at End of Holding Period 7000 5000 7 3000 7 1000 7 4000 7 3000 5000 090 095 100 105 110 Spot Exchange Rate at End of Holding Period Profit Breakeven exchange rate So 7 C 09825 7 00385 09440 22 Calls and Stock The Covered Call Covered call writing refers to owning the portfolio and writing call options The key word to describe this strategy is ceiling writing calls places a ceiling on the value of your Chapter 6 43 EndofChapter Solutions Chapter 6 44 portfolio Covered call writing reduces the expected return and standard deviation of a portfolio as well as negative skewness The following probability density function illustrates this strategy Probability L r Change in Portfolio Value Puts and Stock The Protective Put Portfolio insurance with options involves owning the portfolio and buying put options The key word to describe this strategy is floor buying puts places a floor on the value of your portfolio Protective put buying reduces the expected return and standard deviation of a portfolio as well as positive skewness The following probability density function illustrates this strategy Probability L Terminal portfolio value Synthetic Puts and Calls This is a reverse conversion Sell put 475 Buy synthetic put Buy call 7525 Sell stock 165125 Net cash in 164625 Invest at 535 to grow to 16462510535 125 7 16571 Amount owed at expiration 165 Net gain 071 Buy a Call a 1 Max0 ST 7X 7 C7 TC where TC transaction costs 1t ST 7X 7 0005X 7 0005ST 7 C 7 001C 1ltX EndofChapter Solutions 1t7C7001C Buy 21 Put b 1 Max0 X 7 ST 7 P 7 TC 1 X 7 ST 7 P 7 0005X 7 00055T 7 P7 001P Calls and Stock c 1 ST 7 So 7 Max0 ST 7X C 7 TC 1 7 S0 C 7 000550 7 0005X 7 001C 1 ST 7 SO C 7 00055T 7 0005 So 7 001C Puts and Stock d 1 ST 7 SO Max0 X 7 ST 7 P 7 TC 1 X 7 So 7P 7 0005X 7 000550 7 001P Chapter 6 45 Endof Chapter Solutions 1 Chapter 2 4 CHAPTER 2 STRUCTURE OF OPTIONS MARKETS ENDOFCHAPTER QUESTIONS AND PROBLEMS Options a Homeowners insurance is a put option In the event of a loss the insurance company pays you a portion of the value of the house The rest is like a deductible The premium on the put is the insurance premium b The guaranteed tuition arrangement is a call option granted to you by the school If you enroll now you can quotpurchasequot the education at a fixed price up through the next four years The premium on the option is the fact that you have to enroll in this particular school for your freshman year Of course you do not have to exercise the option by continuing to enroll thereafter c This is not an option because no one has the opportunity to forego exercise It is actually a forward contract If the lease were cancelable over the period during which the rental rate was fixed it would be a call option Exercising an Option An American option can be exercised at any time up through the expiration date A European option can be exercised only on the expiration date An American option is equivalent to a European option with the additional feature that it can be exercised early Option Price Quotations The option is on ATampT stock It expires in January If it is an exchangelisted option it expires the Saturday following the third Friday in January The option is a call with an exercise price of 65 a share In other words the option gives the right to buy ATampT stock at 65 a share up to the expiration day in January Contract Size a One contract would now cover 110 shares with an exercise price of 60 110 or 5455 This would be rounded to the nearest eighth for 5412 b Buyers and writers of outstanding contracts are credited with two contracts for every one previously owned or written The exercise price is changed to 1212 The contract size is still 100 c One contract would now cover 10043 or 133 shares with an exercise price of 8534 or 6375 Note In the context of options a 4for3 stock split is the same as a 33 percent stock dividend d No changes to any contract terms Expiration Dates Jan cycle Feb cycle March cycle a Feb Mar Apr Jul Feb Mar May Aug Feb Mar Jun Sep b Jul Aug Oct Jan Jul Aug Nov Feb Jul Aug Sep Dec c Dec Jan Apr Jul Dec Jan Feb May Dec Jan Mar Jun Position and Exercise Limits Short puts and long calls are both strategies designed to profit in a bullish market Thus they are considered to be quoton the same side of the market quot Option Traders The market maker is an independent operator whose objective is to buy options at one price and sell them for a higher price A broker is in business to generate commissions on each transaction A broker does not have to try to guess where the market is going or whether he can earn the bidask spread CBOE rules allow an individual to be both a market maker and a floor broker but not on the same day The reason is the potential for a conflict of interest For example suppose a situation arises in which the trader has to decide whether to execute a personal transaction or a customer transaction Whichever transaction is EndofChapter Solutions Chapter 2 5 done will bring large profits to the holder of the position The trader could obviously be tempted to put personal interests ahead of the customer39s interests The practice of trading as both a market maker and a floor broker is called dual trading Order Book Of cial Consider a limit order to buy an option at no more than 3 If there is no offer to sell for 3 or less the OBO takes the limit order adds it to the other limit orders and makes the highest bids known to the traders If any market maker or broker is willing to lower the ask price to 3 or below the OBO executes the order Because the price may not fall to 3 the limit order may never be filled Other Option Trading Systems In the market maker system an individual trader who is not a broker is required to be a market maker That is the trader must be willing to quote a bid and an ask price on certain options In the specialist system there is an individual called the specialist who is charged with making a market in certain options In addition there are registered option traders who trade on their own but are not required to make a market The market maker system puts the role of the specialist and registered option trader into one person the market maker Exchanges using the specialist system claim that it has the advantage of specialized expertise in keeping the market fair and orderly Proponents of the market maker system argue that because the specialist is a monopolist the cost to the public is much higher than under the market maker system which encourages market makers to compete with each other for the public39s business Mechanics of Trading Since each contract covers 100 shares your 20 calls cover 2000 shares Thus your premium is 4500 You pay your premium to your broker Your broker39s firm must clear its option trades through a clearing firm which is a member of the Options Clearing Corporation OCC Your broker39s firm sends the money to the clearing firm which deposits it with the OCC The clearing firm does not actually have to deposit your money with the OCC It is allowed to consolidate its accounts and using a predetermined formula it deposits the required amount with the clearinghouse Index Options If a call stock option is exercised the writer delivers the stock to the buyer and receives the exercise price If a put is exercised the buyer delivers the stock to the writer and receives the exercise price If a call index option is exercised the writer pays the buyer the difference between the stock price and the exercise price For a put the writer pays the buyer the difference between the exercise price and the stock price The major advantage of exercising an index option rather than a stock option is not having to handle the stock This results in significantly lower transaction costs Option Price Quotations Besides the fact that stock and option prices are already dated by the time they appear in newspapers these prices are not synchronized The prices shown are only the prices of the last trade The last trade of the stock may not have taken place at the same time as the last trade of the option In addition the stock and option markets do not even close at the same time Moreover the prices appearing in the newspapers do not indicate whether the last trade was at a bid price or an ask price Also printed newspapers provide prices for only the most active options though more information can usually be obtained from the newspapers web sites Web sites of the exchanges provide much more current information and in some cases include information not provided by the newspapers such as the bid and ask prices Mechanics of Trading An option position can be terminated by simply executing an offsetting order in the market For example suppose in January you bought a lLicrosoft March 90 call for 5 38 In the middle of February it is selling for 6 14 and you would like to take your profit You simply sell a Microsoft March 90 call which offsets your long position An option can also be closed by exercising it You would simply notify your broker that you want to buy the stock at the exercise price if a call or sell it at the exercise price if a put The third way an option position can be terminated is by expiring outofthe money If it is not advantageous to exercise it by the expiration the option simply expires and your position is terminated In the overthecounter market you can certainly exercise the option or have it expire outofthemoney While you can effectively offset a position by opening up a new but opposite contract the procedure is technically somewhat different than in the exchangelisted options market In the EndofChapter Solutions latter market the contracts cancel each other and no further obligation is incurred In the overthecounter market both contracts remain in force and consequently each is subject to default on the part of the writer Other Types of Options Among the optionlike instruments are warrants convertibles and callable bonds A warrant is an option offered by the firm on its own stock A convertible is a bond or preferred stock that can be converted into common stock at a fixed rate at the holder39s discretion Callable bonds are bonds that can be retired early at a specific price at the discretion of the issuing firm In addition firms issue options similar to warrants to executives and employees Finally we should note that stock itself is like a call option held by the stockholders and written by the bondholders Real Options Real options are options that corporations hold when they invest in certain projects and includes options to expand projects contract projects temporarily shut them down terminate them or sell them to other companies These options do not trade in open markets like exchangelisted and overthe counter options but they possess the characteristics of ordinary options such as having an exercise price and an expiration Transaction Costs in Option Trading Floor trading and clearing fees run from 050 to 100 These represent the costs of paperwork involved in processing the trade as well as the exchange39s overhead Commissions which re ect the cost of the labor involved in arranging the trade vary and depend on the type of broker discount or full service The bidask spread is the cost of providing liquidity to the market The public and floor brokers representing the public incur all of these costs while market makers incur floor trading and clearing fees and may incur the bidask spread if they have to deal with other market makers instead of the public Transactions in the OTC market do not generally incur commissions and floor trading and clearing fees They do incur costs of paperwork and in particular the legal expenses of laying out the rights of each party Since transactions in the OTC market are generally executed through dealers they incur the dealer39s bidask spread OvertheCounter Options Markets Exchangetraded options are regulated by the Securities and Exchange Commission There is essentially no regulation of OTC option transactions Firms that trade in the OTC market however are typically regulated by the National Association of Securities Dealers or if they are banks by banking regulators APPENDIX 2A QUESTIONS AND PROBLEMS l a 10025000 700 17000 b 10025000 5500 5000 300 8000 This amount exceeds the minimum of 300 015000 800 c 10025000 5000 4500 300 8000 This amount exceeds the minimum of 300 014500 750 d 10025000 700 17000 e The stock price exceeds the exercise price so only l0004505 or 22500 can be borrowed The call premium however can be applied so the investor must come up with only 27500 7000 20500 f 20500 10000 transactions Onehundred percent margin must be posted on all option purchase APPENDIX 2B QUESTIONS AND PROBLEMS Chapter 2 6 EndofChapter Solutions Chapter 2 450 600 150 The 150 loss applies against other taxable income and reduces taxes by 150028 42 650 600 50 The tax is 50028 14 The stock is treated as having been purchased for 25 6 31 The taxable gain is 3500 3100 400 The tax is 400028 112 The call would be exercised You deliver the stock and receive 25 for it The sale price of the stock for tax purposes is 25 6 31 You purchased the stock at 30 The tax is 3100 3000028 28 The taxable gain is 600 350 250 The tax is 250028 70 Your loss is 10015 12 300 This is netted against other gains for a tax saving of 300031 93 The aftertax profit is 300 93 207 You exercise the call and receive 10044135 425 1635 Your profit is 1635 1500 135 The tax is l350244 3294 The aftertax profit is 135 3294 10206 The call expires worthless Your loss is 1500 This is netted against other gains for tax savings of1500031 465 The aftertax profit is 1500 465 1035 Taxes are paid at the end of the year on all trading profits whether the positions are closed out or not Thus in a and b if the end of the year came before you sold or exercised the call you would owe taxes on any profits or be able to deduct any losses accumulated up to that time This would be a wash sale You replaced the stock with a call option within the 61day period The loss on the stock is not deductible for tax purposes This would be a wash sale because you acquired the call within a 61day period surrounding the sale of the stock It does not matter that you acquired the call before you sold the stock This is not a wash sale The wash sale rule pertains only to cases where the stock is sold at a loss The rule prohibits deducting the loss In this case the stock was sold at a gain so the wash sale ru e as no effect 7 EndofChapter Solutions CHAPTER 7 ADVANCED OPTION STRATEGIES ENDOFCHAPTER QUESTIONS AND PROBLEMS 1 Chapter 7 46 Option Spreads Basic Concepts Simple long or short positions expose the trader to considerable risk This is especially true for short positions By taking an opposite position in another option ie executing a spread the trader is able to keep the risk to a more manageable level Combined positions of options and stocks do the same thingithe option serves as a hedge for the stock or the stock serves as a hedge for the option A position in both options and stocks however is harder to execute since the options and stock trade in different markets This results in time delays in getting the trades executed On the other hand a spread can be executed much faster with both transactions done almost simultaneously in the same market Bull Spreads Since the stock price is closer to the higher exercise price than to the lower exercise price the short call at the higher exercise price has the greater time value Therefore holding the position longer could result in a greater time value decay on the short call than on the long call This will occur only however if the stock price does not move down The crossover point indicated in Figure 72 in the chapter is a critical stock price below which the shorter holding period is preferred If it appears as if the stock price will fall below the crossover stock price the position should be closed as soon as possible but only if the stock price is not expected to turn back around before the options expire Butter y Spreads Both a straddle and a time spread can be used in this situation A straddle consisting of the purchase of a put and a call with identical exercise prices and expirations would profit if the stock price moved substantially in either direction A time spread consisting of the sale of a longerterm option and the purchase of a shorterterm option would also profit if the stock price moved significantly in either direction This is because a large stock price move will allow repurchase of the longer term option when it has little time value remaining Note that this example is the opposite of the one discussed in the text Also a butter y spread that is short the high and low exercise prices and long two of the middle exercise price could be used in this situation but it would have limited gains if the stock price moved substantially Straddle A protective put provides insurance against a drop in the stock price below the put exercise price The put has a cost that can be offset by selling a call The call exercise price will be above the put exercise price This forces the investor to be willing to give up gains in the stock beyond the call exercise price Thus the stock effectively has a lower limit on its value which is the exercise price of the put and an upper limit on its value which is the exercise price of the call Bull Spread Buy one put with exercise price X1 and sell one put with exercise price X2 The profit equation is n Max0 x1 7 ST 1 iMax0X2 7 sTP2 IfSTltX1 ltx2 nx1 ASTAPI 7X2STP2 x1 7x2 7031 7P2 Recall from Chapter 3 that the difference between exercise prices is less than the difference between premiums Since X1 ltX2 and P1 ltP2 the above profit is negative IfX1ltSTltX2 n7P17X2STP2 EndofChapter Solutions This figure increases dollar for dollar with ST The breakeven is found by setting the profit to zero 7P1 7X2 STP2 0 and solving for ST ST X2 Pl 7P2 If X1 ltX2 ST 1 7 P1 P2 Since P gt P1 this figure is positive The minimum profit occurs in the first profit range ST lt X1 lt X2 and equals X1 7 X2 7 Pl 7 P2 The maximum profit occurs in the third profit range X1 ltX2 lt ST and equals 7 P1 P2 straddle First note that the graph for a short straddle held to expiration is an inverted V If closed out prior to expiration a short straddle which involves the sale of both a put and a call will require the repurchase of both options This means that prior to expiration there will be time values on both the put and the call that will have to be repurchased These time values are highest if the stock price is close to the exercise price So the profit if the stock price is near the exercise price is lower the shorter the holding period The longer the investor can hold the position the less time value that remains on the options however this also gives the stock more time to move substantially and potentially generate a large loss straddle A straddle is a strategy based on the expectation that the market will have high volatility If however everyone else in the market believes that the market will have high volatility then the prices of the call and put will re ect the higher volatility which will result in breakeven points further away from the current stock price A straddle is most likely to be successful when the investor believes that the market will be more volatile than the everyone else in the market believes and the investor s expectations are more accurate Box Spread In internal rate of return IRR problems the return from the investment is compared to the opportunity cost of capital The investment is acceptable if the IRR exceeds the opportunity cost In the box spread problem the investment pays X2 7X1 The initial outlay is C1 7 C2 7 P1 P2 The IR is the value of i that solves X2 7X11i39T C1 icz Pr 132 The opportunity cost is the riskfree rate r If i gt r then the box spread is acceptable If i lt r then the box spread is overpriced and should be reversed Bear Spread Buy the October 170 at 6 Sell the October 165 at 810 n 100Max0ST 7 170 7 6 7Max0ST 7165 810 Chapter 7 47 EndofChapter Solutions Chapter 7 48 Option Value at Expiration S T October 170 October 165 Profit 150 0 2 155 0 0 210 160 0 0 210 165 0 0 210 170 0 5 7290 175 5 10 7290 180 10 15 7290 Note If you use Stratlyz7exls to generate the graphs in this chapter the horizontal axis will be labeled as the Stock Price at Expiration Here it has been changed to Stock Price at End of Holding Period Pro t 150 155 160 165 170 175 180 Stock Price at End of Holding Period Breakeven stock price X1 C1 7 C2 165 810 7 6 16710 Maximum profit 210 1Linimum profit 7290 A bear spread constructed with calls will be subject to the risk of early exercise If the stock price rises sufficiently then the short call can be exercised without the long call being inthemoney Even if the long call is inthemoney and exercised what will happen is that the loss on the upside will be incurred earlier This will not happen however if there are no dividends on the stock during the life of the option Bear Spread rc 00571 0 021 Time to expiration T 7 t 26365 00712 26 days between September 20 and October 16 Plugging into the BlackScholes Merton model we obtain the option values on September 20 for stock prices of150 155 180 Spread value on 920 Value of 170 call on 920 7 Value of 165 call on 920 1 100Spread Value on 920 7 600 810 EndofChapter Solutions Option Value on 920 St October 165 October 170 Profit 150 01914 00488 19574 155 06862 02234 16372 160 18642 07552 991 165 40260 19706 446 170 72362 41480 79882 175 112934 73443 718491 180 158809 113715 724094 Note If you use Stratlyz7exls to generate the graphs in this chapter the horizontal axis will be labeled as the Stock Price at Expiration Here it has been changed to Stock Price at End of Holding Period 300 200 1 5 100 g 0 i i i i i E n 100 7 200 7 300 150 155 160 165 170 175 180 Stock Price at End of Holding Period Approximate breakeven stock price 166 11 Bear Spread The October 160 put price is 45 We need to find an October call that has a price of 45 We use the BlackScholesMerton model with the inputs S0 16513 rc 00571 T 102365 02795 0 021 Inserting various exercise prices into the model we find that at X 17483 the call price is about 450 Thus we buy a put with an exercise price of 160 and sell a call with an exercise price of 17483 The cost of the collar up front is the initial value of the stock 16513 Time to expiration T 7 t 26365 00712 26 days between September 20 and October 16 Plugging into the BlackScholes Merton model we obtain the option values on September 20 for stock prices of150 155 180 Value of collar 100 Value of stock Value of October 160 put 7 Value of October 17483 call Values of stock put and call are on September 20 1t 100Value of collar 7 16513 Option Value on 920 S October 160 Put October 17483 Call Profit 150 99702 00109 751707 155 61092 00637 740845 160 32545 02686 721441 165 14795 08536 4959 170 05679 21324 33055 175 01834 43582 56952 180 00499 75734 73455 Chapter 7 49 EndofChapter Solutions Note If you use Stratlyz7exls to generate the graphs in this chapter the horizontal axis will be labeled as the Stock Price at Expiration Here it has been changed to Stock Price at End of Holding Period Profit 150 155 160 165 170 175 180 Stock Price at End of Holding Period Maximum profit 00 1Linimum profit This clearly occurs when the stock is at zero Then the call is at zero and the put goes to its maximum value the present value of its exercise price of 160 which is 160e390 0571mm 1593508 So you would have invested 16513 and ended up with 1593508 for a loss of 578 or 578 overall Approximate breakeven stock price 164 12 Butter y Spreads Sell October 160 at 1110 Buy two October 16539s at 810 each Sell October 170 at 6 1r 1002Max0ST 7 165 7 810 7Max0 ST 7160 1110 7Max0 ST 7170 6 Option Value at Expiration ST October 160 October 165 October 170 Profit 150 0 0 0 90 155 0 0 0 90 160 0 0 0 90 165 5 0 0 7410 170 10 5 0 90 175 15 10 5 90 180 20 15 10 90 Note If you use Stratlyz7exls to generate the graphs in this chapter the horizontal axis will be labeled as the Stock Price at Expiration Here it has been changed to Stock Price at End of Holding Period Chapter 7 50 EndofChapter Solutions Pro t 150 155 160 165 170 175 180 Stock Price at End of Holding Period Breakevens x1 c1 izcz c3 1601110728106 716090 2x2 7x1 icl 2c2 7C3 7 216571607 1110281076 16910 Maximum profit 90 1Linimum profit 7110 13 Calendar Spreads Buy August 170 at 325 Sell October 170 at 6 August 170 T 7 t 20365 00548 rc 00535 October 170 T 7 t 76365 02082 rc 00571 Based on 20 days between August 1 and August 21 and 76 days between August 1 and October 16 Plugging into the BlackScholesMerton model we obtain the option values on August 1 for stock prices at 150 155 180 Spread value on 81 Value of August 170 Call on 81 7 Value of October 170 Call on 81 1 100Spread Value on 81 7 325 6 Option Value on 81 August 170 October 170 Profit 150 00164 08844 18820 155 01083 17301 11282 160 04793 30639 1653 165 15084 49781 77197 170 35836 75118 711783 175 68145 106446 710801 180 109577 143064 75987 Note If you use Stratlyz7exls to generate the graphs in this chapter the horizontal axis will be labeled as the Stock Price at Expiration Here it has been changed to Stock Price at End of Holding Period Chapter 7 51 EndofChapter Solutions Pro t U1 0 150 155 160 165 170 175 180 Stock Price at End of Holding Period Maximum profit 275 if ST 0 profit 7100325 7 6 275 lLinimum profit 711753 Approximate breakeven stock prices 161 and 181 14 Time Value Decay Using the BlackScholes Merton model with re 00571 0 021 So 16513 X 165 and time to expiration as indicated below we obtain the following values for the call on the dates indicated Time value Option Value 7 Intrinsic Value 165137165 013 Date 715 731 815 831 915 930 1016 Time to Expiration Time Value 02548 81228 02110 72876 01699 64410 01260 54416 00849 43673 00438 30368 00000 00000 Time Value 000 1 1 1 1 1 02548 02110 01699 01260 00849 00438 00000 Time to Expiration 15 Box Spread Using a spreadsheet for the BlackScholesMerton model August 160 Chapter 7 52 EndofChapter Solutions So 16513 X 160 rc 00535 T 46365 01260 The delta Nd1 is found to be 07092 October 160 So 16513 X 160 rc 00571 T 102365 02795 The delta Nd1 is found to be 06856 If we buy the August 160 and write the October 160 the hedge ratio is the delta of the August call divided by the delta of the October call 0709206856 10344 So sell 10344 October calls for each August call Suppose we buy 1000 August calls and sell 1034 October calls If the stock price increases by one dollar the August calls increase by 07091 each for a total gain of 709 The October calls increase by 06855 each for a total loss since we are short of 103406855 709 The procedure is valid only for very small changes in the stock price Otherwise the hedge could have significant gamma risk 16 straddle Buy the October 165 call at 810 Buy the October 165 put at 675 n 100Max0ST 7 165 7 810 Max0165 7 ST 7 675 Option Value at Expiration ST October 165 Call October 165 Put Profit 150 155 0 10 7485 160 0 5 7985 165 0 0 71485 170 5 0 7985 175 10 0 7485 180 15 0 15 Note If you use Stratlyz7exls to generate the graphs in this chapter the horizontal axis will be labeled as the Stock Price at Expiration Here it has been changed to Stock Price at End of Holding Period Chapter 7 53 Endof Chapter Solutions 500 500 1000 Pro t 1500 2000 150 155 160 165 170 175 180 Stock Price at End of Holding Period Breakevens XCP 165810675 17985 X7C7P 16578107675 15015 1Linimum profit 71485 Maximum profit 00 17 straddle X 165 rc 00571 0 021 T 7 t 26365 00712 based on 26 days between 920 and 1016 Plugging into the BlackScholesMerton model we obtain the option values on September 20 for stock prices 150 155 180 Straddle value on 920 Value of October 165 Call on 920 Value of October 165 Put on 9 20 1t 100Straddle Value on 920 7 810 7 675 Option Value at End of Holding Period S October 165 Call October 165 Put Profit 150 01911 145217 71372 155 06856 100162 741482 160 18633 61939 767928 165 40250 33555 774695 170 72353 15658 760489 175 112927 06233 729340 180 158803 02109 12412 Note If you use Stratlyz7exls to generate the graphs in this chapter the horizontal axis will be labeled as the Stock Price at Expiration Here it has been changed to Stock Price at End of Holding Period Chapter 7 54 EndofChapter Solutions Profit 200 200 400 600 800 150 155 160 165 170 175 180 Stock Price at End of Holding Period Approximate breakeven stock prices 178 and 150 18 straddle Buy two October 165 calls at 810 each Buy one October 165 put at 675 1r 1002Max0sT 7 165 7 810 MaxosT 71657 675 ST 150 155 160 165 170 175 180 October 165 Call Option Values at Expiration October 165 Put Profit 0 15 7795 0 10 71295 0 5 71795 0 0 72295 5 0 71295 10 0 7295 15 0 705 Note If you use Stratlyz7exls to generate the graphs in this chapter the horizontal axis will be labeled as the Stock Price at Expiration Here it has been changed to Stock Price at End of Holding Period Pro t 150 155 160 165 170 175 180 Stock Price at End of Holding Period The graph shows the straddle the dashed line overlaid with the strap The strap provides a higher profit on the upside and a larger loss on the downside Breakevens Chapter 7 55 Endof Chapter Solutions XCP2165 8106752 176475 X7P72C 1657675 72810 14205 1Linimum profit 71002810 675 72295 19 straddle Sell two August 170 puts at 75 Sell one August 170 call at 325 1 10072Max0170 7 ST 7 75 7 Max0ST 7 170 325 Option Value at Expiration ST August 170 Call August 170 Put Profit 150 0 20 72175 155 0 15 71175 160 0 10 7175 165 0 5 825 170 0 0 1825 175 5 0 1325 180 10 0 825 Note If you use Stratlyz7exls to generate the graphs in this chapter the horizontal axis will be labeled as the Stock Price at Expiration Here it has been changed to Stock Price at End of Holding Period 3000 2000 a 1000 E 0 1 1 1 1 1 E n 1000 2000 3000 150 155 160 165 170 175 180 Stock Price at End of Holding Period Note that only one breakeven point appears in the graph because of the range of stock prices chosen Breakevens X C 2P 170 325 275 18825 X 7P 7 C2 170 7 75 7 3252 160875 1Linimum profit When ST 0 1r C 2P 7 2X 100325 275 7 2170 731175 20 Box Spreads Note If the continuously compounded rate is 00535 then the discrete rate is e005 71 7 0055 Chapter 7 56 EndofChapter Solutions 22 Chapter 7 57 Buy August 160 call at 810 Sell August 170 call at 325 Buy August 170 put at 75 Sell August 160 put at 275 The net premium is 810 7 325 75 7 275 960 The present value of the future payoff is 170 7160105539 125 7 9933 Where 0 1260 is the time to expiration from July 6 to August 21 Thus the net present value is 9933 7 960 0333 This spread is underpriced so buy it That is buy the August 160 call and 170 put and sell the August 170 call and 160 put This will generate a negative cash flow up front of 960 At expiration you will receive 10 but the present value of 10 is more than the initial cash ow at T VT max0X2 7 ST 7 max0X1 7 ST expiration max0 ST 7 X1 7 2max0 ST 7X2 T max0ST 7X1 7max0ST 7X2 HV X27X17C12C27C3 nV C17C2 C172C2C3 s X1 C1 72C2 C3 and sX1C17C2 Collars straddles The following table provides a summary of collar and straddle strategies EndofChapter Solutions Collar Strategies Straddle at VT ST VT explmtlon max0X1 7ST max 0 S X nV7 23 straddle Buy August 165 put at 475 Buy August 170 call at 325 n 100Max0165 7 ST 7 475 Max0ST 7 170 7 325 Option Value at Expiration ST August 165 Put August 170 Call Profit 150 15 0 700 155 10 0 200 160 5 0 7300 165 0 0 7800 170 0 0 7800 175 0 5 7300 180 0 10 200 Note If you use Stratlyz7exls to generate the graphs in this chapter the horizontal axis will be labeled as the Stock Price at Expiration Here it has been changed to Stock Price at End of Holding Period 1000 500 5 E 0 l l l l l E n 500 1000 150 155 160 165 170 175 180 Stock Price at End of Holding Period Breakevens 7 Max0 ST 7 170 7 325 Max0 165 7 ST 7 475 Chapter 7 58 Endof Chapter Solutions Chapter 7 59 Ira S 7 7 sT 7170 7 325 7 475 N Breakeven 170 325 475 178 1ltX 7 7325 165 7 ST 7 475 Breakeven 165 7 475 7 325 157 The strangle is similar to a straddle By using a lower exercise price put or higher exercise price call however the cost is lower though the profit is also lower Money Calendar Spreads Buy October 165 call at 810 Sell August 170 call at 325 o 021 On August 1 the October 165 call has T 7 t 76365 02082 and re 00571 On August 1 the August 170 call has T 7 t 20365 00548 and re 00535 Plugging into the BlackScholes Merton model we obtain the option values on August 1 for stock prices of 150 155 180 Value of diagonal spread on 81 Value of October 165 call on 81 7 Value of August 170 Call on 81 Profit 100Value of spread on 81 7 810 325 Option Price at End of Holding Period ST October 165 Call August 170 Call Profit 150 15867 00164 732797 155 28800 01083 720783 160 47666 04793 75627 165 72909 15084 9325 170 104321 35836 19985 175 141154 68145 24509 180 182350 109577 24273 Note If you use Stratlyz7exls to generate the graphs in this chapter the horizontal axis will be labeled as the Stock Price at Expiration Here it has been changed to Stock Price at End of Holding Period 200 7 100 7 100 7 200 7 300 7 Pro t 150 155 160 165 170 175 180 Stock Price at End of Holding Period EndofChapter Solutions Approximate breakeven stock price 1619 Chapter 7 60 Endof Chapter Solutions CHAPTER 9 PRINCIPLES OF PRICING FORWARDS FUTURES AND OPTIONS ON FUTURES END OF CHAPTER QUESTIONS AND PROBLEMS 1 Chapter 9 66 Value of a ForwardFutures Contract Other than the insignificant margin requirement a futures contract requires no initial outlay of funds Immediately after buying the contract and before the price changes the holder of the contract cannot receive anything for selling it Since you cannot receive anything for it it has no value The spot commodity however requires payment and it can be resold immediately for cash Thus it has value equal to whatever price it will fetch in the market the current spot price This result applies equally to a forward contract Concept of Price Versus Value The spot price is affected by the cost of carry and the risk premium The cost of carry is the cost of storing an asset plus the interest foregone by investing funds in the asset The storage costs include the actual direct physical costs of storage The risk premium is the amount by which the expected future price is discounted to compensate the person holding the asset for assuming the risk ForwardFutures Pricing Revisited When there is a negative cost of carry we usually find an explanation in the form of a convenience yield The idea behind it is that spot prices are unusually high owing to a greater shortage of the good now than is expected in the future Thus there is a quotpremiumquot paid to owners of the good which is called the convenience yield It is simply an unusually high spot price that rewards holders of the good and discourages storage Convenience yields are likely to exist on goods with harvests that produce random output and goods that are consumed In addition the goods should be storable They are unlikely to be found on stocks and bonds because their future supplies are fairly well known and they are not really quotconsumedquot though in a minor sense bonds are ForwardFutures Pricing Revisited Contango is a market in which the futures price exceeds the spot price The cost of carry is the difference between the futures price and the spot price It is positive in a contango market In contrast backwardation is a market in which the futures price is less than the spot price In that case the cost of carry is negative A negative cost of carry is usually explained by justifying a convenience yield In such a market the spot price is abnormally high as a result of a shortage of the good The negative cost of carry is a disincentive to store the good The convenience yield refers to the premium in the spot price that re ects the attractiveness of owning the commodity at a time when it is in short supply Commodities and Storage Costs The futures price is the expected spot price of wheat in December Therefore traders expect the spot price of wheat in December to be 364 ForwardFutures Pricing Revisited The futures price will not be the expected spot price in September because the dominance of the long hedgers will induce a risk premium Thus the futures price of 276 is biased low Without information on the magnitude of the risk premium it is impossible to come up with a precise estimate The expected spot price in September however is no less than 276 Of course the actual spot price in September could be far less Early Exercise of Call and Put Option 0n Futures In Chapters 3 4 and 5 we covered American call options on the spot and explained that in the absence of dividends they will not be exercised early They will always sell for at least the lower bound which is higher than the intrinsic value and usually more Call options on the futures however might be exercised early If the price of the underlying instrument is extremely high the call will begin to behave like the underlying instrument For an option on a futures this means that the call will behave like the futures changing almost dollarfordollar with the futures price For an option on the spot the call will behave like the spot changing almost oneforone with the price of the spot Exercise of the futures call will release funds tied up in the call and provide a position in the futures Exercise of the call on the spot does not however release funds since the investor has to purchase the spot instrument EndofChapter Solutions Chapter 9 67 Black Futures Option Pricing Model The Black model is not an American option on futures pricing model and Eurodollar options on futures are American Also the Black model assumes constant interest rates Since Eurodollars are interestsensitive instruments they violate an assumption of the model Black Futures Option Pricing Model A spot option pricing model such as BlackScholes Merton is a model for pricing options on instruments with a cost of carry of re or re 7 5 if there is a dividend yield Since a futures requires no outlay of funds nor does it incur a storage cost it has no cost of carry The cost of carry relevant to the futures price is the cost of carry of the underlying spot instrument An option on a futures is therefore an option on an instrument with a zero cost of carry The futures option pricing model is the same as the spot option pricing model where the spot price is replaced with the futures price and the cost of carry is zero The latter is established by assuming a dividend yield equal to the interest rate Value of a Forward Contract The value of the forward contract can be found by subtracting the present value of the forward price from the current spot price Thus the value of the contract is 52 7 4511039 5 909 This is the correct value of the contract at this point six months into the life of the contract because it is the value of a portfolio that could be constructed at this time to produce the same result six months later That is you could buy the asset costing 52 and take out a loan promising to pay 45 in six months This combination would guarantee that you would receive at time T six months later the value of the asset ST minus the 45 loan repayment which is the value of the forward contract when it expires Value of a Futures Contract The value at the opening is 89970 7 89930 040 In dollars this is 040500 200 An instant before the close the value is 89910 7 89930 4120 In dollars that is 7020500 7100 After the market has closed the contract is markedtomarket the gain or loss is distributed and the value is zero Price of a Futures Contract Consider a futures contract on a stock Say you sell short the stock and buy the futures You receive S0 up front and during the holding period you earn interest of iSO If the stock pays dividends a short seller has to make them up so you would incur a cost of DT where DT is the compound future value of the dividends At expiration you accept delivery of the stock and pay f Thus your profit is So 7 f0T iSo 7 DT Since this must equal zero EndofChapter Solutions Chapter 9 68 f0T SO iSo ADT Spot Prices Risk Premiums and the Carry Arbitrage for Generic Assets a In a market with risk premiums the futures price underestimates the spot price at expiration by the amount of the risk premium Therefore the expected spot price in December is 364 0035 3675 b Arbitrage assures us that whether or not a risk premium exists the futures price equals the spot price plus the cost of carry This is confirmed by noting that the spot price of 35225 plus the cost of carry of ll75 equals the futures price of364 c The answer is apparent in part a The expected price of wheat in December exceeds the futures price by the risk premium d If there is a risk premium holders of long futures contracts expect to sell them for a profit equal to the risk premium Thus the expected futures price at expiration is 364 035 3675 which is also the expected spot price at expiration e Speculators who take long positions in futures earn the risk premium They do so because they are supplying insurance to the hedgers and therefore expect to receive a return in compensation for their willingness to take the risk Stock Indices and Dividends a T 73365 02 f0T 95649600596 3900275lt 2 96265 At 96050 it is underpriced b f0T 956490 0059602 7 527 96236 At 96050 it is overpriced The main difference is compounding of interest Annual compounding results in lower proceeds than continuous hence the annual compounded carrying cost is lower than the continuous compounding Futures Prices and Risk Premia EST 60 Elt1gt 4 0 550 EST Ema 60 f0T EfTT 7 Elt1gt 60 7 4 56 Foreign Exchange a So 00093l3 F 0010475 r 00615 p 00364 With annual compounding the forward rate should be 1 07730365 00093 l31017m 001045 So the forward rate should be 001045 but is actually 0010475 Thus the forward contract is overpriced You should buy the yen in the spot market and sell it in the forward market b With continuous compounding the forward rate should be EndofChapter Solutions 0009313e 075001117303 001050 So the forward rate should be 001050 but is actually 0010475 Thus the forward contract is underpriced You should sell the yen in the spot market and buy it in the forward market 17 PutCall Parity of Options on Futures 100 days between September 12 and December 21 T 100365 02740 CfoTTX 7 Pf0TTX 7 2625 7 325 7 23 rod 7 X1 r39T 42370 7 400102753902740 2352 We can view the futures as overpriced and assume the call and put are correctly priced We sell the futures buy a call and sell a put Payoffs at Expiration l SI Short futures ST fo TD ST f0 D Long call 0 Sr X Short put 0 MI f0T 7 X f0T X f0T 7 X 42370 7 400 2370 The present value of this is 2370l 027502740 2352 The portfolio will cost 2625 7 325 2300 Thus you will earn a present value of 2352 7 23 052 18 Pricing Options on Futures The option s life is January 31 to March 18 so T 46365 01260 a Intrinsic Value Max0 f0 7X 7 x0 48310 7 480 10 b Time Value Call Price 7 Intrinsic Value 695 7 310 385 c Lower bound Max0 f0 7X1 r39T Max0 48310 7 4801028439 126 309 d Intrinsic Value Max0 X 7 f0 7 Max0 480 7 48310 7 0 e Time Value Put Price 7 Intrinsic Value 525 7 0 525 f Lower bound Max0 X 7 f01 r39T Max0 480 7 483101028439 125 Chapter 9 69 EndofChapter Solutions 70 ote t e ower oun a 1es on to uro can uts hl b dppy 1y E p p g C 7 P f0 7X1 r39T 7 525 48310 7 480102843901260 7834 The actual call price is 695 so putcall parity does not hold 19 PutCall Parity of Options on Futures f0 7 gt00 r39T 7 102 7 10011039025 7 195 c7P747175 7225 C 7 P is too high so the call is overpriced andor the put is underpriced or we could assume the futures is underpriced So sell the call buy the put and buy the futures At expiration the payoffs will be fSX f1gtX Short call 0 7fT 7 X Long put X 7 fT 0 Long futures fzig f1f9 X 7 f0 X 7 f0 This is equivalent to a riskfree loan as a lender if X gt f0 or as a borrower if f0 gt X Here f0 gt X so you are a borrower The present value should be X 7 f01 r T 102 7 100110390 25 7195 Thus you sell the call for 4 and buy the put for 7175 for a net in ow of 225 At expiration you pay back 200 20 Black Futures Option Pricing Model First find the continuously compounded riskfree rate rc ln10284 00280 Then price the option d1 lnfOX 522T o T ln48310480 08 22 1260 0841260 02409 dz 7 d1 57 02409708JW72125 Ndl 7 N247 5948 Ndz N217 5832 C equotTfoNd1 XNdzl 7 e0W126 gt483105948 4805832 7 739 The option appears to be underpriced You could sell equotT Ndl 05927 futures and buy one call adjusting the hedge ratio through time and earn an arbitrage profit 21 Black Futures Option Pricing Model P 7 ercT1 Nd2 f equotT1 Nd1 We already know that N024 05948 and N021 05832 Then P 7 480e3900m 12601 7 05832 7 48310e390 18 10 5 11 7 05948 7 430 Chapter 9 70 EndofChapter Solutions Foreign Currencies and Foreign Interest Rates Interest Rate Parity The correct forward price is given by T F0T s0 1665101510216568 r Because the forward price is higher than the model price we will sell the forward contract If transaction costs could be covered you would buy the foreign currency in the spot market at 1665 and sell it in the forward market at 1664 You would earn interest at the foreign interest rate of 2 percent By selling it forward you could then convert back to dollars at the rate of 1664 In other words 1665 would be used to buy 1 unit of the foreign currency which would grow to 102 units the 2 percent foreign rate Then 102 units would be converted back to 1021664 169728 This would be areturn 0f1697281665 71 0019387 or 19 percent which is better than the U S rate Lower Bound of a European Option on Futures f0T 100 X 90 and r 5 The lower bound on a futures option is C f0T T X 2 Max0 15a 7X1 r T 7 Max0 100 7 901 005391 7 95238 The quoted price of 940 violates the lower bound and the quoted price is low Therefore we would buy the futures call option and hedge the resulting risk as illustrated in the following cash flow table contract rT1 rT rT1 rT 95238 24 Stock Indices and Dividends a Find the future value of the dividends 0751125 365 0851123 365 090 7 2608 f0T 7 100019190465 7 2608 7 100226 b Since f0T SO 0 then 0 f0T 7 So so 100226 7100 0226 This is the compound future value of the interest lost minus the compound future value of the dividends 25 ForwardFutures Pricing Revisited Let the spot price be So the futures price be f0T and the margin requirement be M Consider the position of someone who buys the asset and sells a futures contract to form a riskfree hedge Chapter 9 71 EndofChapter Solutions Today Buy the asset paying So and sell the futures by depositing M dollars in a margin account that earns the rate q where q lt r At expiration The accumulated costs of storage and the interest lost on SO dollars add up to 6 When the trader delivers the asset he receives f0T The total amount of cash will be f0T 7 e M interest on M at the rate q Since the transaction is still riskfree the amount initially invested must grow at the riskfree rate to equal this future value however the spot price does not have to be compounded because the interest on it is already included in the cost of carry Thus f0TM1qTlr39T6 SM Solving for f0T gives fS6 M1 1qT1r39T The bracketed term is the difference in interest between the rate q and rate r If q is less than r the whole bracketed term is greater than zero so the futures price will be greater than the spot price plus the cost of carry In other words if the margin account pays interest at less than the riskfree rate the futures price will be greater than the spot price plus the cost of carry The higher futures price compensates for the loss of interest Chapter 9 72 Endof Chapter Solutions
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