contract
Suppose we now execute the following conversion:
short a September 100 calllong a September 100 putlong an underlying contract
If the underlying contract for both June and September is identical, they will cancel out,leaving us with:
long a June 100 call short a September 100 callshort a June 100 put long a September 100 put
These combined long and short synthetic positions taken at the same exercise prices butin different expiration months is known as a jelly rol? or simply a roll. The strategy is
 
3. This very unscientific sounding term seems to have originated among traders on the Chicago BoardOptions Exchange.232 Chapter 11
most common in the stock option market where the underlying for both months is thesame.
What should be the value of a roll in the stock option market? In our example, thesale of the June/September 100 roll, the trader will buy stock at 100 at June expirationand sell the same stock at 100 at September expiration. The value of the roll is the costof holding the stock for the three-month period from June to September. If interest ratesare 8%, the holding costs are:
100 x 3/12 x 8% = 2.00
Therefore, the roll should have a value of two points. Expressed another way, thedifference between the June 100 synthetic (the June 100 call less the June 100 put) andthe September 100 synthetic (the September 100 call less the September 100 put}should be exactly two points.
Now suppose that the stock pays a quarterly dividend of 4 point. Since the traderwill own the stock for the period from June to September, he will receive the 34 pointdividend. The value of the roll is now the two-point carrying cost less the 34 pointdividend, or 1%. The full value of the roll is:
jelly roll = long term synthetic - short-term syntheticcost of carry — expected dividends
where the cost of carry is calculated on the exercise price over the period betweenexpiration months.
In the same way that we evaluated a box as the combination of two vertical spreads,we can also evaluate a jelly roll as the difference between two time spreads:
long a June 100 call short a September 100 callshort a June 100 put long a September 100 put
This is simply a short June/September 100 call time spread and a long June/Septemberput time spread. Since the roll is a combination of the sale of a call (put) time spreadand the purchase of put (call) time spread, the values of the call and put times spreadsshould differ by exactly the value of the roll. If the roll is worth 11/4, the differencebetween the call and put time spreads should also be 11%. If the call time spread istrading for 31/4, the put time spread should be trading for 2. This is logical when werealize that we can rewrite the jelly roll in a slightly different form:
jelly roll = long-term synthetic — short-term synthetic
(long-term call — long-term put) — (short-term call — short-term put)(long-term call — short-term call) — (long-term put — short-term put}carrying costs — expected dividends
The difference between the call and put time spreads should therefore be equal to thecost of carry on the exercise price less the expected dividends.Option Arbitrage 233
Since jelly rolls involve holding stock positions for some period of time, they haverisks similar to those affecting conversions and reversals. If interest rates rise, or thedividend is cut, a jelly rol! will become more valuable. If interest rates fall, or thedividend is increased, a jelly roll will become less valuable.
Finally, we can create a time box by taking opposing synthetic positions at differentexercise prices and expiration dates:
long a June 90 call short a September 100 callshort a June 90 put long a September 100 put
The value of this position is the combined value of purchasing the June 90/100 box andselling the June/September 100 jelly roll. If the June 90/100 box is worth 934, and theJune/September 100 jelly roll is worth 11, the time box should be worth 8!/.
USING SYNTHETICS IN VOLATILITY SPREADS
A trader need not restrict himself to using synthetics only in arbitrage strategies. He canalso execute a volatility or directional strategy using a synthetic. Consider the followingsituation. Dealing in a futures option market where the underlying March futurescontract is trading very actively at 100.00, a trader wants to execute the followingbackspread for a one-point credit:
buy 20 March 105 callssell 10 March 100 calls
That is, the amount he expects to receive for the sale of one March 100 call less theamount he expects to pay for two March 105 calls should equal one point. Suppose thecurrent market in these options is:
Option Bid/AskMarch 100 call 2.70/2.90March 105 call 85/.95
If the trader were able sell the March 100 calls at the ask price of 2.90, he would bewilling to pay the ask price of .95 for the March 105 calls, since 2.90 — (2 x .95) = 1.00.In the same way, if he were able to buy the March 105 calls at the bid price of .85, hewould be willing to sell the March 100 calls at the bid price of 2.70, since 2.70 — (2 x.85} = 1.00.
Suppose that an offer comes into the marketplace to sell March 105 cails at .85.The trader immediately buys these calls and is about to offer the March 100 calls at2.70, when suddenly a bid of 2.80 is made for March 100 puts. Does this affect whatthe trader might do?
Since the new bid is for the March 100 put, and the trader is interested in sellingthe March 100 call, it may seem that the put bid will have no affect on his strategy. Butrecall the synthetic relationship:
synthetic short call = short put + short underlying234 Chapter 11
In the futures market, the value of the call is:call price = put price + futures price — exercise price — carrying costs
If the underlying March futures contract is trading at 100, there is no carrying costassociated with a March 100 synthetic because the difference between the futures priceand exercise price is zero. Therefore:
March 100 call = March 100 put + 100- 100-0March 100 call = March 100 put
In other words, if the trader believes he can trade the March futures contract at 100,then the prices of the March 100 call and put ought to be identical. He originallyintended to sell the March 100 call at 2.70 to complete his backspread. He is now giventhe chance to sell the March 100 call, synthetically, at 2.80. If he sells the March 100put at 2.80, and simultaneously sells the March futures contract at 100.00, he haseffectively sold the March 100 call at 2.80, which is .10 better than he originallyintended.
Whenever a trader ts consideringa strategy, he ought to always ask himself whetherhe can do better by executing some part of his strategy synthetically. Usually he won’tbe able to because synthetic relationships tend to be very efficient. But every now andthen the trader will find that the synthetic position is slightly more favorable. Over acareer of trading even small savings can add up.
Using synthetic equivalents, we can also rewrite some volatility spreads in morefamiliar terms. For example, we said that call and put butterflies with the same exerciseprices are identical. Suppose we take a typical call butterfly and rewrite it in terms of itssynthetic equivalents:
Contract Synthetic Equivalentlong 1 March 95 call long 1 March 95 put/long an underlying contractshort 2 March 100 calls short 2 March 100 puts/short 2 underlying contractslong 1 March 105 call long 1 March 105 put/long an underlying contractSince the long and short underlying contracts in our synthetic equivalent cancel out, weare left with a put butterfly:long 1 March 95 putshort 2 March 100 putslong 1 March 105 put
Using synthetic equivalents, we can also express the following strategy in a morefamiliar form:
long 2 March 100 callsshort an underlying contract
This is simply a long straddle, since we can take one of the March 100 calls and rewriteit synthetically:
long March 100 call = long March 100 put + long underlying contractOption Arbitrage 235
leaving: 3
long 1 March 100 callshort an underlying contract
long 1 March 100 putlong an underlying contract
Since the Jong and short underlying contracts cancel out, we are left with a typical longstraddle: ‘
long 1 March 100 calllong 1 March 100 put
TRADING WITHOUT THEORETICAL VALUES
Synthetic relationships can often enable a trader to make logical trading decisionswithout the aid of values generated by a theoretical pricing model. Usually all that isrequired is an ability to calculate the carrying cost on a position and, in the case of stocks,to determine the dividend payout.
While a trader can’t always be as certain ofa profit as he is with arbitrage strategiessuch as conversions, reversals, boxes, and jelly rolls, there are many logical relationshipsbetween options and underlying contracts, or between options and other options, whichenable a trader to identify potentially profitable trading strategies without the use of atheoretical pricing model.
Consider the following option prices with the underlying price at 101.50:
95 call 100 call 105 call8.00 4.80 1.60
Is there anything wrong with these prices?If the trader purchases one 95 call for 8.00, sells two 100 calls at 4.80, andpurchases one 105 call for 1.60, he has bought the 95/100/105 butterfly for zero since:
(2 x 4.80) — (8.00 + 1.60) = 9.60 - 9.60 =0
Since a butterfly can never be worth less than zero at expiration, and in this case mightbe worth as much as five points, the purchase of this butterfly represents a risklesstrading opportunity.
Another way of reaching the same conclusion is to realize that a butterfly is madeup of two consecutive vertical spreads. In our example, the purchase of the95/100/105 butterfly consists or purchasing the 95/100 vertical call spread and sellingthe 100/105 vertical call spread. We know that as the market rises, a vertical call spreadgoes more deeply into-the-money and becomes more valuable. Therefore, as long asthere is an equal amount between exercise prices, a vertical call spread with lowerexercise prices will always be more valuable than a vertical call spread with higherexercise prices. The opposite is true of vertical put spreads. Those with higher exerciseprices are always more valuable than those with lower exercise prices. If two verticalspreads violate this relationship, a trader can take advantage of the situation by236 Chapter I 1
purchasing the spread which ought to be more valuable, and selling the spread whichought to be less valuable. In our example, the vertical call spreads are:
95/100 spread 8.00 - 4.80 3.20100/105 spread 4.80 — 1.60 3.20
While both spreads are trading at the same price, the 95/100 spread is intuitively morevaluable than the 100/105 spread. Consequently, a trader will try to purchase the95/100 spread and sell the 100/105 spread. If he succeeds, he will have purchased thebutterfly for zero. While there is no guarantee that this will be profitable (the underlyingmarket could finish below 95 or above 105), the strategy still makes sense because thereis no risk associated with it.
Consider a different type of relationship:
Underlying price = 99.75 Interest rate = 095 100 105calls: 6.85 3.70 1,10puts: 2.10 3.95 6.35
First, we might check to make sure that all the synthetic relationships balance. Sincethey do, there is no profit opportunity there. Next, we might check the butterflies. Boththe call and put butterflies are trading for .55. This might or might not be a reasonableprice; without a theoretical pricing model it would be difficult to say. But at least thebutterflies are trading for some positive number, so it is not immediately apparent thatthey represent a good trading opportunity. Is there any other logical relationship thatmight be violated?Look at the prices of the straddles:
95 100 105call + put: 8.95 7.65 74S
We know that a straddle becomes more valuable as the underlying price movesaway from the exercise price, resulting in one side of the straddle, either the call or put,going more deeply into-the- money. With the underlying price at 99.75, the 95 straddleshould be more valuable than the 100 straddle, and this is reflected in the respectiveprices of 8.95 and 7.65. The 105 straddle, being further away from 99.75, should alsobe more valuable than the 100 straddle. But here the prices do not reflect the additionalvalue. The 105 straddle is .30 cheaper than the 100 straddle. Even though we don’tknow the theoretical value of either the 100 or 105 straddle, we do know that the 105straddle is too cheap with respect to the 100 straddle. Therefore, if given the opportu-nity, a trader would attempt to purchase the 105 straddle at 7.45 and sell the 100straddle at 7.65. There is no guarantee that this strategy will be profitable. Theunderlying contract could finish above 105, in which case the 100 straddle will beworth five points more than the 105 straddle. But if we assume that movement israndom, then the sale of the 100 straddle and the purchase of the 105 straddle puts thelaws of probability heavily on the trader’s side.Option Arbitrage 237
Finally, consider this scenario:
Underlying price = 100.75 Interest rate = 095 100 105March calls: 7.50 3.85 2.35June calls: 9.65 5.70 3.30
Is there any apparent mispricing here?
First, we might look at the butterflies. The March butterfly is trading for (7.50 +2.35) — (2 x 3.85} = 2.15. The June butterfly is trading for (9.65 + 3.30) — (2 x 5.70)= 1.55. Since both butterflies are trading for some positive amount, it isn’t immediatelyclear that we should buy or sell either one? What about the prices of the butterflies withrespect to each other? Should the June butterfly, at 1.55, be less expensive than theMarch butterfly, at 2.15? It may seem that the June butterfly, with more time toexpiration, ought to be more expensive. In option evaluation we usually associate moretime with greater value. But if the market remains at 100.75, the price of the butterflywill expand towards 4.25 as time passes. Hence, it is logical that the March butterfly ismore valuable than the June butterfly.
Is there any other relationship we might consider? When we see options withdifferent expiration dates, time spreads certainly come to mind.
95 100 105June call less March call: 2.15 1.85 95
Is there anything wrong with these prices?
We know that a time spread maintains its greatest value with the underlyingcontract at the exercise price. The 100 time spread, being more at-the-money, is tradingfor more than the 105 time spread, as one would expect. The 100 time spread shouldalso be more valuable than the 95 time spread. Here the 95 time spread is trading at agreater price, and, again, if we assume random movement in the underlying contract,this is clearly not logical. Without a theoretical pricing model, we can’t be sure whateach of these spreads is worth. But we do know that the 95 call time spread is tooexpensive with respect to the 100 call time spread. If we sell the former and buy thelatter we can’t be certain of a profit, but we do have the laws of probability on our side.
Note that in each of these examples we made an important assumption:
Regardless of the exact theoretical value, there ought to be a uniformprogression of both individual option prices and spread prices in themarketplace. If this uniform progression is violated, a trader can takeadvantage of the situation by purchasing the option or spread which isrelatively cheap and selling the option or spread which is relativelyexpensive.
While the great majority of option traders work with theoretical values, a goodhabit for the new trader is to quickly check the relationship of option prices and spreadsin the market to ensure that there are no obvious mispricings. The trader can start with238 Chapter I 1
conversions and reversals, then look at vertical spreads and butterflies, and finallyconsider straddles and time spreads. Usually there will not be any obvious mispricing.But if there is, the trader can exploit the situation by purchasing the cheap side of therelationship and selling the expensive.
Two evaluation tables, demonstrating typical progressions of option prices andspreads in a stock option and futures option market, are shown in Figures 11-4 and 11-5.239
‘Zp 4aydeyy ul pessnosip eq
 
nS
sanjen peaids uo esiosexe Apee JO OSJJe OU ‘SeNjeA saTeGou eAeY Aew
speeds ews pure Ajiwed ue] sso] oq ABW sanjeA jeone/0ey) ewos ‘panied aspexe Aes ou UM ‘uesdoiNy aq 0) pauinsse ale suondo je souls ,
69°Po'h+0864sLesolsSS2tSovaSe'Btee'¢esxba'8h1"O@L
Option Arbitrage
xo
Hd
Od6°
+08'Ehog'e
HE'St
xoG EL6h tSLL
geebeh
cee86°
br£go¢
BVstsr
pProlPLY
ZbebLy
oo'OLLVNOLL
lee68
SEEook
ge1e'o
olPrlos"
esl£9°9
ZOOL
s9'9oLeSO}
bp’?6E°%
soeG2’?
eneso’e
PPLrs
os206
466BL
00°46'Soo}
£84c6%
28°}coe
ohs6'c
O¢'S}eg
SeeS6°LE
elbtWL
eve00°6s6
se"hSPs
OL'LPLe
ee £9" 9¢°
99°¢ 9vz 68'}oe"ZI 06'0z p0'pzgp’ Le
06° eg
o6'e ler06’! 00°! LeOP's oe’6} 13'6%ZeL Lek 6L NZ9s" ve’
09° 92°
oe'r Po'p16° ze 7aplZl POLL 89°12
06 $8 08O=puepiniq = '%00'8 = eyes ISeuA}U
speaids own indspeeds oun |IeO
sa|ppejs eunrsaleynq aunrsyeonaa ynd eunrs{Baed eo eunc
synd ounrsjeo ounr
sojppeys yoresailaynd YoIeWS|BOMaA ynd yoseyySIBOMOA [120 YOreyy
sind yoreys|j20 YoreW,
%Se = ANNeion
S¥8eM 92 = eUNC ‘syeeM E1 = YURW ‘uONeuIdXe 0) SUL,
00} = sould yooISpb} eanbiy
nc oD ee6Voc
ov'v
6b
bob
00°P
vOvs
1srae
$0
ce9g"
ee'e90°
so’e80°es
SLee
Lera
orLe
ose60°
69°?19
BesWe
ele02es
99°ce
gL€a
ss9s
962or
267466°
sehgt"
oP'lerts
9s"oy
og6e
6s6S"
ees
LL
bey
boy
orl
sres
Or6s
eses
06'2LL
96°P6'L
ee"gL"
op"bPih6P
be
‘ZL seldeyy ul pessnosip aq jm senfed peeids uo asjolexeAuee Jo eye eu ‘Aued UeU] Sse] aq ABW SONIPA JEOle1OSU] eos ‘poyued esfojexe Ayes ou YM ‘UeedoiNny 9q 0} pelunsse ale suondo Je souls ,
byoP’Oc’eol’be
esepeeel
eloeSF
Syaem Z| = Aine
beBL
OLe20°
eecee
IL'e20°
Z0°PO'sLe
%00'8 = 3781 JSas9]U]
eyaperZu1sleelecO'rO°so"£0s66EoF
‘syaom p = AY
speroids own indspeeds owl |[eO
sejppeas Aneselpeyng Ainesjeomed ynd AineSfEOMOA eo Ainesind Aine
syeo Aine
salppens Aewsalpoyng AeyySfeoeA nd AewS[BONIDA |/eo Aeyysind Ae;
sifeo Aew
!%G) = AUINEIOA‘UONeIGXe 0} OUI Los’ = goud sainjny
S-LE eaunBi4
 
24012%
Early Exerciseof American Options
Thus far we have assumed that all option strategies involve holding a position toexpiration. Since the great majority of exchange traded options are American, carryingwith them the right of early exercise, it will be worthwhile to consider some of thecharacteristics of American options. Specifically, we will want to answer two questions:
1. Given the opportunity, under what circumstances might a trader considerexercising an American option prior to expiration?
2. How much more should a trader be willing to pay for an American option overan equivalent European option?
In order for early exercise to be desirable, there must be some positive cash flowwhich will result from early exercise, and the value of this cash must be greater thanthe insurance value of the option. Because the cash flow which results from the exerciseof a stock option is significantly different from the cash flow which results from theexercise of a futures option, the conditions under which eatly exercise will be desirableare different. For this reason, we will look at these two situations separately.
FUTURES OPTIONS
From previous discussions (see Chapter 6) we know that in evaluating an option thereare some factors which make the option more valuable and there are some factors whichmake the option less valuable. Consider a futures option. We might list the factors whichaffect its value as follows:
option value = intrinsic value + volatility value — interest rate value
Since the intrinsic value and volatility components can never be less than zero, thesefactors always enhance the option’s value. As either rises, the option value rises. Onlythe interest rate component might affect the option’s value negatively. As interestconsiderations rise, the option’s value falls. If the negative effects of interest rates aregreater than the positive effects of volatility, it might be possible for an option, if it is
241242 Chapter 12
European, to be worth less than parity.! In such a case, if the option were an Americanoption it would become an early exercise candidate.
For example, suppose a certain futures contract is trading at 100. Suppose also thatwe own an 80 call which will expire in two months, and that this option is subject tostock-type settlement. When we evaluate the option, we find that it has a theoreticalvalue of 20 and a delta of 100. This means that the option has essentially the samecharacteristics as the futures contract. If the option is American, and we want tomaintain the same delta position, we have three possible choices:
1. Hold the option2. Exercise the option3. Sell the option and buy a futures contract
Which of these choices is best?
If we choose the first alternative and hold the position, there is no change to ouraccount. The position we take home is identical to the one with which we begin thenext trading day.
If we choose the second alternative and exercise the 80 call, it is as if we had goneinto the futures market and purchased a futures contract at 80. Since we now have afutures position, we are subject to futures-type settlement. If the futures contract whichwe purchased at 80 settles at 100, 20 points will be credited to our account—20 pointson which we can earn interest. How much interest will we earn? That depends on thelevel of interest rates as well as the amount of time remaining to expiration. If interestrates are 6%, with two months remaining to expiration, we will earn:
20 x 2/12 x 6% = .20
Through early exercise of the option we will earn an additional .20 in interest over thetime remaining to expiration. Since no trader wants to throw away an additional profit,exercising the option early makes better sense than simply holding the position.
What about the third choice, selling the option ourselves and buying a futurescontract? If the 80 call is trading at exactly its intrinsic value of 20 points, there is nodifference between exercising the call, or selling the call and buying a futures contract.Both strategies will result in a 20-point credit, and over the time remaining to expirationwe will earn the same interest of .20. But suppose the 80 call is trading at a price otherthan 20. Suppose it is trading at 19.50. Clearly, this price is not possible in a marketwhere the options are American and can be exercised at any time. Purchasing an 80call for 19.50 is equivalent to purchasing the futures contract for 99.50. If the futurescontract is trading at 100, this means the call is trading at less than parity. If this wereindeed true, all traders would buy the call, sell the futures contract, and immediatelyexercise the call. This would result in a riskless profit of 50, the amount by which the
1. Note that a European option with a theoretical value less than parity will have a positive theta. Its valuewill gradually rise to parity as expiration approaches.Early Exercise of American Options 243
option was below parity. In a market where options are subject to early exercise, nooption ought to be trading at less than parity.
What if the option is trading for more than parity, say 20.50? If we sell the 80 callfor 20.50 and buy the futures contract for 100, we will have the same position we wouldhave had through early exercise. But now, instead 20 points, we will have 20.50 pointscredited to our account, We will realize an additional profit of 50 through the sale ofthe option, as well as the slight additional interest earnings which go with it.
A trader who exercises a futures option early does so to capture the interest on theoption’s intrinsic value. This intrinsic value will be credited to his account only if theoption is subject to stock-type settlement. When he abandons the option throughexercise, he receives the variation resulting from settlement of the futures contract. If,however, futures options are subject to futures-type settlement, as they are on manynon-U.S. exchanges, no cash flow results from the early exercise of an option. With an80 call trading at 20 and the underlying futures contract trading at 100, if the option issubject to futures-type settlement when exercised, 20 points will be credited to thetrader’s account from the settlement of the futures contract. But 20 points willsimultaneously be debited from his account when the 20-point value of the optiondisappears. Since the cash credits and debits exactly offset each other, no additionalinterest earnings can accrue.
In futures markets where the options are subject to futures-type settlement there isnever any economic justification for early exercise, and early exercise can never be anoptimal choice. A trader will always be better off either holding the option or selling it,rather than exercising it early.
STOCK OPTIONS
Early Exercise of Calls for the Dividends
We can separate a stock option call into its components, as we did with a futures option.Now we have the additional consideration of dividends:
call value = intrinsic value + interest rate value + volatility value — dividend value
Since the intrinsic value, interest rate, and volatility components can never be less thanzero, these factors always enhance the call’s value. As any one of them rises, the callvalue rises. Only the dividend component might affect the option’s value negatively. Asthe dividend rises, the call option’s value falls. If the underlying stock pays no dividend,or no dividend is expected prior to expiration of the option, a call option can never havea value less than parity (intrinsic value). If, however, the negative effects of the dividendare greater than the positive effects of interest rates and volatility, it might be possiblefor a call, if it is European, to be worth less than parity.
For example, suppose a certain stock is trading at 100 and that the stock will goex-dividend two points tomorrow. Suppose also that there is a 90 call available whichwill expire in two weeks. When we evaluate the option, we find that it has a theoreticalvalue of 10 and a delta of 100. This means that the option has essentially the same244 Chapter 12
characteristics as the stock. If the option is American, and we want to maintain the samedelta position, we have three possible choices:
1. Hold the option2. Exercise the option3. Sell the option and buy stock
Which of these choices is best?
Suppose we simply hold the option. Certainly we will maintain our delta position.But what will happen tomorrow when the stock gives up its dividend? If the stock opensunchanged, it will open ex-dividend at 98,2 since the two-point dividend will bededucted from its price. Since the option has a value of parity, it will open not at 10,the previous day's parity price, but at 8, today’s parity price. In other words, if we holdthe option we can be certain of losing two points on