2 Payoff from a bear spread
Stock price Payoff from Payoff from Totalrange long cail option short call option payoffS7 > Ky Sp — Ky K, — Sr —(K2 — Ki)Ki < Sr < Ky 0 Ki — Sr —(Sr — Ki)Sr< Ki 0 0 0
Example 9.2 An investor buys for $1 a call with a strike price of $35 and sells for $3 a call with astrike price of $30. The payoff from this bear spread strategy is —$5 if the stock price is above $35 andzero if it is below $30. If the stock price is between $30 and $35, the payoff is —(S; — 30). Theinvestment generates $3 — $1 = $2 up front. The profit is therefore as follows:
Stock price range Profit
Sr < 30 +230 < S7 < 35 32 — SrSr > 35 —3
Like bull spreads, bear spreads limit both the upside profit potential and the downside risk. Bearspreads can be created using puts instead of calls. The investor buys a put with a high strike priceand sells a put with a low strike price, as illustrated in Figure 9.5. Bear spreads created with putsrequire an initial investment. In essence, the investor has bought a put with a certain strike priceand chosen to give up some of the profit potential by selling a put with a lower strike price. Inreturn for the profit given up, the investor gets the price of the option sold.
Butterfly Spreads
A butterfly spread involves positions in options with three different strike prices. It can be created bybuying a call option with a relatively low strike price, K,; buying a call option with a relatively highstrike price, K;; and selling two call options with a strike price, K,, halfway between K, and K3.
 
Figure 9.5 Bear spread created using put optionsTrading Strategies Involving Options 191
 
 
Figure 9.6 Butterfly spread using call options
Generally K> is close to the current stock price. The pattern of profits from the strategy is shown inFigure 9.6. A butterfly spread leads to a profit if the stock price stays close to K, but gives rise to asmaii loss if there is 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 strategy requires asmall investment initially. The payoff from a butterfly spread is shown in Table 9.3.
Suppose that a certain stock is currently worth $61. Consider an investor who feels that asignificant price move in the next six months is unlikely. Suppose that the market prices of six-monthcalls are as follows:
 
Strike price ($) Call price (S)55 1060 765 5
The investor could create a butterfly spread by buying one call with a $55 strike price, buyingone 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 six 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 stockprice is between $56 and $64, a profit is made. The maximum profit, $4, occurs when the stockprice in six months is $60.
Table 9.3 Payoff from a butterfly spread
 
Stock price Payoff from Payoff from Payoff from Totalrange first long call second long call short calls _payoff*Sr < Ky 0 0 0 0K, < Sr < K2 Sr — Ky 0 0 Sr— KiKy < Sp < K3 Sp —K, 0 —2(S7 — Ky) K3—SrSr > K3 S;— Ky Sr — K3 —2(Sr — K2) 0
 
* These payoffs are calculated using the relationship K2 = 0.5(K, + K3).192 CHAPTER 9
Profit
 
 
Figure 9.7 Butterfly spread using put options
Butterfly spreads can be created using put options. The investor buys a put with a low strikeprice, buys a put with a high strike price, and sells two puts with an intermediate strike price, asillustrated in Figure 9.7. The butterfly spread in the example just considered would be created bybuying a put with a strike price of $55, buying a put with a strike price of $65, and selling two putswith a strike price of $60. If all options are European, the use of put options results in exactly thesame spread as the use of call options. Put-call parity can be used to show that the initialinvestment is the same in both cases.
A butterfly spread can be sold or shorted by following the reverse strategy. Options are sold withstrike prices of K, and K3, and two options with the middle strike price K, are purchased. Thisstrategy produces a modest profit if there is a significant movement in the stock price.
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 differentexpiration dates.
A calendar spread can be created by selling a call option with a certain strike price and buying alonger-maturity call option with the same strike price. The longer the maturity of an option, themore 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 theshort-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 9.8. Thepattern is similar to the profit from the butterfly spread in Figure 9.6. The investor makes a profit ifthe 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 orsignificantly below this strike price.
To understand the profit pattern from a calendar spread, first consider what happens if the stockprice is very low when the short-maturity option expires. The short-maturity option is worthlessand the value of the long-maturity option is close to zero. The investor therefore incurs a loss thatis close to the cost of setting up the spread initially. Consider next what happens if the stockprice, S;, is very high when the short-maturity option expires. The short-maturity option costs theTrading Strategies Involving Options 193
t Profit
 
 
 
Figure 9.8 Calendar spread created using two calls
investor Sy — K, and the long-maturity option (assuming early exercise is not optimal) is worth alittle more than S; — K, where K is the strike price of the options. Again, the investor makes a netloss that is close to the cost of setting up the spread initially. If $; is close to K, the short-maturityoption costs the investor either a small amount or nothing at all. However, the long-maturityoption 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 bullishcalendar spread involves a higher strike price, whereas a bearish calendar spread involves a lowerstrike price.
Calendar spreads can be created with put options as well as call options. The investor buys along-maturity put option and sells a short-maturity put option. As shown in Figure 9.9, the profitpattern is similar to that obtained from using calls.
A Profit‘x
 
 
Figure 9.9 Calendar spread created using two puts194 CHAPTER 9
A reverse calendar spread is the opposite to that in Figures 9.8 and 9.9. The investor buys a short-maturity option and sells a long-maturity option. A small profit arises if the stock price at theexpiration 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 shortposition in another call. In the case of bull and bear spreads, the calls have different strike pricesand the same expiration date. In the case of calendar spreads, the calls have the same strike priceand different expiration dates. In a diagonal spread both the expiration date and the strike price ofthe calls are different. This increases the range of profit patterns that are possible.
9.3_| COMBINATIONS
 
A combination is an option trading strategy that involves taking a position in both calls and putson 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 strikeprice and expiration date. The profit pattern is shown in Figure 9.10. The strike price is denotedby K, If the stock price is close to this strike price at expiration of the options, the straddle leads toa loss. However, if there is a sufficiently large move in either direction, a significant profit willresult. The payoff from a straddle is calculated in Table 9.4.
A straddle is appropriate when an investor is expecting a large move in a stock price but does notknow in which direction the move will be. Consider an investor who feels that the price of a certainstock, currently valued at $69 by the market, will move significantly in the next three months. Theinvestor could create a straddle by buying both a put and a call with a strike price of $70 and anexpiration date in three months. Suppose that the call costs $4 and the put costs $3. If the stockprice stays at $69, it is easy to see 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
| Profit
 
Figure 9.10 A straddleTrading Strategies Involving Options 195
Range of Payoff from Payoff from Total
 
stock price call put payoffSr <K 0 K-—Sr K ~ Sy
Sp>K Sy —-—K 0 Sr-K
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.
A straddle seems to be a natural trading strategy to use when a big jump in the price of acompany’s stock is expected, for example, when there is a takeover bid for the company or whenthe outcome of a major lawsuit is expected to be announced soon. However, this is not necessarilythe case. If the general view of the market is that there will be a big jump in the stock price soon,that view will be reflected in the prices of options. An investor will find options on the stock to besignificantly more expensive than options on a similar stock for which no jump is expected. For astraddle to be an effective strategy, the investor must believe that there are likely to be bigmovements in the stock price and these beliefs must be different from those of most other marketparticipants.
The straddle in Figure 9.10 is sometimes referred to as a bottom straddle or straddle purchase. Atop straddle or straddle write is the reverse position. It is created by selling a call and a put with thesame exercise price and expiration date. It is a highly risky strategy. If the stock price on theexpiration date is close to the strike price, a significant profit results. However, the loss arising froma large move in either direction is unlimited.
Strips and Straps
A strip consists of a long position in one call and two puts with the same strike price and expirationdate. A strap consists of a long position in two calls and one put with the same strike price and
Profit Profit
(a) (b)
Figure 9.11 Profit patterns from (a) a strip and (b) a strap196 CHAPTER 9
 
 
 
Figure 9.12 A strangle
expiration date. The profit patterns from strips and straps are shown in Figure 9.11. In a strip theinvestor is betting that there will be a big stock price move and considers a decrease in the stockprice to be more likely than an increase. In a strap the investor is also betting that there will be abig stock price move. However, in this case, an increase in the stock price is considered to be morelikely than a decrease.
Strangles
in a strangle, sometimes called a bottom vertical combination, an investor buys a put and a cali withthe same expiration date and different strike prices. The profit pattern that is obtained is shown inFigure 9.12. The call strike price, K>, is higher than the put strike price, K,. The payoff function fora strangle is calculated in Table 9.5.
A strangle is a similar strategy to a straddle. The investor is betting that there will be a large pricemove, but is uncertain whether it will be an increase or a decrease. Comparing Figures 9.12and 9.10, we see that the stock price has to move farther in a strangle than in a straddle for theinvestor to make a profit. However, the downside risk if the stock price ends up at a central value isless 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 movefor a profit to be realized.
The sale of a strangle is sometimes referred to as a top vertical combination. It can be appropriatefor an investor who feels that large stock price moves are unlikely. However, as with sale of astraddle, it is a risky strategy involving unlimited potential loss to the investor.
Table 9.5 Payoff from a strangle
 
Range of Payoff from Payoff from Totalstock price call put payoffSr < Ki 0 Ki -—Sr K, — SrKy < Sr < Ky 0 0 0
5, > Ky Sy — Ky 0 Sp — KpTrading Strategies Involving Options 197
9.4 OTHER PAYOFFS
 
 
This chapter has demonstrated just a few of the ways in which options can be used to produce aninteresting relationship between profit and stock price. If European options expiring at time T wereavailable with every single possible strike price, any payoff function at time T could in theory beobtained. The easiest illustration of this involves a series of butterfly spreads. Recall that a butterflyspread is created by buying options with strike prices K, and K3 and selling two options with strikeprice K, where K,; < K, < K3 and K; — K) = K, — K,. Figure 9.13 shows the payoff from abutterfly spread. The pattern could be described as a spike. As K, and K3 move closer together, thespike becomes smaller. Through the judicious combination of a large number of very small spikes,any payoff function can be approximated.
Payoff
K, Ky Ks Sp
Figure 9.13 Payoff from a butterfly spread
SUMMARY
 
 
A number of common trading strategies involve a single option and the underlying stock. Forexample, 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 sellinga 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 moreputs. A bull spread can be created by buying a call (put) with a low strike price and selling a call(put) with a high strike price. A bear spread can be created by buying a call (put) with a high strikeprice and selling a call (put) 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. Acalendar 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 ashort position in another option such that both the strike price and the expiration date aredifferent.
Combinations involve taking a position in both calls and puts on the same stock. A straddlecombination involves taking a long position in a call and a long position in a put with the samestrike price and expiration date. A strip consists of a long position in one call and two puts with thesame strike price and expiration date. A strap consists of a long position in two calls and one putwith the same strike price and expiration date. A strangle consists of a long position in a call and aput with different strike prices and the same expiration date. There are many other ways in whichoptions can be used to produce interesting payoffs. It is not surprising that option trading hassteadily increased in popularity and continues to fascinate investors.198 CHAPTER 9
SUGGESTIONS FOR FURTHER READING
 
 
Bookstaber, R. M., Option Pricing and Strategies in Investing, Addison-Wesley, Reading, MA, 1981.Degler, W. H., and H. P. Becker, “19 Option Strategies and When to Use Them,” Futures, June 1984.McMillan, L. G., Options as a Strategic Investment, New York Institute of Finance, New York, 1992.
Scott, J. S., and L. H. Ederington, “Option Spread and Combination Trading,” Working Paper,University of Otago, March 2002
Slivka, R., “Call Option Spreading,” Journal of Portfolio Management, 7 (Spring 1981), 71-76.Welch, W. W., Strategies for Put and Call Option Trading, Winthrop, Cambridge, MA, 1982.
Yates, J. W., and R. W. Kopprasch, “Writing Covered Call Options: Profits and Risks,” Journal ofPortfolio Management, 6 (Fall 1980), 74-80.
QUESTIONS AND PROBLEMS(ANSWERS IN SOLUTIONS MANUAL)
 
 
9.1. What is meant by a protective put? What position in call options is equivalent to a protective put?
9.2. Explain two ways in which a bear spread can be created.
9.3. When is it appropriate for an investor to purchase a butterfly spread?
9.4. Call options on a stock are available with strike prices of $15, $174, and $20 and expiration datesin three months. Their prices are $4, $2, and $4, respectively. Explain how the options can be usedto create a butterfly spread. Construct a table showing how profit varies with stock price for thebutterfly spread.
9.5. What trading strategy creates a reverse calendar spread?
9.6. What is the difference between a strangle and a straddle?
9.7. 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 fromthe strangle?
9.8. Use put—call parity to relate the initial investment for a bull spread created using calls to the initialinvestment for a bull spread created using puts.
9.9. Explain how an aggressive bear spread can be created using put options.9.10. 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 tablethat shows the profit and payoff for both spreads.
9.11. Use put-call parity to show that the cost of a butterfly spread created from European puts isidentical to the cost of a butterfly spread created from European calls.
9.12. Acall 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 wouldthe straddle lead to a loss?
9.13. Construct a table showing the payoff from a bull spread when puts with strike prices K, and Ky(Ky > Ky) are used.
9.14. An investor believes that there will be a big jump in a stock price, but is uncertain as to the direction.Identify six different strategics the investor can follow and explain the differences among them.Trading Strategies Involving Options 199
9.15.
9.16,
9.18.
How can a forward contract on a stock with a particular delivery price and delivery date becreated from options?
A box spread is a combination of a bull call spread with strike prices K, and K» and a bear putspread with the same strike prices. The expiration dates of all options are the same. What are thecharacteristics of a box spread?
. What is the result if the strike price of the put is higher than tho strike price of the call in a
strangle?
One Australian dollar is currently worth $0.64. A one-year butterfly spread is set up usingEuropean call options with strike prices of $0.60, $0.65, and $0.70. The risk-free interest rates inthe United States and Australia are 5% and 4%, respectively, and the volatility of the exchangerate is 15%. Use the DerivaGem software to calculate the cost of setting up the butterfly spreadposition. Show that the cost is the same if European put options are used instead of European calloptions.
ASSIGNMENT QUESTIONS
 
 
9.19.
9.20,
9.21.
9.22.
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 thebutterfly spread lead tu a loss?A diagonal spread is created by buying a call with strike price Kz and exercise date T, and selling acall with strike price K, and exercise date T; (J, > T;). Draw a diagram showing the profit when(a) K, > K, and (b) K, < Kj.Draw a diagram showing the variation of an investor’s profit and loss with the terminal stockprice 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 current stock price.
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 thefollowing positions. In each case provide a table showing the relationship between profit and finalstock price. Ignore the impact of discounting.a. A bull spread using European call options with strike prices of $25 and $30 and a maturityof six monthsb. A bear spread using European put options with strike prices of $25 and $30 and a maturityof six months. A butterfly spread using European call options with strike prices of $25, $30, and $35 andamaturity of one yeard. A butterfly spread using European put options with strike prices of $25, $30, and $35 and amaturity of one yearA straddle using options with a strike price of $30 and a six-month maturity
A cteanole aeine antone with stike areee of aid and a <ixemonth mattyA SUfaigie Usiig Options with strike prices aL $30 and $35 and a six-month maturity.
o
moCHAPTER 10
INTRODUCTION TO BINOMIALTREES
A useful and very popular technique for pricing a stock option involves constructing a binomialtree. This is a diagram that represents different possible paths that might be followed by the stockprice over the life of the option. In this chapter we will take a first look at binomial trees and theirrelationship to an important principle known as risk-neutral valuation. The approach we adopthere is similar to that in an important paper published by Cox, Ross, and Rubinstein in 1979.
The material in this chapter is intended to be introductory. More details on the use of numericalprocedures involving binomial trees are given in Chapter 18.
10.1 A ONE-STEP BINOMIAL MODEL
 
We start by supposing that we are interested in valuing a European call option to buy a stock for $21in three months. A stock price is currently $20. We make a simplifying assumption that at the end ofthree months the stock price will be either $22 or $18. This means that the option will have one oftwo values at the end of the three months. If the stock price turns out to be $22, the value of theoption will be $1; if the stock price turns out to be $18, the value of the option will be zero. Thesituation is illustrated in Figure 10.1.
It turns out that an elegant argument can be used to price the option in this situation. The onlyassumption needed is that no arbitrage opportunities exist. We set up a portfolio of the stock and
Stock price = $22Option price = $1
Stock price = $20
Stock price = $18
Option price = $0
Figure 10.1 Stock price movements in numerical example
200{Introduction to Binomial Trees 201
the option in such a way that there is no uncertainty about the value of the portfolio at the end ofthe three months. We then argue that, because the portfolio has no risk, the return it earns mustequal the risk-free interest rate. This enables us to work out the cost of setting up the portfolio andtherefore the option’s price. Because there are two securities (the stock and the stock option) andonly two possible outcomes, it is always possible to set up the riskless portfolio.
Consider a portfolio consisting of a long position in A shares of the stock and a short position inone call option. We calculate the value of A that makes the portfolio riskless. If the stock pricemoves up from $20 to $22, the vaiue of the shares is 22A and the vaiue of the option is i, so thatthe total value of the portfolio is 224 — 1. If the stock price moves down from $20 to $18, thevalue of the shares is 18A and the value of the option is zero, so that the total value of theportfolio is 18A. The portfolio is riskless if the value of A is chosen so that the final value of theportfolio is the same for both alternatives. This means
22A—1=18AorA= 0.25
A riskless portfolio is therefore
Long: 0.25 sharesShort: 1 option
If the stock price moves up to $22, the value of the portfolio is22 x 0.25-1=4.5If the stock price moves down to $18, the value of the portfolio is18 x 0.25=4.5
Regardless of whether the stock price moves up or down, the value of the portfolio is always 4.5 at- the end of the life of the option.
Riskless portfolios must, in the absence of arbitrage opportunities, earn the risk-free rate ofinterest. Suppose that in this case the risk-free rate is 12% per annum. It follows that the value of
the portfolio today must be the present value of 4.5, or4.5 PPI? = 4,367
The value of the stock price today is known to be $20. Suppose the option price is denoted by f.The value of the portfolio today is20x 0.25- f=5-—fIt follows that5 — f = 4.367orf = 0.633
This shows that, in the absence of arbitrage opportunities, the current value of the option must be
0.633. If the value of the option were more than 0.633, the portfolio would cost less than 4.367 to* set up and would earn more than the risk-free rate. If the value of the option were less than 0.633,
shorting the portfolio would provide a way of borrowing money at less than the risk-free rate.202 CHAPTER 10
 
A Generalization
We can generalize the argument just presented by considering a stock whose price is Sy and anoption on the stock whose current price is f. We suppose that the option lasts for time 7 and thatduring the fife of the option the stock price can either move up from Sp to a new level Sou or downfrom Sy to a new level Sod, where « > 1 and d < 1. The proportional increase in the stock pricewhen there is an up movement is w — 1; the proportional decrease when there is a down movementis | ~ d. If the stock price moves up to Su, we suppose that the payoff from the option is /,; if thestock price moves down to Spd, we suppose the payoff from the option is f,. The situation isillustrated in Figure 10.2.
As before, we imagine a portfolio consisting of a long position in A shares and a short positionin one option. We calculate the value of A that makes the portfolio riskless. If there is an upmovement in the stock price, the value of the portfolio at the end of the life of the option is
SouA — fyIf there is a down movement in the stock price, the value becomes
SodA — faThe two are equal whenSpud — fy = SpdA — fyor
fu — faA= 10.1Sou — Sod (10.1)In this case, the portfolio is riskless and must earn the risk-free interest rate. Equation (10.1) showsthat A is the ratio of the change in the option price to the change in the stock price as we movebetween the nodes.Tf we denote the risk-free interest rate by r, the present value of the portfolio is
(Soud — fue”The cost of setting up the portfolio isSoA — fIt follows thatSpA ~ f = (Spud ~ fyen™
SoufuSafSodSa
Figure 10.2 Stock and option prices in a general one-step treeIntroduction to Binomial Trees 203
orf= SpA ~ (Sud — fer"
Substituting for A from equation (16.1) and simplifying reduces this equation to
 
f=e" ph tQ-p) fal (10.2)whereedp= (10.3)
Equations (10.2) and (10.3) enable an option to be priced using a one-step binomial model.In the numerical example considered previously (see Figure 10.1), w= 1.1, d=0.9, r= 0.12,T =0.25, f, = 1, and f, = 0. From equation (10.3),
eh l2X4/12_ 9.9
pap = 0.6523
and, from equation (10.2),f = 70 ?025(9 6523 x 1 +.0.3477 x 0) = 0.633
The result agrees with the answer obtained earlier in this section.
irrelevance of the Stock’s Expected Return
The option-pricing formula in equation (10.2) does not involve the probabilities of the stock pricemoving up or down. For example, we get the same option price when the probability of an upwardmovement is 0.5 as we do when it is 0.9. This is surprising and seems counterintuitive. It is naturalto assume that, as the probability of an upward movement in the stock price increases, the valuc ofacall option on the stock increases and the value of a put option on the stock decreases. This is notthe case.
The key reason is that we are not valuing the option in absolute terms. We are calculating itsvalue in terms of the price of the underlying stock. The probabilities of future up or downmovements are already incorporated into the price of the stock. It turns out that we do not needto take them into account again when valuing the option in