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#### 73323241-10-Trading-Strategies

Course: MATH 174, Spring 2011

School: UCLA

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rading T Str"ategies Involving Options We discussed the profit pattern from an investment in a single stock option in Chapter 8. In tllis chapter we cover more fully the range of profit patterns obtainable using options. We assume that the underlying asset is a stock. Similar results can be obtained for other underlying assets, such as foreign currencies, stock indices, and futures contracts. The options used...

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rading T Str"ategies Involving Options We discussed the profit pattern from an investment in a single stock option in Chapter 8. In tllis chapter we cover more fully the range of profit patterns obtainable using options. We assume that the underlying asset is a stock. Similar results can be obtained for other underlying assets, such as foreign currencies, stock indices, and futures contracts. The options used in the strategies we discuss are European. American options may lead to slightly different outcomes because of the possibility of early exercise. In the first section we consider what happens when a position in a stock option is combined with a position in the stock itself. We then move on to exanline the profit patterns obtained when an investment is made in two or more different options on the same stock. One of the attractions of options is that they can be used to create a wide range of different payoff functions. (A payofffunction is the payoff as a function of the stock 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 trading strategy will ignore the time value of money. The profit will be shown as the final payoff minus the initial cost. (In theory, it should be calculated as the present value of the final payoff nlinus the initial cost.) 10.1 STRATEGIES INVOLVING A SINGLE OPTION AND A STOCK There are a number of different trading strategies involving a single option on a stock and the stock itself. The profits from these are illustrated in Figure 10.1. In this figure and in other figures throughout this chapter, the dashed line shows the relationship between profit and the stock price for the individual securities constituting the portfolio, whereas the solid line shows the relationship between profit and the stock price for the whole portfolio. In Figure 1O.1(a), the portfolio consists of a long position in a stock plus a short position 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 becomes necessary if there is a sharp rise in the stock price. In Figure 10.1 (b), a short position in a stock is combined with a long position in a call option. This is the reverse of writing 224 CHAPTER 10 Profit patterns (a) long position in a stock combined with short position in a call; (b) short position in a stock combined with long position in a call; (c) long position in a put combined with long position in a stock; (d) short position in a put combined with short position in a stock. Figure 10.1 Profit Profit ; ; ; ; ; ; ;;Long ;; Stock ,, ,, ,, ,, ,, ....; ;; Long ,, , " ,, ,, ........ ,, K ; ; ,, , Long ; Stock;; Lana ',Put" , , ., ,, , ; ; ; ; ; ; ; ; ; ; ;;K ; , ,------------- (c) " Short ',Stock ,, (b) Profit ,, ,, '. (a) ,, Call ; ST Short " Call ; ; Profit " , Short ',Stock , " Short Put ", ; ;------------- v ',K ,, , ,, ,, ,, ,, ,, , (d) a covered call. In Figure 10.I(c), the investment strategy involves buying a put option on a stock and the stock itself. The approach is sometimes referred to as a protective put strategy. In Figure 10.I(d), a short position in a put option is combined with a short position 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 patterns discussed 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, the 225 Trading Strategies Involving Options put--eall parity relationship is p + 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 European call, K is the strike price of both call and put, r is the risk-free interest rate, T is the time to maturity of both call and put, and D is the present value of the dividends anticipated during the life of the options. Equation (10.1) shows that a long position in a put combined with a long position in the stock is equivalent to a long call position plus a certain amount (= Ke- rT + D) of cash. This explains why the profit pattern in Figure 1O.1(c) is similar to the profit pattern from a long call position. The position in Figure 10.1 (d) is the reverse of that in Figure 1O.1(c) and therefore leads to a profit pattern similar to that from a short call position. Equation (l0.1) can be rearranged to become So - c = Ke -rT +D- P In other words, a long position in a stock combined with a short position in a call is equivalent to a short put position plus a certain amount (= Ke- rT + D) of cash. This equality explains why the profit pattern in Figure 1O.1(a) is similar to the profit pattern from a short put position. The position in Figure 1O.1(b) is the reverse of that in Figure 1O.1(a) and therefore leads to a profit pattern similar to that from a long put position. 10.2 SPREADS A spread trading strategy involves taking a position in two or more options of the same type (i.e., two or more calls or two or more puts). Bull Spreads One of the most popular types of spreads is a bull spread. This can be created by buying a call option on a stock with a certain strike price and selling a call option on the same Figure 10.2 Profit from bull spread crea.ted using call options. Profit K1 ,/ / / / / / ______________ J / / ,, ,, ,, ,, '. 226 CHAPTER 10 Table 10.1 Stock price range Payoff from a bull spread created using calls. Payoff ji'OI11 long call option Payoff ji'OI11 short call option Total payoff ST ~ K2 KI < ST < K2 ST:::;; K\ stock with a higher strike price. Both options have the same expiration date. The strategy is illustrated in Figure 10.2. The profits from the two option positions taken separately are shown by the dashed lines. The profit from the whole strategy is the sum of 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 always less 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 of the 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 different circumstances. 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 price on the expiration date lies between the two strike priCes, the payoff is ST - K \. If the stock 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 the payoff. A bull spread strategy limits the investor's upside as well as downside risk. The strategy can be described by saying that the investor has a call option with a strike price equal to K\ and has chosen to give up some upside potential by selling a call option with strike price K 2 (K2 > K\). In return for giving up the upside potential, the investor gets the Figure 10.3 Profit from bull spread created using put options. Profit "" " "" "" "" "" , "" " .,.-------------"" "" "" "" "" "" 227 Trading Strategies Involving Options price 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 and have a small probability of giving a relatively high payoff (= K 2 - K 1). As we move from type 1 to type 2 and from type 2 to type 3, the spreads become more conservative. Example 10.1 An investor buys for \$3 a call with a strike price of \$30 and sells for \$1 a call with a strike price of \$35. The payoff from this bull spread strategy is \$5 if the stock price 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 the strategy is \$3 - \$1 = \$2. The profit is therefore as follows: Stock price range Profit ST ~ 30 30 < ST < 35 ST ~ 35 ST- 32 3 -2 Bull spreads can also be created by buying a put with a low strike price and selling a put with a high strike price, as illustrated in Figure 10.3. Unlike the bull spread created from calls, 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 Spreads An investor who enters into a bull spread is hoping that the stock price will increase. By contrast, an investor who enters into a bear spread is hoping that the stock price will decline. Bear spreads can be created by buying a put with one strike price and selling a put with another strike price. The strike price of the option purchased is greater than the strike price of the option sold. (This is in contrast to a bull spread, where the strike Figure 10.4 Profit from bear spread created using put options. Profit "" "" "" " ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... -"-"" " "" "" "" "",---------------- 228 CHAPTER 10 Table 10.2 Payoff from a bear spread created with put options. Stock price range Payofffrom long put option Payoff}i'om short put option Sr ~ K2 K j < Sr < K2 Sr ~ K j o K2 -Sr K2 -Sr o o Total payoff -(Kj - Sr) price of the option purchased is always less than the strike price of the option sold.) In Figure IDA, the profit from the spread is shown by the solid line. A bear spread created from puts involves an initial cash outflow because the price of the put sold is less than the 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 a lower strike price. In return for the profit given up, the investor gets the price of the option sold. Assume that the strike prices are K j and K 2 , with K j < K2 Table 10.2 shows the payoff that will be realized from a bear spread in different circumstances. If the stock price is greater than K 2 , the payoff is zero. If the stock price is less than Kj, the payoff is K 2 - K 1 If the stock price is between K 1 and K 2, the payoff is K 2 - Sr. The profit is calculated by subtracting the initial cost from the payoff. Example 10.2 An investor buys for \$3 a put with a strike price of \$35 and sells for \$1 a put with a strike price of \$30. The payoff from this bear spread strategy is zero if the stock price 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 is therefore as follows: Stock price range Profit Sr ~ 30 30 < Sr < 35 +3 33 - Sr Sr~35 -2 Like bull spreads, bear spreads limit both the upside profit potential and the downside risk. Bear spreads can be created using calls instead of puts. The investor buys a call with a high strike price and sells a call with a low strike price, as illustrated in Figure 10.5. Bear spreads created with calls involve an initial cash inflow (ignoring margin requirements). Box Spreads A box spread is a combination of a bull call spread with strike prices K j and K 2 and a bear put spread with the same two strike prices. As shown in Table 10.3 the payoff from a box spread is always K 2 - K j The value of a box spread is therefore always the present value of this payoff or (K2 - K1)e- rr . If it has a different value there is an arbitrage opportunity. If the market price of the box spread is too low, it is profitable to 229 Trading Strategies Involving Options Figure 10.5 Profit from bear spread created using call options. Profit -------------,, ",- ,, ,, ,, ,, ,, "" ,, ,, " "" " " "" "" " "" '. buy the box. This involves buying a call with strike price K(, buying a put with strike price K 1, selling a call with strike price K 1, and selling a put with strike price K 1 If the market price of the box spread is too high, it is profitable to sell the box. This involves buying a call with strike price K 1 , buying a put with strike price K(, selling a call with strike price K(, and selling a put with strike price K 1 . It is important to realize that a box-spread arbitrage only works with European options. Most of the options that trade on exchanges are American. As shown in Business Snapshot 10.1, inexperienced traders who treat American options as European are liable to lose money. Butterfly Spreads A butterfly spread involves positions in options with three different strike prices. It can be created by buying.a call option with a relatively low strike price, K(, buying a call option with a relatively high strike price, K 3 , and selling two call options with a strike price, 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 spread leads to a profit if the stock price stays close to K1 , but gives rise to a small loss if there is a significant stock price move in either direction. It is therefore an appropriate strategy for an investor who feels that large stock price moves are unlikely. The strategy requires a small investment initially. The payoff from a butterfly spread is shown in Table 10.5. Table 10.3 Stock price range ST ~ K1 Kl < ST < K1 ST ::( Kl Payoff from a box spread. Payofffrom bull call spread Payoff from bear put spread Total payoff K1- K l K1- K l K1- K l 230 CHAPTER 10 Business Snapshot 10.1 . Losing Money with Box Spreads Suppose that a stock has a price of \$50 and a volatility of 30%. No dividends are expected and the risk-free rate is 8%. A trader offers you the chance to sell on the CBOE 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 payoff is certain to be \$5 in 2 months. By selling the box spread for \$5.10 and investing the funds (or 2 months you would have more than enough funds to meet the \$5 payoff in 2 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 payoff from the box spread is calculated on the assumption that the options comprising the box 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 costs 0.96 - 0.26 = \$0.70. (This is the same for both European and American options because, as we saw in Chapter 9, the price of a European call is the same as the price of an American call when there are no dividends.) A bear put spread with the same strike prices costs9A6 - 5.23 = \$4.23 if the options al:e European and 10.00 - 5.44 = \$4.56 if they are American. The combined value of both spreads if they are created with European options is 0.70 + 4.23 = \$4.93. This is the theoretical box spread price calculated above. The combined value of buying both spreads if they are American is 0.70 + 4.56 = \$5.26. Selling a box spread created \vith American options for \$5.10 would not be a good trade. You would realize this almost immediately as the trade involves selling a \$60 strike put and this \vould be exercised against you almost as soon as you sold it! Suppose that a certain stock is currently worth \$61. Consider an investor who feels that a significant price move in the next 6 months is unlikely. Suppose that the market prices of 6-month calls are as follows: Strike price (\$ ) Call price (\$) 55 60 65 10 7 5 Values of 2-month European and American options on a non-dividend-paying stock. Stock price = \$50; interest rate = 8% per annum; and volatility = 30% per annum. Table 10.4 Option type Strike price European option price American option price Call Call Put Put 60 55 60 55 0.26 0.96 9046 5.23 0.26 0.96 10.00 5.44 Tl~adilli Strategies Involving Options Figure 10.6 231 Profit 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. It costs \$10 + \$5 - (2 x \$7) = \$1 to create the spread. If the stock price in 6 months is greater than \$65 or less than \$55, the total payoff is zero, and the investor incurs a net loss of \$1. If the stock price is between \$56 and \$64, a profit is made. The maximum profit, \$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 low strike price, buys a put with a high strike price, and sells two puts with an intermediate strike price, as illustrated in Figure 10.7. The butterfly spread in the example just considered would be created by buying a put with a strike price of \$55, buying a put with a strike price of \$65, and selling two puts with a strike price of \$60. If all options are European, the use of put options results in exactly the same spread as the use of call options. Put-eall parity can be used to show that the initial investment is the same in both cases. A butterfly spread can be sold or shorted by following the reverse strategy. Options are sold with prices strike of K 1 and K 3, and two options with the middle strike price K 2 are purchased. This strategy produces a modest profit if there is a significant movement in the stock price. Table 10.5 Payoff from a butterfly spread. Stock price range Payoff from first long call Payoff from second long call Payofffrom short calls ST < K, K, < ST < K2 K 2 < ST < K3 ST> K3 0 ST-K, ST-K, ST-KI 0 0 0 0 0 ST- K 3 -2(ST - K2) -2(ST - K 2) * These payoffs are calculated using the relationship K2 = O.5(K] + K3). Total payoff* 0 ST-K, K3 -ST 0 232 CHAPTER 10 Figure 10.7 Profit from butterfly spread using put options. Profit --------------------\ \ \ \ \ " ,,"" \:'->""------------------------7-------( \ ,,"" \ --------------" \\ \ \ \ \ Calendar Spreads up to now we have assumed that the options used to create a spread all expire at the same time. We now move on to calendar spreads in which the options have the same strike price and different expiration dates. ,A calendar spread can be created by s~lling a call option with a certain strike price and buying a longer-maturity call option with the same strike, price. The longer the maturity of an option, the more expensive it usually is. A calendar spread therefore usually requires an initial investment. Profit diagrams for calendar spreads are usually produced so that they show the profit when the short-maturity option expires on the assumption that the long-maturity option is sold at that time. The profit pattern for a calendar spread produced from call options is shown in Figure 10.8. The pattern is Figure 10.8 Profit from calendar spread created using two calls. , Profit , ,, / / / -----------------,, ,, "" "" " ... // ,, "" ,, / "" ,, ,, ,, ,, , " 233 Trading Strategies Involving Options Profit from a calendar spread created using two puts. Figure 10.9 Profit "" " "" "" "" " "" "- -"-, , / / /~--------------- ST / / / / / / / / / / / / / / similar to the profit from the butterfly spread in Figure 10.6. The investor makes a profit if the stock price at the expiration of the short-maturity option is close to the strike price of the short-maturity option. However, a loss is incurred when the stock price is significantly above or significantly below this strike price. To understand the profit pattern from a calendar spread, first consider what happens if the stock price is very low when the short-maturity option expires. The short-maturity option is worthless and the value of the long-maturity option is close to zero. The investor 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-maturity option 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 spread initially. If ST is close to K, the short-maturity option costs the investor either a small amount or nothing at all. However, the long-maturity option is still quite valuable. In this 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 calendar spread involves a lower strike price. Calendar spreads can be created with put options as well as call options. The investor buys a long-maturity put option and sells a short-maturity put option. As shown in Figure 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 investor buys a short-maturity option and sells a long-maturity option. A small profit arises if the stock price at the expiration of the short-maturity option is well above or well below the strike price of the short-maturity option. However, a significant loss results if it is close to the strike price. Diagonal Spreads Bull, bear, and calendar spreads can all be created from a long position in one call and a short position in another call. In the case of bull and bear spreads, the calls have 234 CHAPTER 10 Figure 10.10 Profit from a straddle. Profit different strike prices and the same expiration date. In the case of calendar spreads, the calls have the same strike price and different expiration dates. In a diagonal spread both the expiration date and the strike price of the calls are different. This increases the range of profit patterns that are possible. 10.3 COMBINATIONS A combination is an option trading strategy that involves taking a position in both calls and puts on the same stock. We will consider straddles, strips, straps, and strangles. Straddle One popular combination is a straddle, which involves buying a call and put with the same strike price and expiration date. The profit pattern is shown in Figure 10.10. The strike price is denoted by K. If the stock price is close to this strike price at expiration of the options, the straddle leads to a loss. However, if there is a sufficiently large move in either direction, a significant profit will result. The payoff from a straddle is calculated in Table 10.6. A straddle is appropriate when an investor is expecting a large move in a stock price but does not know in which direction the move will be. Consider an investor who feels that the price of a certain stock, currently valued at \$69 by the market, will move significantly in the next 3 months. The investor could create a straddle by buying both a put and a call with a strike price of \$70 and an expiration date in 3 months. Suppose that the call costs \$4 and the put costs \$3. If the stock price stays at \$69, it is easy to see Table 10.6 Payoff from a straddle. Range of stock price Payoff from call Payoff ji'om put Total payoff ST ~ K ST> K 0 ST-K K-ST 0 K-S T ST-K 235 Tl'ading Strategies Involving Options that the strategy costs the investor \$6. (An up-front investment of \$7 is required, the call expires worthless, and the put expires worth \$1.) If the stock price moves to \$70, a loss of \$7 is experienced. (This is the worst that can happen.) However, if the stock price jumps up to \$90, a profit of \$13 is made; if the stock moves down to \$55, a profit of \$8 is made; and so on. As discussed in Business Snapshot 10.2 an investor should carefully consider whether the jump that he or she anticipates is already reflected in option prices before putting on a straddle trade. The straddle in Figure 10.10 is sometimes referred to as a bottom straddle or straddle purchase. A top straddle or straddle write is the reverse position. It is created by selling a call 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 profit results. However, the .loss arising from a large move is unlimited. Figure 10.11 Profit from a strip and a strap. Profit Profit Strip Strap 236 CHAPTER 10 Figure 10.12 Profit from a strangle. Profit ,, - ~T~-_- -_- - -:_ / / -Strips and Straps A strip consists of a long position in one call and two puts with the same strike price and expiration date. A strap consists of a long position in two calls and one put with the same strike price and expiration date. The profit patterns from strips and straps are shown in Figure 10 .11. In a strip the investor is betting that there will be a big stock price move and considers a decrease in the stock price to. be more likely than an increase. 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 than a decrease. Strangles In a strangle, sometimes called a bottom vertical combination, an investor buys a put and a call with the same expiration date and different strike prices. The profit pattern that is obtained is shown in Figure 10.12. The call strike price, K 2 , is higher than the put strike price, 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 a large 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 farther in a strangle than in a straddle for the investor to make a profit. However, the downside risk 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 strike prices are. The farther they are apart, the less the downside risk and the farther the stock price has to move for a profit to be realized. Table 10.7 Payoff from a strangle. Range of stock price Payoff from call Payoffjrom put Total payoff ST ~ K j Kl < ST < K2 0 0 ST- K2 KI -ST 0 0 KI-ST 0 ST- K 2 ST ~ K2 Trading Strategies Involving Options Figure 10.13 237 Payoff from a butterfly spread. The sale of a strangle is sometimes referred to as a top vertical combination. It can be appropriate 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 potential loss to the investor. 10.4 OTHER PAYOFFS This chapter has demonstrated just a few of the ways in which options can be used to produce an interesting relationship between profit and stock price. If European options expiring at time T were available with every single possible strike price, any payoff function at time T could in theory be obtained. The easiest illustration of this involves a series of butterfly spreads. Recall that a butterfly spread is created by buying options with strike prices K 1 and K 3 and selling two options with strike price K 2 , where K 1 < K 2 < K 3 and K 3 - K 2 = K 2 - K 1 Figure 10.13 shows the payoff from a butterfly spread. 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 of very small spikes, any payoff function can be .approximated. SUMMARY A number of common trading strategies involve a single option and the underlying stock. For example, writing a covered call involves buying the stock and selling a call option 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 two or more puts. A bull spread can be created by buying a call (put) with a low strike price and selling a put (call) with a high strike price. A bear spread can be created by buying a put (call) with a high strike price and selling a put (call) with a low strike price. A butterfly spread involves buying calls (puts) with a low and high strike price and selling two calls (puts) with some intermediate strike price. A calendar spread involves selling a call (put) with a short time to expiration and buying a call (put) with a longer time to expiration. A diagonal spread involves a long position in one option and a short position in 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. A straddle combination involves taking a long position in a call and a long position in a CHAPTER 10 238 put with the same strike price and expiration date. A strip consists of a long position in one call and two puts with the same strike price and expiration date. A strap consists of a 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 and the same expiration date. There are many other ways in which options can be used to produce interesting payoffs. It is not surprising that option trading has steadily . increased in popularity and continues to fascinate investors. FURTHER READING Bharadwaj, A. and J. B. Wiggins. "Box Spread and Put-eall Parity Tests for the S&P Index LEAPS Markets," Journal of Derivatives, 8, 4 (Summer 2001): 62-7l. Chaput, J. S., and L. H. Ederington, "Option Spread and Combination Trading," Journal of Derivatives, 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 of Derivatives, 8, 3 (Spring 2001): 63-75. Ronn, A. G. and E.1. Ronn. "The Box-Spread Arbitrage Conditions," Review of Financial Studies, 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 a protective 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, and expiration dates in 3 months. Their prices are \$4, \$2, and \$~, respectively. Explain how the options can be used to create a butterfly spread. Construct a table showing how profit 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 \$45 costs \$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 to the 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 bear spread? 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 European puts is identical to the cost of a butterfly spread created from European calls. Tradiizg Strategies Involving Options 239 10.12. A call with a strike price of \$60 costs \$6. A put with the same strike price and expiration date costs \$4. Construct a table that shows the profit from a straddle. For what range of stock prices would the straddle lead to a loss? A 10.13. Construct a table showing the payoff from a bull spread when puts with strike prices K I and 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 to the direction. Identify six different strategies the investor can follow and explain the differences among them. 10.15. How can a forward contract on a stock with a particular delivery price and delivery date be created from options? 10.16. "A box spread comprises four options. Two can be combined to create a long forward position and two can be combined to create a short forward position." Explain this statement. . 10.17. What is the result if the strike price of the put is higher than the strike price of the call in a strangle? 10.18. One Australian dollar is currently worth \$0.64. A I-year butterfly spread is set up using European call options with strike prices of \$0.60, \$0.65, and \$0.70. The risk-free interest rates in the United States and Australia are 5% and 4% respectively, and the volatility of the exchange rate is 15%. Use the DerivaGem software to calculate the cost of setting up the butterfly spread position. Show that the cost is the same if European put options are used instead of European call options. Assignment Questions 10.19. 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 butterfly spread can be created. Construct a table showing the profit from the strategy. For what range 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 T2 and selling a call with strike price K I and exercise date TI , where T2 > TI Draw a diagram 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 terminal stock 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 options In each case, assume that the call option has an exercise price equal to the current stock price. 10.22. Suppose that the price of a non-dividend-paying stock is \$32, its volatility is 30%, and the risk-free rate for all maturities is 5% per annum. Use DerivaGem to calculate the cost of setting up the following positions: (a) A bull spread using European call options with strike prices of \$25 and \$30 and a maturity of 6 months 240 CHAPTER 10 (b) A bear spread using European put options with strike prices of \$25 and \$30 and a maturity 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 maturity In each case provide a table showing the relationship between profit and final stock price. Ignore the impact of discounting.

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UCLA - MATH - 174
T ERPrope:rties ofStock OptionsIn this chapter we look at the factors affecting stock option prices. We use a number ofdifferent arbitrage arguments to explore the relationships between European optionprices, American option prices, and the underlyin
UCLA - MATH - 174
T ERMechanics ofOptions MarketsWe introduced options in Chapter 1. This chapter explains how options markets areorganized, what terminology is used, how the contracts are traded, how marginrequirements are set, and so on. Later chapters will examine
UCLA - MATH - 174
S vvapsThe first swap contracts were negotiated in the early 1980s. Since then the market hasseen phenomenal growth. Swaps now occupy a position of central importance in theover-the-counter derivatives market.A swap is an agreement between two compani
UCLA - MATH - 174
T ERIntere:st RateFuturesSo far we have covered futures contracts on commodities, stock indices, and foreigncurrencies. We have seen how they work, how they are used for hedging, and howfutures prices are set. We now move on to consider interest rate
UCLA - MATH - 174
T ERInterest RatesInterest rates are a factor in the valuation of virtually all derivatives and will featureprominently in much of the material that will be presented in the rest of this book. Inthis chapter we cover some fundamental issues concerned
UCLA - MATH - 174
I ntroductionIn the last 25 years derivatives have become increasingly important in the world offinance. Futures and options are now traded actively on many exchanges throughoutthe world. Many different types of forward contracts, swaps, options, and o
UCLA - MATH - 174
CONTENTS IN BRIEFList of Business SnapshotsList of Technical NotesPrefaceVI. Introductionv 2. Mechanics of futures marketsv 3. Hedging, strategies using futuresv 4. Interest ratesv' 5. Determination of forward and futures pricesv6. Interest rate
Lee - ACCT - 403
COURSE SCHEDULEACCT 403Textbook: Hoyle, et al., Advanced Accounting (2011).WEEK/MODULEREADING &amp; STUDY1Hoyle, et al.: chs. 16172 Presentationslecture notes2ASSIGNMENTSPOINTSCourse Requirements ChecklistClass IntroductionsGroup DB Forum 10*
UCLA - MATH - 131a
Analysis Qual: Key Ideas and Problems To-DoWill Feldman, Brent Nelson, Nick Cook, Alan Mackey(Now with more Folland Problems!)(Red is To-Do)Spring 1970:R2. f Lebesgue measurable and continuous at 1. g is in L^1. Evaluate the limit as n goes toinfini
Lee - ACCT - 403
COURSE SYLLABUSACCT 403ADVANCED ACCOUNTING IICOURSE DESCRIPTIONAdvanced problems, involving government and non-profit organizations, estates and trusts,financially distressed entities, translation and consolidation of foreign entities and segmentrep
UCLA - MATH - 131a
Mathematics 170A HW3 Due Tuesday, January 31, 2012.Problems 31,34,39,41,49,52 on pages 60-69.E1 . Suppose the events A, B, C are pairwise independent and satisfy11351P (A) = , P (B ) = , P (C ) = , P (A B C ) = .23448Are A, B, C independent? E
UCLA - MATH - 131a
Mathematics 170A HW2 Due Tuesday, January 24, 2012.Problems 14,16,17,18,19,21,22,23 on pages 56-58.C. A community has m families with children. The largest familyhas k children. For i = 1, ., k , there are ni families with i children, son1 + n2 + + nk
Lee - BUSI - 400
COURSE SCHEDULEBUSI 400Textbook: David, Strategic Management: Concepts and Cases (2011).WEEK/MODULEREADING &amp; STUDY1David: chs.1 &amp; 10OverviewLecture NotesASSIGNMENTSPOINTSCourse Requirements ChecklistIndividual Assurance of Learning Ex. 1012
UCLA - MATH - 131a
Mathematics 170A HW1 Due Tuesday, January 17, 2012.Problems 1, 2, 5, 6, 7, 8, 9, 10 on pages 5354.A. Show that if A and Bn are events, thenA Bn = (A Bn )n=1n=1in two dierent ways:(a) Directly, without using De Morgans laws.(b) Using the result of
Lee - BUSI - 400
COURSE SYLLABUSBUSI 400STRATEGIC PLANNING/BUSINESS POLICYCOURSE DESCRIPTIONThis capstone course for all business majors seeks to integrate the concepts, techniques, andknowledge of all areas of business administration. Its focus is strategic manageme
UCLA - MATH - 131a
151A HW 8 SolutionsWill Feldman1. (a) Calculus.(b) e in both cases by calculus.(c) For composite trapezoid rule the error bound is (for h = 1/N the grid length)1f (x)dx 0h(f (0) +2N 12f (kh) + f (1) k=1h2 max[0,1] |f ( )|.12So in order to
Lee - BUSI - 400
THE EXTERNAL ASSESSMENT1.2.3.4.5.6.7.8.9.10.11.The Nature of an External AuditEconomic ForcesSocial, Cultural, Demographic, and Environmental ForcesPolitical, Governmental, and Legal ForcesTechnological ForcesCompetitive ForcesCompetitiv
UCLA - MATH - 131a
151A HW 7 SolutionsWill Feldman1. This is not dicult just plug the values given in the problem into the second dierence quotient formulaf (x) f (x + h) 2f (x) + f (x h)h2for h = 0.1, 0.01. The interesting part is that the approximation for h = 0.1 i
Lee - BUSI - 400
1.Explain how to conduct an external strategicmanagement audit?.n effective approach for conducting an external strategic-management auditconsists of four basic steps:(1) select key variables,(2) select key sources of information,(3) use forecastin
UCLA - MATH - 131a
151A HW 6 SolutionsWill Feldman1. Let f C n+1 ([a, b]) and p be the interpolating polynomial at the n + 1 equally spaced node pointsa = x0 , ., xn = b with (b a)/n = h = xj +1 xj . Recall that we have the error bound|f (x) p(x)| |f (n+1) ( )| n |x x
Lee - BUSI - 400
MISP2-MMU-TMI3411-WRYPage 1 of 10Date Due: Fri 24 Nov 2006Course: Multimedia Information Strategic Planning (MISP)Course Code: TMI3411Multimedia University, Cyberjaya, Trimester 2, Session 2006/2007.ANSWERS MISP2 ASSIGNMENT NO. 2MISP2-A2Q1 Converge
UCLA - MATH - 131a
151A HW 5 SolutionsWill Feldman1. See book for similar problems.2. Using the Newton forward dierence formula given in the problem statementnskPn (x0 + sh) = f (x0 ) +k=1kf ( x0 )we want to calculate ( 2 Pn )(10) given values for ( k Pn )(0) for
Lee - BUSI - 342
Module 5, DB3RE: Choose a performance management method (i.e. Critical Incident, MBO, BOS, etc.)and use the Internet to locate the website of a company which has recently introduced anew performance management system. Then, assess from the information
UCLA - MATH - 131a
Math 151A HW3 SolutionsWill Feldman1. (a) This is a translation of ln x(b) Noting that f (x) = ln (x + 2) is monotone and continuous we have that ln 2 x ln 4 for x [0, 2] and by the intermediate value theorem every value in[ln 2, ln 4] [0, 2] is obta
UCLA - MATH - 131a
Math 151A HW2 SolutionsWill Feldman1. For part (c) use the error estimate for bisection method |xn x | 2n |a b| where(a, b) is the initial interval and xn = (an + bn )/2 where (an , bn ) is the interval at thenth stage. So if you want error less than
UCLA - MATH - 131a
Math 151A HW1 SolutionsWill Feldman1. We want to nd the third order Taylor polynomial for the function f (x) =1+xabout x0 = 0. Recall the formula for the Taylor polynomial of order n centered at x0is given bynPn (x) =k=0f (k) (x0 )(x x0 )k .k!
UCLA - MATH - 131a
Math 131AProblem Set #3Due February 1Exercises 9.1(b), 9.4, 9.9, 9.10, 9.11, 9.14 from the textbook. (In 9.14, you can assumethe result of 9.12 even though I didnt assign it.)Additional Problem: Suppose that (sn ) and (tn ) are sequences such that, f
UCLA - MATH - 131a
Math 131AProblem Set #2Due January 25Exercises 7.4, 8.1(b)(c), 8.2(b)(e), 8.4, 8.7(a), 8.9, 8.10 from the textbookAdditional Problem 1: (Uniqueness of Limits) We were a bit casual in lecture whenspeaking of the number L being the limit of the sequenc
UCLA - MATH - 131a
Math 131AProblem Set #1Due January 20Exercises 1.8, 1.11, 2.4, 3.5, 3.8, 4.1-4.4(a)(b)(e)(k)(v), 4.11, 4.12, 4.14 from thetextbookAdditional Problem 1: (More Practice with Induction) Prove, by induction, that if p(x)is a polynomial of degree n 1 wit
UCLA - MATH - 131a
The Real Numbers as Decimal ExpansionsEven though dening the real numbers as innite decimal expansions makesproving that they are a complete ordered eld dicult, most people think ofthem that way. So lets dene a real number to be + or an integer, follow
UCLA - MATH - 131a
CardinalityThe introduction of cardinal numbers by Georg Cantor in the 1870s made itpossible to compare the size of innite sets, and say that some are larger thanothers. The crucial denition for this is that two sets are the same size havethe same car
UCLA - MATH - 131a
Mathematics 131A/1Welcome to Analysis! This course will be devoted to the theoretical basis forcalculus as stated in the title of our textbook, Elementary Analysis: The Theoryof Calculus by Kenneth A Ross. The topics that I plan (hope) to cover are: Th
UCSB - ECON - 100 B
HOMEWORK 2,OPERATIONS RESEARCHJanuary 18, 2012.Name: Due on Monday, January 23. Write the number of the assignment at the top of the rst page of yourassignment. Dont forget to write down your name. Write the complete statement of each question as
UCSB - ECON - 100
Adam CaveGeometry 300Classifying Models of Incidence Geometry up to IsomorphismThis paper will focus on classifying all the possibilities of four and five pointmodels of incidence geometry up to isomorphism. The idea for this paper arose whileworking
CUHK - SOCI - 1001
Chapter 3 - Culture andS ociety:CULTURE S HOCKVivian DuongChaewon KimDavid LawSOCI1001B; Term 1 2011-2012http:/indiequill.files.wordpress.com/2008/06/lightning1.jpgCulture Shock, defined:&quot;The phrase were one feels disorientated,uncertain, out of
CUHK - SOCI - 1001
Chapter 3 - Culture and Society:CULTURE SHOCKVivian DuongChaewon KimDavid LawSOCI1001B; Term 1 2011-2012http:/indiequill.files.wordpress.com/2008/06/lightning1.jpgCulture Shock, defined:&quot;The phrase were one feels disorientated,uncertain, out of p
CUHK - SOCI - 1001
Van DuongSOCI 1001B Intro to SociologyDue October 11, 2011Term 1, 2011-2012Biography: History of the MakingIn the Sociological Imagination, C. Wright Mills (1959) discusses in the chapter, ThePromise, the relationship between history of the world an
CUHK - ANTH - 2350
Duong 1Van Vivian DuongDec. 3, 2011ANT 2350: Meanings of LifeFall 2011 Due Dec. 12, 2011Take-Home Final Examination1. Briefly compare Confucius, Calvin, Condorcet, Zhuangzi, and Huxley as to their sensesof what an ideal human society would look lik
CUHK - ANTH - 2350
Van Vivian DuongStudent ID# 1155014746ANTH 2350 Meanings of LifeTake-Home Midterm ExamTerm 1, Fall 2011 - Due October 31, 20111. How does capitalism shape the ikigai of work and family in at least three differentsocieties today?As mentioned in the
CUHK - MEA - 4
Duong 1Van Vivian DuongMAEG4030 Heat TransferAssignment 2Fall 2011Due Nov. 16, 20111.a. Solve the above problem using Matlab pdepe function. Describe your approach (thedefinitions of the various input parameters of the function) and include your M
CUHK - MEA - 4
Duong 1Van Vivian DuongMAEG4030 Heat TransferAssignment 2Fall 2011Due Nov. 16, 20111.a. Solve the above problem using Matlab pdepe function. Describe your approach (thedefinitions of the various input parameters of the function) and include your M
CUHK - MEA - 4
Van Vivian DuongMAEG4030 Heat TransferAssignment 2Fall 2011Due Nov. 16, 20111.a. Solve the above problem using Matlab pdepe function. Describe your approach (thedefinitions of the various input parameters of the function) and include your Matlabco
CUHK - MEA - 4
heatloss vs. thickness4.50E+084.00E+083.50E+08he a t los s ( J)thickness (m) heatloss (J)0.0013931200000.0021965600000.0031310400000.004982800000.005786240000.006655200000.007561600000.008491400000.009436800000.01393120003.00E+08
CUHK - MEA - 4
MAE2010ComputerAidedDraftingMAE2010ProjectDescriptionandRequirementPartA)ProposalSubmissiondue:17Oct,2011(5pm)PartB)WrittenReportdue:21Nov,2011(5pm)IntroductionEachstudentisrequiredtojoinagroupofthreeorfourstudentstocarryoutadesignproject. Eachgroup
CUHK - MEA - 4
Van DuongMay 31, 2011Bio 1; Section 03DHW Assignment# 1The Scientific Method: Organic BananasObservations:I bought whole bananas a few days ago when they were still green and unripe. They weremarked as organic bananas grown on a Doles farm and sold
CUHK - MEA - 2
MAE 2010 Course Project Group (ERB105 Dec 5)Group No.1MemberMemberMemberMemberNg Yan WaiZhang Pei XinMok Siu Cheong2Cheuk Tsz YingTam Shek HeiLam Wing HangCheung Ka Chun3Chan Tsz KitTse Wing NamCheung Lok YinJin Linyuan4Wong Man HoIV
UCM - BIO - 1
Things to go over:o Mutation Errors Point mutation- Silent mutation no change in amino acid sequence; redundancy of genetic code- Missense mutation base substitutions change the genetic code such that one amino acid substitutesfor another in a protein
UCM - BIO - 1
Lecture 16 Animals and EcologyChordates Reptiles alligators, turtles, snakes, lizards, tuataras Body covered with hard, keratinized scales Protect animal from desiccation and from predators Fertilization internal, female lays shelled egg Amniotic eg
UCM - BIO - 1
Van DuongSummer 2011Bio 1Due June 27, 2011HW Assignment 3 &amp;4Research Article Critique: Biological Effects of Offshore Oil ProductionI. SummaryIn the research article titled, Biomarkers in Natural Fish Populations Indicate AdverseBiological Effects
George Mason - CHEMISTRY - 211
Purpose:The purpose of this experiment was to use a spectrometer to determine the concentration of aunknown nickel solution. This was achieved by making a series of standard solutions andplotting a calibration curve and then using this information to d
George Mason - CHEMISTRY - 212
Calculations:Vml at equivalence point for HCl:Theoretical endpoint for HCl: 25.0mlPercent error: 3.6%Vml at equivalence point for acetic acid:Theoretical endpoint for acetic acid: 25.0mlPercent error: 0.4%pKa = pH so Ka for acetic acid from titrati
UCM - BIO - 1
Van DuongJune 6, 2011Bio 1; Section 03HW Assignment# 2Designing an ExperimentFor homework assignment one, the objective and goal was to understand and utilize thescientific method. We were assigned to observe something biological, write down ourobs
George Mason - CHEMISTRY - 212
Conclusion:The purpose of this lab was to determine the mass present of an unknown sample of a coppercontaining compound made from a known mass of a copper containing compound, copperglycinate. This was accomplished by performing a titration with thios
CUHK - SOCI - 1001
From: Van Thuc Duong3125 Kermath DriveSan Jose, CA 95132To:Superior Court of Merced, County of MercedMerced Traffic Division Traffic School670 West 22nd Street, Room 6Merced, CA 95340Case No. TRM081519Citation No. 81121NPSubject: Request to take
George Mason - CHEMISTRY - 212
Tables and Graphs:Table 1:Crystal Violet Absorbance Versus Time NaOH=0.0250M Trial 1% Transmittance(%T)Time (min.)A=-log(%T/100)InA1/A168.20.166-1.796.02272.60.139-1.977.19376.80.115-2.178.72480.20.0958-2.3510.4583.60.0778
CUHK - SOCI - 1001
Duong, Van Thuc; TRM081519Defendants Name: Van Thuc DuongCase Number: TRM081519Citation Number: 81121NPSpecifics: CVC 22349(a), Exceed 65 mphStatement of Facts:I respectfully submit this written declaration to the Court pursuant to CVC 40902. I plea
George Mason - CHEMISTRY - 212
Calculations:Cu(s) | Cu2+(aq) | Ag+(aq) | Ag(s)The half reactions:Cu(s) Cu2+(aq) + 2e- oxidation (black) E = -0.339Ag+(aq) + e- Ag(s) reduction (red) E = 0.7993VCu(s) + 2Ag+(aq) Cu2+(aq) + 2Ag(s) Ecell = +0.463Concentration: silver 0.1M, copper 0.1M
CUHK - SOCI - 1001
Duong, Van Thuc; 77899LKDefendants Name: Van Thuc DuongCase/Citation No.: 77899LKSpecifics: VC 22349(a), Exceed 65 mphSummary:I respectfully submit this written declaration to the Court pursuant to CVC 40902. I pleadnot guilty to the charge of viola
George Mason - CHEMISTRY - 212
Calculations:Initial concentration (NCS)1- solution #4:M2 = =Initial concentration of Fe3+ solution #4:Equilibrium concentration [Fe(NCS)]2+ =0.231= 4720.9x + 0.0008x= 4.88x10-5Equilibrium concentration Fe3+ (M) =3.6 x 10-4 4.88x10-5 = 3.1 x 10-4
George Mason - CHEMISTRY - 212
Purpose:In this calorimetry experiment, the heat of formation of magnesium oxide was determined usingcalorimetry and Hesss law. The results obtained during the experiment were compared to theknown Hf of MgO was used to calculate the percent error of th
George Mason - CHEMISTRY - 212
Purpose:Identification of an unknown metal sample, this unknown metal sample will be reacted with astrong aqueous acid; this will result in gas production. Collecting the sample of gas will takeplace over water and then using the Ideal Gas Law and stoi