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Unformatted text preview: 10.1 Trading Strategies
Invoiving Options We discussed the proﬁt pattern from an investment in a single stock option in
Chapter 8. In this 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. Amer
ican 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 examine 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 payoff function 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 minus the initial cost.) 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 10.1(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 nan 224 CHAPTER 10 Figure 10.1 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; (0) long
position in a put combined with long position in a stock; ((1) short position in a put
combined with short position in a stock. Proﬁt ll’Long
ll Stock I (b) Proﬁt \ Short (C) (d) a covered call. In Figure 10.1(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 10.1(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 Trading Strategies Involving Options 10.2 225 put—call parity relationship is p+S0=c+KeTrT + D (10.1) where p is the price of a European put, S0 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 riskfree 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’” + D) of ' cash. This explains why the proﬁt pattern in Figure 10.1(c) is similar to the proﬁt pattern from a long cali position. The position in Figure 10.1(d) is the reverse of that in
Figure 10.1(0) and therefore leads to a proﬁt pattern similar to that from a short call
position. Equation (10.1) can be rearranged to become Soc=K6"'T+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 (2 Ke‘” + D) of cash. This
equality explains why the proﬁt pattern in Figure 10.1(a) is similar to the proﬁt
pattern from a short put position. The position in Figure 10.1(b) is the reverse of
that in Figure 10.1(a) and therefore leads to a proﬁt pattern similar to that from a
long put position. 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 226 CHAPTER 10 Table 10.1 Payoff from a bull spread created using calls. Stock price Payoﬁ’ from Payoﬂ from Total
range long, call option short call option payoﬂ
ST>K2 ST‘"K1 “(Sr—K2) Kz—Ki
K1<ST<K2 ST~K1 0 ST—K1
ST g K1 0 0 0 ‘ 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 K1 is the strike price of the call option bought, K2 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 payoﬂ‘ 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 K2 — K1. If the stock price
on the expiration date lies between the two strike prices, the payoff is ST — K1. 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 K2 (K2 > K1). In return for giving up the upside potential, the investor gets the Figure 10.3 Proﬁt from bull spread created using put options. T Proﬁt Trading Strategies Involving Options 227 price of the option with strike price K2. 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 payoﬁ" (= K2 — K1). 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 S 30 —2
30<ST<35 87—32
ST 2 35 3 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 upfront cash ﬂow to the investor
(ignoring margin requirements) and a payoﬁ‘ 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 buyinga 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 1 0.4 Proﬁt from bear spread created using put options. Proﬁt 228 ' CHAPTER 10 Table 10.2 Payoﬁ" from a bear spread created with put options. Stock price Payoﬂ from Payoﬂﬁ‘om Total
range longput option short put option payoﬂ
ST 2 K2 0 0 0
K1<ST<K2 Rig51" 0 Kg—ST Sr<K1 Kz“Sr "(Kl"ST) K2"'K1 price of the option purchased is always less than the strike price of the option sold.) In
Figure 10.4, 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 K1 and K2, with K; < 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 K2, the payoff is zero. If the stock price is less than K1, the payoff is
K2 — K1. If the stock price is betWCen K I and K2, the payoff is K2 — ST. 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 payoif 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
ST g 30 +3 30<ST<35 33—87'
57 2 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 K1 and K2 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 K3 — K1. The value of a box spread is therefore always the
present value of this payoff or (K2 ~— K 1)e‘rT. 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 Trading Strategies Involving Options 229 Figure 1 0.5 Proﬁt from bear spread created using call options. buy the box. This involves buying a call with strike price K1, buying a put with strike
price K2, selling a call with strike price K2, and selling a put with strike price K ,. 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 K2, buying a put with strike price K1, selling a call with
strike price K1, and selling a put with strike price K2. 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 showu in
Business Snapshot 10.1, inexperienced traders who treat American options as European
are liable to lose money. Butterfly Spreads A buttelﬁy spread involves positions in options with three different strike prices. It can
be created by buyinga call option with a relatively low strike price, K1, buying a call
option with a relatively high strike price, K3, and selling two call options with a strike
price, K2, halfway between K1 and K3. Generally K2 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 K2, 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 1 0.3 Payoff from a box spread. Stock price Payoﬂ from Payoﬁ’ from Total
range bull call spread bear put spread payoﬂ
ST>K2 Kg—Kl 0 KgKl
K1<ST<K2 ST—Kl Kg—ST K2—K1 ST<K1 0 Kz—Ki KZ—Kl «:55; 174.23 230 CHAPTER 10 Business SnapshotJOJgymLosing MbncyzwithBoxSpreads _ Suppose that a stockhas a‘price of $50 and a volatility of, 30%. No dividends are
expected and the risk—free rate is 8%. A traderroffers you the chanceto sell on the
CBOE a 2~rnonth box“ spread where the Strike ‘pricesai'e $55 ﬁ'and' $60'for $5.10.
Should you do the trade? , " ' r _ '_ ff ,' My ‘ 4‘ ‘ V' V ,, The trade certainly. sounds attractive. In this case K1 55,’ K: t“ 60, and the payoff
is certain to be $5 1112 months. By selling the box [spread forr$5L10jand inveSting the
funds for 2 monthsyou would h’avemore than enough funds torneet the $5 payoff in
2 months. The theoretical value of the box spreadltoday is SVXfefQ‘VVGS’FZ/‘z:$4.93. ' Unfortunately there is a snag. CBOE‘stook optiOns are American and the $5payolf
from the boxspread is Calculated on" the'rassumptimi that'the options CompriSingthe
‘box are European. Option prices for this example'(calculated using DerivaGem) are;
shown in Table [10.4.24 bull call spread where‘the strike prices are $55'and $60'COS’ES
0.96— 0.26: $0.70. (This is the SSame, fOr both 'VEuropean'andgAinerican“ options
because, as we saw in Chapter 9, the price Of a‘EurOPean callis the same as theipriceof '
an American'cgill Whent'there are'no 'dividye‘_nds.)‘A bearVp'utlspreadgwiththe Same strike
prices c'osts9.46~‘ 5.23 $34.23 ifthe Options atesur'opeananaiooo 51447: $4.56
if they are American. ‘iThe'combined value of Eboth‘spreads Vifi’theytare‘created with
European Options is 017044.23 This‘iiis:IthejtheoretiCal :bonilspread priced
calculated abOve. The combined valueOf buyingboth spreads if theyar'e' American is
0704—456”: $5.26. Selling a hex spread'i‘creatediiwith‘ American’omions‘ for $5.10
would not be a goodtrade; You Would realiize this'alnios‘t immediately, as the trade
involves selling a $60 strike putand this would'beexercised“ against “you almost as 80011
as Yousold’it! ' ' * V ' ‘ ' V' 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~rnonth calls are as follows: Strike price (3 ) c511 price (S) 55 10
60 7
65 5 Table 1 0.4 Values of 2month European and American options
on a non~dividend~paying stock. Stock price = $50; interest rate
= 8% per annum; and volatility 2 30% per annum. Option Strike European American
type price option price ' option price
Call 60 0.26 0.26
Call 55 0.96 0.96
Put 60 9.46 10.00 5.23 5.44 PM 55 Trading Strategies Involving Options 231 Figure 10.6 Proﬁt from butterﬂy‘spread using call options. 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) 2 $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—call 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 strike prices of K1 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 Payoﬁ’ from Payoﬂ from Payoﬁ” from Total range ﬁrst long call second long call slzort calls payoff *
ST < K1 0 0 0 O
K1<ST<K2 ST—K1 0 0 ST—Kl
K3 <ST< K3 ST—K1 0 —2(ST~K2) K3~ST
S7 > K3 ST—Kl ST—K3 ~2(ST~K2) 0 * These payoffs are calculated using the relationship K; = 0.5(K1 + K3). W _, _, ._ __ ,. ., _, , .. _. .,. ._ __ , ,, , . _ , _ r, “*MNM,. 232 CHAPTER 10 Figure 1 0.7 Proﬁt from butterﬂy spread using put options. 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 selling a call option with a certain strike price
and buying a longermaturity 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 1 0.8 Proﬁt from calendar spread created using two calls. Trading Strategies Involving Options 233 Figure 10.9 Proﬁt from a calendar spread created using two puts. Proﬁt \ 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 shortmaturity 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 shortmaturity option expires. The shortmaturity
option is worthless and the value of the longmaturity 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 shortmaturity
option expires. The short—maturity option costs the investor ST —— K, and the long—
maturity 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 shortmaturity 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 longmaturity put option and sells a shortmaturity 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 shortmaturity option and sells a longmaturity option. A small proﬁt arises if
the stock price at the expiration of the shortmaturity option is well above or well below
the strike price of the shortmaturity 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 10.3 CHAPTER 10 Figure 1 0.10 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. COMBINATIONS A cmnbination 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 aaMmeﬂaWMMMWWLemu/NW;r ““mw" "‘"7 "" * "V ' “V” “'V‘i‘fw‘y "hermetic #9.“:vgmxg WA ﬂing.» Newman», Jams/M WW.” chwa/«MJ: u mambo.” We“ Table 10.6 Payoff from a straddle. Range of Payoff Payoﬂ Total stock price from call from put payoﬂ
ST s K I 0 K — ST K  ST
$7 > K STK 0 ST—K Trading Strategies Itmolvitzg Options 235 Business Snapshot 10.2 How .to Make Money‘fromt Trading straddles Suppose that a big move is expected in a company’s stock pricebecausekthere isia
takeover bid for the company or the outcome of a major lawsuit involying' the
company is about to be announced. Should you trade a straddle? ’ 'V‘ , j ’ A straddle seems a natural trading strategy in this case. However; if your View of the
company’s situation is muchthe same as that, of ,othermarketparticipants, thisView
will be reﬂected in the priCes of options. Options on the stockwill be signiﬁcantly more,
expensive than options on a similarstock for? which nojumpis expected The V4shaped
proﬁt pattern'from the straddle in Figure 10.10 will have mOVed downward, so'that a
bigger move in thestock price is necessarnyr you to make a prom.) 3 7 .V , if . : I For a straddle to be an‘effective strategy, you milst believe that thereareli , rely togbe'
big movements in the ,stockryprice [and thesebeliefs must, be diﬁ‘erent _from'thOSeg_of'
most iother investOrs. Market prices incorporate the beliefs of market participants. To
make money from any investment strategy, you must takea :view that isdiﬁierentyfrom'
most of the rest of the market—and you must be right! L i L V ' ‘ ‘ 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, theloss arising from a large move is unlimited. Figure 10.1 1 Proﬁt 236 CHAPTER 10
Figure 10.12 Proﬁt from a strangle.
Proﬁt 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 stockvprice 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, K2, is higher than the put strike
price, K1. 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. ,_ a, , _,. , L, _, film, ,M _,, W... v ., MN, W, VNWMVHW ,,_/, r, *1 Table 10.7 Payoﬁ“ from a strangle. Range of Payoﬂ Payoﬂ from T 0m!
stock price from call put payoﬂ
ST<K1 0 Kl—ST K1~ST
K; < ST < K; 0 0 0
ST  K2 5:72K2 Srer 0 1w . MW we MM... WWW, “WNW r "Mr WWW ,; WWW,“ Trading Strategies Involving Options 10.4 237 Figure 10.1 3 Payoﬂ‘ from a butterﬂy spread. Payoff K1 K2 K3 S r ,,,,,_~,,,,mzw,s 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. 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 K1 and K3 and selling two options with strike price K3, Where
K1 < K2 < K3 and K3 — K2 2 K2 — K]. Figure 10.13 shows the payoff from a butterﬂy
spread. The pattern could be described as a spike. As K \ and K3 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 238 CHAPTER 10 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—Call Parity Tests for the S&P Index
LEAPS Markets,” Journal of Derivatives, 8, 4 (Summer 2001): 62—71. Chaput, J. S., and L. H. Ederington, “Option Spread and Combination Trading,” Journal of
Derivatives, 10, 4 (Summer 2003): 70~88. V McMillan, L. G. Options as a Strategic Investment. 4th edn., Upper Saddle River: PrenticeHall,
2001. Rendleman, RJ. “Covered Call Writing from an Expected Utility Perspective,” Journal of
Derivatives, 8, 3 (Spring 2001): 63—75. Ronn, A. G. and El. Ronn. “The Box—Spread Arbitrage Conditions,” Review of Financial
Studies, 2, l (1989): 91108. Questions And Problems (Answers in solutions Manual) 10.1. 10.2.
10.3.
10.4. 10.5.
10.6.
10.7.
10.8.
10.9. 10.10. 10.11. What is meant by a protective put? What position in call options is equivalent to a
protective put? Explain two ways in which a bear spread can be created. I
When is it appropriate for an investor to purchase a butterﬂy spread? 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. What trading strategy creates a reverse calendar spread?
What is the difference between a strangle and a straddle? 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? Use put—call parity to relate the initial investment for a bull spread created using calls to
the initial investment for a bull spread created using puts. Explain how an aggressive bear spread can be created using put options. 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 payoiT for both spreads. Use putcall 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. Trading Strategies Involving Options 10.12. 10.13. 10.14. 10.15. 10.16. 10.17. 10.18. 239 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? .5 Construct a table showing the payoff from a bull spread when puts with strike prices K1
and K2, with K2 > K1, are used. 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. How can a forward contract on a stock with a particular delivery price and delivery date
be created from options? “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. What is the result if the strike price of the put is higher than the strike price of the call in
a strangle? One Australian dollar is currently worth $0.64. A l—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. 10.20. 10.21. 10.22. 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 the butterﬂy spread lead to a loss? A diagonal spread is created by buying a call with strike price K2 and exercise date T2
and selling a call with strike price KI and exercise date T1, where T2 > T1. Draw a
diagram showing the proﬁt when (a) K2 > K1 and (b) K2 < K1. 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. Suppose that the price of‘a nondividendpaying stock is $32, its volatility is 30%, and the riskfree rate for all maturities is 5% per annum. Use DerivaGern to calculate the costof 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 (0) A butterﬂy Spread using European call options with strike prices of $25, $30, and
$35 and a maturity of 1 year ((1) A butterﬂy spread using European put options with strike prices of $25, $30, and
$35 and a maturity of 1 year A (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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 Spring '11
 Mr.singh
 Management

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