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**Unformatted text preview: **T rading Str"ategiesInvolving OptionsWe discussed the profit pattern from an investment in a single stock option inChapter 8. In tllis chapter we cover more fully the range of profit patterns obtainableusing options. We assume that the underlying asset is a stock. Similar results can beobtained for other underlying assets, such as foreign currencies, stock indices, andfutures contracts. The options used in the strategies we discuss are European. American options may lead to slightly different outcomes because of the possibility of earlyexercise.In the first section we consider what happens when a position in a stock option iscombined with a position in the stock itself. We then move on to exanline the profitpatterns obtained when an investment is made in two or more different options on thesame stock. One of the attractions of options is that they can be used to create a widerange of different payoff functions. (A payofffunction is the payoff as a function of thestock price.) If European options were available with every single possible strike price,any payoff function could in theory be created.For ease of exposition the figures and tables showing the profit from a tradingstrategy will ignore the time value of money. The profit will be shown as the finalpayoff minus the initial cost. (In theory, it should be calculated as the present value ofthe final payoff nlinus the initial cost.)10.1STRATEGIES INVOLVING A SINGLE OPTION AND A STOCKThere are a number of different trading strategies involving a single option on a stockand the stock itself. The profits from these are illustrated in Figure 10.1. In this figureand in other figures throughout this chapter, the dashed line shows the relationshipbetween profit and the stock price for the individual securities constituting theportfolio, whereas the solid line shows the relationship between profit and the stockprice for the whole portfolio.In Figure 1O.1(a), the portfolio consists of a long position in a stock plus a shortposition in a call option. This is known as writing a covered call. The long stock position"covers" or protects the investor from the payoff on the short call that becomesnecessary if there is a sharp rise in the stock price. In Figure 10.1 (b), a short positionin a stock is combined with a long position in a call option. This is the reverse of writing224CHAPTER 10Profit patterns (a) long position in a stock combined with short positionin a call; (b) short position in a stock combined with long position in a call; (c) longposition in a put combined with long position in a stock; (d) short position in a putcombined with short position in a stock.Figure 10.1ProfitProfit;;;;;;;;Long;; Stock,,,,,,,,,,....;;; Long,,,",,,,........,,K;;,,,Long ;Stock;;Lana',Put",,.,,,,;;;;;;;;;;;;K;,,-------------(c)"Short',Stock,,(b)Profit,,,,'.(a),,Call;STShort "Call;;Profit",Short',Stock,"ShortPut",;;-------------v',K,,,,,,,,,,,,,,(d)a covered call. In Figure 10.I(c), the investment strategy involves buying a put option ona stock and the stock itself. The approach is sometimes referred to as a protective putstrategy. In Figure 10.I(d), a short position in a put option is combined with a shortposition in the stock. This is the reverse of a protective put.The profit patterns in Figures 10.1 have the same general shape as the profit patternsdiscussed in Chapter 8 for short put, long put, long call, and short call, respectively.Put-call parity provides a way of understanding why this is so. From Chapter 9, the225Trading Strategies Involving Optionsput--eall parity relationship isp + So = c + Ke -rT + D(10.1)where p is the price of a European put, So is the stock price, c is the price of a Europeancall, K is the strike price of both call and put, r is the risk-free interest rate, T is the timeto maturity of both call and put, and D is the present value of the dividends anticipatedduring the life of the options.Equation (10.1) shows that a long position in a put combined with a long position inthe stock is equivalent to a long call position plus a certain amount (= Ke- rT + D) ofcash. This explains why the profit pattern in Figure 1O.1(c) is similar to the profitpattern from a long call position. The position in Figure 10.1 (d) is the reverse of that inFigure 1O.1(c) and therefore leads to a profit pattern similar to that from a short callposition.Equation (l0.1) can be rearranged to becomeSo - c = Ke -rT+D-PIn other words, a long position in a stock combined with a short position in a call isequivalent to a short put position plus a certain amount (= Ke- rT + D) of cash. Thisequality explains why the profit pattern in Figure 1O.1(a) is similar to the profitpattern from a short put position. The position in Figure 1O.1(b) is the reverse ofthat in Figure 1O.1(a) and therefore leads to a profit pattern similar to that from along put position.10.2 SPREADSA spread trading strategy involves taking a position in two or more options of the sametype (i.e., two or more calls or two or more puts).Bull SpreadsOne of the most popular types of spreads is a bull spread. This can be created by buyinga call option on a stock with a certain strike price and selling a call option on the sameFigure 10.2Profit from bull spread crea.ted using call options.ProfitK1,//////______________ J//,,,,,,,,'.226CHAPTER 10Table 10.1Stock pricerangePayoff from a bull spread created using calls.Payoff ji'OI11long call optionPayoff ji'OI11short call optionTotalpayoffST ~ K2KI < ST < K2ST:::;; K\stock with a higher strike price. Both options have the same expiration date. Thestrategy is illustrated in Figure 10.2. The profits from the two option positions takenseparately are shown by the dashed lines. The profit from the whole strategy is the sumof the profits given by the dashed lines and is indicated by the solid line. Because a call- price always decreases as the strike price increases, the value of the option sold is alwaysless than the value of the option bought. A bull spread, when created from calls,therefore requires an initial investment.Suppose that K\ is the strike price of the call option bought, K 2 is the strike price ofthe call option sold, and ST is the stock price on the expiration date of the options.Table 10.1 shows the total payoff that will be realized from a bull spread in differentcircumstances. If the stock price does well and is greater than the higher strike price,the payoff is the difference between the two strike prices, or K 2 - K\. If the stock priceon the expiration date lies between the two strike priCes, the payoff is ST - K \. If thestock price on the expiration date is below the lower strike price, the payoff is zero.The profit in Figure 10.2 is calculated by subtracting the initial investment from thepayoff.A bull spread strategy limits the investor's upside as well as downside risk. The strategycan be described by saying that the investor has a call option with a strike price equal toK\ and has chosen to give up some upside potential by selling a call option with strikeprice K 2 (K2 > K\). In return for giving up the upside potential, the investor gets theFigure 10.3Profit from bull spread created using put options.Profit""""""""""", "" ".,.-------------""""""""""""227Trading Strategies Involving Optionsprice of the option with strike price~2'Three types of bull spreads can be distinguished:1. Both calls are initially out of the money.2. One call is initially in the money; the other call is initially out of the money.3. Both calls are initially in the money.The most aggressive bull spreads are those of type 1. They cost very little to set up andhave a small probability of giving a relatively high payoff (= K 2 - K 1). As we movefrom type 1 to type 2 and from type 2 to type 3, the spreads become more conservative.Example 10.1An investor buys for $3 a call with a strike price of $30 and sells for $1 a call witha strike price of $35. The payoff from this bull spread strategy is $5 if the stockprice is above $35, and zero if it is below $30. If the stock price is between $30 and$35, the payoff is the amount by which the stock price exceeds $30. The cost of thestrategy is $3 - $1 = $2. The profit is therefore as follows:Stock price rangeProfitST ~ 3030 < ST < 35ST ~ 35ST- 323-2Bull spreads can also be created by buying a put with a low strike price and selling a putwith a high strike price, as illustrated in Figure 10.3. Unlike the bull spread created fromcalls, bull spreads created from puts involve a positive up-front cash flow to the investor(ignoring margin requirements) and a payoff that is either negative or zero.Bear SpreadsAn investor who enters into a bull spread is hoping that the stock price will increase. Bycontrast, an investor who enters into a bear spread is hoping that the stock price willdecline. Bear spreads can be created by buying a put with one strike price and selling aput with another strike price. The strike price of the option purchased is greater thanthe strike price of the option sold. (This is in contrast to a bull spread, where the strikeFigure 10.4Profit from bear spread created using put options.Profit"""""""...... ...... ...... ...... ......... ......... ...... ...-"-""""""""""",----------------228CHAPTER 10Table 10.2Payoff from a bear spread created with put options.Stock pricerangePayofffromlong put optionPayoff}i'omshort put optionSr ~ K2K j < Sr < K2Sr ~ K joK2 -SrK2 -SrooTotalpayoff-(Kj - Sr)price of the option purchased is always less than the strike price of the option sold.) InFigure IDA, the profit from the spread is shown by the solid line. A bear spread createdfrom puts involves an initial cash outflow because the price of the put sold is less thanthe price of the put purchased. In essence, the investor has bought a put with a certain- strike price and chosen to give up some of the profit potential by selling a put with alower strike price. In return for the profit given up, the investor gets the price of theoption sold.Assume that the strike prices are K j and K 2 , with K j < K2 Table 10.2 shows thepayoff that will be realized from a bear spread in different circumstances. If the stockprice is greater than K 2 , the payoff is zero. If the stock price is less than Kj, the payoff isK 2 - K 1 If the stock price is between K 1 and K 2, the payoff is K 2 - Sr. The profit iscalculated by subtracting the initial cost from the payoff.Example 10.2An investor buys for $3 a put with a strike price of $35 and sells for $1 a put witha strike price of $30. The payoff from this bear spread strategy is zero if the stockprice is above $35, and $5 if it is below $30. If the stock price is between $30 and$35, the payoff is 35 - ST. The options cost $3 - $1 = $2 up front. The profit istherefore as follows:Stock price rangeProfitSr ~ 3030 < Sr < 35+333 - SrSr~35-2Like bull spreads, bear spreads limit both the upside profit potential and the downsiderisk. Bear spreads can be created using calls instead of puts. The investor buys a callwith a high strike price and sells a call with a low strike price, as illustrated inFigure 10.5. Bear spreads created with calls involve an initial cash inflow (ignoringmargin requirements).Box SpreadsA box spread is a combination of a bull call spread with strike prices K j and K 2 and abear put spread with the same two strike prices. As shown in Table 10.3 the payoff froma box spread is always K 2 - K j The value of a box spread is therefore always thepresent value of this payoff or (K2 - K1)e- rr . If it has a different value there is anarbitrage opportunity. If the market price of the box spread is too low, it is profitable to229Trading Strategies Involving OptionsFigure 10.5Profit from bear spread created using call options.Profit-------------,,",-,,,,,,,,,,"",,,,""""""""""""'.buy the box. This involves buying a call with strike price K(, buying a put with strikeprice K 1, selling a call with strike price K 1, and selling a put with strike price K 1 If themarket price of the box spread is too high, it is profitable to sell the box. This involvesbuying a call with strike price K 1 , buying a put with strike price K(, selling a call withstrike price K(, and selling a put with strike price K 1 .It is important to realize that a box-spread arbitrage only works with Europeanoptions. Most of the options that trade on exchanges are American. As shown inBusiness Snapshot 10.1, inexperienced traders who treat American options as Europeanare liable to lose money.Butterfly SpreadsA butterfly spread involves positions in options with three different strike prices. It canbe created by buying.a call option with a relatively low strike price, K(, buying a calloption with a relatively high strike price, K 3 , and selling two call options with a strikeprice, K 1, halfway between K 1 and K 3 Generally K 1 is close to the current stock price.The pattern of profits from the strategy is shown in Figure 10.6. A butterfly spreadleads to a profit if the stock price stays close to K1 , but gives rise to a small loss if thereis a significant stock price move in either direction. It is therefore an appropriatestrategy for an investor who feels that large stock price moves are unlikely. The strategyrequires a small investment initially. The payoff from a butterfly spread is shown inTable 10.5.Table 10.3Stock pricerangeST ~ K1Kl < ST < K1ST ::( KlPayoff from a box spread.Payofffrombull call spreadPayoff frombear put spreadTotalpayoffK1- K lK1- K lK1- K l230CHAPTER 10Business Snapshot 10.1 . Losing Money with Box SpreadsSuppose that a stock has a price of $50 and a volatility of 30%. No dividends areexpected and the risk-free rate is 8%. A trader offers you the chance to sell on theCBOE a 2-month box spread where the strike prices are $55 and $60 for $5.10.Should you do the trade?The trade certainly sounds attractive. In this case K} = 55, K 2 = 60, and the payoffis certain to be $5 in 2 months. By selling the box spread for $5.10 and investing thefunds (or 2 months you would have more than enough funds to meet the $5 payoff in2 months. The theoretical value of the box spread today is 5 x e-O.08x2/12 = $4.93.Unfortunately there is a snag. CBOE stock options are American and the $5 payofffrom the box spread is calculated on the assumption that the options comprising thebox are European. Option prices for this example (calculated using DerivaGem) are_ shown in Table lOA. A bull call spread where the strike prices are $55 and $60 costs0.96 - 0.26 = $0.70. (This is the same for both European and American optionsbecause, as we saw in Chapter 9, the price of a European call is the same as the price ofan American call when there are no dividends.) A bear put spread with the same strikeprices costs9A6 - 5.23 = $4.23 if the options al:e European and 10.00 - 5.44 = $4.56if they are American. The combined value of both spreads if they are created withEuropean options is 0.70 + 4.23 = $4.93. This is the theoretical box spread pricecalculated above. The combined value of buying both spreads if they are American is0.70 + 4.56 = $5.26. Selling a box spread created \vith American options for $5.10would not be a good trade. You would realize this almost immediately as the tradeinvolves selling a $60 strike put and this \vould be exercised against you almost as soonas you sold it!Suppose that a certain stock is currently worth $61. Consider an investor who feelsthat a significant price move in the next 6 months is unlikely. Suppose that the marketprices of 6-month calls are as follows:Strike price ($ )Call price ($)5560651075Values of 2-month European and American optionson a non-dividend-paying stock. Stock price = $50; interest rate= 8% per annum; and volatility = 30% per annum.Table 10.4OptiontypeStrikepriceEuropeanoption priceAmericanoption priceCallCallPutPut605560550.260.9690465.230.260.9610.005.44Tl~adilli Strategies Involving OptionsFigure 10.6231Profit from butterfly' spread using call options.Profit\\\\\Kj"",,"\~"~-----------------------7-------("-------------_/,,"\\\\\\\\The investor could create a butterfly spread by buying one call with a $55 strike price,buying one call with a $65 strike price, and selling two calls with a $60 strike price. Itcosts $10 + $5 - (2 x $7) = $1 to create the spread. If the stock price in 6 months isgreater than $65 or less than $55, the total payoff is zero, and the investor incurs a netloss of $1. If the stock price is between $56 and $64, a profit is made. The maximumprofit, $4, occurs when the stock price in 6 months is $60.Butterfly spreads can be created using put options. The investor buys a put with a lowstrike price, buys a put with a high strike price, and sells two puts with an intermediatestrike price, as illustrated in Figure 10.7. The butterfly spread in the example justconsidered would be created by buying a put with a strike price of $55, buying a putwith a strike price of $65, and selling two puts with a strike price of $60. If all optionsare European, the use of put options results in exactly the same spread as the use of calloptions. Put-eall parity can be used to show that the initial investment is the same inboth cases.A butterfly spread can be sold or shorted by following the reverse strategy. Optionsare sold with strike prices of K 1 and K 3, and two options with the middle strike price K 2are purchased. This strategy produces a modest profit if there is a significant movementin the stock price.Table 10.5Payoff from a butterfly spread.Stock pricerangePayoff fromfirst long callPayoff fromsecond long callPayofffromshort callsST < K,K, < ST < K2K 2 < ST < K3ST> K30ST-K,ST-K,ST-KI00000ST- K 3-2(ST - K2)-2(ST - K 2)* These payoffs are calculated using the relationship K2 = O.5(K]+ K3).Totalpayoff*0ST-K,K3 -ST0232CHAPTER 10Figure 10.7Profit from butterfly spread using put options.Profit--------------------\\\\\",,""\:'->""------------------------7-------(\,,""\--------------"\\\\\\Calendar Spreadsup to now we have assumed that the options used to create a spread all expire at thesame time. We now move on to calendar spreads in which the options have the samestrike price and different expiration dates.,A calendar spread can be created by s~lling a call option with a certain strike priceand buying a longer-maturity call option with the same strike, price. The longer thematurity of an option, the more expensive it usually is. A calendar spread thereforeusually requires an initial investment. Profit diagrams for calendar spreads are usuallyproduced so that they show the profit when the short-maturity option expires on theassumption that the long-maturity option is sold at that time. The profit pattern for acalendar spread produced from call options is shown in Figure 10.8. The pattern isFigure 10.8Profit from calendar spread created using two calls.,Profit,,,///-----------------,,,,"""""...//,,"",,/"",,,,,,,,,"233Trading Strategies Involving OptionsProfit from a calendar spread created using two puts.Figure 10.9Profit"""""""""""" "--"-, ,///~---------------ST//////////////similar to the profit from the butterfly spread in Figure 10.6. The investor makes a profitif the stock price at the expiration of the short-maturity option is close to the strikeprice of the short-maturity option. However, a loss is incurred when the stock price issignificantly above or significantly below this strike price.To understand the profit pattern from a calendar spread, first consider what happensif the stock price is very low when the short-maturity option expires. The short-maturityoption is worthless and the value of the long-maturity option is close to zero. Theinvestor therefore incurs a loss that is close to the cost of setting up the spread initially.Consider next what happens if the stock price, ST, is very high when the short-maturityoption expires. The short-maturity option costs the investor ST - K, and the longmaturity option is worth close to ST - K, where K is the strike price of the options.Again, the investor makes a net loss that is close to the cost of setting up the spreadinitially. If ST is close to K, the short-maturity option costs the investor either a smallamount or nothing at all. However, the long-maturity option is still quite valuable. Inthis case a significant net profit is made.In a neutral calendar spread, a strike price close to the current stock price is chosen.A bullish calendar spread involves a higher strike price, whereas a bearish calendarspread involves a lower strike price.Calendar spreads can be created with put options as well as call options. The investorbuys a long-maturity put option and sells a short-maturity put option. As shown inFigure 10.9, the profit pattern is similar to that obtained from using calls.A reverse calendar spread is the opposite to that in Figures 10.8 and 10.9. The investorbuys a short-maturity option and sells a long-maturity option. A small profit arises ifthe stock price at the expiration of the short-maturity option is well above or well belowthe strike price of the short-maturity option. However, a significant loss results if it isclose to the strike price.Diagonal SpreadsBull, bear, and calendar spreads can all be created from a long position in one call anda short position in another call. In the case of bull and bear spreads, the calls have234CHAPTER 10Figure 10.10Profit from a straddle.Profitdifferent strike prices and the same expiration date. In the case of calendar spreads, thecalls have the same strike price and different expiration dates.In a diagonal spread both the expiration date and the strike price of the calls aredifferent. This increases the range of profit patterns that are possible.10.3 COMBINATIONSA combination is an option trading strategy that involves taking a position in both callsand puts on the same stock. We will consider straddles, strips, straps, and strangles.StraddleOne popular combination is a straddle, which involves buying a call and put with thesame strike price and expiration date. The profit pattern is shown in Figure 10.10. Thestrike price is denoted by K. If the stock price is close to this strike price at expiration ofthe options, the straddle leads to a loss. However, if there is a sufficiently large move ineither direction, a significant profit will result. The payoff from a straddle is calculatedin Table 10.6.A straddle is appropriate when an investor is expecting a large move in a stock pricebut does not know in which direction the move will be. Consider an investor who feelsthat the price of a certain stock, currently valued at $69 by the market, will movesignificantly in the next 3 months. The investor could create a straddle by buying both aput and a call with a strike price of $70 and an expiration date in 3 months. Supposethat the call costs $4 and the put costs $3. If the stock price stays at $69, it is easy to seeTable 10.6Payoff from a straddle.Range ofstock pricePayofffrom callPayoffji'om putTotalpayoffST ~ KST> K0ST-KK-ST0K-S TST-K235Tl'ading Strategies Involving Optionsthat the strategy costs the investor $6. (An up-front investment of $7 is required, the callexpires worthless, and the put expires worth $1.) If the stock price moves to $70, a lossof $7 is experienced. (This is the worst that can happen.) However, if the stock pricejumps up to $90, a profit of $13 is made; if the stock moves down to $55, a profit of $8is made; and so on. As discussed in Business Snapshot 10.2 an investor should carefullyconsider whether the jump that he or she anticipates is already reflected in option pricesbefore putting on a straddle trade.The straddle in Figure 10.10 is sometimes referred to as a bottom straddle or straddlepurchase. A top straddle or straddle write is the reverse position. It is created by selling acall and a put with the same exercise price and expiration date. It is a highly risky strategy.If the stock price on the expiration date is close to the strike price, a significant profitresults. However, the .loss arising from a large move is unlimited.Figure 10.11Profit from a strip and a strap.ProfitProfitStripStrap236CHAPTER 10Figure 10.12Profit from a strangle.Profit,,-~T~-_--_-- -:_//-Strips and StrapsA strip consists of a long position in one call and two puts with the same strike priceand expiration date. A strap consists of a long position in two calls and one put with thesame strike price and expiration date. The profit patterns from strips and straps areshown in Figure 10 .11. In a strip the investor is betting that there will be a big stockprice move and considers a decrease in the stock price to. be more likely than anincrease. In a strap the investor is also betting that there will be a big stock-price move.However, in this case, an increase in the stock price is considered to be more likely thana decrease.StranglesIn a strangle, sometimes called a bottom vertical combination, an investor buys a put anda call with the same expiration date and different strike prices. The profit pattern that isobtained is shown in Figure 10.12. The call strike price, K 2 , is higher than the put strikeprice, K j The payoff function for a strangle is calculated in Table 10.7.A strangle is a similar strategy to a straddle. The investor is betting that there will be alarge price move, but is uncertain whether it will be an increase or a decrease.Comparing Figures 10.12 and 10.10, we see that the stock price has to move fartherin a strangle than in a straddle for the investor to make a profit. However, the downsiderisk if the stock price ends up at a central value is less with a strangle.The profit pattern obtained with a strangle depends on how close together the strikeprices are. The farther they are apart, the less the downside risk and the farther thestock price has to move for a profit to be realized.Table 10.7Payoff from a strangle.Range ofstock pricePayofffrom callPayoffjromputTotalpayoffST ~ K jKl < ST < K200ST- K2KI -ST00KI-ST0ST- K 2ST~K2Trading Strategies Involving OptionsFigure 10.13237Payoff from a butterfly spread.The sale of a strangle is sometimes referred to as a top vertical combination. It can beappropriate for an investor who feels that large stock price moves are unlikely.However, as with sale of a straddle, it is a risky strategy involving unlimited potentialloss to the investor.10.4 OTHER PAYOFFSThis chapter has demonstrated just a few of the ways in which options can be used toproduce an interesting relationship between profit and stock price. If European optionsexpiring at time T were available with every single possible strike price, any payofffunction at time T could in theory be obtained. The easiest illustration of this involves aseries of butterfly spreads. Recall that a butterfly spread is created by buying optionswith strike prices K 1 and K 3 and selling two options with strike price K 2 , whereK 1 < K 2 < K 3 and K 3 - K 2 = K 2 - K 1 Figure 10.13 shows the payoff from a butterflyspread. The pattern could be described as a spike. As K 1 and K 3 move closer together,the spike becomes smaller. Through the judicious combination of a large number ofvery small spikes, any payoff function can be .approximated.SUMMARYA number of common trading strategies involve a single option and the underlyingstock. For example, writing a covered call involves buying the stock and selling a calloption on the stock; a protective put involves buying a put option and buying the stock.The former is similar to selling a put option; the latter is similar to buying a call option.Spreads involve either taking a position in two or more calls or taking a position in twoor more puts. A bull spread can be created by buying a call (put) with a low strike priceand selling a put (call) with a high strike price. A bear spread can be created by buying aput (call) with a high strike price and selling a put (call) with a low strike price. Abutterfly spread involves buying calls (puts) with a low and high strike price and sellingtwo calls (puts) with some intermediate strike price. A calendar spread involves selling acall (put) with a short time to expiration and buying a call (put) with a longer time toexpiration. A diagonal spread involves a long position in one option and a short positionin another option such that both the strike price and the expiration date are different.Combinations involve taking a position in both calls and puts on the same stock. Astraddle combination involves taking a long position in a call and a long position in aCHAPTER 10238put with the same strike price and expiration date. A strip consists of a long position inone call and two puts with the same strike price and expiration date. A strap consists ofa long position in two calls and one put with the same strike price and expiration date.A strangle consists of a long position in a call and a put with different strike prices andthe same expiration date. There are many other ways in which options can be used toproduce interesting payoffs. It is not surprising that option trading has steadily.increased in popularity and continues to fascinate investors.FURTHER READINGBharadwaj, A. and J. B. Wiggins. "Box Spread and Put-eall Parity Tests for the S&P IndexLEAPS Markets," Journal of Derivatives, 8, 4 (Summer 2001): 62-7l.Chaput, J. S., and L. H. Ederington, "Option Spread and Combination Trading," Journal ofDerivatives, 10, 4 (Summer 2003): 70-88..McMillan, L. G. Options as a Strategic Investmelll. 4th edn., Upper Saddle River: Prentice-Hall,200l.Rendleman, R. J. "Covered Call Writing from an Expected Utility Perspective," Journal ofDerivatives, 8, 3 (Spring 2001): 63-75.Ronn, A. G. and E.1. Ronn. "The Box-Spread Arbitrage Conditions," Review of FinancialStudies, 2, 1 (1989): 91-108.Questions And Problems (Answers in Solutions Manual):.zt'10.1. What is meant by a protective put? What position in call options is equivalent to aprotective put?10.2. Explain two ways in which a bear spread can be created.10.3. When is it appropriate for an investor to purchase a butterfly spread?IDA. Call options on a stock are available with strike prices of $15, $17~, and $20, andexpiration dates in 3 months. Their prices are $4, $2, and $~, respectively. Explain howthe options can be used to create a butterfly spread. Construct a table showing howprofit varies with stock price for the butterfly spread.10.5. What trading strategy creates a reverse calendar spread?10.6. What is the difference between a strangle and a straddle?10.7. A call option with a strike price of $50 costs $2. A put option with a strike price of $45costs $3. Explain how a strangle can be created from these two options. What is the.pattern of profits from the strangle?10.8. Use put-eall parity to relate the initial investment for a bull spread created using calls tothe initial investment for a bull spread created using puts.10.9. Explain how an aggressive bear spread can be created using put options.10.10. Suppose that put options on a stock with strike prices $30 and $35 cost $4 and $7,respectively. How can the options be used to create (a) a bull spread and (b) a bearspread? Construct a table that shows the profit and payoff for both spreads.10.11. Use put-eall parity to show that the cost of a butterfly spread created from Europeanputs is identical to the cost of a butterfly spread created from European calls.Tradiizg Strategies Involving Options23910.12. A call with a strike price of $60 costs $6. A put with the same strike price and expirationdate costs $4. Construct a table that shows the profit from a straddle. For what range ofstock prices would the straddle lead to a loss?A10.13. Construct a table showing the payoff from a bull spread when puts with strike prices K Iand K 2 , with K 2 > K b are used.10.14. An investor believes that there will be a big jump in a stock price, but is uncertain as tothe direction. Identify six different strategies the investor can follow and explain thedifferences among them.10.15. How can a forward contract on a stock with a particular delivery price and delivery datebe created from options?10.16. "A box spread comprises four options. Two can be combined to create a long forwardposition and two can be combined to create a short forward position." Explain thisstatement..10.17. What is the result if the strike price of the put is higher than the strike price of the call ina strangle?10.18. One Australian dollar is currently worth $0.64. A I-year butterfly spread is set up usingEuropean call options with strike prices of $0.60, $0.65, and $0.70. The risk-free interestrates in the United States and Australia are 5% and 4% respectively, and the volatility ofthe exchange rate is 15%. Use the DerivaGem software to calculate the cost of setting upthe butterfly spread position. Show that the cost is the same if European put options areused instead of European call options.Assignment Questions10.19. Three put options on a stock have the same expiration date and strike prices of $55, $60,and $65. The market prices are $3, $5, and $8, respectively. Explain how a butterflyspread can be created. Construct a table showing the profit from the strategy. For whatrange of stock prices would the butterfly spread lead to a loss?10.20. A diagonal spread is created by buying a call with strike price K 2 and exercise date T2and selling a call with strike price K I and exercise date TI , where T2 > TI Draw adiagram showing the profit when (a) K 2 > K I and (b) K 2 < K I .10.21. Draw a diagram showing the variation of an investor's profit and loss with the terminalstock price for a portfolio consisting of:(a) One share and a short position in one call option(b) Two shares and a short position in one call option(c) One share and a short posItion in two call options(d) One share and a short position in four call optionsIn each case, assume that the call option has an exercise price equal to the currentstock price.10.22. Suppose that the price of a non-dividend-paying stock is $32, its volatility is 30%, andthe risk-free rate for all maturities is 5% per annum. Use DerivaGem to calculate thecost of setting up the following positions:(a) A bull spread using European call options with strike prices of $25 and $30 and amaturity of 6 months240CHAPTER 10(b) A bear spread using European put options with strike prices of $25 and $30 and amaturity of 6 months(c) A butterfly spread using European call options with strike prices of $25, $30, and$35 and a maturity of 1 year(d) A butterfly spread using European put options with strike prices of $25, $30, and$35 and a maturity of 1 year(e) A straddle using options with a strike price of $30 and a 6-month maturity(f) A strangle using options with strike prices of $25 and $35 and a 6-month maturityIn each case provide a table showing the relationship between profit and final stock price.Ignore the impact of discounting....

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