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ToLeona,forhersupportandencouragementthroughoutmycareer.

ToEddie,whocontinuallymakesmeproudtobea

father.

Contents

Preface

1FinancialContractsBuying andSellingNotional

Value of aForwardContractSettlementProceduresMarketIntegrity

2ForwardPricingPhysicalCommodities(Grains,

EnergyProducts,PreciousMetals,etc.)StockBonds andNotesForeignCurrenciesStock andFuturesOptionsArbitrage

DividendsShortSales

3ContractSpecificationsandOptionTerminology

ContractSpecificationsOption PriceComponents

4ExpirationProfitandLoss

ParityGraphs

5TheoreticalPricingModels

TheImportance ofProbabilityA SimpleApproach

The Black-ScholesModel

6VolatilityRandomWalks andNormalDistributionsMean andStandardDeviationForward Price

astheMeanofaDistributionVolatility as aStandardDeviationScalingVolatility forTimeVolatility andObservedPriceChangesA Note onInterest-Rate

ProductsLognormalDistributionsInterpretingVolatilityData

7RiskMeasurementITheDeltaTheGammaTheThetaTheVegaTheRho

Interpretingthe RiskMeasures

8DynamicHedgingOriginalHedge

9RiskMeasurementIIDeltaTheta

VegaGammaLambda(Λ)

10IntroductiontoSpreading

WhatIsaSpread?

OptionSpreads

11Volatility

Spreads

Straddle

Strangle

Butterfly

Condor

RatioSpread

ChristmasTree

CalendarSpread

TimeButterfly

Effect of Changing InterestRatesandDividends

DiagonalSpreads

Choosing an AppropriateStrategy

Adjustments

SubmittingaSpreadOrder

12BullandBearSpreads

NakedPositions

BullandBearRatioSpreads

BullandBearButterfliesandCalendarSpreads

VerticalSpreads

13RiskConsiderations

VolatilityRisk

PracticalConsiderations

HowMuchMarginforError?

DividendsandInterest

WhatIsaGoodSpread?

14

Synthetics

SyntheticUnderlying

SyntheticOptions

Using Synthetics in aSpreadingStrategy

Iron Butterflies and IronCondors

15Option

Arbitrage

OptionsonFutures

LockedFuturesMarkets

OptionsonStock

ArbitrageRisk

16EarlyExerciseof

AmericanOptions

ArbitrageBoundaries

Early Exercise of CallOptionsonStock

EarlyExerciseofPutOptionsonStock

Impact of Short Stock onEarlyExercise

EarlyExerciseofOptionsonFutures

Protective Value and EarlyExercise

PricingofAmericanOptions

EarlyExerciseStrategies

EarlyExerciseRisk

17Hedging

withOptions

ProtectiveCallsandPuts

CoveredWrites

Collars

ComplexHedgingStrategies

HedgingtoReduceVolatility

PortfolioInsurance

18TheBlack-ScholesModel

n(x)andN(x)

AUsefulApproximation

TheDelta

TheTheta

Maximum Gamma, Theta,

andVega

19BinomialOptionPricing

ARisk-NeutralWorld

ValuinganOption

TheDelta

TheGamma

TheTheta

VegaandRho

TheValuesofuandd

GammaRent

AmericanOptions

Dividends

20Volatility

Revisited

HistoricalVolatility

VolatilityForecasting

Implied Volatility as aPredictorofFutureVolatility

ForwardVolatility

21PositionAnalysis

Some Thoughts on MarketMaking

StockSplits

22StockIndexFuturesandOptions

WhatIsanIndex?

StockIndexFutures

StockIndexOptions

23ModelsandtheRealWorld

MarketsAreFrictionless

Interest Rates Are ConstantovertheLifeofanOption

Volatility Is Constant overtheLifeoftheOption

TradingIsContinuous

ExpirationStraddles

Volatility Is Independent ofthe Price of the UnderlyingContract

Underlying Prices atExpiration Are LognormallyDistributed

SkewnessandKurtosis

24VolatilitySkews

ModelingtheSkew

SkewnessandKurtosis

SkewedRiskMeasures

ShiftingtheVolatility

Skewness and KurtosisStrategies

ImpliedDistributions

25VolatilityContracts

RealizedVolatilityContracts

ImpliedVolatilityContracts

TradingtheVIX

Replicating a VolatilityContract

Volatility ContractApplications

Afterword:A FinalThought

AGlossaryofOption

Terminology

BSomeUsefulMath

Rate-of-ReturnCalculations

Normal Distributions andStandardDeviation

Volatility

Index

Preface

It probably seems strangeforanauthortowait20yearsto revise a professionalpublication, especially onethathasbeencontinuously inprint over the entire period.To those of you who were

hoping for at least onerevision in the interveningyears, I can only offer myapology and the excuse thatother obligations preventedme from undertaking such arevision.

Much has changed inoption markets over the last20 years. Most markets arenow fully electronic, and thedays of floor trading areclearlynumbered.Onlyinthe

United States do option-trading floors still exist, andeven those are inevitablygiving way to electronictrading. Twenty years ago,organized option marketsexisted only in the majorindustrialized nations. But asthe importance of derivativesasbothaninvestmentvehicleand a risk-management toolhas become widelyrecognized, new optionmarkets have opened in

countries around the world.Options are now traded notonlyontraditionalproducts—stocks, interest rates,commodities, and foreigncurrencies—but also on abewildering array of newproducts—real estate,pollution, weather, inflation,and insurance. Manyexchanges have also addedvariations on traditionalproducts—short-term andmidcurve options, flex

options, options on spreads,and implied and realizedvolatilitycontracts.

Not only has there beena dramatic increase in thenumber of option markets,but the traders in thosemarkets have becomeincreasingly sophisticated.When this text was firstpublished, knowledgeabletraderscouldonlybefoundatfirms that traded derivatives

professionally—market-making firms, hedge funds,investment banks, and otherproprietary trading firms.Now, many retail customershave a level of knowledgeequaltothatofaprofessionaltrader. At the same time,universities are adding orexpanding programs infinancial engineering. Inmany cases, those whochooseacareerinderivativestrading have already had in-

depth exposure to themathematics of optionpricing.

Whilemuchhaschangedinthelast20years,muchhasalso remained the same.There is still a core body ofmaterial that a seriousoptiontrader needs to master, andthiscorematerialismuchthesame as it has always been.The previous edition of thistextwasanattempttopresent

thismaterialinamannerthatwaseasilyaccessibleandthatdid not require a familiaritywith advanced mathematics.This edition retains thatapproach. Although somepresentations may have beenchanged in the interest ofimproving an explanation orclarifying a concept, all themajor topics from theprevious edition have beenretained.

So what’s new in thisedition? As in the firstedition, an attempt has beenmade to explain importantconcepts in the simplestpossible manner using anintuitive rather thanmathematical approach.However,itisalsotruethatafull understanding of manyoption concepts requires afamiliarity with moreadvanced mathematics.Consequently, some

explanations have beenexpanded to include adiscussion of the relevantmathematics. But even thesediscussions tend to avoidmathematical concepts withwhich many readers areunlikelytobefamiliar.Manychapters have also beenexpanded to include a moredetailed discussion of therelevant topics. In addition,there are several completelynew chapters covering

forward pricing, riskdynamics, the Black-Scholesmodel, binomial optionpricing, and volatilitycontracts.

As with any livinglanguage, marketterminology, and morespecifically, optionterminology, has changedover time. Some terms thatwere commonwhen the firstedition appeared have gone

out of favor or disappearedcompletely. Other terms thatdid not previously exist havegainedwideacceptance.Thisis reflected in small changestothevocabularyusedinthistext.

Itisalmostimpossibletokeep up with the amount ofinformation that is availableon options.Not only do newbooks appear with greaterfrequency, but the Internet

has enabled traders to findrelevant source materialalmost instantaneously. Forthis reason, the Bibliographyhas been eliminated. Thisshouldnotbeconstruedasanattempt todiscourage readersfrom consulting othersources.Thisbookrepresentsonlyoneapproach tooptions—that of a professionaltrader.Manyexcellentoptionbooks are available, and anyaspiring option trader will

wanttoconsultabroadrangeoftextsinordertounderstandthe many different ways onecanapproachoptionmarkets.For those who are interestedin themathematics of optionpricing,thistextisinnowaymeant to take the place of agood university textbook onfinancialengineering.

Nothing in this text isreally new, and all theconcepts will be familiar, in

one formor another, tomostexperienced option traders.The presentation representsmybestattempt,asanoptioneducator, to present theseconceptsinaclearandeasilyaccessible manner. Thematerial isbasednotonlyonwhat I have personallylearned throughoutmycareerbut also on the knowledgeand experiences of manyotherswithwhomIhavebeenprivileged to work. In

particular,mycolleaguesTimWeithers and SamuelKadziela offered manyhelpful comments andinsights and in some casesrescued me fromembarrassing errors. Anyremaining errors, of whichthere are almost certainly afew,arestrictlymyown.

I make no claim tohaving found a magic secretto successful option trading.

Anyone seeking such aformula will have to lookelsewhere.Thesecret,ifthereisone,isinlearningasmuchas possible, applying in thereal world what has beenlearned, and analyzing bothone’s successes and one’sfailures.

SheldonNatenberg

FinancialContracts

My friend Jerry lives in asmalltown,thesametowninwhich he was born andraised. Because Jerry’s

parents are no longer aliveandmanyofhis friendshaveleft, he is seriously thinkingofpackingupandmoving toa larger city. However, Jerryrecently heard that there is aplantobuildamajorhighwaythatwillpassveryclosetohishometown. Because thehighway is likely to bringnew life to the town, Jerry isreconsidering his decision tomove away. It has alsooccurred to Jerry that the

highway may bring newbusinessopportunities.

For many years, Jerry’sfamily was in the restaurantbusiness,andJerryisthinkingofbuildingarestaurantatthemain intersection leadingfrom the highway into town.If Jerry does decide to buildtherestaurant,hewillneedtoacquire land along thehighway. Fortunately, Jerryhas located a plot of land,

currently owned by FarmerSmith, that is ideally suitedfor the restaurant. Becausethe landdoesnot seem tobein use, Jerry is hoping thatFarmer Smith might bewillingtosellit.

If Farmer Smith isindeed willing to sell, howcanJerryacquire the landonwhichtobuildhisrestaurant?First,JerrymustfindouthowmuchFarmerSmithwantsfor

the land.Let’s say$100,000.IfJerrythinksthatthepriceisreasonable, he can agree topay this amount and, inreturn, take ownership of theland. In this case, Jerry andFarmer Smith will haveentered into a spot or cashtransaction.

In a cash transaction,both parties agree on terms,followed immediately by anexchange of money for

goods. The trading of stockon an exchange is usuallyconsidered to be a cashtransaction: the buyer andseller agree on the price, thebuyerpaystheseller,andtheseller delivers the stock. Theactions essentially take placesimultaneously. (Admittedly,on most stock exchanges,there is a settlement periodbetween the time theprice isagreed on and the time thestock is actually delivered

and payment is made.However, the settlementperiod is relatively short, sofor practical purposes mosttraders consider this a cashtransaction.)

However, it has alsooccurred to Jerry that it willprobablytakeseveralyearstobuild the highway. BecauseJerrywantstheopeningofhisrestaurant to coincide withthe opening of the highway,

he doesn’t need to beginconstructionontherestaurantfor at least another year.There is no point in takingpossession of the land rightnow—it will just sit unusedfor a year. Given hisconstruction schedule, Jerryhas decided to approachFarmer Smithwith a slightlydifferent proposition. Jerrywill agree to Farmer Smith’spriceof$100,000,buthewillproposetoFarmerSmiththat

theycomplete the transactionin one year, at which timeFarmer Smith will receivepayment, and Jerry will takepossessionoftheland.Ifbothpartiesagreetothis,JerryandFarmer Smith will haveentered into a forwardcontract. In a forwardcontract, the parties agree onthe termsnow,but theactualexchangeofmoneyforgoodsdoesnottakeplaceuntilsomelater date, the maturity or

expirationdate.If Jerry and Farmer

Smith enter into a forwardcontract, it’sunlikelythat thepriceFarmerSmithwillwantfor his land in one year willbe the same price that he isasking today. Because boththe payment and the transferof goods are deferred, theremay be advantages ordisadvantagestoonepartyorthe other. Farmer Smithmay

point out that if he receivesfull payment of $100,000rightnow,hecandeposit themoney inhisbankandbeginto earn interest. In a forwardcontract, however, he willhave to forego any interestearnings.As a result, FarmerSmithmay insist that he andJerry negotiate a one-yearforward price that takes intoconsideration this loss ofinterest.

Forward contracts arecommon when a potentialbuyer requires goods in thefuture or when a potentialsellerknows that a supplyofgoods will be ready for salein the future. A bakery mayneed a periodic supply ofgrain to support operations.Some grain may be requirednow, but the bakery alsoknows that additional grainwill be required at regularintervals in the future. In

order to eliminate the risk ofrisinggrainprices,thebakerycan buy grain in the forwardmarket—agreeing on a pricenow but not taking deliveryor making payment untilsome later date. In the sameway, a farmer who knowsthat hewill have grain readyforharvestata laterdatecansell his crop in the forwardmarket to insure againstfallingprices.

Whenaforwardcontractis traded on an organizedexchange, it is usuallyreferred to as a futurescontract. On a futuresexchange, the contractspecifications for a forwardcontract are standardized tomoreeasily facilitate trading.The exchange specifies thequantityandqualityofgoodsto be delivered, the date andplace of delivery, and themethod of payment.

Additionally, the exchangeguaranteestheintegrityofthecontract. Should either thebuyerorthesellerdefault,theexchange assumes theresponsibilityof fulfilling thetermsoftheforwardcontract.

The earliest futuresexchanges enabled producersand users of physicalcommodities—grains,precious metals, and energyproducts—to protect

themselves against pricefluctuations. More recently,many exchanges haveintroduced futures contractson financial instruments—stocks and stock indexes,interest-rate contracts, andforeign currencies. Althoughthere is still significanttrading in physicalcommodities, the total valueof exchange-traded financialinstruments now greatlyexceedsthevalueofphysical

commodities.Returning to Jerry, he

finds that he has a newproblem.Thegovernmenthasindicated its desire to buildthe highway, but thenecessary funds have not yetbeen authorized. With manyother public works projectscompeting for a limitedamount of money, it’spossible that the entirehighway project could be

canceled. If this happens,Jerry intends to return to hisoriginalplanandmoveaway.Inordertomakeaninformeddecision, Jerry needs time toseewhatthegovernmentwilldo.Ifthehighwayisactuallybuilt,JerrywantstopurchaseFarmer Smith’s land. If thehighway isn’t built, Jerrywantstobeabletowalkawaywithoutanyobligation.

Jerry believes that he

willknowforcertainwithinayear whether the highwayprojectwillbeapproved.Asaresult, Jerry approachesFarmer Smith with a newproposition.JerryandFarmerSmith will negotiate a one-year forward price for theland, but Jerrywill have oneyear to decidewhether to goaheadwiththepurchase.Oneyear from now, Jerry caneither buy the land at theagreed-on forward price, or

he can walk away with noobligationorpenalty.

There is much that canhappen over one year, andwithout some inducementFarmer Smith is unlikely toagree to this proposal.Someone may make a betterofferfortheland,butFarmerSmith will be unable toaccept the offer because hemust hold the land in theevent that Jerry decides to

buy. For the next year,Farmer Smith will be ahostage to Jerry’s finaldecision.

Jerry understandsFarmer Smith’s dilemma, sohe offers to negotiate aseparate payment tocompensateFarmerSmithforthis uncertainty. In effect,Jerry is offering to buy theright todecideat a laterdatewhethertopurchasetheland.

Regardless of Jerry’s finaldecision, Farmer Smith willget to keep this separatepayment. IfJerryandFarmerSmith can agree on thisseparate payment, as well asthe forward price, they willenter intoanoptioncontract.An option contract gives oneparty the right to make adecisionatalaterdate.Inthisexample,Jerryisthebuyerofa call option, giving him theright todecideat a laterdate

whethertobuy.FarmerSmithisthesellerofthecalloption.

Decidingwhethertobuythe land for his restaurant isnot Jerry’s only problem.Heowns a house that heinheritedfromhisparentsandthat he was planning to sellpriortomovingaway.Beforehearing about the highwayproject, Jerry had put up a“ForSale”signinfrontofthehouse, and a young couple,

seeing the sign, showedenough interest in the houseto make an offer. Jerry wasseriously consideringaccepting the offer, but thenthehighwayprojectcameup.NowJerrydoesn’tknowwhattodo.Ifthegovernmentgoesahead with the highway andJerry goes ahead with hisrestaurant, he wants to keephishouse. Ifnot,hewants tosell the house. Given thesituation, Jerrymightmakea

proposaltothecouplesimilarto that which he made toFarmer Smith. Jerry and thecouple will agree on a pricefor the house, but Jerry willhave one year in which todecide whether to actuallysellthehouse.

Like Farmer Smith, thecouple’s initial reaction islikely to be negative. If theyagreetoJerry’sproposal,theywill have tomake temporary

housing arrangements for thenextyear.Iftheyfindanotherhouse they like better, theywon’t be able to buy itbecause they mighteventually be required topurchase Jerry’s house.Theywill spend the next year inhousing limbo, a hostage toJerry’sfinaldecision.

As with Farmer Smith,Jerry understands thecouple’s dilemma and offers

to compensate them for theirinconvenience by paying anagreed-on amount.Regardless of Jerry’s finaldecision, the couple will getto keep this amount. If Jerryand the couple can agree onterms, Jerry will havepurchased a put option fromthecouple.Aputoptiongivesone party the right to decidewhethertosellatalaterdate.

Perhaps the most

familiar type of optioncontractisinsurance.Inmanywaysaninsurancecontract isanalogous to a put option.Ahomeowner who purchasesinsurancehastherighttosellallorpartofthehomebacktothe insurance company at alaterdate.Ifthehomeshouldburn to the ground, thehomeowner will inform theinsurance company that henowwishes to sell the homeback to the insurance

company for the insuredamount. Even though thehome no longer exists, theinsurance company is payingthe homeowner as if it wereactuallypurchasingthehome.Of course, if the house doesnot burndown, perhaps evenappreciating in value, thehomeowner is under noobligationtosellthepropertytotheinsurancecompany.

As with an insurance

contract, the purchase of anoption involves the paymentofapremium.Thisamountisnegotiatedbetween thebuyerand the seller, and the sellerkeepsthepremiumregardlessofanysubsequentdecisiononthepartofthebuyer.

Many terms of aninsurancecontractaresimilarto the terms of an optioncontract. An option, like aninsurance contract, has an

expiration date. Does ahomeownerwantasix-monthinsurancepolicy?Aone-yearpolicy? The insurancecontract may also specify anexerciseprice,howmuch theholder will receive if certainevents occur. This exerciseprice,whichmayalsoincludea deductible amount, isanalogous to an agreed-onforwardprice.

The logic used to price

option contracts is alsosimilar to the logic used toprice insurance contracts.What is theprobabilitythatahousewill burn down?Whatis the probability thatsomeone will have anautomobileaccident?Whatisthe probability that someonewill die? By assigningprobabilities to differentoccurrences, an insurancecompany will try todetermineafairvaluefor the

insurance contract. Theinsurance company hopes togenerate a profit by sellingthecontracttothecustomerata price greater than its fairvalue. In the same way,someone dealing withexchange-traded contractsmay also ask, “What is theprobability that this contractwill go up in value?What isthe probability that thiscontract will go down invalue?” By assigning

probabilities to differentoutcomes, itmaybepossibleto determine the contract’sfairvalue.

In laterchapterswewilltake a closer look at howforwards,futures,andoptionsare priced. For now, we canseethattheirvaluesarelikelyto depend on or be derivedfrom the value of someunderlying asset. When myfriend Jerry wanted to enter

into a one-year forwardcontract tobuythelandfromFarmer Smith, the value ofthe forward contract derivedfrom(amongotherthings)thecurrent value of the land.When Jerry was consideringbuying a call option fromFarmer Smith, the value ofthat option derived from thevalueoftheforwardcontract.When Jerry was consideringsellinghishouse,thevalueofthe put option derived from

the current value of thehouse. For this reason,forwards,futures,andoptionsare commonly referred to asderivative contracts or,simply,derivatives.

There is one othercommon type of derivativescontract. A swap is anagreement to exchange cashflows. The most commontype, aplain-vanilla interest-rateswap,isanagreementto

exchange fixed interest-ratepayments for floatinginterest-rate payments. But aswap can consist of almostany type of cash-flowagreement between twoparties. Because swaps arenot standardized andtherefore most often tradedoffexchanges,inthistextwewill restrictourdiscussion tothemostcommonderivatives—forwards, futures, andoptions.

BuyingandSelling

We usually assume that inorder to sell something, wemust first own it. For mosttransactions,thenormalorderis to buy first and sell later.However, in derivativemarkets, the order can bereversed. Instead of buyingfirst andselling later,wecansell first and buy later. Theprofit that results from a

purchase and sale is usuallyindependent of the order inwhich the transactions occur.We will show a profit if weeitherbuyfirstata lowpriceand sell later at a high priceorsellfirstatahighpriceandbuylateratalowprice.

Sometimeswemaywantto specify theorder inwhichtrades take place. The firsttrade to take place, eitherbuying or selling, is an

openingtrade,resultinginanopen position. A subsequenttrade, reversing the initialtrade, is a closing trade. Awidely used measure oftrading activity in exchange-traded derivative contracts isthe amount of open interest,the number of contractstraded on an exchange thathavenotyetbeenclosedout.Logically,thenumberoflongand short contracts that havenot been closed out must be

equalbecauseforeverybuyertheremustbeaseller.

If a trader first buys acontract (an opening trade),he is long thecontract. If thetrader first sells a contract(alsoanopening trade),he isshort the contract. Long andshort tend to describe aposition once it has beentaken,buttradersalsorefertotheactofmakinganopeningtrade as either going long

(buying) or going short(selling).

A long position willusually result in a debit (wemust pay money when webuy),andashortpositionwillusually result in a credit (weexpect to receive moneywhen we sell). We will seelater that thesetermsarealsoused when trading multiplecontracts, simultaneouslybuying some contracts andsellingothers.Whenthetotaltraderesultsinadebit, it isalongposition;when it results

in a credit, it is a shortposition.

Thetermslongandshortmay also refer to whether atrader wants the market toriseor fall.A traderwhohasa long stock market positionwants the stock market torise.Atraderwhohasashortposition wants the market tofall.However,whenreferringto derivatives, the terms canbeconfusingbecauseatrader

whohasbought,orislong,aderivative may in fact wantthe underlyingmarket to fallin price. In order to avoidconfusion, we will refer toeitheralongorshortcontractposition (we have eitherboughtorsoldcontracts)oralongor shortmarketposition(we want the underlyingmarkettoriseorfall).

NotionalValueofaForwardContract

Becauseaforwardcontractis an agreement to exchangemoney for goods at somelater date, when a forwardcontract is initiallytraded,nomoney changes hands.Becausenocashflowresults,in a sense, there is no cashvalue associated with thecontract. But a forward

contractdoeshaveanotionalvalue or nominal value. Forphysical commodities, thenotional value of a forwardcontract is equal to thenumber of units to bedelivered at maturitymultipliedbytheunitprice.Ifa forward contract calls forthedeliveryof1,000units ata price of $75 per unit, thenotionalvalueofthecontractis$75×1,000=$75,000.

For some forwardcontracts,physicaldeliveryisnot practical. For example,manyexchangestradefuturescontracts on stock indexes.Butitwouldbeimpracticaltoactuallydeliverastockindexbecause it would require thedelivery of all stocks in theindex in exactly the rightproportion, which in somecases might mean deliveringfractional shares. Forfinancial futures, where the

contractisnotsettledthroughphysical delivery, thenotionalvalue isequal to thecash price of the index orinstrument multiplied by apoint value. A stock indexthat is trading at 825.00 andthathasapointvalueof$200hasanotionalvalueof825.00×$200=$165,000.

The point value of astock index or similarcontract is set by the

exchange so that thecontracthas a notional value that isdeemed reasonable fortrading. If the point value isset too high, trading in thecontractmaybe tooriskyformost market participants. Ifthepointvalueissettoolow,transaction costs may beprohibitive because it mayrequire trading a largenumber of contracts toachievethedesiredresult.

SettlementProcedures

What actually happenswhen a contract is traded onan exchange? The settlementprocedure—the manner inwhich the transfer of moneyandownershipofacontractisfacilitated—depends on therulesoftheexchangeandthetypeofcontracttraded.

Consider a trader who

buys 100 shares of a $50stock on an exchange. Thetotalvalueofthestockis100×$50=$5,000,andthebuyeris required to pay the sellerthis amount. The exchange,acting as intermediary,collects $5,000 from thebuyer and transfers thismoney to the seller. At thesame time, the exchangetakes delivery of the sharesfrom the seller and transfersthese to the buyer. This is

essentially a cash transactionwith the exchange makingbothdeliveryandpayment.

Suppose that the stockthatwasoriginallypurchasedat$50persharesubsequentlyrises to $60. How will thebuyer feel? Hewill certainlybe happy and may mentallyrecordaprofitof$1,000(100shares times the$10 increaseper share). But he can’tactually spend this $1,000

because the profit isunrealized —it only appearson paper (hence the termpaper profit). If the buyerwantstospendthe$1,000,hewill have to turn it into arealizedprofitbygoingbackinto the marketplace andselling his 100 shares tosomeone else at $60 pershare. This stock-typesettlement requires full andimmediate payment, and allprofits or losses are

unrealized until the positionisclosed.

Now consider whathappens when a futurescontract is traded on anexchange. Because a futurescontractisaforwardcontract,there is no immediateexchange of money forgoods. The buyer pays nomoney,andthesellerreceivesnone. But by entering into aforward contract, both the

buyer and the seller havetaken on future obligations.At contract maturity, theseller is obligated to deliver,and the buyer is obligated topay. The exchange wants toensure that both parties liveuptotheseobligations.Todothis, the exchange collects amargin deposit from eachparty that itholdsas securityagainst possible default bythe buyer or seller. Theamount of margin is

commensurate with the riskto the exchange and dependson the notional value of thecontract, as well as thepossibility of pricefluctuations over the life ofthe futures contract. Anexchange will try to setmargin requirements highenough so that the exchangeis reasonably protectedagainstdefaultbutnotsohighthatitinhibitstrading.

For example, considerthe futures contract callingfordeliveryof1,000unitsofacommodityataunitpriceof$75.Thenotionalvalueofthecontract is $75,000. If theexchange has set a marginrequirement for the contractat $3,000, when the contractis traded, both the buyer andseller must immediatelydeposit $3,000 with theexchange.

What happens if theprice of the commoditysubsequently rises to $80?Nowthebuyerhasaprofitof$5×1,000=$5,000,whereasthe seller has a loss of equalamount. As a result, theexchange will now transfer$5,000 from the seller’saccount to the buyer’saccount. This daily variationcredit or debit results fromfluctuationsinthepriceofthefuturescontractaslongasthe

position remains open.Futures-type settlement,where there is an initialmargin deposit followed bydaily cash transfers, is alsoknown as margin andvariationsettlement.

A futures trader cancloseoutapositioninoneoftwoways.Priortomaturityofthe futures contract, he canmake an offsetting trade,selling out the futures

contractheinitiallyboughtorbuying back the futurescontract he initially sold. Ifthepositionisclosedthroughanoffsettingpurchaseorsale,a final variation payment ismade,andthemargindepositisreturnedtothetrader.

Alternatively, a tradermay choose to carry thepositiontomaturity,atwhichtime physical settlement willtake place. The seller must

makedelivery, and thebuyermustpayanamountequal tothe current value of thecommodity. After deliveryandpaymenthavebeenmade,the margin deposits will bereturned to the respectiveparties. In our example, theoriginal trade price was $75.Ifthepriceofthecommodityatmaturity is $90, the buyermust pay $90 × 1,000 =$90,000.

It may seem that thebuyer has paid $15more perunit than the original tradepriceof$75.Butrecallthatasthe futures contract rose inprice from $75 to $90, thebuyer was credited with $15in the form of variation. Thetotalpricepaid, the$90 finalprice less the $15 variation,was indeed equal to theagreed-on price of $75 perunit.

Futurescontractssuchasstock indexes, which are notsettled through physicaldelivery, can also be carriedtomaturity.Inthiscase,thereisonefinalvariationpaymentbasedontheunderlyingindexprice at maturity. At thattime, themargindeposits arealso returned to the parties.Thesetypesoffutures,whereno physical delivery takesplace atmaturity, are said tobecash-settled.

A futures trader mustalways have sufficient fundsto cover the marginrequirementsforanytradeheintends to make. But heshould also have sufficientfunds to cover any variationrequirements. If the positionmoves against him and hedoes not have sufficientfunds, he may be forced toclosethepositionearlierthanintended.

There is an importantdistinction between marginand variation. Margin1 ismoney collected by theexchange to ensure that atrader can fulfill futurefinancial obligations shouldthemarketmoveagainsthim.Even though deposited withthe exchange, margindeposits still belong to thetrader and can therefore earninterest for the trader.

Variation is a credit or debitthat results from fluctuationsin the price of a futurescontract.Avariationpaymentcaneitherearninterest,ifthevariationresultsinacredit,orlose interest, if the variationresultsinadebit.

Examples of the cashflowsandprofitor loss for aseries of stock and futurestradesareshowninFigures1-1 and 1-2, respectively. In

eachexample,weassumethatthe opening trade was madeat the first day’s settlementpricesothatthereisnoprofitandloss(i.e.,aP&Lofzero)at the end of day 1. Forsimplicity, we have alsoignored any interest earnedon credits or interest paid ondebits.

Figure1-1Stock-typesettlement.

Figure1-2Futures-typesettlement.

We make this veryimportantdistinctionbetweenstock-type settlement andfutures-type settlementbecause some contracts aresettled like stock and somecontracts are settled likefutures. It shouldcomeasnosurprise that stock is subjectto stock-type settlement andfuturesaresubject tofutures-type settlement. But whatabout options? Currently, all

exchange-traded options inNorth America, whetheroptions on stock, stockindexes, futures, or foreigncurrencies, are settled likestock. Options must be paidfor immediately and in full,and all profits or losses areunrealized until the positionis liquidated. In stock optionmarkets, this is both logicaland consistent because boththe underlying contract andoptions on that contract are

settled using identicalprocedures.However,onU.S.futures options markets, theunderlying contract is settledone way (futures-typesettlement),whiletheoptionsare settled in a differentway(stock-type settlement). Thiscan sometimes causeproblems when a trader hasbought or sold an option tohedge a futures position.Even if the profits from theoption position exactly offset

the losses from the futuresposition, the profits from theoption position, because theoptionsare settled like stock,areunrealized.But the lossesfromthefuturespositionwillrequire an immediate cashoutlay to cover variationrequirements. If a trader isunaware of the differentsettlementprocedures,hecanoccasionally find himselfwith unexpected cash-flowproblems.

The settlement situationon most exchanges outsideNorth America has beensimplified by making optionand underlying settlementprocedures identical. If theunderlyingissubjecttostock-type settlement, then theoptionsontheunderlyingaresubject to stock-typesettlement. If the underlyingis subject to futures-typesettlement, then the optionsare subject to futures-type

settlement. Under thismethod,atraderisunlikelytohave a surprise variationrequirementonapositionthathethinksiswellhedged.

In this text, whenpresenting option examples,wewillgenerallyassume thesettlementconventionusedinNorth America, where alloptions are subject to stock-typesettlement.

MarketIntegrity

Anyone who enters into acontract to buy or sell wantsto be confident that thecounterparty will fulfill hisresponsibilities under thetermsofthecontract.Abuyerwants to be sure that theseller will deliver; a sellerwants to be sure that thebuyer will pay. No one willwanttotradeinamarketplace

if there is a real possibilitythat the counterparty mightdefault on a contract. Toguarantee the integrity of anexchange-traded contract,exchanges assume theresponsibility for bothdelivery and payment.Whena trade is made on anexchange, the link betweenbuyer and seller isimmediately broken andreplacedwith two new links.The exchange becomes the

buyerfromeachseller.If thebuyer defaults, the exchangewill guarantee payment. Theexchange also becomes theseller to each buyer. If theseller defaults, the exchangewillguaranteedelivery.

To protect itself againstpossibledefault,anexchangewill establish aclearinghouse. Theclearinghouse may be adivisionof theexchangeora

completely independententity and is responsible forprocessing and guaranteeingall trades made on theexchange.2Theclearinghouseassumes the ultimateresponsibilityforensuringtheintegrity of all exchange-tradedcontracts.3

Figure1-3Theclearingprocess.

The clearinghouse ismadeupofmemberclearingfirms. A clearing firmprocesses trades made byindividual traders and agreesto fulfill any financialobligation arising from thosetrades. Should an individualtrader default, the clearingfirmguaranteesfulfillmentofthat trader’s responsibilities.No individual may trade onan exchange without first

becoming associated with aclearingfirm.

As part of itsresponsibilities, a clearingfirmwill collect the requiredmargin from individualtraders and deposit thesefunds with theclearinghouse.4 In somecases, the clearinghousemaypermit a clearing firm toaggregate the positions of alltraders at the firm. Because

some traders will have longpositions while other traderswill have short positions inthe same contract, theclearinghousemayreducethemargin deposits requiredfrom theclearing firm.At itsdiscretion, and depending onmarket conditions, a clearingfirm may require anindividual trader to depositmoremoneywiththeclearingfirm than is required by theclearinghouse.

The current system ofguarantees—individualtrader, clearing firm, andclearinghouse—has proveneffective in ensuring theintegrity of exchange-tradedcontracts. Althoughindividual traders andclearing firms occasionallyfail, a clearinghouse hasnever failed in the UnitedStates.

1Amarginrequirementforaprofessionaltraderonanequityoptionsexchangeissometimesreferredtoasahaircut.2IntheUnitedStates,thetwolargestderivativesclearinghousesaretheOptionsClearingCorporation,responsibleforprocessingallequityoptiontrades,andtheCMEClearingHouse,responsibleforprocessingalltradesmadeonexchangesfallingwithintheCMEGroup.Forinstrumentsotherthanderivatives,suchasstockandbonds,theDepositoryTrustandClearingCorporationprovidesclearingservicesformanyU.S.exchanges.3Althoughtheexchangeand

clearinghousemaybeseparateentities,forsimplicity,wewilloccasionallyusethetermsinterchangeably.4Wenotedearlierthat,intheory,thereisnolossofinterestassociatedwithamargindeposit.Inpractice,theamountofinterestpaidonmargindepositswillvarybyclearingfirmandistypicallynegotiatedbetweentheclearingfirmandtheindividualcustomer.

ForwardPricing

What should be the fairprice for a forward contract?We can answer this questionby considering the costs andbenefits of buying nowcompared with buying on

some future date. In aforward contract, the costsand benefits are noteliminated; they are simplydeferred. They shouldtherefore be reflected in theforwardprice.

forwardprice=currentcashprice+costsofbuyingnow–

benefitsofbuyingnow

Let’s return to ourexample from Chapter 1

wheremyfriendJerrywantedto acquire land on which tobuild a restaurant. He wasconsidering both a cashpurchase and a one-yearforwardcontract. Ifheentersinto a forward contract,whatshould be a fair one-yearforwardpricefortheland?

IfJerrywantstobuytheland right now, he will havetopayFarmerSmith’saskingprice of $100,000. However,

in researching the feasibilityof a one-year forwardcontract,Jerryhaslearnedthefollowing:

1. The cost ofmoney, whetherborrowing orlending,1 iscurrently 8.00percentannually.2. The owner ofthe land must pay

$2,000inrealestatetaxes; the taxes aredueinninemonths.3.Thereisasmalloilwellonthelandthat pumps oil attherateof$500permonth; the oilrevenue isreceivable at theendofeachmonth.

If Jerry decides to buythe land now, what are the

costs compared with buyingthe landoneyear fromnow?First, Jerry will have toborrow $100,000 from thelocal bank. At a rate of 8percent, the one-year interestcostswillbe

8%×$100,000=$8,000

IfJerrybuysthelandnow,hewill also be liable for the$2,000 in property taxes duein nine months. In order to

paythetaxes,hewillneedtoborrow an additional $2,000from the bank for theremainingthreemonthsoftheforwardcontract

$2,000+($2,000×8%×3/12)=$2,000+$40=

$2,040

The total costs of buyingnow are the interest on thecash price, the real estatetaxes, and the interest on the

taxes

$8,000+$2,040=$10,040

Whatare thebenefitsofbuyingnow?IfJerrybuystheland now, at the end of eachmonth he will receive $500worth of oil revenue. Overthe 12-month life of theforward contract, he willreceive

12×$500=$6,000

Additionally, Jerry canearn interest on the oilrevenue. At the end of thefirstmonth,hewillbeabletoinvest$500 for11months at8 percent. At the end of thesecondmonth,hewillbeabletoinvest$500for10months.The total interest on the oilrevenueis

($500×8%×11/12)+($500×8%×10/12)+…+($500×

8%×1/12)=$220

The total benefits ofbuying now are the oilrevenue plus the interest ontheoilrevenue

$6,000+$220=$6,220

If there are no otherconsiderations, a fair one-year forward price for thelandoughttobe

Assuming that Jerry andFarmer Smith agree on allthese calculations, it shouldmake no difference to eitherpartywhetherJerrypurchasesthe land now at a price of$100,000 or enters into aforward contract to purchasethelandoneyearfromnowat

a price of $103,820. Thetransactions are essentiallythesame.

Traders in forward orfutures contracts sometimesrefer to the basis, thedifference between the cashprice and the forward price.Inourexample,thebasisis

$100,000–$103,820=–$3,820

In most cases, the basis

will be a negative number—the costs of buying nowwilloutweigh the benefits ofbuyingnow.However,inourexample, the basis will turnpositive if the price of oilrises enough. If one year’sworthofoilrevenue,togetherwiththeinterestearnedontherevenue, is greater than the$10,040 cost of buying now,theforwardpricewillbelessthan the cash price.Consequently, the basis will

bepositive.Howshouldwecalculate

the fair forward price forexchange-traded futurescontracts? This depends onthe costs and benefitsassociated with a position inthe underlying contract. Thecosts and benefits for somecommonly traded futures arelistedinthefollowingtable:

PhysicalCommodities(Grains,EnergyProducts,PreciousMetals,etc.)

If we buy a physicalcommodity now, we willhave to pay the current pricetogether with the interest onthisamount.Additionally,wewill have to store the

commodity until maturity ofthe forward contract. Whenwe store the commodity, wewould also bewise to insureit against possible loss whileinstorage.If

C =commodityprice2t = time tomaturityoftheforwardcontract

r = interestrates = annualstorage costspercommodityuniti = annualinsurancecostspercommodityunit3

then the forward price Fcanbewrittenas

F=C×(1+r×t)+(s×t)+(i×t)

Initially, it may seemthat there are no benefits tobuying a physicalcommodity, so the basisshouldalwaysbenegative.Anormal or contangocommodity market is one inwhich long-term futurescontracts trade at a premiumto short-term contracts. Butsometimes the opposite

occurs—a futures contractwill trade at a discount tocash. If the cash price of acommodity is greater than afutures price, the market isbackward or inbackwardation. This seemsillogical because the interestandstoragecostswillalwaysbe positive. However,consider a company thatneeds a commodity to keepits factory running. If thecompany cannot obtain the

commodity, it may have totake the very costly step oftemporarily closing thefactory. The cost of suchdrastic action may, in thecompany’s view, beprohibitive. In order to avoidthis, the company may bewilling to pay an inflatedpricetoobtainthecommodityright now. If commoditysupplies are tight, the pricethatthecompanymayhavetopay could result in a

backward market—the cashpricewill begreater than theprice of a futures contract.The benefit of being able toobtain a commodity rightnow is sometimes referred toasaconvenienceyield.

It can be difficult toassign an exact value to theconvenience yield. However,ifinterestcosts,storagecosts,and insurance costs areknown, a trader can infer the

convenience yield byobserving the relationshipbetween the cash price andfutures prices. For example,consider a three-monthforward contract on acommodity

Three-monthforward priceF=$77.40Interest rate r=8percentAnnualstorage

costss=$3.00Annualinsurancecostsi=$0.60

What should be the cashpriceC?If

If the cash price in themarketplace is actually$76.25,theconvenienceyield

oughttobe$1.25.Thisistheadditional amount users arewilling topay for thebenefitof having immediate accesstothecommodity.

Stock

If we buy stock now, wewill have to pay the currentprice together with theinterest on this amount. Inreturn, we will receive any

dividends that the stock paysover the life of the forwardcontract together with theinterest earned on thedividendpayments.If

S=stockpricet = time tomaturityoftheforwardcontractr = interestrate over thelife of the

forwardcontract

di = each dividendpayment expectedprior tomaturity ofthe forwardcontractti = time remainingto maturity aftereach dividendpaymentri = the applicableinterest rate (the

forwardrate4)fromeach dividendpayment tomaturity of theforwardcontract

then the forward price Fcanbewrittenas

Example

StockpriceS=$67.00Time tomaturity t = 8monthsInterest rate r=6.00percentSemiannualdividendpayment d =$0.33Time to nextdividend

payment = 1month

Fromthis,weknowthat

then a fair eight-month

forward price for the stockshouldbe

Except for long-termstockforwardcontracts,therewill usually be a limitednumber of dividendpayments, and the amount ofinterestthatcanbeearnedoneach payment will be small.For simplicity, we will

aggregateall thedividendsDexpected over the life of theforward contract and ignoreany interest that can beearnedon thedividends.Theforwardprice fora stockcanthenbewrittenas

F=[S×(1+r×t)]–D

An approximate eight-month forward price shouldbe

67.00×(1+0.06×8/12)–(2×0.33)=69.02

Bondsandnotes

If we treat the couponpayments as if they weredividends, we can evaluatebond and note forwardcontracts ina similarmannerto stock forwards. We mustpay the bond price togetherwith the interest cost on that

price. In return, we willreceive fixed couponpayments on which we canearninterest.If

B=bondpricet = time tomaturityoftheforwardcontractr = interestrate over thelife of theforward

contractci = each couponexpected prior tomaturity of theforwardcontractti = time remainingto maturity aftereach couponpaymentri = applicableinterest rate fromeach couponpayment to

maturity of theforwardcontract

then the forward price Fcanbewrittenas

ExampleBondpriceB=$109.76Time to

maturityt=10monthsInterest rate r=8.00percentSemiannualcouponpayment c =5.25percentTime to nextcouponpayment = 2months

Fromthis,weknowthat

then a fair 10-monthforward price for the bondshouldbe

ForeignCurrencies

With foreign-currencyforward contracts, we mustdeal with two different rates—the domestic interest ratewemustpayonthedomesticcurrency to buy the foreigncurrency and the foreigninterest rate we earn if wehold the foreign currency.Unfortunately, if we beginwith the spot exchange rate,

add the domestic interestcosts, and subtract theforeign-currency benefits, weget an answer that isexpressed in different units.To calculate a foreign-currency forward price, wemust first express the spotexchange rateS as a fraction—the cost of one foreign-currency unit in terms ofdomestic-currency units Cddivided by one foreign-

currencyunitCf

Suppose that we have adomesticraterdandaforeignrate rf. What should be theforward exchange rate at theendoftimet?IfweinvestCfat rf and we invest Cd at rf,the exchange rate at time toughttobe

For example, suppose that€1.00=$1.50.Then

thenthesix-monthforwardpriceis

StockandFuturesoptions

In this text we will focusprimarily on the two mostcommonclassesofexchange-tradedoptions—stockoptionsand futures options.5Although there is sometradinginoptionsonphysicalcommodities, bonds, andforeign currencies in theover-the-counter (OTC)market,6almostallexchange-traded options on theseinstruments are futures

options. A trader inexchange-traded options oncrude oil is really tradingoptions on crude oil futures.A trader in exchange-tradedbondoptionsisreallytradingoptionsonbondfutures.

For both stock optionsandfuturesoptions,thevalueof the optionwill depend onthe forward price for theunderlyingcontract.Wehavealreadylookedattheforward

priceforastock.Butwhat istheforwardpriceforafuturescontract?Afuturescontractisaforwardcontract.Therefore,theforwardpriceforafuturescontractisthefuturesprice.Ifathree-monthfuturescontractistradingat$75.00,thethree-month forward price is$75.00.Ifasix-monthfuturescontract is trading at $80.00,the six-month forward priceis$80.00.Insomeways, thismakes options on futures

easier to evaluate thanoptions on stock because noadditional calculation isrequired to determine theforwardprice.

Arbitrage

Ifasked todefine the termarbitrage, a trader mightdescribe it as “a trade thatresults in a riskless profit.”Whetherthereissuchathing

asarisklessprofitisopenfordebate because there isalmostalwayssomethingthatcan go wrong. For ourpurposes, we will definearbitrage as the buying andselling of the same or veryclosely related instruments indifferent markets to profitfromanapparentmispricing.

For example, consider acommodity that is trading inLondonatapriceof$700per

unitandtradinginNewYorkat a price of $710 per unit.Ignoringtransactioncostsandany currency risk, thereseems to be an arbitrageopportunitybypurchasingthecommodity in London andsimultaneously selling it inNewYork.Will thisyieldanarbitrageprofitof$10?Orarethere other factors that mustbe considered? Oneconsideration might betransportation costs. The

buyer in New York willexpect delivery of thecommodity.Ifthecommodityis purchased in London, andif it costsmore than $10 perunit to ship the commodityfrom London to New York,any arbitrage profit will beoffset by the transportationcosts. Even if transportationcosts are less than $10, thereare also insurance costsbecause no one will want torisklossofthecommodityin

transit,eitherbyairorbysea,from London to New York.Of course, anyone trading acommodity professionallyought to know thetransportation and insurancecosts.Consequently,itwillbeimmediatelyobviouswhetheranarbitrageprofitispossible.

In a foreign-currencymarket, a tradermay attempttoprofitbyborrowingalow-interest-rate domestic

currency and using this topurchase a high-interest-rateforeign currency. The traderhopes to pay a low interestrate and simultaneously earnahighinterestrate.However,thistypeofcarrytradeisnotwithout risk. The interestrates may not be fixed, andover the life of the strategy,the interest rate thatmust bepaidonthedomesticcurrencymay rise while the interestratethatcanbeearnedonthe

foreign currency may fall.Moreover, the exchange rateis not fixed. At some point,the trader will have to repaythedomesticcurrencythatheborrowed. He expects to dothiswiththeforeigncurrencythathenowowns.Ifthevalueof the foreign currency hasdeclined with respect to thedomesticcurrency,itwillcosthim more to repurchase thedomestic currency and repaythe loan. The carry trade is

sometimes referred to asarbitrage, but in fact, itentailssomanyrisks that thetermisprobablymisapplied.

Because cash marketsand futures markets are soclosely related, a commontype of cash-and-carryarbitrage involves buying inthecashmarket,sellinginthefutures market, and carryingthepositiontomaturity.

Returning to our

previousstockexample:

StockpriceS=$67.00Time tomaturity t = 8monthsInterest rate r=6.00percentExpecteddividendsD =0.66

Ignoring interest on the

dividend, the calculatedeight-monthforwardpriceis

67.00×(1+0.06×8/12)–0.66=69.02

Suppose that there is amarket in forward contractson this stock and that theprice of an eight-monthforward contract is $69.50.Whatwill a trader do? If thetrader believes that thecontractisworthonly$69.02,

he will sell the forwardcontract at $69.50 andsimultaneously buy the stockfor $67.00. The cash-and-carry arbitrage profit shouldbe

69.50–69.02=0.48

Toconfirmthis,wecanlistall the cash flows associatedwith the transaction, keepingin mind that at maturity thetrader will deliver the stock

and in return receive theagreed-on forward price of69.50.

Fluctuationsinthepriceofeitherthestockorthefutures

contract will not affect theresults. Both the initial stockprice ($67.00) and the priceto be paid for the stock atmaturity ($69.50) are fixedandcannotbechanged.

Eventhoughfluctuationsin the stock or futures pricedo not represent a risk, otherfactors may affect theoutcome of the strategy. Ifinterest rates rise, the interestcosts associated with buying

the stock will rise, reducingthe potential profit.7Moreover, unless thecompany has actuallyannounced theamountof thedividend, the expecteddividend payment might bean estimate based on thecompany’s past dividendpayments. If the companyunexpectedly cuts thedividend, the arbitrage profitwillbereduced.

Given the apparentmispricing of the futurescontract, a trader mightquestion his own evaluation.Is$69.02anaccurateforwardprice? Perhaps the interestrate of 6 percent is too low.Perhapsthedividendof$0.66istoohigh.

We initially made ourcalculations by solving forFin terms of the spot price,time, interest rates, and

dividends

F=[S×(1+r×t)]–D

If we know the forwardpriceFbutaremissingoneoftheothervalues,wecansolvefor thatmissing value. If weknow the forwardprice, timetomaturity, interest rate, anddividend,wecansolveforS,the implied spot price of theunderlyingcontract

If we know everythingexcept the interest rate r, wecan solve for the impliedinterestrate

If we know everythingexcept the dividend D, wecan solve for the implied

dividend

D=[S×(1+r×t)]–F

Implied values are animportant concept, one thatwe will return to frequently.If a trader believes that acontract is fairly priced, theimplied valuemust representthe marketplace’s consensusestimateofthemissingvalue.

Returning to our eight-month forward contract,

suppose that we believe thatall values except the interestrateareaccurate.What is theimpliedinterestrate?

If we know all valuesexcept the dividend, theimplieddividendis

D=[S×(1+r×t)]–F=[67.00×(1+0.06×8/12)]–

69.50=0.18

If two dividends areexpected over the life of theforward contract, themarketplace seems to expecttwopaymentsof$0.09each.

Dividends

In order to evaluate

derivativecontractson stock,a trader may be required tomakeanestimateofastock’sfuturedividendflow.Atraderwill usually need to estimatethe amount of the dividendand the date on which thedividend will be paid. Tobetter understand dividends,it may be useful to definesome important terms in thedividendprocess.

Declared Date. The date

on which a companyannouncesboththeamountofthe dividend and the date onwhich the dividend will bepaid. Once the companydeclares the dividend, thedividendriskiseliminated,atleast until the next dividendpayment.

RecordDate. The date onwhich the stock must beownedinordertoreceivethedividend. Regardless of the

date on which the stock ispurchased, ownership of thestock does not becomeofficial until the settlementdate, the date on which thepurchaser of the stockofficiallytakespossession.Inthe United States, thesettlement date for stock isnormally three business daysafter the trade is made(sometimesreferredtoasT+3).

Ex-Dividend Date (Ex-Date).Thefirstdayonwhichastock is tradingwithout therights to the dividend. In theUnitedStates,thelastdayonwhich a stock can bepurchased inorder to receivethedividendisthreebusiness

daysprior to the recorddate.The ex-dividend date is twobusiness days prior to therecorddate.

On the ex-dividend date,quotes for the stock willindicate that the stock istradingex-div, and all quoteswill be posted with theamount of the dividenddeducted from the stockprice.Ifastockclosesontheday prior to the ex-dividenddate at apriceof$67.50andopens the following day (theex-dividenddate)atapriceof$68.25,andtheamountofthedividendis$0.40,thepriceof

thestockwillread

68.25+1.15ex-div0.40

If the stock had openedunchanged, the price wouldhavebeen thepreviousday’sprice of $67.50 less thedividendof$0.40,or$67.10.With the stock at $68.25, itspriceincreaseis$1.15.

PayableDate.Thedateonwhich the dividend will be

paid to qualifyingshareholders (those owningsharesontherecorddate).

The amount of thedividend can often beestimated from thecompany’s past dividendpayments. Ifacompanypaysquarterly dividends, as iscommonintheUnitedStates,andhaspaidadividendof25cents for the last10quarters,then it is reasonable to

assume that in the future thecompanywillcontinuetopay25cents.

We have generallyignored the interest that canbe earned on dividends, so itmay seem that the date onwhich the dividend will bepaid is not really important.If, however, the date onwhich the dividend will bepaid is expected to fall closeto the maturity date of a

derivative contract, a slightmiscalculation of thedividend date cansignificantlyalterthevalueofthederivative.

ShortSales

Manyderivativesstrategiesinvolve buying and sellingeither stock or futurescontracts. Except for thesituation when a market is

locked,8 there are norestrictions on the buying orselling of futures contracts.Therearealsono restrictionsonthepurchaseofstockoronthe sale of stock that isalready owned. However,there may be situations inwhich a trader will want tosell stock short, that is, sellstockthathedoesnotalreadyown.Thetraderhopestobuyback the stock at a later date

atalowerprice.Depending on the

exchange or local regulatoryauthority, there may bespecial rules specifying theconditionsunderwhich stockcanbesoldshort.Inallcases,however, a traderwhowantsto sell stock short must firstborrow the stock. This ispossible because manyinstitutions that hold stockmaybewillingtolendoutthe

stocktofacilitateashortsale.A brokerage firm holding aclient’s stock may bepermittedunderitsagreementwiththeclienttolendoutthestock. This does not meanthat one can always borrowstock. Sometimes it will bedifficultorevenimpossibletoborrow stock, resulting in ashort-stocksqueeze.Butmostactively traded stocks can beborrowed with relative ease,with the borrowing usually

facilitated by the trader’sclearingfirm.

Consider a trader whoborrows 900 shares of stockfrom a brokerage firm inordertosellthestockshortatapriceof$68pershare.Thepurchaser will pay the trader$68 × 900, or $61,200, andthe trader will deliver theborrowed stock. Thepurchaser of the stock doesnot care whether the stock

was sold short or long(whether the seller borrowedthe stock or actually ownedit).Asfaras thepurchaser isconcerned, he is now theownerofrecordofthestock.

Borrowed stock musteventually be returned to thelender, in this case thebrokerage firm. As securityagainst this obligation, thebrokerage firm will hold the$61,200 proceeds from the

sale.Because the$61,200, intheory, belongs to the trader,the firm will pay the traderinterest on this amount. Atthe same time, the trader isobligated to pay thebrokeragefirmanydividendsthataccrueovertheshort-saleperiod.

Howdoes thebrokeragefirmasthelenderprofitfromthis transaction? The lendingfirm profits because it pays

the trader only a portion ofthe full interest on the$61,200. The exact amountpaidtothetraderwilldependon how difficult it is toborrowthestock.Ifthestockis easy to borrow, the tradermayreceiveonlyslightlylessthantheratehewouldexpectto receive on any ordinarycash credit. However, ifrelatively few shares areavailable for lending, thetrader may receive only a

fractionofthenormalrate.Inthemostextremecase,wherethe stock is very difficult toborrow, the trader mayreceiveno interestatall.Theratethatthetraderreceivesonthe short sale of stock issometimes referred to as theshort-stockrebate.

We can make adistinction between the longraterlthatappliestoordinaryborrowing and lending and

the short rate rs that appliestotheshortsaleofstock.Thedifference between the longandshort rates represents theborrowingcostsrbc

rl–rs=rbc

In a previous examplewe determined the forwardpriceforastock

StockpriceS=

$67.00Time tomaturity t = 8monthsInterest rate r=6.00percentExpecteddividendpayment D =$0.66

Ignoring interest on thedividends, the eight-monthforwardpriceis

67.00×(1+0.06×8/12)–0.66=69.02

If the price of an eight-month forward contract is$69.50, there is an arbitrageopportunity by selling theforward contract andpurchasing the stock.Suppose that instead theeight-month forward contractistradingatapriceof$68.75.Now there seems to be anarbitrage opportunity by

purchasing the forwardcontractandsellingthestock.Indeed, if a trader alreadyowns the stock, this willresult in a profit of $69.02 –$68.75 = $0.27. If, however,thetraderdoesnotownstockandmust sell the stock shortin order to execute thestrategy, he will not receivethe full interest of 6 percent.Ifthelendingfirmwillretain2percent inborrowingcosts,the trader will only receive

the short rate of 4 percent.Theforwardpriceisnow

67.00×(1+0.04×8/12)–0.66=68.13

If the trader attempts toexecute the arbitrage bysellingthestockshort,hewilllosemoneybecause

68.13–68.75=–0.62

Atraderwhodoesnotown

thestockcanonlyprofitiftheforward price is less than$68.13 or more than $69.02.Between these prices, noarbitrageispossible.

Whatinterestrateshouldapply to option transactions?Unlikestock,anoptionisnota deliverable security. It is acontract that is createdbetweenabuyerandaseller.Evenifatraderdoesnotownaspecificoption,heneednot

“borrow” the option in orderto sell it. For this reason,wealways apply the ordinarylong rate to the cash flowresulting from either thepurchaseorsaleofanoption.

1Atthispoint,wewillassumethatthesameinterestrateappliestoalltransactions,whetherborrowingorlending.Admittedly,foratrader,theinterestcostofborrowingwillalmostalwaysbehigherthantheinterestearnedwhenlending.2Inthischapteronly,wewilluseacapitalCtorepresentthepriceofacommodity.Inallotherchapters,Cwillrefertothepriceofacalloption.3Forphysicalcommodities,bothstorageandinsurancecostsusuallyarequotedtogetherasoneprice.4Theforwardrateistherateofinterestthatisapplicablebeginningonsomefuturedateforaspecifiedperiodof

time.Forwardratesareoftenexpressedinmonths

1×5forwardrateAfour-monthratebeginninginonemonth3×9forwardrateAsix-monthratebeginning

inthreemonths4×12forwardrateAneight-monthratebeginninginfourmonths

Aforward-rateagreement(FRA)isanagreementtoborroworlendmoneyforafixedperiod,beginningonsomefuturedate.A3×9FRAisanagreementtoborrowmoneyforsixmonths,

butbeginningthreemonthsfromnow.

5Later,inChapter22,wewillalsolookatstockindexfuturesandoptions.6TheOTCmarket,orover-the-countermarket,isatermusuallyappliedtotradingthatdoesnottakeplaceonanorganizedexchange.7Ifmoneyhasbeenborrowedorlentatafixedrate,thereisnointerest-raterisk.However,mosttradersborrowandlendatavariablerate,resultingininterest-rateriskoverthelifeoftheforwardcontract.8Somefuturesexchangeshavedailypricelimitsforfuturescontracts.Whenafuturescontractreachesthislimit,the

marketissaidtobelockedorlockedlimit.Ifthemarketiseitherlimituporlimitdown,nofurthertradingmaytakeplaceuntilthepricecomesoffthelimit(someoneiswillingtosellatapriceequaltoorlessthantheuplimitorbuyatapriceequaltoorhigherthanthedownlimit).

ContractSpecifications

andOptionTerminology

Everyoptionmarketbringstogether tradersand investorswith different expectationsand goals. Some enter themarket with an opinion onwhich direction prices willmove. Some intend to useoptions to protect existingpositions against adverseprice movement. Some hopeto take advantage of pricediscrepanciesbetweensimilaror relatedproducts.Someactas middlemen, buying and

selling as an accommodationto other market participantsandhopingtoprofitfromthedifference between the bidpriceandaskprice.

Even thoughexpectationsandgoalsdiffer,everytrader’seducationmustinclude an understanding ofoption contract specificationsand a mastery of theterminology used in optionmarkets. Without a clear

understandingofthetermsofan option contract and therights and responsibilitiesunder that contract, a tradercannothopetomakethebestuseofoptions,norwillhebeprepared for the very realrisks of trading. Without afacility in the language ofoptions, a trader will find itimpossible to communicatehisdesiretobuyorsellinthemarketplace.

ContractSpecifications

Thereareseveralaspectstocontractspecifications.

TypeIn Chapter 1, we

introduced the two types ofoptions. A call option is theright to buy or take a long

positioninanassetatafixedpriceonorbeforeaspecifieddate.Aputoptionistherighttosellortakeashortpositioninanasset.

Note the differencebetween an option and afutures contract. A futurescontractrequiresdeliveryatafixed price. The buyer andseller of a futures contractboth have clearly definedobligations that they must

meet. The seller must makedelivery, and the buyermusttakedelivery.Thebuyerofanoption, however, has achoice.Hecanchoosetotakedelivery (a call) or makedelivery (a put). If the buyerofanoptionchoosestoeithermake or take delivery, theseller of the option isobligated to take the otherside. In option trading, allrights lie with the buyer andallobligationswiththeseller.

UnderlyingThe underlying asset or,

more simply, the underlyingis the security or commoditytobeboughtorsoldundertheterms of the option contract.If an option is purchaseddirectly fromabankorotherdealer, the quantity of theunderlying can be tailored tomeet the buyer’s individualrequirements. If theoption ispurchased on an exchange,

thequantityoftheunderlyingis set by the exchange. Onstock option exchanges, theunderlying is typically 100shares of stock.1 The ownerof a call has the right to buy100shares;theownerofaputhas the right to sell 100shares. If, however, thepriceof an underlying stock iseither very lowor very high,an exchange may adjust thenumber of shares in the

underlying contract in ordertocreateacontractsizethatisdeemed reasonable fortradingontheexchange.2

On all futures optionsexchanges, the underlying isuniformly one futurescontract.Theownerof a callhas the right to buy onefuturescontract;theownerofaputhastherighttosellonefutures contract. Most often,the underlying for an option

on a futures contract is thefutures month thatcorresponds to the expirationmonth of the option. Theunderlying for an Aprilfutures option is an Aprilfutures contract; theunderlying for a Novemberfuturesoption isaNovemberfuturescontract.However,anexchangemayalsochoose tolist serial options on futures—option expirations wherethere is no corresponding

futures month. When afutures option has nocorresponding futuresmonth,theunderlyingcontract is thenearest futures contractbeyond expiration of theoption.

For example, manyfinancial futuresare listedonaquarterlycycle,withtradingin March, June, September,and December futures. Theunderlying for a March

option is a March futurescontract; theunderlyingforaJune option is a June futurescontract. If there are alsoserialoptions,then

Theunderlying fora January orFebruaryoption is aMarch futurescontract.The

underlying foran April orMay option isa June futurescontract.The

underlying fora July orAugust optionisaSeptemberfuturescontract.The

underlying foran October orNovemberoption is aDecemberfuturescontract.

Some interest-ratefutures markets [e.g.,Eurodollars at the ChicagoMercantile Exchange, ShortSterling and Euribor at theLondon International

Financial Futures Exchange],in addition to listing long-term options on a long-termfuturescontract,mayalsolistshort-term options on thesame long-term futurescontract. A March futurescontract maturing in twoyears may be the underlyingfor a March option expiringin two years. But the samefutures contract may also bethe underlying for a Marchoption expiring in one year.

Short-term options on long-term futures are listed asmidcurve options. Theoptions can be one-yearmidcurve(ashort-termoptionon a futures contract with atleast one year to maturity),two-year midcurve (a short-term option on a futurescontract with at least twoyears to maturity), or five-year midcurve (a short-termoption on a futures contractwith at least five years to

maturity).

ExpirationDateorExpiryThe expiration date is the

date on which the owner ofanoptionmustmakethefinaldecision whether to buy, inthecaseofacall,ortosell,inthe case of a put. Afterexpiration, all rights andobligations under the option

contractceasetoexist.On many stock option

exchanges, the expirationdateforstockandstockindexoptions is the thirdFriday ofthe expiration month.3 Ofmore importance to mosttradersisthelasttradingday,the last business day prior toexpirationonwhichanoptioncan be bought or sold on anexchange. For most stockoptions, expiration day and

the last trading day are thesame, the third Friday of themonth. However, GoodFriday, a legal holiday inmany countries, occasionallyfalls on the third Friday ofApril. When this occurs, thelast trading day is theprecedingThursday.

When stock optionswereintroducedintheUnitedStates, trading in expiringcontractsendedatthecloseof

business on the third Fridayofthemonth.However,manyderivative strategies requirecarrying an offsetting stockposition to expiration, atwhichtimethestockpositionis liquidated. Consequently,stockexchangesfoundthatasthe close of tradingapproached on expirationFriday, theywere facedwithlarge orders to buy or sellstock. These large ordersoften had the effect of

disrupting trading ordistortingpricesatexpiration.

Toalleviate theproblemof large order imbalances atexpiration, some derivativeexchanges, working with thestockexchangesonwhichtheunderlying stocks weretraded, agreed toestablishanexpiration value for aderivatives contract based onthe opening price of theunderlying contract rather

than the closing price on thelast trading day. This AMexpiration is commonly usedfor stock index contracts.Options on individual stocksare still subject to thetraditional PM expiration,where the value of an optionis determined by theunderlying stock price at theclose of trading on the lasttradingday.

Although the expiration

date for stock options isrelatively uniform, theexpiration date for futuresoptions can vary, dependingontheunderlyingcommodityor financial instrument. Forfutures on physicalcommodities, such asagricultural or energyproducts,deliveryatmaturitymay take several days. As aconsequence, options onfutures for physicalcommoditieswilloftenexpire

several days or even weeksprior to the maturity of thefutures contract, mostcommonlyinthemonthpriorto the futures month. Anoption on a March futurescontract will expire inFebruary;anoptiononaJulyfuturescontractwillexpireinJune; an option on aNovember futures contractwill expire in October. Atraderwillneedtoconsulttheexchange calendar to

determine the exactexpiration date, which is setbyeachindividualexchange.

ExercisePriceorStrikePriceTheexerciseorstrikeprice

is the price at which theunderlying will be deliveredshouldtheholderofanoptionchoosetoexercisehisrighttobuy or sell. If the option is

exercised,theownerofacallwill pay the exercise price;the owner of a put willreceivetheexerciseprice.

The exercise pricesavailable for trading on anoption exchange are set bytheexchange,usuallyatequalintervals and bracketing thecurrent price of theunderlying contract. If theprice of the underlyingcontract is 62 when options

are introduced, the exchangemaysetexercisepricesof50,55, 60, 65, 70, and 75. At alater date, as the price of theunderlying moves up ordown, the exchange can addadditional exercise prices. Ifthe price of the underlyingrisesto70,theexchangemayaddexercisepricesof80,85,and 90. Additionally, if theexchange feels that it willfurther facilitate trading, itcan introduce intermediate

exercise prices—52½, 57½,62½,67½.

As an example of anexchange-traded option, thebuyer of a crude oil October90 call on the New YorkMercantile Exchange has therighttotakealongpositioninoneOctobercrudeoilfuturescontract for 1,000 barrels ofcrudeoil(theunderlying)ataprice of $90 per barrel (theexercise price) on or before

the October expiration (theexpirationdate).Thebuyerofa General Electric March 30put on the Chicago BoardOptions Exchange has theright to take a short positionin 100 shares of GeneralElectric stock (theunderlying) at a price of $30per share (the exercise price)onorbeforeMarchexpiration(theexpirationdate).

Option contract

specifications are furtheroutlinedinFigure3-1.

Figure3-1Optioncontractspecifications.

ExerciseandAssignmentThebuyerofacalloraput

option has the right toexercise that option prior toits expiration date, therebyconverting the option into along underlying position inthe case of a call or a shortunderlying position in thecase of a put. A trader whoexercisesacrudeoilOctober

90 call has chosen to take along position in one Octobercrude oil futures contract at$90 per barrel.A traderwhoexercisesaGEMarch30puthas chosen to take a shortposition in 100 shares ofGEstock at $30 per share.Oncean option is exercised, therights and obligationsassociated with the optioncease to exist, just as if theoption had been allowed toexpire.

A trader who intends toexercise an option mustsubmit an exercise notice toeithertheselleroftheoption,ifpurchasedfromadealer,ortotheexchange,iftheoptionwas purchased on anexchange. When a validexercise notice is submitted,the seller of the option hasbeenassigned.Dependingonthe type of option, the sellerwillberequiredtotakealongor short position in the

underlying contract at theoption’sexerciseprice.

Onceacontracthasbeentraded on an exchange, thelinkbetweenbuyerandselleris broken,with the exchangebecoming thecounterparty toalltrades.Still,whenatraderexercises an option, theexchange must assignsomeone toeitherbuyorselltheunderlyingcontractat theexercise price.Howdoes the

exchangemakethisdecision?The party who is assignedmust be someone who hassold the option and has notclosed out the positionthrough an offsetting trade.Beyond this, the exchange’sdecision on who will beassigned is essentiallyrandom, with no traderhaving either a greater orlesser probability of beingassigned.

New traders sometimesbecome confused aboutwhether the exercise andassignment result in a longposition (buying theunderlying contract) or ashort position (selling theunderlying contract). Thefollowingsummarymayhelp:ifyou

Depending on theunderlying contract,when anexchange-traded option isexercised,itcansettleinto

1. The physicalunderlying2. A futuresposition

3.Cash

SettlementintothePhysicalUnderlyingIf a call option settles into

the physical underlying, theexerciser pays the exerciseprice and in return receivestheunderlying.Ifaputoptionsettles into the physicalunderlying, the exerciserreceives the exercise priceandinreturnmustdeliverthe

underlying. Stock optionsalwayssettleintothephysicalunderlying.

You exercise oneJanuary110callonstock.

You mustpay 100 ×$110 =$11,000.You receive

100 shares of

stock.You are assignedon six April 40callsonstock.

You receive600 × $40 =$24,000.You must

deliver 600shares ofstock.

You exercise twoJuly 60 puts on

stock.You receive

200 × $60 =$12,000.You must

deliver 200shares ofstock.

You are assignedonthreeOctober95putsonstock.

You mustpay300×$95

=$28,500.You receive300 shares ofstock.

Note that the cash flowresultingfromsettlementintothe physical underlyingdepends only on the exerciseprice. In our examples,whetherthepriceofthestockat exercise is $10 or $1,000,the exerciser of a call pays

only the exercise price, notthestockprice.Theexerciserof a put receives only theexerciseprice.Ofcourse, theprofit or loss resulting fromthe option trade will dependon both the stock price andthe price originally paid fortheoption.But thecash flowwhen the option is exercisedisindependentofthese.

Settlementintoa

FuturesPositionIf an option settles into a

futuresposition,itisjustasifthe exerciser is buying orselling the futurescontract atthe exercise price. Theposition is immediatelysubject to futures-typesettlement,requiringamargindepositandaccompaniedbyavariationpayment.

An underlying futurescontractiscurrentlytradingat

85.00 with a point value of$1,000. Margin requirementsare$3,000percontract.

You exercise oneFebruary80call.

Youimmediatelybecome longone futurescontract at apriceof80.You must

deposit withthe exchangethe requiredmargin of$3,000.You will

receive avariationcreditof (85–80)×$1,000 =$5,000.

You are assignedonsixMay75calls.

Youimmediatelybecome shortsix futurescontracts at apriceof75.You must

deposit withthe exchangethe requiredmargin of 6 ×$3,000 =$18,000.

You willhave avariation debitof (75–85)×$1,000×6=–$60,000

You exercise fourAugust100puts.

Youimmediatelybecome shortfour futurescontracts at a

priceof100.You must

deposit withthe exchangethe requiredmargin of 4 ×$3,000 =$12,000.You will

receive avariationcreditof(100–85)×$1,000 × 4 =

$60,000.You are assignedon two November95puts.

Youimmediatelybecome longtwo futurescontracts at apriceof95.You must

deposit withthe exchange

the requiredmargin of 2 ×$3,000 =$6,000.You will

have avariation debitof (85–95)×$1,000×2=–$20,000.

SettlementintoCashThis type of settlement is

used primarily for indexcontracts where delivery oftheunderlyingcontract isnotpractical. If exercise of anoption settles into cash, nounderlying position results.Thereisacashpaymentequalto thedifferencebetween theexercise price and theunderlyingpriceattheendofthetradingday.

An underlying index isfixedattheendofthetrading

dayat300.Theexchangehasassigned a value of $500 toeachindexpoint.

You exercise threeMarch250calls.

You haveno underlyingposition.Your

account willbe creditedwith (300 –

250)×$500×3=$75,000.

You are assignedon seven June 275calls.

You haveno underlyingposition.Your

account willbe debited by(275 – 300) ×$500 × 7 =

$87,500.You exercise twoSeptember 320puts.

You haveno underlyingposition.Your

account willbe creditedwith (320 –300)×$500×2=$20,000.

You are assignedon four December340puts.

You haveno underlyingposition.Your

account willbe debited by(300 – 340) ×$500 × 4 =$80,000.

ExerciseStyleIn addition to the

underlying contract, exerciseprice, expiration date, andtype, an option is furtheridentified by its exercisestyle, either European orAmerican.AEuropeanoptioncan only be exercised atexpiration. In practice, thismeans that the holder of aEuropean option must makethe final decision whether to

exercise or not on the lastbusiness day prior toexpiration. In contrast, anAmerican option can beexercisedonanybusinessdaypriortoexpiration.

The designation of anoption’s exercise style aseither European orAmericanhas nothing to do withgeographic location. Manyoptions traded in the UnitedStates are European, and

many options traded inEurope are American.4Generally, options on futuresand options on individualstocks tend to be American.OptionsonindexestendtobeEuropean.

OptionPriceComponents

As in any competitive

market, an option’s price, orpremium, is determined bysupply and demand. Buyersand sellersmake competitivebids and offers in themarketplace.Whenabidandoffer coincide, a trade ismade.

Thepremiumpaidforanoption can be separated intotwo components—theintrinsic value and the timevalue.Anoptionhasintrinsic

value if it enables the holderof theoption to buy lowandsellhighorsellhighandbuylow, with the intrinsic valuebeing equal to the differencebetweenthebuyingpriceandthe selling price. With anunderlyingcontracttradingat$435, the intrinsic value of a400callis$35.Byexercisingthe option, the holder of the400 call can buy at $400. Ifhe then sells at the marketprice of $435, $35 will be

credited to his account.Withanunderlyingcontracttradingat$62,theintrinsicvalueofa70 put is $8. By exercisingthe option, the holder of theputcansellat$70.Ifhethenbuys at the market price of$62, he will show a totalcreditof$8.

A call will only haveintrinsic value if its exerciseprice is less than the currentmarket price of the

underlying contract becauseno one would choose to buyhighandsell low.Aputwillonlyhaveintrinsicvalueifitsexercise price is greater thanthe current market price ofthe underlying contractbecausenoonewouldchoosetoselllowandbuyhigh.Theamount of intrinsic value isthe amount by which theexercisepriceis less thanthecurrent underlying price inthe case of a call or the

amountbywhichtheexerciseprice is greater than thecurrent underlying price inthe case of a put. No optioncan have an intrinsic valuelessthanzero.IfSisthespotprice of the underlyingcontractandX istheexerciseprice,then

Call intrinsicvalue =maximum ofeither0orS–

X.Put intrinsicvalue =maximum ofeither0orX–S.

Note that the intrinsicvalue is independent of theexpiration date. With theunderlying contract at $83, aMarch 70 call and aSeptember 70 call both havean intrinsic value of $13. A

June 90 put and aDecember90put both have an intrinsicvalueof$7.

Usually, an option’sprice in themarketplace willbe greater than its intrinsicvalue. The time value,sometimesalsoreferred toastheoption’s timepremium orextrinsic value, is theadditional amount ofpremiumbeyond the intrinsicvalue that traders are willing

to pay for an option.Marketparticipantsarewillingtopaythis additional amountprimarily because of theprotective characteristicsaffordedbyanoptionoveranoutrightlongorshortpositionintheunderlyingcontract.

An option’s premium isalwayscomposedofpreciselyitsintrinsicvalueanditstimevalue. Examples of intrinsicvalue and time value are

shown in Figure 3-2. If a$400 call is trading at $50withtheunderlyingtradingat$435, the time value of thecallmustbe$15because theintrinsic value is $35. Thetwocomponentsmustaddupto theoption’s totalpremiumof$50.Ifa$70putonastockis trading for $11 with thestocktradingat$62,thetimevalue of the put must be $3because the intrinsic value is$8.Again, the intrinsicvalue

and the time value must adduptotheoption’spremiumof$11.

Figure3-2Intrinsicvalueandtimevalue.

Eventhoughanoption’spremiumisalwayscomposedof its intrinsic value and itstime value, one or both ofthese components can bezero. If the option has nointrinsicvalue,itspriceinthemarketplace will consistsolely of time value. If theoption has no time value, itsprice will consist solely ofintrinsic value. In the lattercase, traders say that the

optionistradingatparity.Although an option’s

intrinsic value can never beless than zero, it is possibleforaEuropeanoptiontohavea negative time value. (MoreaboutthisinChapter16whenwe look at the early exerciseof American options.) Whenthis happens, the option cantrade for less than parity.Usually,however,anoption’spremium will reflect some

nonnegative amount of timevalue.

IntheMoney,AttheMoney,andOutoftheMoneyDepending on the

relationship between anoption’s exercise price andthe price of the underlyingcontract, options are said to

be in the money, at themoney,andoutofthemoney.Anyoptionthathasapositiveintrinsicvalueissaidtobeinthemoney by the amount ofthe intrinsic value. With astock at $44, a $40 call is inthemoney by $4.A $55 puton the same stock is in themoney by $11. An optionwithnointrinsicvalueissaidtobeoutofthemoney,anditsprice consists solely of timevalue. In order to be in the

money, a call must have anexercise price lower than thecurrent price of theunderlyingcontract,andaputmust have an exercise pricehigher than the current priceof the underlying contract.Note that if a call is in themoney, a put with the sameexercisepriceandunderlyingcontract must be out of themoney.Conversely,iftheputis in the money, a call withthe same exercise pricemust

be out of the money. In ourexamples with the stock at$44,the$40putisoutofthemoneyby$4andthe$55callisoutofthemoneyby$11.

Finally,anoptionwhoseexercise price is equal to thecurrent price of theunderlyingcontract is said tobeatthemoney.Technically,such an option is also out ofthe money because it has nointrinsic value.Tradersmake

the distinction between at-the-money and out-of-the-moneyoptionsbecause,asweshall see, at-the-moneyoptions often have veryspecific and desirablecharacteristics, and suchoptions tend to be the mostactivelytraded.

If we want to be veryprecise,foranoptiontobeatthemoney, its exercise pricemust be exactly equal to the

current price of theunderlying contract.However, for exchange-traded options, the term iscommonlyapplied to thecalland putwhose exercise priceis closest to the currentpriceof the underlying contract.With a stock at $74 and $5betweenexerciseprices($65,$70, $75, $80, etc.), the $75call and the $75 put are theat-the-money options. Theseare the call and put options

withexercisepricesclosesttothe current price of theunderlying contract. In-, at-,andout-of-the-moneyoptionsareoutlinedinFigure3-3.

Figure3-3In-,at-,andout-of-the-moneyoptions.

AutomaticExerciseAt expiration an in-the-

money option will alwayshave some intrinsic value. Atrader can capture this valuebyeithersellingtheoptioninthe marketplace prior toexpiration or exercising theoption and immediatelyclosing the underlyingposition. When exchange-traded options were firstintroduced, anyone wishing

to exercise an option wasrequired to formally submitan exercise notice to theexchange. If someone forgotto submit an exercise noticefor an in-the-money option,the option would expireunexercised, and the traderwouldlosetheintrinsicvalue.This is an outcome that norationalpersonwouldaccept.Unfortunately, in the earlydays of option trading, thisoccurred occasionally for

various reasons: perhaps thetrader was unaware that hewas required to submit anexercise notice, perhaps thetrader was out ofcommunication with theexchange and was thereforeunable to submit an exercisenotice, or perhaps there wasan error on the part of theclearing firm in processingtheexercisenotice.

To avoid a situation

whereanin-the-moneyoptionexpires unexercised, whichwould be an embarrassmentto both the individual traderand the exchange, mostexchanges have instituted anautomatic exercise policy.The exchange will exerciseonbehalfoftheoptionholderany in-the-money option atexpiration,evenifanexercisenotice has not beensubmitted. The criteria forautomatic exercise may vary

fromoneexchangetoanotherandmayalsovarydependingonwhoholds theoption.Forexample, because oftransaction costs, it may notbe economically worthwhileto exercise an option that isonly very slightly in themoney. Therefore, theexchange may automaticallyexerciseonlyoptionsthatarein the money by somepredetermined amount. If theautomatic exercise threshold

is 0.05, then an option mustbe in the money by at least0.05inorderfortheexchangeto exercise the option. If theoption is in the money by0.03, a trader may stillexercise the option but mustdo so by submitting anexercise notice. On theopposite side, if theoption isinthemoneyby0.06,atraderwho feels that the option isnot worth exercising maysubmit a do not exercise

notice. Otherwise, theexchange will automaticallyexercise the option on thetrader’sbehalf.

Because professionaltraders and retail customershavedifferentcoststructures,the exchange may have adifferent automatic exercisethreshold for eachparty.Thethreshold may be 0.05 forretailcustomersbutonly0.02for professionals. To

determine who is aprofessionaltraderandwhoisnot,anexchangewillusuallyspecify the criteria necessaryfor inclusion in eachcategory.

OptionMarginingDepending on the

exchange and the type ofunderlying contract, optionscanbesubjecttoeitherstock-

type settlement or futures-type settlement. However,onceanoptiontradeismade,thereareadditional risks thatthe clearinghouse mustconsider. Is the risk to anoption position limited orunlimited? If unlimited, howshould the clearinghouseprotectitself?

When the risk of anoptionpositionis limited, themarginthatmustbedeposited

with the clearinghouse willnever be greater than themaximumrisktotheposition.The buyer of an option cannever have risk greater thanthe premium paid for theoption,and theclearinghousewill never require a margindeposit greater than thisamount. Even if an optionposition is very complex, aslong as there is a maximumrisktotheposition,therewillalso be a maximum margin

requirement.Some option positions,

however,haveunlimitedrisk.For such positions, theclearinghouse must considerthe risk associated with awide variety of outcomes.Once this is done, theclearinghouse can require amargindepositcommensuratewiththeperceivedriskoftheposition. Unlike futuresmargining, where the

clearinghouse sets a fixedmargindeposit foreachopenfutures position, there is nosinglemethodofdeterminingthe margin for a complexoption position.However, allmethods are risk-based,requiring an analysis of theposition’s risk under a broadrangeofmarketconditions.IntheUnitedStates,theOptionsClearing Corporation hasdeveloped its own risked-based margining system for

stock and index options. Themost widely used marginingsystem on futures exchangesis the Standard PortfolioAnalysis of Risk (SPAN)system developed by theChicago MercantileExchange. Both marginingsystems create an array ofpossible outcomes withrespecttoboththeunderlyingpriceandtheperceivedspeedwith which the underlyingprice can change. The

clearinghouse then uses thisarray to determine areasonable marginrequirement.5

1Onehundredsharesissometimesreferredtoasaroundlot.Anordertobuyorsellfewerthan100sharesisanoddlot.2Manyexchangesalsopermittradinginflexoptions,wherethebuyerandsellermaynegotiatethecontractspecifications,includingthequantityoftheunderlying,theexpirationdate,theexerciseprice,andtheexercisestyle.3Intheearlydaysofoptiontrading,exchange-tradedoptionsoftenexpiredonanonbusinessday,typicallyonaSaturday.Thisgavetheexchangeanextradaytoprocessthepaperworkassociatedwithexpiringoptions.4Itdoesappearthatthefirstoptions

tradedintheUnitedStatescarriedwiththemtherightofearlyexercise—hencethetermAmericanoption.5AdescriptionofSPANmarginingcanbefoundathttp://www.cmegroup.com/clearing/risk-management.Adescriptionoftherisk-basedmarginingsystemusedbytheOptionsClearingCorporationcanbefoundathttp://www.optionsclearing.com/risk-management/margins/.

ExpirationProfitandLoss

The trader who enters anoption market for the firsttime may find himselfsubjected to a form of

contract shock. Unlike atrader in equities or futures,whosechoicesarelimitedtoasmallnumberof instruments,an option trader must oftendeal with a bewilderingassortmentofcontracts.Withseveral expiration months,with multiple exercise pricesavailable in eachmonth, andwith both calls and puts ateach exercise price, it is notunusual for an option tradertobe facedwithwhatat first

seems like an overwhelmingnumberofdifferentcontracts.With so many choicesavailable,atraderneedssomelogical method of decidingwhich options actuallyrepresentprofitopportunities.Whichshouldhebuy?Whichshouldhesell?Whichshouldheavoid?Thechoicesare sonumerous that a prospectiveoption trader might beinclined to give up infrustration.

Tobegin,a tradermightask a very obvious question:whatisanoptionworth?Thequestionmaybeobvious,butthe answer, unfortunately, isnot,becauseoptionpricescanbeaffectedbymanydifferentmarket forces. However,there is one time in anoption’s life when everyoneought to be able to agree onthe option’s value. Atexpiration,anoptionisworthexactly its intrinsic value:

zero if it isoutof themoneyor thedifferencebetween theunderlying price and theexercise price if it is in themoney.

Following is a series ofunderlying prices and thevalue at expiration for twooptions, a $95 call and $110put:

For the 95 call, if theunderlyingpriceatexpiration

is95orbelow,thecallisoutof the money and thereforeworthless. If, however, theunderlying price rises above95,the95callwillgointothemoney, gaining one point invalue for each point that theunderlying price rises above95. For the 110 put, if theunderlying price is 110 orabove, the put is out of themoney and thereforeworthless. But if theunderlying price falls below

110,the110putgoesintothemoney, gaining one point invalue for each point declineintheunderlyingprice.

ParityGraphs

For someone who hasboughtanoption,theintrinsicvalue represents a credit, orpositive value. The buyer oftheoptionwillbeabletobuylow and sell high. For

someone who has sold anoption, the intrinsic valuerepresentsadebit,ornegativevalue.Theselleroftheoptionwillbeforcedtobuyhighandsell low. We can use anoption’s intrinsic value todraw a graph of the value ofan option position atexpirationasafunctionoftheprice of the underlyingcontract. Figure 4-1 showssuch a graph for a long callposition. Below the exercise

price,theoptionhasnovalue.Above theexerciseprice, theoption gains one point invalue for each point increaseintheunderlyingprice.

Figure4-1Longcall.

Figure 4-2 shows thevalue of a short call positionat expiration. Now, if theoption is in the money, thevalue of the position isnegative. For every point theunderlying rises above theexercise price, the positionlosesonepointinvalue.

Figure4-2Shortcall.

We can create the sametype of expiration graphs forlong and short put positions,as shown in Figures 4-3 and4-4. For a long or short put,the value of the position iszeroiftheunderlyingpriceisabove the exercise price. Fora longput, thepositiongainsone point for each pointdecline in the underlyingprice. For a short put, theposition loses one point in

value for each point declineintheunderlyingprice.

Figure4-3Longput.

Figure4-4Shortput.

A parity graphrepresents the value of anoption position at expiration,paritybeinganothernameforintrinsic value. Because oftheir shapes, traderssometimes refer to the fourbasicparitygraphs (longandshort call and long and shortput) as the hockey-stickdiagrams.

The four basic paritygraphs highlight one of the

mostimportantcharacteristicsof option trading. Buyers ofoptions have limited risk(they can never lose morethan the price of the option)andunlimitedprofitpotential.Sellers of options havelimited profit potential (theycannevermakemorethantheprice of the option) andunlimitedrisk.1

Given the apparentlyunbalanced risk-reward

tradeoff, new option traderstend to have the samereaction: why would anyonedo anything other than buyoptions? The purchase of anoption results in a positionwith limited risk andunlimited profit, whichcertainly seems moredesirable than the limitedprofit and unlimited risk thatresult from the sale of anoption. Yet, in every optionmarket, thereare traderswho

are willing to sell options.Why are they willing to dothis in the face of thisapparently unbalanced risk-reward tradeoff? The answerhas to do with not just thebest and worst that canhappen but also with thelikelihood of thoseoccurrences. It’s true thatsomeonewho sells anoptionis exposed to unlimited risk,butiftheamountreceivedfortheoptionisgreatenoughand

the perceived risk is lowenough, a trader might bewilling to take that risk. Inlaterchapterswewillsee thevery important roleprobability plays in optionpricing.

SlopeFromtheparitygraphs,we

canseethatifanoptionisoutof the money, its value is

unaffected by changes in theprice of the underlyingcontract.Iftheoptionisinthemoney, it will either gain orlose value as the underlyingpricechanges.The slope of the graph is

the change in value of theoption position with respecttochangesin thepriceof theunderlying contract, oftenexpressedasafraction

We can summarize theslopes of the basic positionsasfollows:

In addition to paritygraphsforindividualoptions,

we can also create paritygraphs for positionsconsistingofmultipleoptionsbyaddinguptheslopesoftheindividualoptions.Figure4-5is the parity graph of aposition consisting of a longcalland longputat the sameexercise price. We cancalculate the total slopes asfollows:

Figure4-5(a)Longcallandlongputatthesameexerciseprice.(b)Combinedposition.

Thecombinedpositionwillgain value if the underlyingprice moves in eitherdirection away from theexerciseprice.Thepositionistypical of many optionstrategies that may besensitive to themagnitude ofmovement in the underlyingcontract rather than thedirectionofmovement.

Many option strategiesinvolve combining options

with the underlying contract,so we will also want toconsider the slope of anunderlying position. Asshown in Figure 4-6, theslope of a long underlyingpositionisalways+1,andtheslope of a short underlyingposition is always –1. Theslopesareconstantregardlessof the underlying price. Thisis an important distinctionbetween an option positionand an underlying position.

Because of the insurancefeature of an option, theparity graph of an optionposition will always bend attheexerciseprice.

Figure4-6Longandshortunderlyingposition.

Figure 4-7 shows theparitygraphofapositionthatcombines two long calloptions at the same exerciseprice and with a shortunderlying contract. Belowthe exercise price, the totalslope is –1 (0 for the out-of-the-money calls, –1 for theshort underlying). Above theexerciseprice, the total slopeis +1 (+2 for the in-the-money calls, –1 for the short

underlying contract). Thisparitygraphisidenticaltotheposition inFigure4-5,whichmust mean that the sameoption strategy can beconstructed inmore thanoneway. This is an importantcharacteristic of options thatwewilllookatinmoredetailinChapter14.Note also thatthelocationoftheunderlyingposition is irrelevant to theparity graph. Regardless ofthe price of the underlying,

the slope is always either +1foralongunderlyingpositionor –1 for a short underlyingposition.

Figure4-7(a)Longtwocallsandshortanunderlyingcontract.(b)Combinedposition.

Figure 4-8 is the paritygraphofalongcallandshortputatthesameexerciseprice.Below the exercise price, thetotal slope is +1 (0 for thelong out-of-the-money call,+1fortheshortin-the-moneyput). Above the exerciseprice, the total slope is also+1 (+1 for the in-the-moneycall, 0 for the out-of-the-moneyput).Theslopeof theentire position is always +1,

exactly the same as a longunderlyingcontract.

Figure4-8(a)Longcallandshortputatthesameexerciseprice.(b)Combinedposition.

If a position consists ofmany different contracts,including underlyingcontracts and calls and putsoverawiderangeofexerciseprices, the parity graph forthe position may be quitecomplex. But the procedurefor constructing the graph isalways the same: determinetheslopesofthegraphbelowthe lowest exercise price,above the highest exercise

price, and between all theintermediate exercise prices,and then connect all the linesegments.

Considerthisposition:

What should the parity

graphlooklike?To determine the slopes

ofacomplexposition,itmaybehelpfultoconstructatableshowing the slopes of theindividual contracts over allintervals.Wecanthenaddupthe individual slopes to getthe total slope over eachinterval.

The entire parity graphis shown inFigure 4-9.Notethatfor thisgraphthere isnoy-axis. For complex graphswhereoptionsareboughtandsold at many differentexerciseprices,itmaynotbepossibletopositionthegraphalong the y-axis.Nevertheless,theparitygraphtells us something about thecharacteristicsoftheposition.Here we can see that the

potential profit on thedownside, as well as thepotentiallossontheupside,isunlimited.

Figure4-9

ExpirationProfitandLossAparitygraphmay tell us

the characteristics of anoption position at expiration,but an equally importantconsideration will be theprofitorlossthatresultsfromthe position. Whether theposition makes or losesmoney will depend on the

prices at which the contractsare bought and sold. Thepurchase of options willcreate a debit, whereas thesale of options will create acredit. For a simple optionposition, theexpirationprofitandloss(P&L)graphwillbethe parity graph shifteddownward by the amount ofany debit or upward by theamountofanycredit.

Consider the following

option prices with theunderlyingcontracttradingatapriceof98.00:

Figure 4-10 shows theparity graph of a long 100call position. If the option ispurchased at a price of 3.50,

we can construct theexpiration P&L graph byshifting the entire paritygraph down by this amount.Iftheunderlyingisanywherebelow 100 at expiration, theoptionwill beworthless, andthe position will lose 3.50.With the underlying above100,theslopeofthegraphis+1; the option will gain onepoint in value for each pointincrease in the price of theunderlying. We can also see

thatthereisabreakevenpriceat which the option positionwill be worth exactly 3.50.Logically, thismust occur atan underlying price of103.50.

Figure4-10Longa100callatapriceof3.50.

Figure 4-11 shows theparitygraphofashort95putposition. If theoption is soldat a price of 2.25, we canconstruct the expiration P&Lgraph by shifting the entiregraph up by this amount.With an underlying priceanywhere above 95 atexpiration, theoptionwill beworthless, and the positionwill show a profit of 2.25.With an underlying price

below 95, the slope of thegraph is+1; thepositionwilllose one point for each pointdecline in the price of theunderlying. The breakevenpriceforthepositionis92.75,thepriceatwhich the95putwillbeworthexactly2.25.

Figure4-11Shorta95putatapriceof2.25.

The relative expirationvalueoflongoptionpositionsat different exercise prices—95, 100, and 105—is shownin Figure 4-12. The samerelative value for long putpositions is shown in Figure4-13. Calls with lowerexercise prices have greatervalues (i.e., they enable theholder to buy at a lowerprice), whereas puts withhigher exercise prices have

greater values (i.e., theyenable the holder to sell at ahigherprice).

Figure4-12Longa95call–6.25;longa100call–3.50;longa105call–1.75.

Figure4-13Longa95put–2.25;longa100put–4.50;longa105put–7.75.

For more complexpositions, it may not beimmediately clear whetherthe position will result in acredit or debit. In this case,we can construct anexpirationP&Lgraphbyfirstdetermining the slopesof thegraph over all the intervals.Then we can calculate theP&L at one point, and fromthis one P&L point, we canuse the slopes to determine

theP&Latallotherpoints.

Consider the followingposition

The slopes of the positionare

It is usually easiest todetermine the P&L at anexerciseprice,solet’suse95.The P&L at an underlyingpriceof95is

Figure 4-14 shows the

entire expiration P&L graphfor the position. Below 95,theslopeofthegraphis0,sothe P&L is always –3.00.Between 95 and 105, theslope is +1, so the P&L at105 (10 points higher) is –3.00+10.00=+7.00.Above105, theslope is–2,with theposition losing one point foreach point increase in thepriceoftheunderlying.

Figure4-14

The position has twobreakeven prices, onebetween95 and 105 and oneabove105.WithaP&Lof–3.00 at 95 and a slope of +1between95and105, thefirstbreakevenis

95.00+(3.00/1)=98.00

With a P&L of +7.00 at105andaslopeabove105of–2,thesecondbreakevenis

105.00+(7.00/2)=108.50

Finally, let’s go back tothe parity-graph positionshowninFigure4-9.Supposethat we are told that atexpirationwithanunderlyingprice of 62.00 the positionwill show a profit of 2.10.What will be the P&L atexpiration if the underlyingprice is 81.50? Using theslopes,wecanworkourwayfrom62.00to81.50

At an underlying price of81.50, thepositionwill show

a loss of 13.40.We can alsosee that there are threebreakeven prices for theposition:

All critical points for theposition are shown in Figure4-15.

Figure4-15

1Admittedly,intraditionalstockandcommoditymarkets,aputdoesnotrepresentunlimitedprofitpotentialtothebuyernorunlimitedrisktothesellerbecausetheunderlyingcontractcannotfallbelowzero.Butforpracticalpurposesmosttradersthinkofbothcallsandputsashavingunlimitedpotentialvalue.

TheoreticalPricingModels

In Chapter 4, weconsidered the value of anoption and the profit or lossresulting from an option

strategy at the moment ofexpiration. From theexpiration profit and loss(P&L) graphs, we can seeclearly that the direction inwhich an underlying contractmoves can be an importantconsideration in choosing anoptionstrategy.Atraderwhobelieves that the underlyingmarketwill risewillbemoreinclinedeithertobuycallsorsell puts. A trader whobelieves that the underlying

marketwill fallwill bemoreinclinedeither tobuyputsorsell calls. In each case, thedirectional movement in theunderlying market willincrease the likelihood thatthestrategywillbeprofitable.

However, an optiontrader has an additionalproblem that we might callthe “speed” of themarket. Ifwe ignore interest anddividend considerations, a

trader who believes that astockwillriseinpricewithina specified period can bereasonably certain ofmakingaprofit ifhe is right.Hecansimplybuythestock,waitforit to reach his target price,and then sell the stock at aprofit.

Thesituationisnotquitesosimpleforanoptiontrader.Supposethatatraderbelievesthat a stockwill rise in price

from $100, its present price,to $115 within the next fivemonths. Suppose also that a$110 call expiring in threemonthsisavailableatapriceof $2. If the stock rises to$115 by expiration, thepurchaseofthe$110callwillresult in a profit of $3 ($5intrinsic value minus the $2costoftheoption).Butisthisprofit a certainty?What willhappen if the price of thestockremainsbelow$110for

the next three months andonly reaches $115 after theoption expires? Then theoption will expire worthless,andthetraderwilllosehis$2investment.

Perhapsthetraderwoulddo better to purchase a $110callthatexpiresinsixmonthsrather than three months.Now he can be certain thatwhenthestockreaches$115,thecallwillbeworthat least

$5 in intrinsic value. Butwhat if the price of the six-month option is $6? In thiscase, the trader still mightshow a loss. Even if theunderlying stock reaches thetarget price of $115, there isno guarantee that the $110call will ever be worthmorethanits$5intrinsicvalue.

A trader in anunderlying market isinterested almost exclusively

in the direction in which themarket will move. Althoughan option trader is alsosensitive to directionalconsiderations, he must alsogivesomethoughttohowfastthemarket is likely tomove.If a trader in the underlyingstock and an option trader inthe same market take longmarket positions in theirrespective instruments andthemarketdoes in factmovehigher, the stock trader is

assured of a profit,while theoption trader may show aloss. If the market fails tomove sufficiently fast, thefavorable directional movemay not be enough to offsetthe option’s loss in timevalue.Aspeculatorwilloftenbuy options for theirseemingly favorable risk-reward characteristics(limited risk, unlimitedreward), but if he purchasesoptions, not onlymust he be

right about market direction,he must also be right aboutmarket speed. Only if he isright on both counts can heexpect to make a profit. Ifpredicting the correct marketdirectionisdifficult,correctlypredictingdirectionandspeedis probably beyond mosttraders’capabilities.

The concept of speed iscrucial in trading options.Indeed, many option

strategiesdependonlyon thespeed of the underlyingmarket and not at all on itsdirection.Ifatraderishighlyproficient at predictingdirectional moves in theunderlying market, he isprobably better advised totrade the underlyinginstrument. Only when atrader has some feel for thespeedcomponentcanhehopeto successfully enter theoptionmarket.

TheimportanceofProbability

One can never be certainabout future marketconditions, so almost alltradingdecisionsarebasedonsome estimate of probability.We often express ouropinions about probabilityusing words such as likely,good chance, possible, andprobable. But in an option

evaluation we need to bemore specific. We need todefineprobabilityinwaythatwillenableustodothetypesof calculations required tomake intelligent decisions inthemarketplace.Ifwecandothis, we will find thatprobability and the choice ofstrategygohandinhand.Ifatraderbelievesthatastrategyhasaveryhighprobabilityofprofit and a very lowprobabilityofloss,hewillbe

satisfied with a smallpotential profit because theprofit is likely to be quitesecure.On the other hand, ifthe probability of profit isvery low, the trader willdemand a large profit whenmarket conditions developfavorably. Because of theimportance of probability inthe decision-making process,it will be worthwhile toconsider some simpleprobabilityconcepts.

ExpectedValueSuppose thatwe are given

the opportunity to roll a six-sided die, and each time weroll,wewill be paid a dollaramount equal to the numberthat comes up. If we roll aone,wearepaid$1;ifwerollatwo,wearepaid$2;andsoonuptosix,inwhichcaseweare paid $6. If we are giventheopportunitytorollthediean infinite number of times,

onaverage,howmuchdoweexpecttoreceiveperroll?

We can calculate theanswer using some simplearithmetic. There are sixpossible numbers, each withequal probability. If we addup the six possible outcomes1+2+3+4+5+6=21anddividethisbythesixfacesonthe die, we get 21/6 = 3½.That is, on average, we canexpect to get back $3½ each

time we roll the die. This isthe average payback, orexpected value. If we mustpay for the privilege ofrolling the die, what is areasonable price? If wepurchase the chance to rollthe die for less than $3½, inthe long run, we expect toshowaprofit.Ifwepaymorethan$3½,inthelongrun,weexpecttoshowaloss.Andifwe pay exactly $3½, weexpect to break even. Note

the qualifying phrase in thelongrun.Theexpectedvalueof$3½is realisticonly ifweare allowed to roll the diemany,many times. Ifwe areallowedtorollonlyonce,wecannot be certain of gettingback$3½.Indeed,onanyoneroll, it is impossible to getback$3½becauseno faceofthediehas exactly3½spots.Nevertheless, if we pay lessthan$3½forevenonerollofthe die, the laws of

probability are on our sidebecause we have paid lessthantheexpectedvalue.

In a similar vein,consider a roulette bet. Theroulette wheel has 38 slots,numbers 1 through 36 and 0and 00.1 Suppose that acasino allows a player tochoose a number. If theplayer’snumbercomesup,hereceives $36; if any othernumbercomesup,hereceives

nothing.Whatistheexpectedvalue for this proposition?There are 38 slots on theroulette wheel, each withequal probability, but onlyoneslotwillreturn$36totheplayer. If we divide the oneoutcome where the playerwins $36 by the 38 slots onthewheel,theresultis$36/38=$0.9474,orabout95cents.A player who pays 95 centsfor theprivilegeofpickinganumber at the roulette table

can expect to approximatelybreakeveninthelongrun.

Ofcourse,nocasinowillletaplayerbuysuchabetfor95 cents. Under thoseconditions, the casino wouldmake no profit. In the realworld,aplayerwhowants topurchasesuchabetwillhavetopaymorethantheexpectedreturn, typically $1. The 5-cent difference between the$1priceofthebetandthe95-

cent expected valuerepresentstheprofitpotential,oredge, to the casino. In thelong run, foreverydollarbetat the roulette table, thecasino can expect to keepabout5cents.

Given the precedingconditions, any playerinterested in making a profitwould rather switch placeswith the casino. Then hewouldhave a5-cent edgeon

hissidebysellingbetsworth95centsfor$1.Alternatively,theplayerwouldliketofindacasino where he couldpurchasethebetfor less thanitsexpectedvalueof95cents,perhaps 88 cents. Then theplayer would have a 7-centedgeoverthecasino.

TheoreticalValueThe theoretical value of a

proposition is the price onewouldbewilling topaynowtojustbreakeveninthelongrun.Thus far, theonly factorwe have considered indetermining the value of aproposition is the expectedvalue. We used this conceptto calculate the 95-cent fairpricefortheroulettebet.

Suppose that in ourroulette example the casinodecides to change the

conditions slightly. Theplayermaynowpurchasetheroulette bet for its expectedvalue of 95 cents, and asbefore, ifheloses, thecasinowill immediately collect his95 cents. Under the newconditions, however, if theplayer wins, the casino willsendhimhis$36winningsintwo months. Will both theplayer and the casino stillbreak even on theproposition?

Wheredidtheplayergetthe95centsthathebetattheroulette wheel? In theimmediate sense, he mayhave taken it out of hispocket. But a closerexamination may reveal thathewithdrew themoney fromhis bank prior to visiting thecasino. Because he won’treceive his winnings for twomonths, he will have to takeinto consideration the twomonths of interest he would

haveearnedhadheleftthe95cents in the bank. Thetheoreticalvalueof thebet isreally thepresentvalueof itsexpected value, the 95 centsexpectedvaluediscountedbyinterest. If interest rates are12 percent annually, thetheoreticalvalueis

95cents/(1+0.12×2/12)≈93cents

Even if the player

purchases the bet for itsexpected return of 95 cents,he will still lose 2 centsbecauseoftheinterestthathecould have earned for twomonths if he had left hismoney in the bank. Thecasino,ontheotherhand,willtakethe95cents,putit inaninterest-bearing account, andat the end of two monthscollect 2 cents in interest.Under the new conditions, ifaplayerpays93centsforthe

roulette bet today andreceives his winnings in twomonths, neither he nor thecasino can expect to makeanyprofitinthelongrun.

The two most commonconsiderations in optionevaluation are the expectedreturn and interest. Theremay, however, be otherconsiderations. Suppose thatthe player is a good client,andthecasinodecidestosend

him a 1-cent bonus a monthfrom now. He can add thisadditional payment to theprevious theoretical value of93 cents to get a newtheoretical value of 94 cents.Thisissimilartothedividendpaid to owners of stock in acompany. In fact, dividendscan be an additionalconsideration in evaluatingboth stock and options onstock.

If a casino is sellingroulette bets that have anexpectedvalueof95centsfora price of $1, does thisguaranteethat thecasinowillmake a profit? It does if thecasino can be certain ofstaying in business for the“longrun”becauseoverlongperiodsof time thegoodandbad luck will tend to evenout.Unfortunately,beforethecasinoreachesthelongrun,itmust survive the short run.

It’spossiblethatsomeonecanwalkuptotheroulettewheel,make a series of bets, andhave their number come up20 times in succession.Clearly, this isveryunlikely,but the laws of probabilitysay that it could happen. Iftheplayer’sgoodluckresultsin the casino going out ofbusiness, the casino willneverreachthelongrun.

The goal of option

evaluation is to determine,through the use of atheoreticalpricingmodel, thetheoreticalvalueofanoption.The trader can thenmake anintelligent decision whethertheprice of theoption in themarketplace is either too lowor too high and whether thetheoretical edge is sufficienttojustifymakingatrade.Butdetermining the theoreticalvalue is only half theproblem.Becauseanoption’s

theoretical value is based onthelawsofprobability,whichare only reliable in the longrun, the trader must alsoconsider thequestionof risk.Even ifa traderhascorrectlycalculated an option’stheoreticalvalue,howwillhecontrol the short-term badluck that goes with anyprobability calculation? Weshall see that in the realworld,anoption’s theoreticalvalue is always open to

question. For this reason, atrader’sabilitytomanageriskisat leastas importantashisability to calculate atheoreticalvalue.

AWordonModelsWhat is a model?We can

thinkofamodelasa scaled-downormoreeasilymanagedrepresentation of the realworld. The model may be a

physicalone,suchasamodelairplane or architecturalmodel, or it may be amathematical one, such as aformula.Ineachcase,weusethe model to betterunderstand the world aroundus.However,itisunwise,andsometimes dangerous, toassumethatthemodelandthereal world that it representsare identical in every way.We may have an excellentmodel,butitisunlikelytobe

an exact replica of the realworld.

Allmodels,iftheyaretobe effective, require us tomake certain priorassumptions about the realworld. Mathematical modelsrequire the input of numbersthat quantify theseassumptions. If we feedincorrect data into a model,we can expect an incorrectrepresentation of the real

world. As every model userknows, “Garbage in, garbageout.”

These generalobservations about modelsare no less true for optionpricing models. An optionmodelisonlysomeone’sideaof how an option might beevaluated under certainconditions.Becauseeitherthemodel itself or the data thatwefeedintothemodelmight

be incorrect, there is noguarantee that model-generated values will beaccurate.Norcanwebesurethatthesevalueswillbearanylogical resemblance to actualpricesinthemarketplace.

A new option trader islike someone entering a darkroom for the first time.Without any guidance, hemay grope around, hopingthat he eventually findswhat

he is looking for. The traderwho is armed with a basicunderstanding of theoreticalpricing models enters thesameroomwithacandle.Hecan make out the generallayout of the room, but thedimness of the candleprevents him fromdistinguishing every detail.Moreover, some of what heseesmay be distorted by theflickering of the candle. Inspite of these limitations, a

trader is more likely to findwhathe is lookingforwithasmall candle than with noilluminationatall.2

The real problems withtheoretical pricing modelsarise after the trader hasacquiredsomesophistication.As he gains confidence, hemaybegintoincreasethesizeof his trades. When thishappens,hisinabilitytomakeout every detail in the room,

as well as the distortionscaused by the flickeringcandle flame, take onincreased importance.Now amisinterpretation of what hethinks he sees can lead tofinancialdisasterbecauseanyerror in judgment will begreatlymagnified.

Thesensibleapproachisto make use of a model, butwithafullawarenessofwhatit can and cannot do.Option

traders will find that atheoretical pricing model isan invaluable tool tounderstanding the pricing ofoptions. Because of theinsightsgainedfromamodel,the great majority ofsuccessful option traders relyon some type of theoreticalpricing model. However, anoptiontrader,ifheistomakethe best use of a theoreticalpricingmodel,mustbeawareofitslimitationsaswellasits

strengths.Otherwise, hemaybe no better off than thetradergropinginthedark.3

ASimpleApproach

How might we adapt theconcepts of expected valueand theoretical value to thepricing of options? Consideranunderlyingcontract thatatexpirationcantakeononeoffive different prices: $80,

$90, $100, $110, or $120.Assume,moreover, that eachof the five prices is equallylikely with 20 percentprobability. The prices andprobabilities are shownFigure5-1.

Figure5-1

What will be theexpected value for thiscontract at expiration?Twenty percent of the time,the contract will be worth

$80; 20 percent of the time,the contract will be worth$90; and so on, up to the 20percent of the time, thecontractisworth$120:

(20%×$80)+(20%×$90)+(20%×100)+(20%×$110)+(20%×$120)=$100

Atexpiration,theexpectedvalueforthecontractis$100.

Now consider theexpected value of a 100 call

using the same underlyingprices and probabilities. Wecanmoreeasilyseethevalueof the call by overlaying theparity graph for the call onour probability distribution.Thishasbeendone inFigure5-2.Iftheunderlyingcontractis at $80, $90, or $100, thecallisworthless.If,however,the underlying contract is at$110or$120,theoptionwillbeworthitsintrinsicvalueof$10and$20,respectively:

(20%×0)+(20%×0)+(20%×0)+(20%×$10)+

(20%×$20)=$6

Figure5-2

Ifwewant to develop atheoretical pricing modelusingthisapproach,wemightpropose a series of possibleprices and probabilities forthe underlying contract atexpiration. Then, given anexercise price, we cancalculatetheintrinsicvalueoftheoptionateachunderlyingprice, multiply this value byitsassociatedprobability,addup all these numbers, and

thereby obtain an expectedvalue for the option. Theexpected value for a call atexpirationis

whereeachSi isapossibleunderlying price atexpiration, and pi is theprobability associated withthatprice.Theexpectedvalue

foraputis

In the foregoingexample, we used a simplescenario with only fivepossiblepriceoutcomes,eachwith identical probability.Obviously, this is not veryrealistic.Whatchangesmightwemake todevelopamodelthat more accurately reflects

therealworld?Foronething,we need to know thesettlement procedure for theoption.Iftheoptionissubjectto stock-type settlement, wemustpaythefullpriceof theoption. If the100callhasanexpected value of $6 atexpiration, the theoreticalvalue will be the presentvalue of this amount. Ifinterest rates are 12 percentannually (1 percent permonth) and the option will

expire in two months, thetheoreticalvalueoftheoptionis

Whatotherfactorsmightwe consider? We assumedthat all five price outcomeswere equally likely. Is this arealisticassumption?Supposethat you were told that onlytwo prices were possible at

expiration,$110and$250. Ifthe current price of theunderlyingcontractiscloseto$100, which do you think ismore likely? Experiencesuggests that extreme pricechanges that are far awayfrom today’s price are lesslikelythansmallchangesthatremainclosetotoday’sprice.Forthisreason,$110ismorelikelythan$250.Perhapsourprobability distribution oughtto reflect this by

concentrating theprobabilities around thecurrent price of theunderlying contract. Onepossibledistributionisshownin Figure 5-3. Using thesenew probabilities, theexpected value for the 100callisnow

(0%×0)+(20%×0)+(0%×0)+(20%×$10)+(10%×

$20)=$4

Figure5-3

If, as before, the option issubject to stock-typesettlement, the theoreticalvalueis

Note that the newprobabilities did not changethe expected value for theunderlying contract. Becausethe probabilities are

symmetricalaround$100,theexpected value for theunderlying contract atexpirationisstill$100.

No matter how weassign probabilities, we willwant to do so in such awaythattheexpectedvaluefortheunderlying contractrepresents themost likely, oraverage, value at expiration.Whatisthemostlikelyfuturevalue for the underlying

contract? In fact, there is noway to know. But we mightask what the marketplacethinks the most likely valueis.Recallwhatwouldhappenif the theoretical forwardpricewere different from theactual price of a forwardcontract in the marketplace.Everyone would execute anarbitrage by either buying orselling the forward contractand taking the oppositeposition in the cash market.

In a sense, the marketplacemust think that the forwardpriceisthemostlikelyfuturevalue for the underlyingcontract. If we assume thatthe underlying market isarbitrage-free, the expectedvalue for the underlyingcontractmustbeequal to theforwardprice.

Suppose in our examplethattheunderlyingcontractisa stock that is currently

tradingat$100and thatpaysno dividend prior toexpiration. The two-monthforwardpriceforthestockis

$100×[1+(0.12×2/12)]=$100×1.02=$102

If $102 is the expectedvalueforthestock,insteadofassigning the probabilitiessymmetrically around $100,wemaywant to assign themsymmetrically around $102.

This distribution is shown inFigure5-4.Nowtheexpectedvalueforthe100callis

(10%×0)+(20%×0)+(40%×$2)+(20%×$12)+

(10%×$22)=$5.40

Figure5-4

andthetheoreticalvalueis

Intheexamplesthusfar,we have assumed asymmetrical probabilitydistribution. But as long astheexpectedvalueisequaltothe forwardprice, there isnorequirement that theprobabilities be assignedsymmetrically. Figure 5-5

shows a distribution wherethepriceoutcomesareneithercentered around the forwardpricenoraretheprobabilitiessymmetrical.Nonetheless,theexpected value for theunderlying contract is stillequalto$102

(6%×83)+(15%×90)+(39%×$99)+(33%×$110)+(7%×$123)=4.98+13.5+38.61+36.30+8.61=

$102

Figure5-5

Using these probabilities,the theoretical value of the100callis

Theforwardpriceoftheunderlying contract plays acentral role in all optionpricingmodels.ForEuropeanoptions, the current price ofthe underlying contract is

important only insofar as itcan be turned into a forwardprice.Becauseofthis,traderssometimes make thedistinction between optionsthat are at the money (theexercise price is equal to thecurrent underlying price) andoptions which are at theforward (theexercisepriceisequal to the forward price atexpiration).Inmanymarkets,at-the-forwardoptionsarethemost actively traded, and

such options are often usedbytradersasabenchmarkforevaluating and trading otheroptions.

Even if we assume anarbitrage-free market in theunderlying contract, we stillhave a major hurdle toovercome. In our simplifiedmodel,weassumedthattherewere only five possible priceoutcomes. In the real world,however, thereareaninfinite

number of possibilities. Toenable our model to moreclosely approximate the realworld, we would like toconstruct a probabilitydistribution with everypossible price outcometogether with its associatedprobability. This may seeman insurmountable obstacle,butwewillseeinsubsequentchapters how we mightapproximate such aprobabilitydistribution.

We can now summarizethe necessary steps indevelopingamodel:

1.Proposeaseriesofpossiblepricesatexpiration for theunderlyingcontract.2. Assign aprobability to eachpossible price withthe restriction thatthe underlyingmarket is arbitrage-

free—the expectedvalue for theunderlying contractmustbeequaltotheforwardprice.3.Fromthepricesand probabilities insteps 1 and 2, andfrom the chosenexercise price,calculate theexpected value oftheoption.

4. Lastly,depending on theoption’s settlementprocedure,calculatethepresentvalueoftheexpectedvalue.

TheBlack-ScholesModel

Oneofthefirstattemptstodescribe traded options in

detailwasapamphletwrittenby Charles Castelli andpublishedinLondonin1877,“The Theory of Options inStocks and Shares.”4 Thispamphlet included adescription of somecommonly used hedging andtrading strategies such as the“call-of-more” and the “call-and-put.” Today, thesestrategies are known as acovered-writeandastraddle.

The origins of modernoption pricing theory aremost often ascribed to theyear 1900, when Frenchmathematician LouisBachelier published TheTheory of Speculation, thefirst attempt to use highermathematics to price optioncontracts.5 AlthoughBachelier’s treatise was aninterestingacademic study, itresulted in little practical

application because therewere no organized optionmarkets at that time.However,in1973,concurrentwith the opening of theChicago Board OptionsExchange, Fischer Black, atthe time associated with theUniversity of Chicago, andMyron Scholes, associatedwith the MassachusettsInstituteofTechnology,builtontheworkofBachelierandother academics to introduce

the first practical theoreticalpricing model for options.6The Black-Scholes model,7with its relatively simplearithmetic and limitednumber of inputs, most ofwhichwereeasilyobservable,proved an ideal tool fortraders in the newly openedU.S.optionmarket.Althoughother models havesubsequentlybeen introducedto overcome some of its

original weaknesses, theBlack-Scholesmodelremainsthe most widely used of alloptionpricingmodels.

In its original form, theBlack-Scholes model wasintended to evaluateEuropean options (no earlyexercise permitted) on non-dividend-paying stocks.Shortly after its introduction,realizing that many stockspay dividends, Black and

Scholes added a dividendcomponent. In 1976, FischerBlack made slightmodificationstothemodeltoallow for the evaluation ofoptionson futures contracts.8In 1983, Mark Garman andSteven Kohlhagen of theUniversity of California atBerkeley made additionalmodificationstoallowfortheevaluation of options onforeign currencies.9 The

futures version and theforeign-currency version areknown formally as theBlackmodel and the Garman-Kohlhagen model,respectively. However, theevaluation method in eachvariation, whether theoriginalBlack-Scholesmodelfor stock options, the Blackmodel for futures options, orthe Garman-Kohlhagenmodel for foreign currencyoptions,issosimilarthatthey

have all come to be knownsimply as the Black-Scholesmodel. The various forms ofthemodeldifferonly inhowthey calculate the forwardprice of the underlyingcontract and the settlementprocedurefortheoptions.Anoption trader will simplychoose the form appropriatetotheoptionsandunderlyinginstrument in which he isinterested.

Givenitswidespreaduseand its importance in thedevelopment of other pricingmodels, we will, for themoment, restrict ourselves toa discussion of the Black-Scholesmodelanditsvariousforms. In later chapters wewill consider the question ofearly exercise. We will alsolook at alternative methodsfor pricing options when wequestion some of the basicassumptions in the Black-

Scholesmodel.Thereasoningthatledto

the development of theBlack-Scholes model issimilar to the simpleapproach we took earlier inthis chapter for evaluatingoptions. Black and Scholesworked originally with callvalues, butputvalues canbederived in much the sameway. Alternatively, we willsee later that for European

options there is a uniquepricing relationship betweenan underlying contract and acall and put with the sameexercise price and expirationdate. This relationship willenable us to derive a putvalue from the companioncallvalueoracallvaluefromthecompanionputvalue.

To calculate an option’stheoretical value using theBlack-Scholes model, we

needtoknow,ataminimum,five characteristics of theoption and its underlyingcontract:

1. The option’sexerciseprice2. The timeremaining toexpiration3. The currentprice of theunderlyingcontract

4. The applicableinterest rate overthe life of theoption5.Thevolatilityofthe underlyingcontract

Thelastinput,volatility,may be unfamiliar to a newtrader.Whilewewill put offa detailed discussion of thisinput to Chapter 6, from ourprevious discussion, we can

reasonablyinferthatvolatilityis related to either the speedof the underlying market orthe probabilities of differentpriceoutcomes.

If we know each of therequired inputs, we can feedthem into the theoreticalpricing model and therebygenerate a theoretical value(seeFigure5-6).

Figure5-6

Black and Scholes alsoincorporated into theirmodelthe concept of a risklesshedge. For every optionposition, there is atheoretically equivalentposition in the underlyingcontract such that, for smallprice changes in theunderlying contract, theoption position will gain orlosevalueatexactlythesamerate as the underlying

position. To take advantageof a theoretically mispricedoption, it is necessary toestablish this riskless hedgeby offsetting the optionposition with a theoreticallyequivalent underlyingposition. That is, whateveroption position we take, wemusttakeanopposingmarketposition in the underlyingcontract. The correctproportion of underlyingcontracts needed to establish

this riskless hedge isdetermined by the option’shedgeratio.

Why is it necessary toestablish a riskless hedge?Recall that in our simplifiedapproach, an option’stheoreticalvaluedependedonthe probability of variousprice outcomes for theunderlying contract. As theprice of the underlyingcontract changes, the

probability of each outcomewill also change. If theunderlying price is currently$100 and we assign a 25percent probability to $120,wemightdroptheprobabilityfor $120 to 10 percent if theprice of the underlyingcontract falls to $90. Byinitiallyestablishingarisklesshedge and then adjusting thehedge as market conditionschange, we are taking intoconsideration these changing

probabilities.In this sense, an option

can be thought of as asubstituteforapositionintheunderlyingcontract.Acall isa substitute for a longposition; a put is a substitutefor a short position.Whetherthe substitute position isbetter than an outrightposition in the underlyingcontract depends on thetheoreticalvalueoftheoption

comparedwithitspriceinthemarketplace. If a call can bepurchased for less than itstheoreticalvalueoraputcanbe sold for more than itsvalue, in the long run, itwillbe more profitable to take along market position bypurchasing calls or sellingputs than by purchasing theunderlying contract. In thesame way, if a put can bepurchased for less than itstheoreticalvalueoracallcan

be sold for more than itsvalue, in the long run, itwillbe more profitable to take ashort market position bypurchasing puts or sellingcalls than by selling theunderlyingcontract.

In laterchapterswewilldiscuss the concept of ariskless hedge in greaterdetail. For now, we simplysummarize the four basicoption positions, their

corresponding marketpositions,andtheappropriatehedges:

For new traders, it maybehelpfultopointoutthatwearealwaysdoingtheoppositewithcalls and theunderlying(i.e., buy calls, sell theunderlying;sellcalls,buy theunderlying) and doing thesame with puts and theunderlying(i.e.,buyputs,buythe underlying; sell puts, sellthe underlying). Especiallywith puts, more than a fewnew traders have initiallydone it backwards, buying

puts and selling theunderlyingorsellingputsandbuying the underlying. This,ofcourse,isnohedgeatall.

Because the theoreticalvalue obtained from atheoretical pricing model isnobetter than the inputs intothe model, a few commentson eachof the inputswill beworthwhile.

ExercisePriceThere shouldneverbeany

doubtabouttheexercisepriceof an option because it isfixed under the terms of thecontract and does not varyover the life of the option.10A March 60 call cannotsuddenlyturnintoaMarch55call. A September 100 putcannot turn into a September110put.

TimetoExpirationAswith theexerciseprice,

anoption’sexpirationdate isfixed and will not vary. AMarch 60 call will notsuddenlyturnintoanApril60call,norwillaSeptember100put turn into an August 100put.Of course, eachday thatpasses brings us closer toexpiration,sointhissensethetime to expiration isconstantly growing shorter.

However,theexpirationdate,like the exercise price, isfixed by the exchange andwillnotchange.

In financialmodels, oneyear is typically the standardunit of time. Therefore, timeto expiration is entered intothe Black-Scholes model asan annualized number. If weexpresstimeintermsofdays,wemustmaketheappropriateadjustment by dividing the

number of days to expirationby 365. However, mostoption-evaluation computerprograms already have thistransformation incorporatedintothesoftware,soweneedonlyenterthecorrectnumberof days remaining toexpiration.

It may seem that wehave a problem in decidingwhatnumberofdaystoenterinto themodel.We need the

amount of time remaining toexpiration for two purposes:(1) to determine thelikelihoodofpricemovementintheunderlyingcontractand(2) to make interestcalculations. For the former,weareonlyinterestedindayson which the price of theunderlying contract canchange. For exchange-tradedcontracts, thiscanonlyoccuronbusiness days.Thismightleadustodropweekendsand

holidays from ourcalculations. On the otherhand, for interest-ratepurposes, we must includeevery day. If we borrow orlend money, we expectinterest to accrue every day,nomatter that some of thosedaysarenotbusinessdays.

However, this is notreally a problem. Indetermining the likelihood ofprice movement in the

underlying contract, weobserve only business daysbecause these are the onlydays onwhich price changescan occur. Then we scalethesevalues toanannualizednumberbeforefeedingit intothe theoreticalpricingmodel.Theresultisthatwecanfeedinto our model the actualnumber of days remaining toexpiration, knowing that themodelwillinterpretallinputscorrectly.

Although traderstypically express time toexpiration in days, a tradermay want to use a differentmeasure. Especially asexpiration approaches, atradermayprefertousehoursor even minutes. In theory,finer time increments shouldyield more accurate values.But there is a practicallimitationtousingverysmallincrements of time. As timepasses, the discrete

increments of time we feedinto a theoretical pricingmodel may not accuratelyrepresent the continuouspassage of time in the realworld. Most traders havelearned through experiencethatasexpirationapproaches,the use of a theoreticalpricing model becomes lessreliable because the inputsbecome less reliable. Indeed,very close to expiration,many traders stop using

model-generated valuesaltogether.

UnderlyingPriceUnlike the exercise price

and time to expiration, thecorrect price of theunderlying contract is notalways obvious. At any onetime, there is abidpriceandan ask price (the bid-askspread), and it may not be

clearwhetherweoughttouseone or the other of theseprices or perhaps some priceinbetween.

Consider an underlyingmarket where the last tradeprice was 75.25 but that iscurrently displaying thefollowingbid-askspread:

75.20–75.40

If a trader is using atheoretical pricing model to

evaluate options on thismarket,what price shouldhefeed into the model? Onepossibility is 72.25, the lasttrade price. Anotherpossibility might be 75.30,the midpoint of the bid-askspread.

Even though we arefocusing on the use oftheoretical pricing models,we should emphasize thatthere is no law that says a

trader must make anydecisions based on orconsistent with a theoreticalpricing model. A trader cansimplybuyorselloptionsandhope that the trade turns outfavorably. But a disciplinedtrader who uses a pricingmodel knows that he isrequired to hedge the optionposition by taking anopposing market position inthe underlying contract.Therefore, the underlying

price that he feeds into thetheoretical pricing modeloughttobethepriceatwhichhe believes he can make theopposing trade. If the traderintends to purchase calls orsell puts, both of which arelongmarketpositions,hewillhedge by selling theunderlying contract. In thiscase, he will want to usesomething close to the bidpricebecausethatisthepriceatwhichhecanprobablysell

the underlying. On the otherhand, if the trader intends tosellcallsorbuyputs,bothofwhich are short marketpositions, he will hedge bypurchasing the underlyingcontract.Nowhewillwanttouse something close to theask price because that is theprice at which he canprobablybuytheunderlying.

In practice, if theunderlying market is very

liquid,with a narrowbid-askspread and many contractsavailable at each price, atraderwhomustmakeaquickdecisionmayverywelluseaprice close to the midpointbecause that probablyrepresents a reasonableestimate of where theunderlying can be bought orsold. But in an illiquidmarket,withaverywidebid-ask spread and only a fewcontracts available at each

price, the trader must giveextra thought to theappropriate underlying price.Insuchamarket,particularlyif the prices are changingrapidly, itmaybedifficult toexecuteevenasmallorderatthequotedprices.

InterestRatesBecause an option trade

may result in either a cash

credit or debit to a trader’saccount, the interestconsiderations resulting fromthiscashflowmustalsoplaya role in option evaluation.This is a function of interestrates over the life of theoption.

Interest rates play tworoles in the theoreticalevaluation of options. First,they may affect the forwardprice of the underlying

contract. If the underlyingcontract is subject to stock-type settlement, as we raiseinterest rates, we raise theforward price, increasing thevalue of calls and decreasingthe value of puts. Second,interest rates may affect thepresentvalueoftheoption.Iftheoptionissubjecttostock-type settlement, as we raiseinterest rates, we reduce thepresent value of the option.Although interest rates may

affect both the forward priceandthepresentvalue,inmostcases, the same rate isapplicable, andweneedonlyinputoneinterestrateintothemodel. If, however, differentratesareapplicable,aswouldbe the case with foreign-currencyoptions(theforeign-currency interest rate playsone role, and the domestic-currency interest rate plays adifferentrole),themodelwillrequire the input of two

interestrates.Thisisthecasewith the Garman-Kohlhagenversion of the Black-Scholesmodel.

Whatinterestrateshoulda trader use when evaluatingoptions? Textbooks oftensuggest using the risk-freerate, the rate that applies tothe most creditworthyborrower. In most markets,thegovernment is consideredthe most secure borrower of

funds, so the yield on agovernment security with amaturityequivalenttothelifeof the option is the generalbenchmark. For a 60-dayoption denominated indollars, we might use theyield on a 60-day U.S.Treasury bill; for a 180-dayoption, we might use theyield on a 180-day U.S.Treasurybill.

Inpractice,noindividual

can borrow or lend at thesame rateas thegovernment,so it seemsunrealistic to usethe risk-free rate. Todetermine a more realisticrate, a tradermight look to afreely traded market ininterest-rate contracts. In thisrespect, traders often useeither the London InterbankOffered Rate (LIBOR)11 orthe Eurocurrency markets todeterminetheapplicablerate.

For dollar-denominatedoptions, Eurodollar futurestraded at the ChicagoMercantile Exchange areoften used to determine abenchmarkinterestrate.

The situation is furthercomplicated by the fact thatmost traders do not borrowand lend at the same rate, sothe correct interest rate will,in theory,dependonwhetherthe trade will create a credit

oradebit.Intheformercase,thetraderwillbeinterestedinthe borrowing rate; in thelatter case, he will beinterested in the lending rate.However, among the inputsinto the model—theunderlying price, time toexpiration, interest rates, andvolatility—interest rates tendto play the least importantrole.Usingaratethat“makessense”isusuallyareasonablesolution. Of course, for very

large positions or for verylong-term options, smallchanges in the interest ratecan have a large impact.Butfor most traders, getting theinterest rate exactly right isusually not a majorconsideration.

DividendsWe did not list dividends

asamodelinputinFigure5-5

becausetheyareonlyafactorin the theoretical evaluationofstockoptionsandthenonlyifthestockisexpectedtopayadividendoverthelifeoftheoption. Inorder toevaluateastockoption, themodelmustaccurately calculate theforward price for the stock.This requires us to estimateboth the amount of thedividend and the date onwhich the dividend will bepaid. In practice, rather than

using the date of dividendpayment, an option trader islikely to focus on the ex-dividend date, the date onwhich the stock is tradingwithout the rights to thedividend.The exact dividendpayment date is important incalculating the interest thatcanbeearnedonthedividendpayment and therebycalculating a more accurateforwardprice.Butforatraderownership of the stock in

order to receive the dividendis the primary consideration.Adeeplyin-the-moneyoptionmay have many of the samecharacteristics as stock, butonly ownership of the stockcarrieswithittherightstothedividend.

In the absence of otherinformation, most tradersassume that a company islikely to continue its pastdividendpolicy.Ifacompany

has been paying a 75-centdividend each quarter, itwillprobably continue to do so.However, until the companyofficially declares thedividend, this is not acertainty. A company mayincrease or reduce itsdividend or omit itcompletely. If there is thepossibility of a change in acompany’sdividendpolicy,atrader must consider itsimpact on option values.

Additionally, if the ex-dividenddateisexpectedjustpriortoexpiration,adelayofseveral days will cause theex-dividend date to fall afterexpiration. For purposes ofoption evaluation, this is thesame as eliminating thedividend entirely. In such asituation,atraderwillneedtomake a special effort toascertain the exact ex-dividenddate.

VolatilityOf all the inputs required

for option evaluation,volatility is themostdifficultfor traders to understand. Atthesametime,volatilityoftenplaysthemostimportantrolein actual trading decisions.Changes in our assumptionsabout volatility can have adramaticeffectonanoption’svalue. And the manner inwhich the marketplace

assesses volatility can haveanequallydramaticeffectonan option’s price. For thesereasons, we will begin adetailed discussion ofvolatilityinChapter6.

1Weassumearoulettewheelwith38slots,asiscustomaryintheUnitedStates.Insomepartsoftheworld,aroulettewheelmayhavenoslotnumbered00.This,ofcourse,changestheprobabilities.2Onemightalsoarguethatatraderwithacandle(i.e.,theoreticalpricingmodel)mightdropthecandleandburndowntheentirebuilding.Financialcrisesseemtooccurwhenmanytradersdroptheircandlesatthesametime.3Someinterestingdiscussiononthelimitationsofmodels:FischerBlack,“TheHolesinBlackScholes,”Risk1(4):30–33,1988;StephenFiglewski,“WhatDoesanOptionPricingModel

TellUsaboutOptionPrices?”FinancialAnalystsJournal,September–October1989,pp.12–15;FischerBlack,“LivingUptotheModel,”Risk3(3):11–13,1990;andEmanuelDermanandPaulWilmott,“TheFinancialModelers’Manifesto”(January2009),http://www.wilmott.com/blogs/paul/index.cfm/2009/1/8/Financial-Modelers-Manifesto.4AphotocopyoftheCastellipamphlet,whichisnowinthepublicdomain,isavailableatbooks.google.com.5SeeLouisBachelier’sTheoryofSpeculation,MarkDavisandAlisonEtheridge,trans.(Princeton,NJ:PrincetonUniversityPress,2006).AtranslationofBachelier’streatisealso

appearsinTheRandomCharacterofStockMarketPrices,PaulCootner,ed.(Cambridge,MA:MITPress,1964).6FischerBlackandMyronScholes,“ThePricingofOptionsandCorporateLiabilities,”JournalofPoliticalEconomy81(3):637–654,1973.7RobertMerton,who,atthetime,was,likeMyronScholes,associatedwithMIT,isalsocreditedwithsomeofthesameworkthatledtothedevelopmentoftheoriginalBlack-Scholesmodel.Hispaper,“TheRationalTheoryofOptionPricing,”appearedintheBellJournalofEconomicsandManagementScience4(Spring):141–183,1973.InrecognitionofMerton’scontribution,themodelissometimesreferredtoas

theBlack-Scholes-Mertonmodel.ScholesandMertonwereawardedtheNobelPrizeinEconomicSciencesin1997;FischerBlack,unfortunately,diedin1995.8FischerBlack,“ThePricingofCommodityContracts,”JournalofFinancialEconomics3:167–179,1976.9MarkB.GarmanandStevenW.Kohlhagen,“ForeignCurrencyOptionValues,”JournalofInternationalMoneyandFinance2(3):239–253,1983.Wearespeakinghereofoptionsonaphysicalforeigncurrencyratherthanoptionsonaforeign-currencyfuturescontract.ThelattermaybeevaluatedusingtheBlackmodelforfuturesoptions.

10Anexchangemayadjusttheexercisepriceofastockoptionastheresultofastocksplitorinthecaseofanextraordinarydividend.Inpracticalterms,thisisonlyanaccountingchange.Thecharacteristicsoftheoptioncontractremainessentiallyunchanged.11TheLondonInterbankOfferedRate(LIBOR)istheratepaidbytheLondonbanksondollardeposits.Assuch,itreflectsthefree-marketinterestratefordollars.LIBORistheunderlyingforEurodollarfuturestradedattheChicagoMercantileExchange.Thevalueofthesecontractsatmaturityisdeterminedbytheaveragethree-monthLIBORratequotedbythelargest

Londonbanks.

Volatility

Whatisvolatility,andwhyis it so important in optionevaluation?Theoptiontrader,likeatraderintheunderlyinginstrument,isinterestedinthedirection of the market. But

unlike a trader in theunderlying,anoptiontraderisalso sensitive to the speedofthemarket. If themarket foranunderlyingcontractfailstomove at a sufficient speed,options on that contract willhavelessvaluebecauseofthereduced likelihood of themarket going through anoption’s exercise price. In asense, volatility is ameasureof the speed of the market.Marketsthatmoveslowlyare

low-volatility markets;markets that move quicklyarehigh-volatilitymarkets.

One might guessintuitively that somemarketsaremorevolatile thanothers.During 2008, the price ofcrude oil began the year at$99perbarrel,reachedahighof$144perbarrelinJuly,andfinished the year at $45 perbarrel. The price rose 58percent and then dropped 69

percent.Yetfewtraderscouldimagine a major stock indexsuch as the Standard andPoor’s (S&P) 500 Indexexhibitingsimilarfluctuationsoverasingleyear.

If we know whether amarket will be relativelyvolatileorrelativelyquietandcan convey this informationtoatheoreticalpricingmodel,any evaluation of options onthat market will be more

accurate than if we simplyignore volatility. Becauseoption models are based onmathematical formulas, wewill need some method ofquantifying this volatilitycomponent so that we canfeed it into the model innumericalform.

RandomWalksandNormalDistributions

Considerforamomentthepinball maze pictured inFigure 6-1. When a ball isdropped into themaze at thetop,itfallsdownward,pulledbygravitythroughaseriesofnails. When the ballencounters eachnail, there isa 50 percent chance that theballwillmoveto theleftanda 50 percent chance that itwill move to the right. Theball then falls down a levelwhere it encounters another

nail.Finally,atthebottomofthe maze, the ball falls intooneofthetroughs.

Figure6-1Randomwalk.

As the ball falls downthroughthemaze,itfollowsarandom walk. Once the ballenters themaze, nothing canbedonetoartificiallyalteritscourse, nor can one predictthe path that the ball willfollowthroughthemaze.

As more balls aredropped into the maze, theymight begin to form adistribution similar to that inFigure6-2.Most of the balls

tendtoclusternearthecenterof the maze, with adecreasing number of ballsending up in troughs fartheraway from the center. Ifmany balls are dropped intothe maze, they will begin toformabell-shapedornormaldistribution.

Figure6-2Normaldistribution.

If an infinite number ofballs were dropped into themaze, the resultingdistribution might beapproximated by a normaldistributioncurvesuchastheone overlaid on thedistribution in Figure 6-2.Such a curve is symmetrical(ifweflipitfromrighttoleft,it looks the same), it has itspeak in the center, and itstails always move down and

awayfromthecenter.Normal distribution

curves are used to describethe likely outcomes ofrandom events. For example,thecurveinFigure6-2mightalso represent the results offlippingacoin15times.Eachoutcome, or trough,represents the number ofheadsthatoccurfromeach15flips.Anoutcomeintrough0represents 0 heads and 15

tails;anoutcomeintrough15represents 15 heads and 0tails.Ofcourse,wewouldbesurprised to flip a coin 15timesandgetallheadsoralltails. Assuming that the coinis perfectly balanced, someoutcome in between, perhaps8headsand7tails,or9headsand 6 tails, seems morelikely.

Suppose that werearrange the nails in our

mazesothateachtimeaballencounters a nail and moveseither left or right, it mustdrop down two levels beforeit encounters another nail. Ifwedropenoughballsintothemaze,wemayendupwithadistribution similar to thecurve inFigure 6-3. Becausethe sideways movement ofthe balls is restricted, thecurvewillhaveahigherpeakand narrower tails than thecurve in Figure 6-2. In spite

of its altered shape, thedistribution is still normal,although one with slightlydifferentcharacteristics.

Figure6-3

Finally, we might againrearrange the nails so thateachtimeaballdropsdownalevel, itmustmove twonailsleftorrightbeforeitcandropdown to a new level. If wedrop enough balls into themaze, we may get adistributionthatresemblesthecurve in Figure 6-4. Thisdistribution, although stillnormal, will have a muchlower peak and spread out

much more quickly than thedistributions in either Figure6-2orFigure6-3.1

Suppose that we nowthink of the ball’s sidewaysmovement as the up anddown price movement of anunderlying contract and theball’s downward movementas thepassageof time. If theprice movement of anunderlyingcontract followsarandom walk, the curves in

Figures 6-2 through 6-4mightrepresentpossiblepricedistributions in a moderate-,low-, and high-volatilitymarket,respectively.

Figure6-4

Earlierinthischapterwesuggested that the theoreticalpricing of options begins byassigning probabilities to thevarious underlying prices.How should theseprobabilities be assigned?One possibility is to assumethat, at expiration, theunderlying prices arenormally distributed. Giventhat there are many differentnormal distributions, how

willourchoiceofdistributionaffectoptionevaluation?

Because all normaldistributionsaresymmetrical,itmayseemthatthechoiceofdistribution is irrelevant.Increased volatility mayincrease the likelihood oflarge upward movement, butthis should be offset by thegreater likelihood of largedownward movement.However, there is an

importantdistinctionbetweenan option position and anunderlying position. Theexpected value for anunderlying contract dependson all possible priceoutcomes. The expectedvalue for an option dependsonly on the outcomes thatresult in the option finishinginthemoney.Everythingelseiszero.

In Figure 6-5, we have

three possible pricedistributions centered aroundthe current price of anunderlying contract. Supposethatwewanttoevaluateacallatahigherexerciseprice.Thevalue of the callwill dependon the amount of thedistributiontotherightoftheexercise price. We can seethataswemove froma low-volatility distribution, to amoderate-volatilitydistribution, to a high-

volatility distribution, agreaterportionofthepossibleprice distribution lies to theright of the exercise price.Consequently, the optiontakes on an increasinglygreatervalue.

Figure6-5

We might also considerthe value of a put at a lowerexercise price. If we assumethatmovementisrandom,thesame high-volatilitydistribution that will causethe call to take on greatervaluewill also cause the putto take on greater value. Inthe case of the put, more ofthedistributionwill lie to theleft of the exercise price.Because our distributions are

symmetrical, in a high-volatility market, all options,whether calls or puts, higheror lowerexerciseprices, takeon greater value. For thesame reason, in a low-volatility market, all optionstakeonreducedvalues.

MeanandStandardDeviation

If we assume a normaldistributionofprices,wewillneed a method of describingthe appropriate normaldistribution to the theoreticalpricing model. Fortunately,all normal distributions canbe fully described with twonumbers—the mean and thestandard deviation. If weknow that a distribution isnormal,andwealsoknowthemeanand standarddeviation,then we know all the

characteristics of thedistribution.

Graphically, we caninterpret the mean as thelocation of the peak of thedistribution and the standarddeviation as a measure ofhow fast the distributionspreadsout.Distributionsthatspreadoutveryquickly,suchastheoneinFigure6-4,havea high standard deviation.Distributions that spread out

very slowly, such as the onein Figure 6-3, have a lowstandarddeviation.

Numerically,themeanissimply the average outcome,a concept familiar to mosttraders. To calculate themean, we add up all theresultsanddividebythetotalnumber of occurrences.Calculation of the standarddeviation is not quite sosimple and will be discussed

later. What is important atthispointistheinterpretationof these numbers, inparticular, what a mean andstandard deviation suggest interms of likely pricemovement.

Let’s go back to Figure6-2 and consider the troughsnumbered 0 to 15 at thebottom. We suggested thatthese numbers mightrepresentthenumberofheads

resulting from 15 flips of acoin. Alternatively, theymight represent the numberof times a ball goes to theright at each nail as it dropsdown through themaze. Thefirst trough is assigned 0because any ball that endsthere must go left at everynail. The last trough isassigned 15 because any ballthat ends theremust go rightateverynail.

Supposethatwearetoldthat the mean and standarddeviation in Figure 6-2 are7.50 and 3.00, respectively.2What does this tell us aboutthe distribution? The meantells us the averageoutcome.Ifweaddupalltheoutcomesand divide by the number ofoccurrences,theresultwillbe7.50.Intermsof thetroughs,the average result will fallhalfway between troughs 7

and8. (Of course, this is notan actual possibility.However, we noted inChapter 5 that the averageoutcomedoes not have to bean actual possibility for anyoneoutcome.)

The standard deviationdeterminesnotonlyhowfastthe distribution spreads out,but it also tells us somethingabout the likelihoodofaballendingupinaspecifictrough

or group of troughs. Inparticular, the standarddeviation tells us theprobability of a ball endingup in a trough that is aspecified distance from themean. For example, we maywant to know the likelihoodofaballfallingdownthroughthemaze and ending up in atroughlowerthan5orhigherthan 10. The answer to thisquestion depends on thenumber of standard

deviationstheballmustmoveaway from the mean. If weknow this,we can determinethe probability associatedwith that number of standarddeviations.

The exact probabilityassociated with any specificnumber of standarddeviations can be found inmost texts on statistics orprobability. Alternatively,such probabilities can be

easily calculated in mostcommonly used computerspreadsheet programs. Foroption traders, the followingapproximations will beuseful:

±1standarddeviationtakesinapproximately68.3percent

(about2/3)ofalloccurrences.±2

standarddeviationstakesinapproximately95.4percent(about19/20)ofall

occurrences.±3

standarddeviationstakesinapproximately99.7percent(about369/370)ofalloccurrences.

Note that each number

of standard deviations ispreceded by a plus or minussign. Because normaldistributionsaresymmetrical,the likelihood of upmovement and downmovement is identical. Theprobability associated witheach number of standarddeviationsisusuallygivenasa decimal value, but afractional approximation isoften useful to traders, andthisappearsinparentheses.

Now let’s try to answerour question about thelikelihoodofgettingaball ina trough lower than 5 orhigher than 10. We candesignatethedividerbetweentroughs7and8as themean,7½. If the standard deviationis 3,what troughs arewithinone standarddeviationof themean? One standarddeviation from the mean is7½ ± 3, or 4½ to10½.Interpreting ½ as the divider

between troughs, we can seethattroughs5through10fallwithin1standarddeviationofthemean.Weknow that onestandard deviation takes inabout two-thirds of alloccurrences, so we canconclude that out of everythree balls we drop into themaze, two should end up introughs 5 through 10.Whateverisleftover,oneoutof every three balls,will endup in one of the remaining

troughs, 0 through 4 and 11through 15. Hence, theanswer to our originalquestion about the likelihoodof getting a ball in a troughlower than 5 or higher than10 is about1chance in3,orabout 33 percent. (The exactanswer is 100% – 68.3% =31.7%.) This is shown inFigure6-6.

Figure6-6

Let’s try anothercalculation, but this time wecanthinkoftheproblemasawager.Supposethatsomeoneoffersus30to1oddsagainstdroppingaballintothemazeand having it end upspecifically in troughs 14 or15.Isthisbetworthmaking?Onecharacteristicofstandarddeviations is that they areadditive. In our example, ifone standard deviation is 3,

then two standard deviationsare 6. Two standarddeviations from the mean istherefore 7½ ± 6, or 1½ to13½.WecanseeinFigure6-6 that troughs 14 and 15 lieoutside two standarddeviations. Because theprobabilityofgettinga resultwithin two standarddeviations is approximately19 out of 20, the probabilityofgettingaresultbeyondtwostandard deviations is 1

chancein20.Therefore,30to1 odds may seem veryfavorable. Recall, however,that beyond two standarddeviations also includestroughs 0 and 1. Becausenormal distributions aresymmetrical, the chances ofgetting a ball specifically introughs14or15mustbehalfof1chancein20,orabout1chancein40.At30to1odds,the bet must be a bad onebecause the odds do not

sufficientlycompensateusfortheriskinvolved.

In Chapter 5, wesuggested that a trulyaccurate theoretical pricingmodel would require us toassign probabilities to aninfinite number of possibleprice outcomes for anunderlying contract. Then, ifwe multiply each priceoutcome by its associatedprobability, we can use the

results to calculate anoption’s theoretical value.The problem is that aninfinitenumberofanythingisnot easy to work with.Fortunately, thecharacteristics of normaldistributions are so wellknown that formulas havebeendeveloped that facilitatethe computation of both theprobabilities associated withevery point along a normaldistribution curve and the

areaundervariousportionsofthe curve. If we assume thatprices of an underlyinginstrument are normallydistributed, these formulasrepresentauniquesetoftoolsto help us solve for anoption’stheoreticalvalue.

Louis Bachelierwas thefirst to make the assumptionthat the prices of anunderlying contract arenormally distributed. As we

shall see, there are logicalproblems with thisassumption. Consequently,over the years, theassumption has beenmodified to make it moreconsistent with real-worldconditions. In its modifiedform,it is thebasisformanytheoretical pricing models,including the Black-Scholesmodel.

ForwardPriceastheMeanofaDistribution

If we decide to assignprobabilities that areconsistent with a normaldistribution, how dowe feedthis distribution into atheoretical pricing model?Because all normaldistributionscanbedescribedby a mean and a standard

deviation, in some way wemust feed these twonumbersintoourpricingmodel.

In Chapter 5, wesuggested that anydistribution ought to becentered around the mostlikely underlying price atexpiration. Although wecannot know exactly whatthat price will be, if weassume that no arbitrageopportunity exists in the

underlyingcontract, a logicalguess is the forward price. Ifwemake theassumption thatthe forward price representsthe mean of a distribution,theninthelongrun,anytrademade at the currentunderlying price will justbreak even. The variousforms of the Black-Scholesmodeldifferprimarilyinhowthey calculate the forwardprice.Dependingon the typeof underlying contract,

whether a stock, a futurescontract, or a foreigncurrency,themodeltakesthecurrent underlying price, thetime to expiration, interestrates, and, in the case ofstocks,dividends to calculatethe forward price. It thenmakes this the mean of thedistribution.

Volatilityasa

StandardDeviation

Inaddition to themean, tofully describe a normaldistribution, we also need astandard deviation.Whenweinput a volatility into atheoreticalpricingmodel,weare actually feeding in astandard deviation. Volatilityis just a trader’s term forstandard deviation. Becausethe Greek letter sigma (σ) is

the traditional notation forstandarddeviation,inthistextwewillusethesamenotationforvolatility.

Atthispoint,itwillhelpif we assign a workingdefinition to volatility,althoughwewilllatermodifythis definition slightly. Forthe present, we will assumethatthevolatilitywefeedintoa pricing model represents aone standard deviation price

change, in percent, over aone-yearperiod.Forexample,consider a contract with aone-yearforwardpriceof100and that we are told has avolatility of 20 percent.(We’ll discuss later wherethis number might comefrom.) With a mean of 100andastandarddeviationof20percent,ifwecomebackoneyear from now, there is a 68percent probability that thecontract will be trading

between 80 and 120 (100 ±20%), a 95 percentprobability that the contractwill be trading between 60and140(100±2×20%),anda99.7percentprobabilitythatthe contract will be tradingbetween40and160(100±3× 20%). These are theprobabilities associated withone, two, and three standarddeviations.

Insteadofspecifyingthe

forward price, suppose thatwe are dealing with a stockthat is currently trading at$100andthathasthesame20percent volatility. In order todetermine the one-yearprobabilities, we must firstdetermine the one-yearforward price because thisrepresents the mean of thedistribution. If interest ratesare 8 percent and the stockpays no dividends, the one-year forward price will be

$108. Now, a one standarddeviationpricechangeis20%× $108 = $21.60. Thus, oneyear from now, we wouldexpect the same stock to betrading between $86.40 and$129.60 ($108 ± $21.60)approximately 68 percent ofthetime,between$64.80and$151.20 ($108±2×$21.60)approximately 95 percent ofthetime,andbetween$43.20and $172.80 ($108 ± 3 ×$21.60) approximately 99.7

percentofthetime.Returningtoourcontract

with a forward price of 100,supposethatwecomebackatthe end of one year and findthat the contract, which wethoughthadavolatilityof20percent,istradingat35.Doesthismeanthatthevolatilityof20 percent was wrong? Aprice change of more thanthreestandarddeviationsmaybe unlikely, but one should

not confuse unlikely withimpossible. Flipping aperfectly balanced coin 15timesmayresultin15heads,even though the odds of thisoccurring are less than onechance in 32,000. If 20percent is the right volatility,the probability that the pricewillfallfrom100to35inoneyear is less than one chancein 1,500. However, onechance in 1,500 is notimpossible, and perhaps this

was the one time in 1,500whenthepricedidindeedendupat35.Ofcourse, it isalsopossible that we had thewrongvolatility.Butwecan’tmake that determinationwithout looking at a largenumber of price changes forthecontractsothatwehavearepresentative pricedistribution.

ScalingVolatilityforTime

Like interest rates,volatility is alwaysexpressedas an annualized number. Ifsomeone says that interestrates are 6 percent, no oneneeds to ask whether thatmeans 6 percent per day, 6percent per week, or 6percent permonth. Everyoneknowsthatitmeans6percent

peryear.Thesame is trueofvolatility.

We might logically askwhatanannualvolatilitytellsus about the likelihood ofprice changes over someshorter period of time.Although interest rates areproportional to time (wesimply multiply the rate bytheamountoftime),volatilityis proportional to the squareroot of time. To calculate a

volatility, or standarddeviation, over some periodof time other than one year,we must multiply the annualvolatilitybethesquarerootoftime,where the timeperiod tisexpressedinyears

Traders typically calculatevolatility for an underlyingcontract by observing pricechanges at regular intervals.

Let’s begin by assuming thatwe plan to observe pricechanges at the end of everyday. Because there are 365days inayear, itmightseemthat prices can change 365times per year. In this text,though, we are focusingprimarily on exchange-tradedcontracts. Because mostexchanges are closed onweekendsandholidays,ifweobserve the price of anunderlyingcontractattheend

of every day, prices cannotreally change 365 times peryear. Depending on theexchange, there are probablysomewhere between 250 and260 trading days in a year.3Because we need the squarerootof thenumberof tradingdays, for convenience, manytraders assume that there are256 trading days in a yeargiven that the square root of256isawholenumber,16.If

we make this assumption,then

To approximate a dailystandard deviation, we candividetheannualvolatilityby16.

Returningtoourcontracttrading at 100 with avolatility of 20percent,whatis a one standard deviation

pricechangefromonedaytothe next? The answer is20%/16 = 1¼%, so a onestandarddeviationdailypricechangeis1¼%×100=1.25.We expect to see a pricechange of 1.25 or lessapproximately two tradingdaysoutofeverythreeandaprice change of 2.50 or lessapproximately 19 tradingdays out of every 20. Onlyone day in 20 would weexpect to see a price change

ofmorethan2.50.Wecandothesametype

of calculation for a weeklystandard deviation. Now wemustaskhowmanytimesperyear prices can change ifwelook at prices once a week.Therearenocompleteweekswhen no trading takes place,so we must make ourcalculations using all 52trading weeks in a year.Therefore,

To approximate aweekly standard deviation,we can divide the annualvolatilityby7.2.Dividingourannualvolatilityof20percentby the square root of 52, orapproximately 7.2, we get20%/7.2 » 2.78. For ourcontract trading at 100, wewould expect to see a pricechange of 2.78 or less two

weeks out of every three, aprice change of 5.56 or less19weeksoutofevery20,andonly one week in 20 wouldwe expect to see a pricechangeofmorethan5.56.

If we want to be asaccurate as possible, whenestimating a daily or weeklystandard deviation, we oughtto begin by calculating theone-dayorone-weekforwardprice.Butforshortperiodsof

time, the forward price is soclosetothecurrentpricethatmost traders assume forconveniencethataone-dayorone-week distribution iscentered around the currentprice.

Suppose that a stock istrading at $45 per share andhasanannualvolatilityof37percent. What is anapproximate one and twostandarddeviationpricerange

fromoneday to thenextandfrom one week to the next?For one day, we can dividethe annual volatility by 16(the square root of 256, thenumber of trading days in ayear)

A one and two standarddeviationdailyprice range isapproximately

$45±$1.04≈$43.96to$46.04(onestandard

deviation)$45±(2×$1.04)≈$42.92to

$47.08(twostandarddeviations)

For one week, we candividetheannualvolatilityby7.2(thesquarerootof52,thenumberoftradingweeksinayear)

A one and two standarddeviation weekly price rangeisapproximately

$45±$2.31≈$42.69to$47.31(onestandard

deviation)$45±(2×$2.31)≈$40.38to

$49.62(twostandarddeviations)

When we scale volatilityfor time, the sameprobabilities still apply.Approximately 68 percent ofthe occurrences will fallwithin one standarddeviation. Approximately 95percent of the occurrenceswill fall within two standarddeviations.

Volatilityand

ObservedPriceChanges

Whymightatraderwanttoestimatedailyorweeklypricechanges from an annualvolatility? Volatility is theone input into a theoreticalpricingmodel that cannot bedirectly observed. Yet manyoption strategies, if they areto be successful, require areasonable assessment of

volatility. Therefore, anoption trader needs somemethod of determiningwhether his expectationsabout volatility are beingrealized in the marketplace.Unlike directional strategies,whose success or failure canbe immediately observedfrom current prices, there isno such thing as a currentvolatility. A trader mustusuallydetermineforhimselfwhether he is using a

reasonable volatility inputinto the theoretical pricingmodel.

Previously,weestimatedthat for a $45 stock with anannual volatility of 37percent, a one standarddeviation price change isapproximately $1.04.Suppose that over five dayswe observe the followingdaily settlement pricechanges:

+$0.98,–$0.65,–$0.70,+$0.25,–$0.85

Are these price changesconsistent with a 37 percentvolatility?

Weexpecttoseeapricechange of more than $1.04(one standard deviation)about one day in three.Overfivedays,wewouldexpecttosee at least one day, andperhaps two days, with apricechangegreaterthanone

standard deviation. Yet,during this five-day period,wedidnotseeapricechangegreaterthan$1.04evenonce.What conclusions can bedrawn from this? One thingseems clear: these five pricechanges do not appear to beconsistent with a 37 percentvolatility.

Before making anydecisions, we ought toconsider any unusual

conditions that might beaffecting the observed pricechanges. Perhaps this was aholidayweek,andas such, itdidnotreflectnormalmarketactivity. If this is ourconclusion, then 37 percentmay still be a reasonablevolatility estimate. On theother hand, ifwe can see nological reason for themarketbeing less volatile thanpredicted by a 37 percentvolatility, then we may

simply be using the wrongvolatility. Ifwe come to thisconclusion,perhapsweoughtto consider using a lowervolatility that is moreconsistent with the observedpricechanges.Ifwecontinueto use a volatility that is notconsistent with the actualprice changes, then we havethe wrong volatility. If wehave thewrongvolatility,wehave thewrong probabilities.And if we have the wrong

probabilities, we aregenerating incorrecttheoretical values, therebydefeating the purpose ofusing a theoretical pricingmodelinthefirstplace.

Admittedly, five days isaverysmallnumberofpricechanges, and it is unlikelythata traderwill relyheavilyonsuchasmallsample.Ifweflip a coin five times and itcomes up heads each time,

wemay not be able to drawany definitive conclusions.But if we flip the coin 50times and it comes up headsevery time, now we mightconclude that there issomething wrong with thecoin. In the same way, mosttraders prefer to see largerprice samplings, perhaps 20days,or50days,or100days,before drawing any dramaticconclusionsaboutvolatility.

Exactlywhatvolatilityisassociatedwith thefivepricechanges in the foregoingexample? Without doingsome rather involvedarithmetic, it is difficult tosay. (The answer is actually27.8 percent.) However, if atrader has some idea of theprice changes he expects, hecan easily see that thechanges over the five-dayperiodarenotconsistentwith

a37percentvolatility.4Wehaveusedthephrase

price change in conjunctionwith our volatility estimates.Exactlywhatdowemeanbythis? Do we mean thehigh/lowduringsomeperiod?Do we mean open-to-closeprice changes? Or is thereanother interpretation?Although various methodshave been suggested toestimate volatility, the most

common method forexchange-tradedcontractshasbeen to calculate volatilitybased on settlement-to-settlement price changes.Usingthisapproach,whenwesay that a one standarddeviation daily price changeis$1.04,wemean$1.04fromone day’s settlement price tothe next day’s settlementprice. The high/low oropen/close price range mayhavebeeneithermoreorless

thanthisamount,butitisthesettlement-to-settlement pricechangeonwhichwefocus.5

ANoteonInterest-RateProducts

For some interest-rateproducts, primarilyEurocurrency interest-ratefutures, the listed contractprice represents the interest

rate associated with thatcontract, expressed as awhole number, subtractedfrom 100.6 If the LondonInterbank Offered Rate(LIBOR),theinterestpaidondollar deposits outside theUnitedStates,is7.00percent,the associated Eurodollarfutures contract traded at theChicago MercantileExchange will be trading at100 – 7.00 = 93.00. If Euro

Interbank Offered Rate(Euribor),theinterestpaidoneuro deposits outside theEuropeanEconomicUnion,is4.50 percent, the associatedEuribor futures contracttraded at the LondonInternational FinancialFutures Exchange will betradingat100–4.50=95.50.Volatility calculations forthesecontractsaredoneusingthe rate associated with thecontract (the rate volatility)

rather than the price of thecontract(thepricevolatility).

If a Eurodollar futurescontract is trading at 93.00with a volatility of 26percent,anapproximatedailyand weekly one standarddeviationpricechangeis

To be consistent, if weindex Eurodollar futurespricesfrom100,wemustalsoindex exercise prices from100. Therefore, a 93.00exercise price in our pricingmodelisreallya7.00percent(100 – 93.00) exercise price.This transformation alsorequiresustoreversethetypeof option, changing calls toputsandputs tocalls.Toseewhy, consider a 93.00 call.For this call to go into the

money, the underlyingcontract must rise above93.00. But this requires thatinterest rates fall below 7.00percent. Therefore, a 93.00callinlistedtermsisthesameas a 7.00 percent put ininterest-rate terms. A modelthat is correctly set up toevaluate options onEurodollar or other types ofindexedinterest-ratecontractswillmake this transformationautomatically. The price of

the underlying contract andthe exercise price aresubtracted from 100, withlistedcallstreatedasputsandlistedputstreatedascalls.

This type oftransformationisnotrequiredfor most bonds and notes.Depending on the couponrate, the prices of theseproducts may range freelywithout upper limit, oftenexceeding 100. Exchange-

traded options on bond andnote futures are thereforemost often evaluated using atraditional pricing model.However, interest-rateproducts present otherproblems that may requirespecializedpricingmodels.

It is possible to take aninstrumentsuchasabondandcalculate the current yieldbased on its price in themarketplace. If we were to

take a series of bond pricesand from these calculate aseries of yields, we couldcalculate the yield volatility,thatis,thevolatilitybasedonthechangeinyield.Wemightthen use this number toevaluate the theoretical valueof an option on the bond,although to be consistent wewould also have to specifytheexerciseprice in termsofyield. Because it is possibleto calculate the volatility of

an interest-rate product usingthese two different methods,interest-rate traders usuallymake a distinction betweenyield volatility (the volatilitycalculated from the currentyield on the instrument) andprice volatility (the volatilitycalculated from the price ofthe instrument in themarketplace).

LognormalDistributions

Thus farwehaveassumedthat the prices of anunderlying instrument arenormallydistributed.Is thisareasonable assumption?Beyond the question of theexactdistributionofpricesinthe real world, the normaldistribution assumption hasone serious flaw. A normal

distribution is symmetrical.For every possible upwardmove in the price of anunderlying instrument, theremust be the possibility of adownward move of equalmagnitude. If we allow forthe possibility of a $50contract rising $75 to $125,we also must allow for thepossibility of the contractdropping $75 to a price of –$25. But negative prices areclearly not possible for

traditional stocks orcommodities.

We have definedvolatility in terms of thepercent changes in the priceof an underlying instrument.In this sense, an interest rateand volatility are similar inthattheybothrepresentarateof return. The primarydifference between interestand volatility is that interestaccrues only at a positive

rate, whereas volatility is acombination of positive andnegativeratesofreturn.Ifweinvest money at a fixedinterest rate, the value of theprincipal will always grow.However, if we invest in anunderlying instrument with avolatility other than zero, theinstrument may go up ordown in price, resulting ineitheraprofit(apositiverateofreturn)oraloss(anegativerateofreturn).

A rate-of-returncalculation must specify notonly the rate that is beingused but also the timeintervals over which thereturns are calculated.Suppose that we invest$1,000 for one year at anannual interest rate of 12percent. How much will wehave at the end of one year?The answer depends on howthe12percentinterestonourinvestmentispaidout.

As interest is paidmorefrequently, even though it ispaid at the same rate of 12percent per year, the totalyield on the investmentincreases. The yield isgreatestwhen interest ispaidcontinuously. In this case, itis justas if interest ispaidatevery possible moment intime.

Although less common,we can do the same type of

calculation using a negativeinterest rate. For example,suppose that we make a badinvestmentof$1,000andlosemoneyatarateof12percentannually (interest rate = –12%). How much will wehaveattheendofayear?Theanswer,again,dependsonthefrequencyatwhichourlossesaccrue.

Inthecaseofanegativeinterest rate, as losses arecompoundedmorefrequently,even though at the same rateof –12 percent per year, thesmaller the total loss, andconsequently, the smaller thenegativeyield.

In the same way thatinterest can be compoundedat different intervals,volatility can also becompounded at different

intervals. The Black-Scholesmodel is a continuous-timemodel. The model assumesthat volatility is compoundedcontinuously, just as if theprice changes in theunderlyingcontract,eitherupor down, are taking placecontinuouslybutatanannualrate corresponding to thevolatility associated with thecontract. When the percentprice changes are normallydistributed, the continuous

compounding of these pricechanges will result in alognormal distribution ofprices at expiration. Such adistribution is shown inFigure 6-7. The entiredistribution isskewed towardthe upside because upsidepricechanges(apositiverateof return) will be greater, inabsolute terms, thandownside price changes (anegative rate of return). Inour interest-rate example, a

continuously compoundedrate of return of +12 percentyields a profit of $127.50after one year, whereas acontinuously compoundedrate of return of –12 percentyieldsalossofonly$113.08.If the 12 percent is avolatility,thenaonestandarddeviation upward pricechangeattheendofoneyearis +$127.50, whereas a onestandarddeviationdownwardprice change is –$113.08.

Eventhoughtherateofreturnis a constant 12 percent, thecontinuous compounding of12 percent yields differentupward and downwardmoves.

Figure6-7

Notealsothelocationofthemean of the distributionsin Figure 6-7. Themean canbethoughtofas the“balancepoint”ofthedistribution.Fora normal distribution, thepeak of the distribution, ormode,andthemeanhavethesame location, exactly in themiddle of the distribution.But in a lognormaldistribution the right tail,which is open-ended, is

longerthanthelefttail,whichis bounded by zero. Becausethere ismore“weight” to therightofthepeak,themeanofthe lognormal distributionmustbelocatedtotherightofthepeak.

Continuous rates ofreturncanbecalculatedusingthe exponential function,7denoted by either exp(x) orex.Intheprecedingexamples,

$1,000×e0.12=$1,127.50and$1,000×e–0.12=$886.92

Nomatterhowlargethenegative interest rate,continuous compoundingprecludesthepossibilityofaninvestmentfallingbelowzerobecause it is impossible tolosemorethan100percentofan investment.Consequently,in a log-normal distribution,the value of the underlyinginstrument is bounded by

zero on the downside.Clearly, this is a morerealistic representation of thereal world than a normaldistribution.

Wecanseetheeffectofusingalognormaldistributionrather than a normaldistribution by consideringthevalueofa90putand110call with a forward price of100 for the underlyingcontract with six months to

expiration and a volatility of30percent

Under a normaldistribution assumption, boththe call and put have exactly

the same value because theyareboth10percentoutofthemoney. But under thelognormal distributionassumption in the Black-Scholes model, the 110 callwill always have a greatervalue than the 90 put. Thevalue of the 110 call canpotentiallyappreciatewithoutlimitbecause thepriceof theunderlying contract has nolimit on the upside. The 90put,however,canonlyriseto

a maximum value of 90because the price of theunderlyingcontractcanneverfallbelowzero.

Of course, the values inthe preceding example aretrue only in theory. There isno law that prevents the 90put from trading at a pricegreater than the 110 call.Indeed, such pricerelationships occur in manymarkets for a variety of

reasons that we will discusslater. However, one possibleexplanation is that themarketplace disagrees withtheassumptionsonwhichthemodel is based. Perhaps themarketplace believes that alognormal distribution is notan accurate representation ofpossible prices. And perhapsthemarketplaceisright!

InterpretingVolatilityData

When traders discussvolatility, even experiencedtradersmayfindthattheyarenot always talking about thesame thing. When a tradersays that the volatility is 25percent, this statement maytakeonavarietyofmeanings.We can avoid confusion insubsequent discussions if we

define some of the differentwaysinwhichtradersrefertovolatility. We can begin bydividing volatility into twocategories—realizedvolatility,whichweassociatewith an underlying contract,and implied volatility, whichweassociatewithoptions.

RealizedVolatilityThe realized volatility is

the annualized standarddeviation of percent pricechanges of an underlyingcontract over some period oftime.8 When we calculaterealized volatility, we mustspecify both the interval atwhich we are measuring thepricechangesandthenumberof intervals to beused in thecalculations.Forexample,wemight talk about the 50-dayvolatility of an underlying

contract. Or we might talkabout the 52-week volatilityof a contract. In the formercase, we are calculating thevolatilityfromthedailypricechanges over a 50-dayperiod.9Inthelattercase,weare calculating the volatilityfrom the weekly pricechanges over a 52-weekperiod.

On a graph of realizedvolatility, each point

represents the volatility overaspecifiedperiodusingpricechanges over a specifiedinterval. If we graph the 50-day volatility of a contract,each point on the graphrepresents the annualizedstandard deviation of thedaily price changes over theprevious50days.Ifwegraphthe 52-week volatility, eachpointon thegraph representsthe annualized standarddeviationof theweeklyprice

changesover theprevious52weeks.

Tradersmayalsorefertorealized volatility in thefuture (future realizedvolatility) and realizedvolatility in the past(historicalrealizedvolatility).The future realized volatilityis what every trader wouldlike to know—the volatilitythat best describes the futuredistribution of price changes

foranunderlyingcontract.Intheory,itisthefuturerealizedvolatility over the life of theoption that we need to inputinto a theoretical pricingmodel. If a trader knows thefuture realized volatility, heknows the right “odds.”When he feeds this numberinto a theoretical pricingmodel, he can generateaccurate theoretical valuesbecause he has the rightprobabilities.Likethecasino,

hemay lose in the short runbecause of bad luck, but inthe long run, with theprobabilities in his favor, thetrader can be reasonablycertainofmakingaprofit.

Clearly, no one knowswhat the future holds.However, if a trader intendsto use a theoretical pricingmodel, he must try to makeanestimateof future realizedvolatility. In option

evaluation, as in otherdisciplines, a good startingpoint is historical data.Whattypically has been thehistoricalrealizedvolatilityofa contract? If, over the past10 years, the volatility of acontract has never been lessthan10percentnormorethan30 percent, a guess for thefuturevolatilityofeither5or40 percent hardly makessense. This does not meanthat either of these extremes

is impossible. But based onpast performance, and in theabsence of any extraordinarycircumstances,aguesswithinthehistoricallimitsof10and30 percent is probably morerealistic than a guess outsidethese limits.Ofcourse,10 to30percentisstillaverywiderange. But at least thehistorical data offers astarting point. Additionalinformation may help tofurthernarrowtheestimate.

As option traders havecome to appreciate theimportanceofvolatilityasaninput into a pricing model,volatility forecasting modelshave been developed in anattempt to more accuratelypredict future realizedvolatility. If a trader hasaccess toavolatility forecastthathebelievesisreliable,hewillwant touse this forecastto make a better decision asto the future realized

volatility. We will put off adiscussion of possibleforecasting methods untillaterchapters.

When we calculatevolatilityoveragivenperiodoftime,westillhaveachoiceof the time intervals overwhich to measure the pricechanges in the underlyingcontract. A trader mightconsider whether the choiceof intervals, even if the

intervalscoverthesametimeperiod, might affect theresults. For example, wemight look at the 250-dayvolatility, the 52-weekvolatility, and the 12-monthvolatility of a contract. Allvolatilities coverapproximately one year, butone is calculated from dailyprice changes, one fromweekly price changes, andone from monthly pricechanges.

For most underlyingcontracts, the interval that ischosen does not seem togreatly affect the result. It ispossible that a contract willmake large daily moves yetfinish the week unchanged.However, this is by far theexception. A contract that isvolatile from day to day islikely to be equally volatilefromweek toweekormonthto month. Figure 6-8 showsthe250-dayrealizedvolatility

of the S&P 500 Index from2003 through 2012, with thevolatility calculated fromprice changes at threedifferent intervals: daily,weekly, and every fourweeks. The graphs are notidentical,buttheydoseemtohave similar characteristics.There is no clear evidencethat using one interval ratherthan another results inconsistently higher or lowervolatility.

Figure6-8S&P500Index250-dayhistoricalvolatility.

ImpliedVolatilityUnlike realized volatility,

which is calculated fromprice changes in theunderlying contract, impliedvolatility is derived from theprice of an option in themarketplace. In a sense, theimplied volatility representsthe marketplace’s consensusof what the future realized

volatility of the underlyingcontract will be over the lifeoftheoption.

Consider a three-month105 call on a stock that paysno dividend. If we areinterested in purchasing thiscall, we might use a pricingmodel to determine theoption’stheoreticalvalue.Forsimplicity, let’s assume thatthe option is European (noearly exercise) and that we

will use the Black-Scholesmodel. In addition to theexercise price, time toexpiration, and type,we alsoneedthepriceofthestock,aninterest rate, and a volatility.Supposethatthecurrentstockprice is 98.50, the three-month interest rate is 6.00percent,andourbestestimateof volatility over the nextthree months is 25 percent.When we feed this data intoour model, we find that the

optionhasatheoreticalvalueof 2.94. However, when wecheck the price of the optionin the marketplace, we findthat the 105 call is tradingvery actively at a price of3.60.Howcanweaccountforthe fact that we think theoption isworth 2.94, but therest of the world seems tothinkthatit’sworth3.60?

This is not an easyquestion to answer because

therearemanyforcesatworkinthemarketplacethatcannotbe easily identified orquantified. But one way wemight try to answer thequestion is by making theassumption that everyonetradingtheoptionisusingthesame theoretical pricingmodel. If we make thisassumption, the cause of thediscrepancy must be adifference of opinion aboutoneormoreoftheinputsinto

the model. Which inputs arethemostlikelycause?

It’sunlikely tobeeitherthe time to expiration or theexercise price because theseinputsare fixed in theoptioncontract. What about theunderlying price of 98.50?Perhaps we incorrectlyestimated the stockpricedueto the width of the bid-askspread. However, for mostactively traded underlying

contracts, it is unlikely thatthe spread will be wideenough to cause adiscrepancy of 0.66 in thevalue of the option. In orderto yield a value of 3.60 forthe 105 call, we wouldactually have to raise thestockpriceto100.16,andthisis almost certainly welloutsidethebid-askspreadforthestock.

Perhaps our problem is

the interest rate of 6.00percent.But interest ratesareusually the least importantofthe inputs into a theoreticalpricing model. In fact, wewould have to make a hugechange in the interest-rateinput, from 6.00 to 13.30percent, toyielda theoreticalvalueof3.60.

This leaves us with onelikely cause for thediscrepancy—the volatility.

In a sense, the marketplaceseems be using a volatilitythat is different from25 percent. To determinewhat volatility themarketplace is using,we canaskthefollowingquestion:ifwe hold all other inputsconstant (i.e., time toexpiration, exercise price,underlyingprice, and interestrates), what volatility mustwe feed into our model toyieldatheoreticalvalueequal

to the price of the option inthe marketplace? In ourexample, we want to knowwhat volatility will yield avalueof3.60forthe105call.Clearly, the volatility has tobehigher than25percent, sowe might begin to raise thevolatility input into ourmodel.Ifwedo,wefindthatat a volatility of 28.50percent, the 105 call has atheoreticalvalueof3.60.Theimplied volatility of the 105

call—the volatility beingimplied to the underlyingcontract through the pricingof the option in themarketplace—is 28.50percent.

Figure6-9

When we solve for theimplied volatility of anoption,we are assuming thatthe theoretical value (theoption’sprice), aswell as allother inputs exceptvolatility,are known. In effect, we arerunning the theoreticalpricing model backwards tosolve for the unknownvolatility. In practice, this iseasiersaidthandonebecausemost theoretical pricing

models do not work inreverse.However, thereareanumber of relatively simplealgorithms that can quicklysolve for the impliedvolatility when all otherinputsareknown.

Implied volatilitydepends not only on theinputs into the theoreticalpricingmodelbutalsoonthetheoretical pricing modelbeingused.Forsomeoptions,

different models can yieldsignificantlydifferentimpliedvolatilities.Problemscanalsoarisewhen the inputs arenotcontemporaneous. If anoption has not traded forsome time and marketconditions have changed,using an outdated optionprice will result in amisleading or inaccurateimpliedvolatility.Supposeinourexample that thepriceof3.60forthe105callreflected

the last trade, but that tradetook place two hours agowhen the underlying stockpricewasactually99.25.Atastock price of 99.25, theimplied volatility of theoption, at a price of 3.60, isactually 26.95 percent. Thisunderscorestheimportanceofaccurate andcontemporaneous inputswhen calculating impliedvolatilities.

Brokerage firms anddata vendors who provideoption analysis for theirclients will typically includeimplied volatility data. Thedatamay incorporate impliedvolatilities for every optionon anunderlying contract, orthedatamaybeintheformofone implied volatility that isrepresentativeofoptionsonaparticular underlying market.In the latter case, the singleimplied volatility is usually

the result of weighting theindividual impliedvolatilitiesby some criteria, such asvolume of options traded oropen interest, or, as is mostcommon, by assigning thegreatest weight to the at-the-moneyoptions.

Implied volatility in themarketplace is constantlychanging because optionprices, as well as othermarket conditions, are

constantlychanging.Itisasifthe marketplace werecontinuously polling all theparticipantstocomeupwithaconsensus volatility for theunderlying contract for eachexpiration. This is not a pollin the true sense because thetraders do not confer witheach other and then vote onthe correct volatility.However, as bids and offersaremade, the price at whichan option is trading will

represent an equilibriumbetween supply and demand.This equilibrium can beexpressed as an impliedvolatility.

While the termpremiumreally refers to an option’sprice, because the impliedvolatility is derived from anoption’s price, traderssometimes use premium andimplied volatilityinterchangeably. If the

current implied volatility ishigh by historical standardsor high relative to the recenthistorical volatility of theunderlying contract, a tradermight say that premiumlevels are high; if impliedvolatilityisunusuallylow,hemight say that premiumlevelsarelow.

New option traders aretaught, quite sensibly, to selloverpriced options and buy

underpriced options. Byselling options at priceshigher than theoretical valueor buying options at priceslowerthantheoreticalvalue,atrader creates a positivetheoretical edge. But howshoulda traderdetermine thedegree to which an option isoverpriced or underpriced?This sounds like an easyquestion to answer. Isn’t theamount of the mispricingequal to the difference

between the option’s priceand its value? The questionarises because there is morethanonewaytomeasurethisdifference. Returning to ourexample of the 105 call, wemight say that with atheoretical value of 2.94 andapriceof3.60,the105callis0.66 overpriced. But involatility terms the option is3.50 volatility pointsoverpriced because itstheoreticalvalueisbasedona

volatility of 25 percent (ourvolatility estimate), while itsprice is based on a volatilityof28.50percent (the impliedvolatility).Given theunusualcharacteristicsofoptions,itisoftenmoreusefulforatraderto consider an option’s pricein terms of implied volatilityratherthantotalpoints.

Implied volatility isoften used by traders tocompare the relative pricing

of options. In our example,the105callistradingat3.60with an implied volatility of28.50percent.Supposethata100 call under the sameconditions is trading at 5.40.Intotalpoints,the100callisclearly more expensive thanthe 105 call (5.40 versus3.60). But if, at a price of5.40, the 100 call has animplied volatility of 27.51percent, most traders willconclude that in theoretical

termsthe100callisalmostafull percentage point lessexpensive (27.51 percentversus28.50percent)thanthe105call.Traders,infact,talkabout implied volatility as ifitwerethepriceofanoption.A trader who buys the 100call at a price of 5.40 mightsaythatsheboughtthecallat27.51 percent. A trader whosellsthe105callatapriceof3.60 might say that he soldthe call at 28.50 percent. Of

course, options are reallybought and sold in theappropriate currency. Butfrom an option trader’s pointofviewthe impliedvolatilityis often a more usefulexpression of an option’sprice than its actual price incurrencyunits.

Even if the impliedvolatility of the100 call is27.51percentandtheimpliedvolatility of the 105 call is

28.50 percent, this does notnecessarilymeanthatatraderoughttobuythe100callandsellthe105call.Atraderalsowill need to consider whatwillhappenifhisestimateofvolatility turns out to beincorrect. If the futurerealizedvolatilityoverthelifeof theoptions turnsout tobe25 percent, both the 100 calland the 105 call areoverpriced, and the sale ofeither option should, in

theory, result in a profit.Butwhat will happen if thetrader’s volatility estimate iswrong,andthefuturerealizedvolatility turns out to be 32percent? Now the sale ofeither option will result in aloss. The consequences ofbeing wrong about volatilityare an importantconsideration, and this issomething we will look atmore closely in subsequentchapters. However, in the

absence of otherconsiderations, the lowerimplied volatility of the 100call suggests that it is likelytobethebettervalue.

Although option tradersmay at times refer to any ofthe various interpretations ofvolatility, two of these standoutinimportance—thefuturerealized volatility and theimplied volatility. The futurerealized volatility of an

underlying contractdetermines the value ofoptions on that contract. Theimplied volatility is areflection of an option’sprice. These two numbers,value and price, are what alltraders, not just optiontraders,areconcernedwith.Ifa contract has a high valueanda lowprice,a traderwillwant to be a buyer. If acontracthasalowvalueandahighprice,a traderwillwant

to be a seller. For an optiontrader, this usually meanscomparing the expectedfuture realized volatilitywiththe implied volatility. Ifimpliedvolatility is lowwithrespecttotheexpectedfuturevolatility, a traderwill preferto buy options; if impliedvolatilityishigh,atraderwillprefer to sell options. Ofcourse, future volatility is anunknown, so a trader willlook at historical and, if

available, forecast volatilityto help in making anintelligent guess about thefuture. In the final analysis,though, it is the futurerealized volatility thatdeterminesanoption’svalue.

A commonly usedanalogy to help new tradersbetter understand the role ofvolatility is to think ofvolatility as being similar tothe weather. Suppose that a

trader living in Chicago getsup on a July morning andmust decide what clothes towear that day. Will heconsider putting on a heavywintercoat?This isprobablynot a logical choice becauseheknowsthathistoricallyitisnot sufficiently cold inChicago in July to warrantwearing a winter coat. Next,hemight turnon theradioortelevision to listen to theweather forecast. The

forecaster is predicting clearskies with very warmtemperatures close to 90°F(32°C). Based on thisinformation, the trader hasdecided that he will wear ashort-sleeve shirt and doesnot need a sweater or jacket.And he certainly won’t needanumbrella.However,justtobe sure, he decides to lookout the window to see whatthe people passing in thestreet are wearing. To his

surprise, everyone iswearinga coat and carrying anumbrella. Through theirchoiceofclothing,thepeopleoutsideare implyingdifferentweather than the forecast.Given the conflictinginformation, what clothesshould the trader wear? Hemustmakesomedecision,butwhom should he believe, theweather forecaster or thepeople in the street? Therecan be no certain answer

because the trader will notknowthefutureweatheruntiltheendoftheday.Muchwilldepend on the trader’sknowledge of localconditions.Perhapsthetraderlives in an area far removedfrom where the weatherforecasteris located.Thenhemust give added weight tolocalconditions.

The decision on whatclothes to wear, like every

trading decision, depends ona great many factors. Notonly must the decision bemadeonthebasisof thebestavailableinformation,butthedecision must also be madewith consideration for thepossibility of error.What arethe benefits of being right?Whataretheconsequencesofbeingwrong?Ifa traderfailsto take an umbrella and itrains, this may be of littleconsequence if the bus picks

him up right outside hisresidence and drops him offright outside his place ofwork.Ontheotherhand,ifhemust walk several blocks inthe rain, he might becomesickandhavetomissseveraldaysofwork.Thechoicesarenevereasy,andonecanonlyhope to make the decisionthat will turn out best in thelongrun.

Changing our

assumptions about volatilitycan often have a dramaticeffect on the value of anoption.Figure6-10showstheprices,theoreticalvalues,andimpliedvolatilitiesforseveralgold options on July 31,2012. Figure 6-11 focusesspecifically on how thesevalueschangeaswe increasevolatility from 14 to 18percent. Looking for themoment at call values,although all the options

increase in value, the 1600call, theat-the-moneyoption,increases the most, risingfrom41.65to51.60,atotalof9.95. At the same time, the1800 call shows the greatestincrease in percent terms. Itsvalue more than triples from0.78 to 3.05, a total increaseof 291 percent. These areimportantprinciples towhichwe will return later but thatareworthstatingnow:

1.Intotalpoints,achange in volatilitywill have a greatereffect on an at-the-money option thanonanequivalentin-the-money or out-of-the-moneyoption.2. In percentterms, a change involatility will havea greater effect on

an out-of-the-money option thanonanequivalentin-the-money or at-the-moneyoption.

Figure6-10Goldeight-week(40tradingdays)historicalvolatility.

Figure6-11

These same principlesapplytoputsaswellascalls.The 1600 put increases themost in total points, risingfrom29.26to39.21,atotalof9.95. The 1400 put increasesthe most in percent terms,from 0.13 to 0.89, or 585percent.

No matter how onemeasures change, in-the-moneyoptions tend tobe theleast sensitive to changes in

volatility.Asanoptionmovesdeeply into the money, itbecomes more sensitive tochanges in the underlyingprice and less sensitive tochangesinvolatility.Becauseit is often volatilitycharacteristics that investorsand traders are looking forwhen theygo intoanoptionsmarket,itshouldnotcomeasa surprise that most of thetrading volume in optionmarketsisconcentratedinat-

the-money and out-of-the-money options, the optionsthat are most sensitive tochangesinvolatility.

In Figures 6-12 and 6-13,wecan see that the sameprinciples apply to longer-term options. The at-the-moneyoptions(theDecember1600 call and put) changemost in total points,whereasthe out-of-the-money options(theDecember 1800 call and

1400 put) change most inpercent terms. As we wouldexpect, the December optionvalues are greater than theOctober option values withthe same exercise price. Butlook at the magnitude of thechanges as we changevolatility. For the sameexerciseprice,intotalpoints,the December (long-term)options always change morethan theOctober (short-term)options. This leads to a third

principle of optionevaluation:

3. A change involatility will haveagreatereffectonalong-term optionthan an equivalentshort-termoption.

Figure6-12

Figure6-13

The reader may havenoticed several interestingpoints in the foregoingfigures. First, althoughimplied volatilities may varyacross exercise prices, callsand puts with the sameexercisepriceandthatexpireat the same time have verysimilar implied volatilities.Second, when we changevolatility, calls andputswiththe same exercise price and

time to expiration change byapproximately the sameamount.Thesecharacteristicsare theresultofan importantrelationship10 between callsandputsat thesameexerciseprice, a relationship that wewillexamineinmoredetailinChapter15.

Finally, we might askhow much the volatility ofgold can change over aneight-week period? Is a 4

percentage point change arealpossibility? In fact, fromFigure 6-14, the eight-weekhistoricalvolatilityforthe3½yearsleadinguptoJuly2012,wecanseethatsuchchangesarenotatalluncommon.

Figure6-14Goldeight-week(40tradingdays)historicalvolatility.

Given its importance, itis not surprising that seriousoption traders spend aconsiderable amount of timethinking about volatility.From the historical, forecast,and implied volatility, atrader must try to make anintelligent decision aboutfuturevolatility.Fromthis,hewill try to choose optionstrategies that will beprofitablewhenheisrightbut

that will not result in aserious loss when he iswrong. Because of thedifficulty in predictingvolatility, a trader mustalwayslookforstrategiesthatwillleavethegreatestmarginfor error. No trader willsurvive very long pursuingstrategies based on a futurevolatility estimate of 20percent if such a strategyresults in a significant losswhen volatility actually turns

out to be 18 or 22 percent.Given theshifts thatoccur involatility, a 2 percentagepointmarginforerrormaybenomarginforerroratall.

We have not yetconcluded our discussion ofvolatility. But beforecontinuing,itwillbeusefultolookatoptioncharacteristics,trading strategies, and riskconsiderations. We will thenbe in a better position to

examine volatility in greaterdetail.

1Thepinballmaze,orquincunx(sometimesalsocalledaGaltonboard),picturedintheseexamplesisoftenusedtodemonstratebasicprobabilitytheory.Examplesofaquincunxinactioncanbefoundatthefollowingwebsites:

http://www.teacherlink.org/content/math/interactive/flash/quincunx/quincunx.htmlhttp://www.mathsisfun.com/data/quincunx.htmlhttp://www.jcu.edu/math/isep/Quincunx/Quincunx.html

2Thereaderwhoisfamiliarwiththemeanandstandarddeviationandwhowouldliketocheckthearithmeticwillfindthattheactualmeanandstandarddeviationare7.49and3.02.Forsimplicity,wehaveroundedtheseto7.50and3.00.3Asmarketsaroundtheworldbecome

moreintegrated,andwiththeadventofelectronictrading,itmaybecomemoredifficulttodetermineexactlywhatfractionofayearonedayrepresents.Dependingonthecontractandexchange,insomecasesitmaybesensibletolookatpriceseveryday,365daysperyear.4Apricechangegreaterthantwostandarddeviationswilloccurabout1timein20.Becausethereareapproximately20tradingdaysinamonth,asanadditionalbenchmark,mosttradersexpecttoseeadailytwostandarddeviationoccurrenceaboutonceamonth.5Alternativemethodsofestimatingvolatilityhavealsobeenproposedwhen

tradingiscontinuousorwhenthereisnowell-defineddailysettlementprice.See,forexample,MichaelParkinson,“TheExtremeValueMethodofEstimatingtheVarianceoftheRateofReturn,”JournalofBusiness53(1):61–64,1980;MarkB.GarmanandMichaelJ.Klass,“OntheEstimationofSecurityPriceVolatilitiesfromHistoricalData,”JournalofBusiness53(1):67–78,1980;andStanBeckers,“VarianceofSecurityPriceReturnsBasedonHigh,Low,andClosingPrices,”JournalofBusiness56(1):97–112,1983.6ThismethodofquotingEurocurrencycontractsisusedsothatmovesinEurocurrencycontractswilltendtomimicmovesinbondprices.Ifinterest

ratesrise,bothbondpricesandEurocurrencyfutureswillfall;ifinterestratesfall,bothbondpricesandEurocurrencyfutureswillrise.7Itwillbeusefulforanoptiontradertobecomefamiliarwiththecharacteristicsoftheexponentialfunction[exorexp(x)]anditsinverse,thelogarithmicfunction[ln(x)].Thesecanbefoundinanyalgebraorfinancetext.8Inordertoturnpricechangesintocontinuouslycompoundedreturns,volatilityismostoftencalculatedusinglogarithmicpricechanges—thenaturallogarithmofthecurrentpricedividedbythepreviousprice.Inmostcases,thereislittlepracticaldifferencebetweenthepercentpricechangesand

logarithmicpricechanges.9Forexchange-tradedcontracts,volatilitycalculationsusingdailyintervalstypicallyincludeonlybusinessdaysbecausethesearetheonlydaysonwhichpricescanactuallychange.Iftherearefivetradingdaysperweek,a50-dayvolatilitycoversaperiodofapproximately10weeks.10Somereadersmayalreadybefamiliarwiththisrelationship—put-callparity.

RiskMeasurementI

Everytraderwhoentersthemarketplace must balancetwo opposing considerations—reward and risk. A trader

hopes that his analysis ofmarket conditions is correctand that this will lead toprofitable trading strategies.But no sensible trader canafford to ignore thepossibility of error. If he iswrong andmarket conditionschange in a way thatadverselyaffectshisposition,howbadlymightthetraderbehurt? A trader who fails toconsider the risks associatedwithhisposition iscertain to

have a short and unhappycareer.

A trader who purchasesstock or a futures contract isconcerned almost exclusivelywith the direction in whichthe market moves. If thetraderhasa longposition,heis at risk from a decliningmarket; if he has a shortposition, he is at risk from arising market. Unfortunately,theriskswithwhichanoption

trader must deal are not sosimple. A wide variety offorces can affect an option’svalue. If a trader uses atheoretical pricing model toevaluate options, any of theinputs into the model canrepresentariskbecausethereis always a chance that theinputs have been estimatedincorrectly.Eveniftheinputsare correct under currentmarket conditions,over time,conditions may change in a

waythatwilladverselyaffectthe value of his optionposition.Becauseofthemanyforces affecting an option’svalue, prices can change inways that may surprise evenexperienced traders. Becausedecisionsoftenmustbemadequickly, and sometimeswithouttheaidofacomputer,much of an option trader’seducation focuses onunderstanding the risksassociated with an option

position and how changingmarket conditions are likelyto change the value of theposition.

Let’s begin bysummarizing some basic riskcharacteristics of options, asshown in Figure 7-1. Thegeneral effect on optionvalues of changes in theunderlying price, volatility,and time to expiration arewelldefinedregardlessofthe

typeofoption.But theeffectofchanginginterestratesmayvary depending on theunderlying contract andsettlementprocedure.

Figure7-1Effectofchangingmarketconditionsonoptionvalues.

A change in interestratescanaffectoptionsintwoways.First,itmaychangetheforward price of theunderlying contract. Second,it may change the presentvalue of the option. In stockoptionmarkets,risinginterestrateswillincreasetheforwardprice, causing call values toriseandputvaluestofall.Atthesametime,higherinterestrates will reduce the presentvalue of both calls and puts.

Put values clearly will fallbecausebothresultshave theeffectof reducingputvalues.For calls, though, the resultshave opposing effects. Thehigher forward price willcause the call to increase invalue, but the higher interestrate will reduce the presentvalueofthecall.Becausetheprice of a stock is alwaysgreater than the price of anoption, the increase in theforward price will always

haveagreatereffect than thereduced present value.Consequently,calloptionsonstocks will rise in value asinterest rates rise and fall asinterest rates fall.Putoptionson stocks will do just theopposite, falling in value asinterestratesriseandrisinginvalueasinterestratesfall.

The value of a stockoption will also depend onwhetheratraderhasalongor

short stock position. If atrader’s option position alsoincludes a short stockposition, he is effectivelyreducing the interest rate bythe borrowing costs requiredtosellthestockshort(seethesection “Short Sales” inChapter 2). This will reducethe forward price, therebylowering the value of callsand raising thevalueofputs.As a consequence, the traderwho iscarryinga short stock

positionoughttobewillingtosell calls at a lower price orbuy puts at a higher price. Ifthe tradereither sellscallsorbuys puts, he will hedge bypurchasing stock, which willoffsethisshortstockposition.

The fact that optionvaluesdependonwhetherthetraderhedgeswithlongstockor short stock presents acomplication that mosttraderswouldprefertoavoid.

Thisleadstoausefulruleforstockoptiontraders:

Whenever possible atrader should avoid a shortstockposition.Asacorollary,manyactive

option traders prefer to carrysome long stock as part oftheir position. Then, if thetrader must sell stock tohedge a position, he will beable to sell the stock longrather than short. The trader

neednotworryaboutusingadifferent interest ratebecauseany long stock transaction isalwayssubjecttothelong,orordinary, interest rate. Norwill he have to worry aboutany regulatory restrictionsontheshortsaleofstock.

Although stock optionsare always assumed to besubject to stock-typesettlement, with immediatecash payment for the option,

the settlement procedure foroptions on futures contractsmay vary depending on theexchange. In the UnitedStates, optionson futures aresubject to stock-typesettlement, while outside theUnited States, options onfutures are usually subject tofutures-typesettlement.Inthelattercase,nomoneychangeshandswhen either the optionor the underlying futurescontract is traded.

Consequently, interest ratesbecome irrelevant—neitherthe forward price nor thepresent value is affected.Options on futures that aresubject to futures-typesettlement are thereforeinsensitive to changes ininterest rates. If, however,optionsonfuturesaresubjectto stock-type settlement,increasing interest rates willleave the forward priceunchangedbutwillreducethe

option’s present value. As aresult, both call and putvalues will decline. Theeffect, however, is usuallysmallbecausethevalueoftheoption, unless it is verydeeplyinthemoney,issmallrelative to the value of theunderlying contract. Futuresoptions are therefore muchless sensitive to changes ininterest rates than options onstocks.

We also might considerthe case of foreign-currencyoptions.1Herethesituationismore complex because thevalueoftheoptionisaffectedby two interest rates—adomestic rate and a foreignrate. Going back to theforward pricing relationshipsinChapter 2, where S is thespot exchange rate, we cansee that theforwardprice fora foreigncurrencywill fall if

we increase the foreign rate(the denominator becomeslarger) and rise if we reducethe foreign rate (thedenominator becomessmaller)

Thismeansthatcallvalueswill fall and put values willriseasweincreasetheforeignrate.

Wecanalsoseethat theforward price for a currencywill rise if we increase thedomestic rate (the numeratorbecomeslarger)andfallifwereduce the domestic rate (thenumerator becomes smaller).But for options that aresubject to stock-typesettlement, an increase in thedomesticratewillalsoreducethe present value of theoption.Aswithstockoptions,the increase in the forward

price will tend to dominate.Therefore,aswe increase thedomesticrate,callvalueswillrise and put values will fall.The effects of changinginterest rates are summarizedinFigures7-2and7-3.

Figure7-2Effectofchanginginterestratesonoptionvalues.

Figure7-3Effectofchangingdividendsonstockoptionvalues.

If we are evaluatingoptionsonstockandthestockisexpectedtopayadividendover the life of the option, a

change in the dividend willalso affect the value of theoptionbecause itwillchangetheforwardpriceofthestock.Increasing the dividend willreduce the forward price,causingcallvaluestofallandput values to rise. Reducingthedividendwillincreasetheforward price, causing callvalues to rise and put valuestofall.

Even if we are familiar

with the general effects ofchanging market conditionson option values, we stillneed to determine themagnitude of the risk. Ifmarket conditions change,will the change in optionvalues be large or small,representingeitheramajororminor risk, or something inbetween? Fortunately, inaddition to the theoreticalvalue, pricing modelsgenerate a variety of other

numbers that enable us todetermine both the directionandmagnitudeofthechange.These numbers, knownvariously as the Greeks(because they are commonlyabbreviated with Greekletters), theriskmeasures, or(for the mathematicallyinclined) the partialderivatives, will not answerall our questions concerningchanging market conditions,but they are an important

startingpointinanalyzingtherisks associated with bothsimple and complex optionpositions.

TheDelta

Thedelta (Δ) isameasureof an option’s risk withrespect to the direction ofmovement in the underlyingcontract. A positive deltaindicates a desire for upward

movement; a negative deltaindicates a desire fordownward movement. Thedelta has several differentinterpretations, any of whichmay be useful to a traderdepending on the types ofstrategiesbeingexecuted.

RateofChangeAt expiration, an option is

worth exactly its intrinsic

value. Prior to expiration,however,thetheoreticalvalueof an option is a curve thatwill approach intrinsic valueas the option goes verydeeplyintothemoneyorveryfaroutof themoney.This isshown in Figure 7-4. As theunderlying price rises, theslopeofthegraphapproaches+1; as the underlying pricefalls, the slope of the graphapproacheszero.Thedeltaofthe call at any given

underlying price is the slopeof the graph—the rate ofchange in the option’s valuewith respect to movement intheunderlyingcontract.

Figure7-4Theoreticalvalueofacall.

Assuming that all othermarket conditions remainunchanged, a call option cannevergainorlosevaluemorequickly than the underlyingcontract, nor can it move inthe opposite direction of theunderlyingmarket. The deltaof a callmust therefore haveanupperboundof1.00ifthecall is very deeply in themoneyanda lowerboundof0ifthecallisveryfaroutof

the money. Most calls willhave deltas somewherebetween0and1.00,changingvalue more slowly thanchanges in the price of theunderlying contract. A callwith a delta of 0.25 willchangeitsvalueat25percentof the rate of change in theprice of the underlyingcontract. If the underlyingrises (falls) 1.00, the optioncan be expected to rise (fall)0.25. A call with a delta of

0.75 will change its value at75 percent of the rate ofchange in the price of theunderlying contract. If theunderlying rises (falls) 0.60,theoptioncanbeexpectedtogain (lose) 0.45 in value. Acallwithadeltacloseto0.50willriseorfallinvalueatjustabout half the rate of changeinthepriceof theunderlyingcontract.

Puts have characteristics

similartocallsexceptthatputvalues move in the oppositedirection of the underlyingmarket.InFigure7-5,wecansee thatwhen the underlyingprice rises, puts lose value;when the underlying pricefalls,putsgainvalue.Forthisreason, puts always havenegativedeltas, ranging from0 for far out-of-the-moneyputs to–1.00 for deeply in-the-money puts.Aswith calldeltas,putdeltasmeasurethe

rate of change in the put’svaluewithrespecttoachangeinthepriceoftheunderlying,but the negative signindicates that thechangewillbeintheoppositedirectionoftheunderlyingcontract.Aputwith a delta of –0.10 willchangeitsvalueat10percentof the rate of change in theprice of the underlyingcontract, but in the oppositedirection. If the underlyingmoves up (down) 0.50, the

put can be expected to lose(gain) 0.05 in value. A putwith a delta of –0.50 willchange its value atapproximatelyhalftherateofthe underlying, but in theoppositedirection.

Figure7-5Theoreticalvalueofaput.

An option position isoften combined with aposition in the underlyingcontract. To determine thetotal risk of a combinedposition, we will need toassign a delta value to theunderlying contract.Logically, a position in theunderlying contract will gainor lose value at exactly therate of change in theunderlying price. Therefore,

regardless of whether theunderlying is stock, a futurescontract, or some otherinstrument, the underlyingcontractalwayshasadeltaof1.00.

Although delta valuesrangefrom0to1.00forcallsandfrom0 to–1.00forputs,it has become commonpractice among many optiontraderstoexpressdeltavaluesas a whole number by

droppingthedecimalpoint,aconvention that we willfollowinthistext.2Usingthisformat,thedeltaofacallwillfall within the range of 0 to100, and the delta of a putwithintherangeof–100to0.An underlying contract willalwayshaveadeltaof100.

HedgeRatioIn Chapter 5, we

introduced the concept of ariskless, or neutral, hedge, aposition that, within a smallprice range,will neither gainnor losevalueas thepriceofthe underlying contractmoves up or down. We candetermine the proper numberof underlying contracts tooption contracts required forsuchahedgebydividing100(the delta of the underlyingcontract) by the option’sdelta.Foracalloptionwitha

deltaof50, theproperhedgeratio is 100/50, or 2/1. Forevery two options purchased(sold), we need to sell (buy)one underlying contract toestablish a neutral hedge. Acalloptionwithadeltaof40requiresthesale(purchase)oftwo underlying contracts forevery five options purchased(sold)because100/40=5/2.

The hedge ratiointerpretation also applies to

puts, except that when webuyputs,weneed tobuy theunderlying contract, andwhenwesellputs,weneedtosell the underlying contract.Aputwithadeltaof–75willrequire thepurchase(sale)ofthreeunderlyingcontractsforeach four puts purchased(sold)because100/–75=4/–3.

A position is neutrallyhedged, or delta neutral, if

the totalof all thedeltas thatmake up the position add upto0.Ifwebuytwocallswithadeltaof50eachandselloneunderlying contract, the totaldeltapositionis

Ifwe sell four putswith adelta of –75 each and sellthree underlying contracts,

thetotaldeltapositionis

Both positions are deltaneutral.3

A position that is deltaneutral has no particularpreference for either upwardor downward movement inthe price of the underlyingcontract. Although a trader

may take whatever deltaposition he feels isappropriate, either bullish(delta positive) or bearish(delta negative), we will seeinChapter8thatatraderwhois trying to capture thetheoreticalvalueofanoptionmuststartwithandmaintainadelta-neutralpositionovertheentirelifeofanoption.

TheoreticalorEquivalentunderlyingPositionMany option traders come

to the option market aftertrading in the underlyingcontract. Futures optiontraders often start theircareers by trading futures;stock option traders oftenstart by trading stock. If atrader has become

accustomed to evaluating hisriskintermsofthenumberofunderlying contracts boughtor sold (either futurescontracts or shares of stock),hecanusethedeltatoequatethe directional risk of anoption position with apositionofsimilarsizeintheunderlyingmarket.

Because an underlyingcontractalwayshasadeltaof100, in terms of directional

risk, each 100 deltas in anoption position istheoretically equivalent toone underlying contract. Atrader who owns an optionwithadeltaof50 is long,orcontrols, approximately halfof an underlying contract. Ifheowns10suchcontracts,heis long 500 deltas or, inequivalent terms, fiveunderlying contracts. If theunderlying is a futurescontract, the trader is

theoretically long five suchcontracts.If theunderlyingisastockcontractconsistingof100 shares of stock, he istheoretically long 500 sharesof stock. The trader has asimilar theoretical position ifhe sells 20 puts with a deltaof–25eachbecause–20×–25=+500.

It is important toemphasize the theoreticalaspect of the delta

interpretationasanequivalenttoanunderlyingposition.Anoption is not simply asurrogate for an underlyingposition. An actualunderlying position is almostexclusively sensitive todirectional moves in theunderlyingmarket.Anoptionposition, while sensitive todirectional moves, is alsosensitive to other changes inmarket conditions.Anoptiontrader who looks only at his

delta position may beignoring other factors thatcould have a far greaterimpact on his position. Thedeltarepresentsanequivalentunderlying position onlyunder very narrowly definedmarketconditions.

Which interpretation—rate of change in thetheoretical value, the hedgeratio, or the equivalentunderlying position—should

atraderuse?Thatdependsonhow the trader intends tousethedelta.A traderwhohasadeltapositionof+500knowsthat he has a position that issimilar to being long fiveunderlying contracts (theequivalent-underlying-position interpretation). If heis a disciplined theoreticaltrader striving to maintain adelta-neutral position, hemust sell five underlyingcontracts (the hedge-ratio

interpretation).Andfinally,ifheisbullishandmaintainshiscurrent delta position of+500, the value of hisposition will change atapproximately five times, or500 percent, of the rate ofchange in the price of theunderlying contract (the rate-of-change interpretation). Ifthe price of the underlyingcontract rises by 2.00, thetrader’s position should gainapproximately 10.00. If the

price of the underlyingcontract falls by 1.25, thetrader’s position should loseapproximately 6.25.Mathematically, all theseinterpretations are the same.A trader will choose a deltainterpretation that isconsistent with his approachtotrading.

Probability

There is one otherinterpretationofthedeltathatis perhaps of less practicaluse, but is still worthmentioning. If we ignore thesignof thedelta (positiveforcalls, negative for puts), thedelta is approximately equalto the probability that theoption will finish in themoney.Acallwithadeltaof25oraputwithadeltaof–25has approximately a 25percentchanceoffinishingin

themoney.Acallwithadeltaof75oraputwithadeltaof–75 has approximately a 75percentchanceoffinishinginthe money. As an option’sdeltamovescloser to100,or–100 for puts, the optionbecomes more and morelikely to finish in themoney.As the deltamoves closer to0, the option becomes lessandlesslikelytofinishinthemoney. This also explainswhy at-the-money options

tend to have deltas close to50. If we assume that pricechanges are random, there ishalfachance that themarketwillrise(theoptiongoesintothemoney)andhalfachancethat the market will fall (theoption goes out of themoney).4

Of course, the delta isonlyanapproximationof theprobability because interestconsiderations and, in the

case of stock options,dividends may distort thisinterpretation. Moreover,mostoptionstrategiesdependnotonlyonwhetheranoptionfinishesinthemoneybutalsobyhowmuch.Ifatradersellsanoptionwithadeltaof10inthebelief that theoptionwillexpire worthless nine timesout of 10, hemay indeed becorrect. But, if on the tenthtime he loses an amountgreater than the total

premium he took in the ninetimes the option expiredworthless,thetradewillresultinanegativeexpectedreturn.Totradeoptionsintelligently,weneed toconsidernotonlyhow often a strategywins orloses but also how much itwins or loses. Everyexperienced trader is willingtoacceptseveralsmall lossesif he can occasionally offsetthese with one big win thatmore than offsets the losses.

In the same way, noexperienced trader will wanttopursueastrategythatleadsto multiple small profits butoccasionally results in adisastrousloss.5

TheGamma

Figure 7-6 shows call andput delta values using thewhole-number format. Eventhoughdeltasrangefrom0to

100 for calls and from –100to 0 for puts, the graphs arenot straight lines. As theunderlyingpricerisesorfalls,the slope of the graphchanges, approaching 0 atboth extremes. If this werenot true, the delta values ofcalls could fall below 0 orrise above 100, and the deltavalues of puts could fallbelow –100 or rise above 0.The slope appears to begreatest when the underlying

price is close to the option’sexerciseprice.

Figure7-6Deltavalues.

The gamma (Γ),sometimes referred to as theoption’scurvature, istherateof change in the delta as theunderlying price changes.The gamma is usuallyexpressed in deltas gained orlost per one-point change intheunderlying,withthedeltaincreasing by the amount ofthe gamma when theunderlying rises and fallingby the amountof thegamma

when the underlying falls. Ifan option has a gammaof 5,foreachpointrise(fall)intheprice of the underlying, theoption will gain (lose) 5deltas.6 If the option initiallyhas a delta of 25 and theunderlyingmoves up (down)one full point, the new deltaof theoptionwillbe30(20).If the underlying moves up(down)anotherpoint,thenewdeltawillbe35(15).7

FromFigure7-6,wecansee that the delta graphs ofboth calls and puts haveessentially the same shapeand that the graphs alwayshave a positive slope. Thissuggests that calls and putswith the same exercise priceand time to expiration havethe same gamma values andthat these values are alwayspositive. This may seemstrange to a new traderwho,becauseof thedelta, tends to

associate positive numberswith calls and negativenumbers with puts. Butregardless ofwhetherwe areworking with calls or puts,wealwaysadd thegammatotheolddeltaastheunderlyingprice rises and subtract thegamma from the old delta asthe underlying price falls.Whenatraderislongoptions,whethercallsorputs,hehasalonggammaposition.

For example, considerboth an at-the-money callwith a delta of 50 and an at-the-moneyputwithadeltaof–50. How will the deltachange as the underlyingprice changes if both optionshave gamma values of 5? Iftheunderlyingpricerisesonefullpoint,weaddthegammaof5 to thecalldeltaof50toget the new delta of 55. Toget the new put delta if theunderlying contract rises one

point,wealsoaddthegammaof5totheputdeltaof–50togetthenewdeltaof–45.Thisis intuitively logical—as theunderlyingprice rises,at-the-money calls move into themoneyandat-the-moneyputsmoveoutofthemoney.Iftheunderlying contract falls onefull point, in both cases wesubtractthegamma,resultinginacalldeltaof50–5=45andaputdeltaof–50–5=–55. Now the call is moving

outof themoneyand theputismovingintothemoney.

Because all optionsindividually have positivegammavalues,wecancreateapositivegammapositionbybuyingoptions,eithercallsorputs, and a negative gammaposition by selling options.For a complex positionconsisting of many differentoptions, we use the sameinterpretation of the gamma

as we do for individualoptions,addingthegammatotheolddeltaastheunderlyingcontract rises and subtractingthe gamma as the marketfalls. A positive gammaposition will gain deltas asthe market rises (we areadding a positive number)and losedeltas as themarketfalls (we are subtracting apositive number).A negativegamma position will behavein just the opposite way,

losing deltas as the marketrises (we are adding anegativenumber)andgainingdeltasas themarketfalls(weare subtracting a negativenumber). Moreover, the rateofchangeinthedeltawillbedeterminedby thesizeof thegammaposition.Newtradersare often advised to avoidlarge gamma positions,particularly negative ones,because of the speed withwhich the directional risk, as

reflected by the delta, canchange.

While the delta is ameasure of how an option’svalue will change if theunderlyingpricechanges,itisimportanttorememberthatitrepresents an instantaneousmeasure. It is only valid forvery small price changes. Ifthe underlying makes asizablemove,anyestimateofthe option’s new value using

a constant delta will becomelessandlessreliable.Wecan,however, improve thisestimate if we also take intoconsiderationthegamma.

Suppose that at priceS1a call has a theoretical valueC,adeltaΔ,andagammaΓ.If thepriceof theunderlyingchanges from S1 to S2, whatshould be the new value ofthe option? One approachmight be to simply multiply

the change in price, S2 – S1,by thedeltaandadd it to theoriginalvalueC

C+[Δ×(S1–S2)]

But this assumes that thedelta is constant, which it isnot. As the underlying pricemovesfromS1toS2,thedeltaoftheoptionisalsochanging.When the underlying pricereaches S2, the new delta of

theoptionwillbe

Δ+(S1–S2)×Γ

Whichdeltashouldweusefor our calculation, theoriginal delta (Δ) or the newdelta [Δ + (S1 – S2) × Γ]?Rather than use either ofthese delta values, we mightlogically use the averagedelta over the price rangeS1–S2

Averagedelta=[Δ+Δ+(S1–S2)×Γ]/2=Δ+(S1–S2)×

Γ/2

This is not a precisesolution because the gammaalso changes as theunderlyingpricechanges,butitwill yield a better estimatethan using a constant delta.Using the average delta, thenew value of the optionshouldbeapproximately8

C+(S1–S2)×[Δ+(S1–S2)×Γ/2]=C+[(S1–S2)×Δ]+

[(S1–S2)2×Γ/2]

This approach appliesequally well to puts, as longas we remember that a putwillhaveanegativedelta.

For example, supposethatatanunderlyingpriceof97.50, a call option has atheoretical value of 3.65, adelta of 40, and a gamma of

2.5.Iftheunderlyingcontractrises to 101.50, what shouldbetheoption’snewvalue?

At the new underlyingprice of 101.50, the delta oftheoptionis

40+4×2.5=50

The average delta as theunderlying price rises from97.50to101.50is

(40+50)/2=45

Using the average delta,the new option value isapproximately

3.65+(4.00×0.45)=5.45

TheTheta

An option’s value ismadeupofintrinsicvalueandtimevalue. As time passes, thetime-value portion graduallydisappearsuntil,atexpiration,

theoptionisworthexactlyitsintrinsic value. This can beseeninFigures7-7and7-8.

Figure7-7Theoreticalvalueofacallastimepasses.

Figure7-8Theoreticalvalueofaputastimepasses.

The theta (Θ), or timedecay, istherateatwhichanoption loses value as timepasses, assuming that allother market conditionsremain unchanged. It isusually expressed as valuelostperoneday’spassageoftime. An option with a thetaof0.05willlose0.05invalueforeachday thatpasseswithno movement in theunderlying contract. If its

theoretical value today is4.00, one day later itwill beworth3.95.Twodays later itwillbeworth3.90.

Almost all options losevalueastimepasses.Forthisreason, it is common toexpressthethetaasanegativenumber,aconventionthatwewill follow in this text. Anoption with a theta of –0.05will lose 0.05 for each daythatpasseswithnochangesin

anyothermarketconditions.Wewill look at theta in

greater detail in Chapter 9.For now, there is oneimportant characteristic oftheta that is worthmentioning: if an option isexactly at themoney as timepasses,thethetaoftheoptionincreases.With threemonthsremaining to expiration, anat-the-money option mayhave a theta of –0.03.

However,withthreeweekstoexpiration,thesameoption,ifit is still at the money, mayhave a theta of –0.06. Andwith threedays toexpiration,the option may have a thetaof –0.16. The theta becomesincreasingly large asexpirationapproaches.

Is iteverpossibleforanoptiontohaveapositivethetasuch that if nothing changes,theoptionwillbeworthmore

tomorrowthanit is today?Infact, thiscanhappenbecauseof the depressing effect ofinterest rates. Consider a 60callonanunderlyingcontractthat is currently trading at100. How much might thiscallbeworthifweknowthatat expiration the underlyingcontract will still be at 100?Atexpiration, theoptionwillbe worth 40, its intrinsicvalue.However, if theoptionis subject to stock-type

settlement, today it will onlybeworththepresentvalueof40, perhaps 39. If theunderlying price remains at100,astimepasses,thevalueof the option must rise from39(itsvaluetoday) to40(itsintrinsic value at expiration).The option in effect hasnegative time value andtherefore a positive theta. Itwillbeworthslightlymoreaseach day passes. This isshowninFigure7-9.

Figure7-9Ifanoptionhasnegativetimevalue,itsthetawillbepositive;astimepasses,thevalueoftheoptionwillrisetowardintrinsicvalue.

Instances of negativetimevalueand,consequently,positive theta are relativelyrare. At a minimum, theoption must be subject tostock-typesettlement, itmustbedeeplyinthemoney,anditmust also be European withno possibility of earlyexercise. If the option wereAmerican, everyone wouldexercise it today in order toearn interest on the intrinsic

value. We will discuss thissituation in greater detailwhenwetakeacloserlookatearly exercise of Americanoptions.

TheVega

Just as option values aresensitive to changes in theunderlying price (delta) andtothepassageoftime(theta),they are also sensitive to

changes in volatility. This isshowninFigures7-10and7-11.Althoughthetermsdelta,gamma,andthetaareusedbyall option traders, there is noone generally accepted termfor the sensitivity of anoption’stheoreticalvaluetoachangeinvolatility.Themostcommonly used term in thetrading community is vega,and this is the term thatwillbeusedinthistext.Butthisisby no means universal.

Because vega is not a Greekletter, a common alternativein academic literature, whereGreek lettersarepreferred, iskappa(K).9

Figure7-10Theoreticalvalueofacallwithchangingvolatility.

Figure7-11Theoreticalvalueofaputwithchangingvolatility.

Thevegaofanoptionisusually expressed as thechange in theoretical valueforeachonepercentagepointchange in volatility. Becauseall options gain value withrising volatility, the vega forbothcallsandputsispositive.If an option has a vega of0.15, for each percentagepoint increase (decrease) involatility,theoptionwillgain(lose) 0.15 in theoretical

value. If the option has atheoretical value of 3.25 at avolatility of 20 percent, thenitwillhaveatheoreticalvalueof 3.40 at a volatility of 21percent and a theoreticalvalueof3.10atavolatilityof19percent.

TheRho

The sensitivity of anoption’stheoreticalvaluetoa

change in interest rates isgiven by its rho (P), usuallyexpressed as the change intheoreticalvalueforeachonepercentage point change ininterestrates.Unliketheothersensitivities, one cannotgeneralize about the rhobecause its characteristicsdepend on the type ofunderlyinginstrumentandthesettlement procedure for theoptions. The general effectshave already been

summarized in Figure 7-2.Note that foreign-currencyoptions that require deliveryof the currency rather thandelivery of a futures contractareaffectedbybothdomesticand foreign interest rates.Hence,suchoptionshavetwointerest-ratesensitivities,rho1(the domestic interest-ratesensitivity) and rho2 (theforeign interest-ratesensitivity). The latter is

sometimes denoted by theGreekletterphi(Φ).

If both the underlyingcontract and the options aresubject to futures-typesettlement, the rhomustbe0because no cash flow resultsfrom either a trade in theunderlyingcontractoratradein the options.When optionson futures are subject tostock-typesettlement,therhoassociatedwithbothcallsand

puts is negative.An increasein interest rateswill decreasethe value of such optionsbecause it raises the cost ofcarrying the options. In thecase of stock options, callswill have positive rho values(an increase in interest rateswill make calls a moredesirable alternative tobuying the stock) and putswillhavenegativerhovalues(an increase in interest rateswill make puts a less

desirablealternativetosellingthestock).

Although changes ininterest rates can affect anoption’stheoreticalvalue,theinterest rate is usually theleast important of the inputsintoapricingmodel.Forthisreason, the rho is usuallyconsidered less critical thanthe delta, gamma, theta, orvega. Indeed, few individualtraders worry about the rho.

However, a firm or traderwho has a very large optionposition should at least beawareoftheinterest-rateriskassociated with the position.As with any risk, if itbecomes too large, itmaybenecessary to take steps toreducetherisk.Becauseofitsrelatively minor importance,in most examples, we willdisregardtherhoinanalyzingoption strategies andmanagingrisk.

We know that the deltaof an underlying contract isalways 100, but what is thegamma, theta, vega, and rhoof an underlying contract?The gamma is the rate ofchange in the delta withrespect to movement in theunderlying contract. But thedelta of an underlyingcontract is always 100regardless of price changes.Therefore, the gamma mustbe 0.An underlying contract

does not decay, so its thetamust also be 0. Nor is theunderlyingcontractsubjecttovolatility considerations, soits vega must be 0. Andfinally, changes in interestrates do not affect the valueof an underlying contract, sotherhomustalsobe0.10Theonly risk measure weassociate with an underlyingcontract is the delta;everything else is 0. The

signsoftheriskmeasuresforan underlying contract, forcalls, and for puts aresummarizedinFigure7-12.

Figure7-12

InterpretingtheRiskMeasures

If a trader has a positionconsisting of only a smallnumber of options, it isprobably not necessary to doadetailedriskanalysis.Inalllikelihood, the trader alreadyhas a fairly clear picture ofthe potential risks and

rewards associated with theposition. However, if theposition becomes morecomplex, with options atdifferentexpirationdatesovera wide range of exerciseprices, it may not beimmediately apparent whatrisks the trader has taken on.A good starting point inanalyzing the risk of apositionistoconsidertheriskmeasures associated with theposition.

Figure 7-13 shows atheoretical evaluation for ahypothetical seriesofoptionson stock, where theunderlying contract is 100shares. Figure 7-14 showsseveral different positionswith the total delta, gamma,theta, vega, and rho for eachposition.Wewillassumethateachpositionwas initiatedatthequotedprices.

Figure7-13

Figure7-14

First, note that all riskmeasures are additive. Todetermine the total riskmeasure for a position, wemultiplyeachriskmeasurebythe number of contracts(using a plus sign for apurchaseandaminussignfora sale), and add everythingup.

Let’sconsidertheriskofPosition 1 in Figure 7-14.Before doing this, however,

we might ask a morefundamental question: whymight someone take such aposition in the first place?Like every trader, an optiontrader wants to make tradesthatresultinaprofit.Tohavethe best chance of achievingthisgoal,anoptiontraderwilltry to create positionswith apositive theoretical edge,either buying options atprices less than theoreticalvalueand/orsellingoptionsat

pricesgreaterthantheoreticalvalue. Although this is not aguarantee that the positionwillshowaprofit,bycreatinga positive theoretical edge,the trader, like a casino, hasthe laws of probabilityworking in his favor.Therefore, a trader shouldfirst consider whether aposition has a positivetheoreticaledge.

InPosition1,wesold10

June 95 calls at a price of8.55,butthetheoreticalvalueof the options was 8.33, sothe sale created a theoreticaledgeof0.22peroption.Whataboutthetheoreticaledgeforthe trade in the underlying?Fromanoptiontrader’spointofview, the theoreticalvalueof an underlying contract issimply the price at which itwastraded.Consequently,thetheoretical edge for anyunderlying trade is always 0.

The position has a totaltheoreticaledgeof+2.20.

The total delta ofPosition 1 is –10. Althoughthis indicates a very slightpreference for downwardmovement, for practicalpurposes, almost all traderswould consider the positiondeltaneutral.

The total gamma of thepositionis–28.Weknowthata positive or negative delta

indicates a desire for upwardor downward movement inthe price of the underlyingcontract, but what does apositive or negative gammaindicate? Consider what willhappentoourdeltapositionifthe underlying stock starts torise. Just as with anindividual option, for eachpoint increase, we add thegammatotheolddeltatogetthe new delta. But we areadding a negative number (–

28).Ifthestockrisesonefullpointto100.50,thedeltawillbe

–10+(–28)=–38

If the stock rises anotherpointto101.50,thedeltawillbe

–38+(–28)=–66

As the market rises, thedelta becomes a larger

negative number. Because anegative delta indicates adesire for downwardmovement, the more themarket rises, the more wewouldlikeittodecline.

Now consider what willhappentoourdeltapositionifthe underlying stock starts tofall. For each point decline,we subtract the gamma fromthedelta.Ifthestockfallsonepoint to 98.50, the newdelta

willbe

–10–(–28)=+18

Ifthestockfallsanotherpoint to 97.50, the delta willbe

+18–(–28)=+46

As the market falls, thedelta becomes a largerpositive number. For thesame reasonwe do not want

thestockpricetorise(wearecreating a larger negativedelta in a rising market), wealso do not want the stockpricetofall(wearecreatingalarger positive delta in afallingmarket). Ifwe do notwant the market to rise andwedonotwantthemarkettofall, there is only onefavorableoutcomeremaining:we must want the market tosit still. In fact, a negativegamma position is a good

indication that a trader eitherwants the underlying marketto sit still ormoveonlyveryslowly. A positive gammapositionindicatesadesireforverylargeandswiftmovesintheunderlyingmarket.

Whereas delta is ameasure of directional risk,gammacanbethoughtofasameasure of magnitude risk.Dowewantmovesofsmallermagnitude (a negative

gamma) or larger magnitude(a positive gamma)?Alternatively, gamma alsocanbethoughtofasthespeedatwhichwewant themarketto move. Do we want theunderlying price to moveslowly(anegativegamma)orquickly (a positive gamma)?Taken together, thedeltaandgamma tell us somethingaboutthedirectionandspeedthat will either help or hurtourposition.InPosition1,we

want a slow (negativegamma) downward (negativedelta)moveintheunderlyingprice. The worst situationwould be a swift upwardmove. Thenwewould be onthe wrong side of both thedirection (delta) and speed(gamma)ofthemarket.

How will we feel aboutour position if the stockremainscloseto99.50?Fromthe negative gamma, we

knowthatwewantthemarketto remain relatively quiet. Ifthe market does what wewant it to do, we ought toexpectourpositiontoshowaprofit.Where will this profitcome from? The profit willcomefromthethetaof+0.34.Foreachdaythatpasseswithno movement in theunderlyingprice, thepositionshould show a profit ofapproximately 0.34. Thisunderscores an important

principle of option riskanalysis:gammaandthetaarealmost always of oppositesign.11Apositivegammawillbeaccompaniedbyanegativetheta, and vice versa.Moreover, themagnitudes ofthe risks will tend tocorrelate.Alargegammawillbe accompanied by a largetheta,butofoppositesign.Asmall gamma will beaccompaniedbyasmalltheta.

Anoption tradercannothaveit both ways. Either marketmovement will help theposition (positive gamma) orthe passage of time (positivetheta)will—butnotboth.

ThevegaofPosition1is–1.70.This indicatesadesirefor declining volatility. Foreach point decline involatility, the value of ourposition, which was initially+2.20, will increase by 1.70;

for each point increase, thevalue will fall by 1.70. Thisseems to correspond to ourgamma risk. If we have anegativegamma,wewantthemarket to remain relativelyquiet. Isn’t this the same assaying we want lowervolatility? Most traders,however, make an importantdistinction between thegammaandvega.Thegammais a measure of whether wewanthigherorlowerrealized

volatility (whether we wantthe underlying contract to bemorevolatileorlessvolatile).The vega is a measure ofwhether we want higher orlower implied volatility.Although thevolatilityof theunderlying contract andchanges in implied volatilityare often correlated, this isnot always the case. In somecases,theunderlyingcontractcan become more volatilewhile implied volatility is

falling. In other cases, theunderlying contract canbecome less volatile whileimplied volatility is rising.WewilllookattheconditionsthatcancausethisinChapter11,wherewelookatsomeofthe common volatilityspreads.

Suppose that we raisethevolatilityof25percent inFigure7-13 to a volatility of26 percent. What should be

our theoretical profit now?Weknow that for each pointincreaseinvolatility,weneedtoaddthevega(–1.70)totheold value (+2.20) to get thenew value. Our theoreticalprofitat26percentwillbe

+2.20+(–1.70)=+0.50

If we raise the volatilityanother percentage point to27 percent, our theoreticaledgeturnsnegative

+0.50+(–1.70)=–1.20

We can see that theposition has a breakevenvolatilityofapproximately

25(%)+(–2.20/–1.70)(%)=25(%)+1.29(%)=27.29(%)

Ofcourse,amorecommonname for the breakevenvolatilityisimpliedvolatility.Although traders mostcommonly associate implied

volatility with individualoptions,wecanalsoapplytheconcept to more complexpositions. The impliedvolatility of a position is thevolatilitythatmustoccuroverthe life of a position suchthat, in theory, the positionwill just break even.We canmake a rough estimate of aposition’s implied volatilityby dividing the totaltheoretical edge by the totalvega and adding this number

to the volatility used toevaluatetheposition.

ThelastriskmeasureforPosition1istherhoof–1.55.For each percentage pointdeclineintheinterestrate,theposition will show anadditional profit of 1.55. Foreach percentage pointincrease in the interest rate,the position profit will bereducedby1.55.Itshouldnotcomeasasurprisethatrhois

negative because the longstock positionwill inevitablydominate the cash flow,resulting in a debit. If theinterest rate falls, itwill costless tocarry thisdebit. If theinterest rate rises, itwill costmore.

The risks and rewardsassociated with each type ofriskmeasure are summarizedin Figure 7-15. The readershouldtakeafewmomentsto

look over the riskcharacteristics of the otherpositions in Figure 7-14.What combination of marketconditions (e.g., changes inunderlying price, time,impliedvolatility,andinterestrate) will most help eachposition? What combinationwillmosthurteachposition?

Figure7-15

The alert reader mayhave noticed something oddabout Position 2: it has anegative theoretical edge.This is not a misprint. Itindicates that if the inputsinto themodelarecorrect, inthelongrun,thestrategywilllose money. Of course, notrader will intentionally puton such a position, but in amarket where conditions areconstantly changing, a

position that initially seemedsensible may under newconditions represent a losingstrategy.When this occurs, atraderwillmake every efforttocloseout theposition.Thelonger the trader holds theposition, themore likely it isthatitwillresultinaloss.12

Onefinalobservationfortheprospective trader:all thenumbers we have discussedin this chapter—the

theoretical value, delta,gamma, theta, vega, and rho—areconstantlychanging,sothe profitability and risksassociated with differentstrategies are constantlychanging. The importance ofanalyzing risk cannot beoveremphasized.Mosttraderswho fail at option trading doso because they fail to fullyanalyze and understand risk.But there is another type oftrader, one who attempts to

analyze every possible risk.Whenthishappens,thetraderfinds it difficult tomake anytrading decisions at all; he isstricken with paralysisthrough analysis. A traderwhoissoconcernedwithriskthat he is afraid to make atradecannotprofit,nomatterhow well he understandsoptions.Whenatraderentersthe marketplace, he haschosen to take on some risk.The delta, gamma, theta,

vega, and rho enable him toidentify risk; they do noteliminaterisk.Theintelligenttrader uses these numbers tohelpdecidebeforehandwhichrisks are acceptable andwhichrisksarenot.

1Wearereferringheretooptionsontheactualforeigncurrencyratherthanoptionsonforeign-currencyfutures.Inthelattercase,thecharacteristicsarethesameasforanyotherfuturesoption.2ThisconventionoriginatedintheU.S.stockoptionmarket,whereitbecamecommonforstockoptiontraderstoequateonedeltawithoneshareofstock.Becausetheunderlyingcontractconsistedof100shares,tradersassignedadeltaof100totheunderlyingcontract.Manyfuturesoptiontradersalsoexpressthedeltausingthiswhole-numberformat.3Itiscustomarytoindicatethepurchaseofacontractorcontractswith

aplussign(alongcontractposition)andthesaleofacontractorcontractswithanegativesign(ashortcontractposition).4Becauseoptionvaluesarebasedontheforwardpriceoftheunderlyingcontract,itisactuallytheat-the-for-wardoptionthattendstohaveadeltaclosestto50.Thisisonereasonwhyoptionsthatareseeminglyoutofthemoneycanhavedeltasgreaterthan50.Withastockat100,oneyeartoexpiration,andaninterestrateof10percent,theforwardpriceforthestockis110.Undertheseconditions,the110callwillhaveadeltacloseto50,whilethe105callwillhaveadeltagreaterthan50.

5Infact,thedeltaisonlyanapproximationoftheprobabilitythatanoptionwillfinishinthemoney.WewillseelaterthattheBlack-Scholesmodelgeneratesanumberthatmorepreciselyreflectsthisprobability.6Infact,thedeltaisonlyanapproximationoftheprobabilitythatanoptionwillfinishinthemoney.WewillseelaterthattheBlack-Scholesmodelgeneratesanumberthatmorepreciselyreflectsthisprobability.7Forsimplicity,weassumeherethatthegammaisconstant.Inreality,thegamma,likeallriskmeasures,willchangeasmarketconditionschange.8Whenusingthedeltatoestimatethe

changeinanoption’svalue,weneedtorememberthatitisreallyapercentvalue,oravaluebetween0and1.00.9Traderstendtopreferthetermvegabecauseitstartswithavandisthereforeaconvenientreminderthatitisassociatedwithvolatility.VegaissometimesabbreviatedwiththeGreekletternu(ν)becauseinwrittenformitissimilartoav.10Atradermightarguethatifinterestratesriseorfall,itmaychangetheforwardprice,whichcan,inturn,affectoptionvalues.But,fromanoptiontrader’spointofview,thevalueofanunderlyingcontractisnotdirectlyaffectedbychangesininterestrates.

11Interestconsiderationsmayoccasionallyresultinapositionwithagammaandthetaofthesamesign.However,insuchacase,themagnitudesofthenumbersarelikelytobeverysmall.12Intheory,atraderwillnevercreateapositionwithanegativetheoreticaledge,atleastasaninitialtrade.However,onceapositionhasbeenestablished,inlightofalargeroverallposition,atraderwillsometimesintentionallyexecuteatradewithanegativetheoreticaledge.Atradermightbewillingtogiveupasmallamountoftheoreticalprofitinordertomaketheremainingpotentialprofitmoresecure.This,ofcourse,isthe

wholeobjectivebehindhedging.

DynamicHedging

From our discussion thusfar, it ought to be obviouswhy serious option tradersuse theoretical pricing

models.First,amodeltellsussomething about an option’svalue. We can compare thisvalue with the price of theoptioninthemarketplaceandfrom this choose anappropriate strategy. Second,once we have taken aposition, the model helps usquantify many of the risksthatoptiontradingentails.Byunderstanding theserisks,wewill be better prepared tominimize our losses when

market conditions moveagainst us andmaximize ourprofits when marketconditionsmoveinourfavor.

In discussing theperformance of a theoreticalpricingmodel, it is importantto remember that all modelsareprobabilitybased.Evenifwe assume that we have alltherightinputsintothemodeland that the model itself iscorrect, there is noguarantee

thatwewillshowaprofitonany one trade. More oftenthan not, the actual resultswill deviate, sometimessignificantly, from what ispredicted by the theoreticalpricingmodel.Itisonlyovermany trades that the resultswill even out so that, onaverage, we achieve a resultclose to thatpredictedby thetheoreticalpricingmodel.

However, option-pricing

theoryalsosuggeststhatforasingle option trade there is amethod by which we canreduce the variations inoutcome so that the actualresults will more closelyapproximatewhatispredictedby the theoretical pricingmodel.Bytreatingthelifeofan option as a series of bets,ratherthanonebet,themodelcanbeusedtoreplicatelong-termprobabilitytheory.

Consider the followingsituation:

Stock price =$97.70Time to Juneexpiration =10weeksInterest rate =6.00percent

Suppose that we are usingatheoreticalpricingmodeltoevaluate June options on this

stock.Wealreadyhave threeinputs into the model—underlying price, time toexpiration, and the interestrate—but we still need threeadditional inputs—exerciseprice, type, and volatility.Given that we can choosefrom among the availableexercise prices and that wecan also choose the type ofoption(eithercallorput),westill lack the oneunobservable input—

volatility. In theory, wewouldliketoknowthefuturerealized volatility of theunderlying stock over thenext 10 weeks. Clearly, wecan never know the future,butlet’simaginethatwehaveacrystalball thatcanpredictthefuture.Whenwelookintoour crystal ball, we see thatthevolatilityofthestockoverthe next 10 weeks will be37.62percent.

TheJune100call,beingveryclosetoatthemoney,islikelytobeactivelytraded,solet’s focus on that option.Feeding our inputs into theBlack-Scholesmodel,wefindthat the June 100 call has atheoretical value of 5.89.When we check its price inthemarketplace,we find thatit is being offered at 5.00.How canwe profit from thisdiscrepancy?

Clearly, our first movewill be to purchase the June100 call because it isunderpriced by 0.89.Canwenow walk away from theposition and come back atexpiration to collect ourmoney? In our previousdiscussion of theoreticalpricingmodels,wenotedthatthe purchase or sale of atheoreticallymispricedoptionrequires us to establish aneutral hedge by taking an

opposing position in theunderlying contract. Whenthis is done correctly, forsmall changes in thepriceofthe underlying contract, theincrease or decrease in thevalue of the option positionwill exactly offset thedecrease or increase in thevalue of the opposingposition in the underlyingcontract. Such a hedge isunbiased, or neutral, withrespect to directional moves

intheunderlyingcontract.In order to establish the

appropriate riskless hedge,we need to determine thedelta of the June 100 call.Using our theoretical pricingmodel,wefindthattheoptionhas a delta of 50. For eachcall we purchase, we mustsell 0.50, or one-half, of anunderlying contract. Becauseit is usually not possible tobuy or sell fractional

underlying contracts, let’sassumethatwebuy100June100 calls and sell 50underlying contracts.1 Wenowhavethefollowingdelta-neutralposition:

Suppose that one week

laterthepriceofthestockhasmoved up to 99.50. At thispoint, we can feed the newmarket conditions into ourtheoreticalpricingmodel:

Stock price =99.50Interest rate =6.00percentTime to Juneexpiration = 9weeksVolatility =

37.62percent

Notethatwehavemadenochange in the interest rate orvolatility. Theoretical pricingmodels typically assume thatthese two inputs remainconstant over the life of theoption.2 Based on the newinputs, we can calculate thenew delta for the June 100call,inthiscase54.

Our delta position is now+400.Wecanthinkofthisasthe end of one bet, withanotherbetabouttobegin.

Whenever we begin anew bet, we are required toreturn to a delta-neutralposition. In our example, it

will be necessary to reduceour position by 400 deltas.There are a number of waysto do this, but to keep ourpresentcalculationsassimpleas possible and to remainconsistentwiththetheoreticalpricingmodel,wewillmakethe necessary trades in theunderlying contract becauseanunderlyingcontractalwayshas a delta of 100. We canreturn to delta neutral byselling 4 underlying

contracts.Ourpositionisnow

We are again delta neutralandabouttobeginanewbet.As before, our new betdependsonlyonthevolatilityof the underlying contract,notitsdirection.

The extra fourunderlying contracts that wesold were an adjustment toour position. In optiontrading, adjustments aretradesthataremadeprimarilyto ensure that a positionremains delta neutral. In ourcase,thesaleofthefourextracontractshasnoeffectonourtheoretical edge because,from an option trader’s pointof view, an underlyingcontract has no theoretical

value. The trade is madesolely for the purpose ofadjustingourhedgetoremaindeltaneutral.

In Chapter 17, we willlook at the use of options toprotectapreexistingposition.Such protective strategiesusually employ a statichedge, whereby opposingmarket positions are taken indifferent contracts, with theentire position being carried

to a fixed maturity date. Tocapture an option’smispricing, the theoreticalpricing model requires us toemploy a dynamic hedgingstrategy. We mustperiodically reevaluate theposition to determine thedeltaof thepositionandthenbuy or sell an appropriatenumber of underlyingcontracts to return to deltaneutral. This procedure mustbe followed over the entire

lifeoftheoption.Because volatility is

assumed to compoundcontinuously, theoreticalpricing models assume thatadjustments are also madecontinuously and that thehedge is being adjusted atevery moment in time. Suchcontinuous adjustments arenotpossibleintherealworldbecause a trader can onlytradeatdiscrete intervals.By

makingadjustmentsatregularintervals, we are conformingas closely as possible to theprinciples of the theoreticalpricingmodel.

The entire dynamichedging process for ourhedge, with adjustmentsmade at weekly intervals, isshown in Figure 8-1. At theendofeachinterval,thedeltaof the June 100 call wasrecalculated from the time

remaining to expiration, thecurrent price of theunderlying contract, aninterest rate of 6.00 percent,and a volatility of 37.65percent.Notethatwedidnotchange the volatility, eventhough other marketconditions may havechanged. Volatility, likeinterest rates, is assumed tobe constant over the life oftheoption.3

Figure8-1

What will we do withour position at the end of 10weeks when the optionsexpire?Atthat time,weplantocloseoutthepositionby

1.Lettinganyout-of-the-moneyoptions expireworthless2. Selling any in-the-money optionsat parity (intrinsic

value) or,equivalently,exercisingthemandoffsetting themagainst theunderlyingcontract3.Liquidatinganyoutstandingunderlyingcontracts at themarketprice

Let’s go through thisprocedure step by step and

seewhatthecompleteresultsofourhedgeare.

OriginalHedge

At June expiration (week10), with the underlyingcontract at 103.85, we canclose out the June 100 callsbyeithersellingthemat3.85or exercising the calls andselling the underlyingcontract. Either method will

result in a credit of 3.85 toour account. Because weoriginally paid 5.00 for eachoption, we will show a lossonouroptionpositionof

100×(3.85–5.00)=100×–1.15=–115.00

As part of our originalhedge, we also sold 50underlyingcontractsat97.70.At expiration, in order toclose out the position, we

were required to buy themback at 103.85, for a loss of6.15 per contract. Our totallossontheunderlyingtradeistherefore

50×(97.70–103.85)=50×–6.15=–307.50

Adding this to our optionloss, the total loss on theoriginalhedgeis

–115.00–307.50=–422.50

This certainly does notappear to have beensuccessful. We expected tomakemoneyon theposition,yet it appears thatwehave asizableloss.

AdjustmentsFortunately, the original

hedge was not our onlytransaction. In order toremain delta neutral over the

10-weeklifeoftheoption,wewere forced to buy and sellunderlying contracts. At theendofweek1,wewerelong400 deltas, so we wererequired to sell fourunderlyingcontractsat99.50.At the end of week 2, wewereshort1,900deltas,sowewere required to buy 19underlyingcontractsat92.75,andsooneachweekuntiltheend of week 10. Atexpiration, with the

underlyingcontractat103.85,we bought in the 22underlying contracts that wewereshortattheendofweek9.

In this example, eachtime the underlying pricerose, our delta positionbecame positive, sowewereforced to sell underlyingcontracts, and each time theunderlying price fell, ourdelta position became

negative, so we were forcedto buy underlying contracts.Because our adjustmentsdepended only on our deltaposition, we were forced todowhateverytraderwantstodo:buylowandsellhigh.

The resultofmakingallthe adjustments required tomaintain a delta-neutralposition was a profit of467.55.(Thereadermaywishto confirm this by adding up

the cash flow from all thetrades in the adjustmentcolumn in Figure 8-1.) Thisprofit more than offset thelosses incurred from theoriginalhedge.

InterestLostontheOptionPositionWe originally bought 100

Juneoptionsatapriceof5.00each,foratotalcashoutlayof

500.00. At the assumedinterest rate of 6.00 percent,the cost of financing theoption purchase for the 10-week (70-day) life of thepositionwas

–500.00×6%×70/365=–5.75

InterestearnedontheStockPosition

To establish our initialhedge,wesold50underlyingstock contracts at a price of97.70 each, for a total creditof 4,885.00. Over the life ofthe hedge, we were able toearn total interest in theamountof

+4,885×6%×70/365=+56.21

Interestonthe

AdjustmentsEachweekwewereforced

to buy or sell underlyingcontracts in order to remaindelta neutral. As a result,there was either a cash debitonwhichwewererequiredtopay interest or a cash crediton which we were able toearn interest.Forexample,atthe end of week 1, we wereforcedtosellfourunderlyingcontracts at a price of 99.50

each, fora totalcreditof4×99.50 = 398.00. The interestearned on this credit for theremainingnineweekswas

+398.00×6%×63/365=+4.12

Attheendofweek2,wewere forced to buy 19underlying contracts at apriceof92.75each,foratotaldebit of 19 × 92.75 =1,762.25.Theinterestcoston

this debit over the remainingeightweekswas

–1,762.25×6%×56/365=–16.22

Adding up the interest onall the adjustments,we get atotalof–5.28.

DividendsTo keep our example

relatively simple, we haveassumed that the stock paysno dividend over the life ofthe option. If the stock wereto pay a dividend, any longstock position resulting fromeither the original hedge ortheadjustmentprocesswouldreceive the dividend. Anyshortstockpositionwouldberequired to pay out thedividend. There also wouldbe an interest considerationon the amount of the

dividend, interest eitherearned or interest lost,between the date of thedividend payment andexpiration. The dividend andthe interest on the dividendwould then become part ofthetotalprofitorloss.

Whatwas the total cashflowresultingfromtheentire10-weekhedge?Thisamount,+90.24,isshowninFigure8-2. Of course, this represents

thecashflowattheendof10weeks.Toobtaintheinitialorpresent value, we need todiscount backwards over 10weeks at an interest rate of6.00 percent.This gives us afinalvalue,ortotalprofitandloss(P&L),of

Figure8-2

How does this finalvalue of 89.21 comparewithour predicted profit or loss?We purchased 100 Juneoptions at a price of 5.00each, but the options had atheoretical value of 5.89, sothetheoreticalprofitwas

100×(5.89–5.00)=+89.00

In our example, theprofitand lossweremadeup

of five components. Two ofthese were positive (theadjustments and the interestearnedon stock),while threewere negative (the originalhedge, the option carryingcosts, and interest on theadjustments). Is this alwaysthe case? Because pricemovement in the underlyingcontract is assumed to berandom, it is impossible todetermine beforehand whichcomponentswillbeprofitable

andwhichwill not. It wouldalso be possible to constructan example where theoriginal hedgewas profitableandtheadjustmentswerenot.The importantpoint is that ifa trader’s inputs are correct,in some combination, he canexpecttoshowaprofitorlossapproximately equal to thatpredicted by the theoreticalpricingmodel.

Of all the inputs,

volatility is theonlyone thatis not directly observable.Where did our volatilityfigureof 37.62percent comefrom? Obviously, it is notpossible to know the futurevolatility. Inourexample the10pricechangesinFigure8-1 do in fact represent anannualizedvolatilityof37.62percent. The completecalculations are given inAppendixB.

In the foregoingexample,weassumedthatthemarket was frictionless, thatno external factors affectedthe total profit or loss. Thisassumption is basic to manyfinancial models. In africtionless market, weassumethat

1. Traders canfreely buy or sellthe underlyingcontract without

restriction.2. Traders canborrow and lend asmuch money asdesired at oneconstant interestrate.3. Transactioncostsarezero.4.Therearenotaxconsequences.

A trader will

immediately realize thatoption markets are notfrictionless because in thereal world, each of theseassumptions is violated to agreater or lesser degree. Inour example, we wererequired to sell stock toinitiate the original hedge. Ifwedidnotownthestock,wewould need to sell short byfirst borrowing the stock andthen making delivery. Insome markets, short sales

may be difficult to executebecause of exchange orregulatory restrictions.Moreover,evenifashortsaleis possible, a trader typicallywill not receive full intereston the proceeds from theshortsale.

Turning to options onfutures, in some markets,there is a daily limit on theamount of allowable pricemovement for a futures

contract. When this limit isreached,themarketislocked,and no further trading cantake place until the price ofthefuturescontractcomesoffits limit. Clearly, in suchmarkets, the underlyingcontract cannot always befreelyboughtorsold.

Concerning interestrates, different rates apply todifferent market participants.The rate that applies to an

individual trader will not bethesameratethatappliestoalarge financial institution.Moreover, even for the sametrader, different rates canapply to differenttransactions. If a traderhas adebitbalance,itwillcosthimmoretocarrythatdebit;ifhehas a credit balance, he willnot earn as much on thatcredit.There isa spread,andperhaps a fairly large one,betweena trader’sborrowing

and lendingrate.Fortunately,theinterest-ratecomponentisusually the least importantofthe inputs into a theoreticalpricing model. Even thoughthe applicable interest ratemay vary from trader totrader, in general, it willcause only minor changes inthe total profit or loss inrelation to the profit or lossresultingfromotherinputs.

Transactioncosts,onthe

otherhand,canbeaveryrealconsideration. If these costsarehigh, thehedge inFigure8-1 might not be a viablestrategy; all the profits couldbeeatenupbybrokerageandexchange fees. Thedesirability of a strategywilldepend not only on thetrader’s initial transactioncosts but also on thesubsequent costs of makingadjustments. The adjustmentcostisafunctionofatrader’s

desiretoremaindeltaneutral.Atraderwhowantstoremaindeltaneutralateverymomentwill have to adjust moreoften, and more adjustmentsmean greater transactioncosts.

If a trader initiates ahedge but adjusts lessfrequently or does not adjustatall,howwillthisaffecttheoutcome?Becausetheoreticalevaluationofoptionsisbased

on the laws of probability, atrader who initiates atheoretically profitable hedgestillhas theoddsonhisside.Althoughhemayloseonanyoneindividualhedge,ifgivena chance to initiate the samehedgerepeatedlyatapositivetheoretical edge, on average,he should profit by theamount predicted by thetheoretical pricing model.The adjustment process issimply a way of smoothing

out the winning and losinghedges by forcing the tradertomakemorebets,alwaysatthe same favorable odds. Atrader who is disinclined toadjustisatgreaterriskofnotrealizing a profit on any onehedge.Adjustmentsdonotinthemselvesalter the expectedreturn;theysimplyreducetheshort-term effects of goodandbadluck.

Based on the foregoing

discussion, a retail customerand a professional trader arelikely to approach optiontrading in a somewhatdifferentmanner,evenifbothunderstandandusethevaluesgenerated by a theoreticalpricingmodel.Aprofessionaltrader,particularlyifheisanexchange member, hasrelatively low transactioncosts. Because adjustmentscosthimverylittleinrelationto the expected theoretical

profit from a hedge, he willbe inclined tomake frequentadjustments. In contrast, aretail customer whoestablishes the same hedgewillbeless inclinedtoadjustor will adjust less frequentlybecause any adjustmentswillreducetheprofitabilityoftheposition. A retail customerwho understands the laws ofprobability will realize thathis position has the samefavorable odds as the

professionaltrader’sposition,butheshouldalsorealizethathisposition ismore sensitiveto the effects of short-termgood and bad luck. Eventhough the retail customermay occasionally experiencelarger losses than theprofessional trader, he willalso occasionally experiencelargerprofits.Inthelongrun,on average, both should endup with approximately the

sameprofit.4Taxes may also be a

factorinevaluatinganoptionstrategy. When positions areinitiated, when they areliquidated, how the positionsoverlap, and the relationshipbetweendifferentinstruments(e.g., options, stock, futures,physical commodities, etc.)may have different taxconsequences. Suchconsequences may affect the

value of a diversifiedportfolio,andfor this reason,portfolio managers must besensitive to the taxramifications of a strategy.Because each trader hasuniquetaxconsiderationsandthis book is intended as ageneral guide to optionevaluation and strategies, wewill simply assume that eachtraderwishestomaximizehispretaxprofitsandthathewillworryabouttaxesafterward.

It may seem like afortunatecoincidencethatthetheoretical P&L in ourexample and the actual P&Lare so close. In fact, theexample in Figure 8-1 wascarefully constructed todemonstratewhythedynamichedging process is soimportant.Intherealworld,itis unlikely that the actualresults from any one hedgewill so closely match thetheoreticalresults.

Figure 8-3 illustrates ingraphic terms the dynamichedging process. Wedeterminedtheinitialdeltaoftheoption(thedottedline)attheunderlyingpriceof97.70and then took an opposingdelta position in theunderlying contract (thedashed line). For very smallmoves in the underlyingprice, the profit from oneposition offset the loss fromthe other position. As the

changes in the underlyingprice in either directionbecome greater, because ofthe option’s curvature (itsgamma), there is amismatchbetween these two positions.With a falling underlyingprice, the rate at which theoption position loses valuebegins to decline; with arising underlying price, therate at which the optionpositiongainsvaluebeginstoincrease. In Figure 8-3a, we

can see this mismatch, orunhedged amount, at anunderlyingpriceof99.50.

Figure8-3

With the underlyingprice at 99.50, we capturedthevalueofthismismatchbyadjusting the position toreturntodeltaneutral.Thisisshown in Figure 8-3b. Werecalculated the delta at thenew underlying price andtookanewopposingpositionin the underlying contract.When the underlying pricefell to92.75, therewasagaina mismatch equal to the

unhedgedamount.By rehedging the

position eachweek,wewereable to capture a series ofprofits resulting from themismatch between theoption’s changing delta andthe fixed delta of theunderlying contract. Ofcourse, while time waspassing, there were alsointerest considerations. Butmost of the option’s value

was determined by theamount earned on therehedging process. In theory,ifweignoreinterest,thesumof all these small profits (theunhedged amounts in Figure8-3) should approximatelyequalthevalueoftheoption

Optiontheoreticalvalue={·}+{·}+{·}+…+{·}+{·}+

{·}

In our example,

rehedging took place atdiscrete intervals, equivalenttomakinga finitenumberofbets, all with the samepositive theoretical edge. Ifwe want to exactly replicatetheoption’stheoreticalvalue,we need to make an infinitenumber of bets. This, intheory, can only be achievedby continuous rehedging ofthepositionateverypossiblemoment in time. If such aprocesswere possible, and if

all theassumptionsonwhichthe model is based wereaccurate, then the rehedgingprocess would indeedreplicate the exact value oftheoption.

Of course, continuousrehedging is not possible intherealworld.Norareallthemodel assumptions entirelyaccurate. Nonetheless, mosttraders have found throughexperience that using a

dynamic hedging strategy,even if only at discreteintervals, is the best way tocapture the differencebetweenanoption’spriceanditstheoreticalvalue.

Given that continuousrehedging is not possible,how often should a traderrehedge? The answer to thisquestionwilldependoneachtrader’s cost structure andrisk tolerance. We have

already noted that a trader’stransactioncostsare likely toaffect thefrequencyatwhichadjustmentsaremade.Highertransaction costs will oftenlead to less frequentadjustments. Ifwe ignore thequestion of transaction costs,there are two commonapproaches to rehedging:rehedgeatregularintervalsorrehedge whenever the deltabecomes unbalanced by apredeterminedamount.

ThepositioninFigure8-1 is an example of the firstapproach—rehedging atregular intervals. Here wemade adjustments to theposition at the end of eachweek. Of course, we mighthavemadeadjustmentsattheendofeachdayoreveneveryhour if we were willing torecalculate the deltas sofrequently. The more oftenrehedging takes place, themorelikelyitisthatthefinal

result will approximate theresults predicted by themodel. In our example, weused weekly intervals for noother reason than 10 linesseemed to fit nicely on thepage.

Most traders do notinsist on maintaining anexactlydelta-neutralposition.Withinlimits,theyarewillingto accept some directionalrisk. The more directional

risk a trader is willing toaccept, the less frequent theadjustments. And the lessfrequent the adjustments, themore likely the actual resultswill differ from the resultspredicted by the theoreticalpricing model. For example,if a trader decides that he iswillingtoacceptadirectionalrisk up to 500 deltas, norehedging would take placeafterweek1(+400deltas).Ifthetrader iswillingtoaccept

adirectional riskup to1,000deltas, no rehedging wouldtakeplaceattheendofweek1(+400deltas),week3(+800deltas), and week 8 (+1,000deltas). And if the trader iswillingtoacceptadirectionalrisk up to 1,500 deltas,5 norehedgingwouldtakeplaceatthe end of week 1 (+400deltas),week3(+800deltas),week 7 (–1,100 deltas), andweek 8 (+1,000 deltas). In

eachcase,becauseofthelessfrequentrehedging,theactualresults are more likely todiffer from the predictedresults.

NotethatafterthehedgeinFigure8-1wasinitiated,nosubsequent tradesweremadein the option market. Thetrader’sonlyconcernwastherealized volatility, or pricefluctuations,intheunderlyingmarket. These price

fluctuations determined thesize and frequency of theadjustments, and in the finalanalysis, it was theadjustments that determinedtheprofitabilityof thehedge.Wemight thinkof thehedgeas a racebetween the loss intime value of the June 100calls and the cash flowresulting from theadjustments, with thetheoretical pricing modelactingasthejudge.Underthe

assumptions of the model, ifoptions are purchased at lessthan theoretical value, theadjustmentswillwintherace;if options are purchased atmore than theoretical value,thelossintimevaluewillwinthe race. The conditions oftheracearedeterminedbytheinputs into the theoreticalpricingmodel.

We assumed in ourexample that the future

volatility was known to be37.62 percent. What will bethe outcome if volatility issomething other than 37.62percent? Suppose, forexample, that volatility turnsout to be higher than 37.62percent. Higher volatilitymeans greater pricefluctuations,resultinginmoreandlargeradjustments.Inourexample, more adjustmentsmean more profit. This isconsistent with the principle

thathighervolatilityincreasesthevalueofoptions.

Whatabouttheopposite,ifvolatility is less than37.62percent? Lower volatilitymeans smaller pricefluctuations, resulting infewer and smalleradjustments.Thiswill reducethe profit. If the volatility islow enough, the adjustmentprofitwilljustoffsettheothercomponents, so the total

profit from thehedgewillbeexactly zero. This breakevenvolatility is identical to theoption’s implied volatility attheoriginaltradeprice.UsingtheBlack-Scholesmodel,wefindthattheimpliedvolatilityoftheJune100callatapriceof 5.00 is 32.40 percent. Atthis volatility, the racebetween profits from theadjustments and the loss inthe option’s time value willend in an exact tie. Above a

volatilityof32.40percent,weexpect the hedge, includingadjustments and interest, toshow a profit; below 32.40percent,we expect thehedgetoshowaloss.

Because we needed tomakeadjustmentstorealizeaprofit,itmayseemthateveryprofitable hedge requires ustomaintain the position untilexpiration. In practice, thismay not be necessary.

Suppose that immediatelyafter we establish the hedge,the implied volatility in theoptionmarket increases from32.40 percent, the impliedvolatilityatwhichweboughtthe June 100 calls, to 37.62percent,therealizedvolatilityoftheunderlyingcontractweexpect over the life of theoption. What will happen tothepriceoftheJune100call?Its price will rise from 5.00(animpliedvolatilityof32.40

percent) to 5.89 (an impliedvolatility of 37.62 percent).Wecanthensellourcallsforan immediate profit of 0.89per option. Of course, if wewant to close out the hedge,wemustalsobuybackthe50underlying contracts that weoriginally sold. What effectwill the change in impliedvolatilityhaveonthepriceofthese contracts? Impliedvolatility is a characteristicassociated with options, not

with underlying contracts.Consequently, we expect theunderlying contract tocontinue to trade at itsoriginal price of 97.70. Bypurchasing our 50outstanding underlyingcontracts at a price of 97.70,wewill realize an immediatetotalprofitfromthehedgeof89.00, exactly the amountpredicted by the theoreticalpricing model. If we can doall this, there is no reason to

hold the position for the full10weeks.

How likely is animmediate reevaluation ofimplied volatility from 32.40to 37.62 percent? Althoughswift changes in impliedvolatility occur occasionally,more often changes occurgradually over a period oftime and are the result ofequally gradual changes inthe volatility of the

underlying contract. As thevolatility of the underlyingcontract changes, optiondemand rises and falls, andthis demand is reflected in acorresponding rise or fall inthe implied volatility. In ourexample, if the price of theunderlying contract begins tofluctuateatavolatilitygreaterthan 32.40 percent, we canexpect implied volatility torise.If impliedvolatilityeverreaches our target of 37.62

percent, we can simply sellour calls and buy ourunderlying contracts, therebyrealizing our expected profitof 89.00 without having tohold the position for the full10 weeks. But option pricesare subject to a wide varietyof market forces, not all ofthem theoretical. There is noguarantee that impliedvolatilitywill ever reevaluateupward to 37.62 percent. Inthiscase,wewillhavetohold

the position and continue toadjustforthefull10weekstorealizeourprofit.

Every trader hopes thatimplied volatility willreevaluate as quickly aspossible toward his volatilitytarget.Itnotonlyenableshimto realize his profits morequickly, but it eliminates theriskofholdingaposition foran extended period of time.Thelongerapositionisheld,

the greater the possibility oferror fromthe inputs into themodel.

Not only might impliedvolatility not reevaluatefavorably,italsomightmoveagainst us, even if the actualvolatility of the underlyingcontract moves in our favor.Suppose that after initiatingour hedge, implied volatilityimmediately falls from32.40to30.35percent.Thepriceof

the June 100 call will fallfrom 5.00 to 4.65, and wewill have an immediate lossof 100 × –0.35 = –35.00.Doesthismeanthatwemadea bad trade and should closeout the position? If thevolatility forecast of 37.65percent turns out to becorrect, the options will stillbe worth 5.89 by expiration.If we hold the position andadjust, we can eventuallyexpect a profit of 89.00

points. Realizing this, weoughttomaintainthepositionas we originally intended.Eventhoughanadversemovein implied volatility isunpleasant, it is somethingwith which all traders mustlearn to cope. Just as aspeculator can rarely hope topick the exact bottom or topat which to take a long orshort position, an optiontradercanrarelyhopetopickthe exact bottom or top in

implied volatility. He musttry to establish positionswhen market conditions arefavorable. But he must alsorealize that conditions mightbecomeevenmorefavorable.If they do, his initial trademay show a temporary loss.This is something a traderlearnstoacceptasapracticalaspectoftrading.

Let’s look at one otherdynamic hedging example,

this time in the form of anoverpriced put in the futuresoption market. Suppose thatcurrentmarket conditionsareasfollows:

Futuresprice=61.85TimetoMarchexpiration =10weeksInterest rate =8.00percent

Again,let’sassumethatweknowthetruevolatilityoftheunderlying contract over the10-weeklifeoftheoption,inthis case 21.48 percent. Inthis example, we will focuson theMarch 60 put, with atheoreticalvalueof1.46butapriceof1.70,equivalenttoanimplied volatility of 23.92percent.

Because the put isoverpriced,wewill begin by

selling 100 March 60 puts,withadeltaof–35each,andsimultaneously selling 35underlying futures contracts.We will then follow adynamic hedging procedureby recalculating theputdeltaat the end of eachweek andbuying or selling futures toremain delta neutral. Atexpiration, we will close outtheentireposition.Theentiredynamic hedging process isshowninFigure8-4.

Figure8-4

The cash flow in thisexample is slightly differentfrom that inour stockoptionexample. Although these areoptions on futures contractsand in many markets aresubject to futures-typesettlement,wewillfollowtheU.S. convention and assumethattheoptionsaresubjecttostock-type settlement,requiring immediate and fullcash payment. Futures,

however, are always subjectto futures-type settlement:thereisnoinitialcashoutlay,but a cash flow, in the formof variation, will resultwhenever the price of thefutures contract changes.When this occurs, there willbe a cash credit, on whichinterest can be earned, or acash debit, on which interestmustbepaid.

AllP&Lcomponentsfor

this example are shown inFigure 8-5. Three of thesecomponents are the same asin the stock option example:the P&L on the originalhedge, the P&L resultingfrom the delta-neutraldynamic hedging process,and the carrying cost on theoptions.However,theintereston the initial stock position,aswell as the interest on theadjustments, has beenreplacedbytheinterestonthe

variationcreditsanddebits.Figure8-5

For example, as part ofour original hedge, we sold35futurescontractsatapriceof 61.85. After week 1, thefutures price declined to60.83. As a result, wereceived a variation paymentof

35×(61.85–60.83)=35.70

Wewere able to earn8.00percentonthisamountforthe

nine weeks (63 days)remainingtoexpiration

35.70×8%×63/365=0.49

Attheendofweek1,inorder to remaindeltaneutral,wewere forced to sell sevenfutures contracts. This,together with our initial saleof 35 futures, left us short atotal of 42 futures. Afterweek2,thefuturespriceroseto 62.78. The result was a

variationdebitof

42×(60.83–62.78)=–81.90

In order to finance thisdebit for the eightweeks (56days)remainingtoexpiration,weincurredaninterestcostof

–81.90×8%×56/365=–1.01

The total interest on allvariation cash flows was –

0.67.The total cash flow of

24.33andthepresentvalueofthisamount,23.96,areshownin Figure 8-5. The predictedtheoreticalprofitwas

100×(1.70–1.46)=24.00

Inbothour stockoptionand futures option examples,wewereableusethedynamichedging process to capturethe difference between the

option’stheoreticalvalueanditsprice. Inasense,dynamichedging enabled us to taketheothersideofthetrade,butattheoption’struetheoreticalvalue.Whenwebought callsin our stock option example,we sold the same calls attheoretical value through thedynamic hedging process.When we sold puts in ourfutures option example, webought the same puts attheoretical value through the

dynamic hedging process.From this,we can deduce animportant principle of optionevaluation:

Intheory,wecanreplicateanoptionpositionthroughadynamichedgingprocess.Thecostofthisreplicationisequaltothesumofallthecashflowsresultingfromthe

dynamichedgingprocess.Thepresentvalueofthissumisequaltotheoption’stheoreticalvalue.

1Theunderlyingcontractformoststockoptionsis100sharesofstock.Theproperhedgeisthereforeequivalenttoselling5,000sharesofstock.2Whetherthisisinfactarealisticassumptionwewillleaveforalaterdiscussion.3Inpractice,asnewinformationbecomesavailable,tradersareconstantlychangingtheiropinionsaboutinterestratesandvolatility.Herewemaketheassumptionofconstantvolatilityandinterestratesinordertobeconsistentwithoptionpricingtheory.4This,ofcourse,ignorestheveryreal

advantagetheprofessionaltraderoftenhasfrombeingabletobuyatthebidpriceandsellattheaskprice.Aretailcustomercanneverhopetomatchtheprofitresultingfromthisadvantage,norshouldhetrytodoso.5Thesedeltanumberswerechosenonlytoillustratetheeffectofrehedgingbasedonapredetermineddeltarisk.Evenadirectionalriskof500deltasmightbemorethanmanytradersarewillingtoaccept.

RiskMeasurementII

Just as an option’stheoretical value is sensitiveto changes in marketconditions, the sensitivities

themselves also change asmarket conditions change.This underscores animportant aspect of optiontrading: nothing remainsconstant. Depending onmarket conditions, the sameposition can exhibit a widerange of risk characteristics.Today’s small risk canbecometomorrow’sbigrisk.

Although it isimpractical to analyze every

potential risk, intelligenttrading of options stillrequires us to consider therisk of a position under awide variety of marketconditions. Every serioustrader’s education mustinclude an understanding ofthe many different ways inwhich the risk of a positioncan change. Having someawareness of how thesensitivities change withchangingmarketconditionsis

vital if we expect tointelligentlymanage the veryreal risks that option tradingentails. In this chapter, wewilltakeacloserlookathowoption risk measures changeas market conditions changeand how this affects thecharacteristicsofaposition.

Delta

Wehavealready lookedat

the sensitivity of the delta toone possible change inmarket conditions. In Figure7-6, we saw that deltachanges as the price of theunderlying contract changesand that this change isrepresented by the option’sgamma. In addition tochanges in the underlyingprice, the delta is alsosensitive to changes involatilityandtime.

Figure 9-1 shows whathappens to thedeltaofacallas volatility changes. Asvolatility increases, the deltaof an out-of-the-money callrises and the delta of an in-the-money call falls, withboth deltas tending toward50.This is logicalbecauseina low-volatility market anout-of-the-moneycallismorelikely to remain out of themoney and therefore have adeltathatiscloserto0,while

an in-the-money call ismorelikelytoremaininthemoneyandthereforehaveadeltathatis closer to 100. In a high-volatilitymarket,wehavetheopposite effect. An out-of-the-money call has a greaterlikelihood of going into themoney; an in-the-money callhas a greater likelihood ofgoing out of the money.Consequently, the deltas ofboth options will movetoward50.

Figure9-1Calldeltavaluesasvolatilitychanges.

Notethat thedeltaofanat-the-money option tends toremainclose to50regardlessof volatility. This is true ingeneral, although changinginterestratesor,inthecaseofstock options, changingdividends may alter theforward price. Becausetheoretical pricing modelsevaluateoptionsinrelationtotheforwardprice,thedeltaofan at-the-money call may in

fact be either more or lessthan50.Eveniftheoptionisexactly at the forward (theexercise price and forwardpricearethesame),acallwillstill have a delta that isslightly greater than 50because of the lognormaldistribution used to evaluatethe option.This is evident inFigure9-1,wherethedeltaofanat-the-moneycall tends toincrease slightly as volatilityincreases.

Because an option’sdelta changes as volatilitychanges, no trader can becertainthatapositionisreallydelta neutral. The deltadepends on the volatility ofthe underlying contract, andthis is something that willoccur in the future over thelife of the option. Thevolatility we use to calculatethedeltaisaguess.Wemightguessright,butwealsomightguesswrong.Andifweguess

wrong, our delta values willbewrong.

Rather than try to guessthe future volatility, manytraders use the implied delta,the delta that results fromusing the implied volatility.Usingthisapproach,thedeltawill change as impliedvolatilitychanges,eveniftheunderlying contract remainsthe same. Consider a traderwho owns 40 call options

with an implied volatility of32 percent and acorresponding implied deltaof 25 each.Because40×25=1,000,tohedgethepositiondelta neutral, the trader willsell 10 underlying contracts.If,however,impliedvolatilityrises to 36 percent, the deltaof the options will tendtoward50.Ifthenewimplieddelta is 30, the trader’s deltaposition is now (40 × 30) –(10 × 100) = +200. The

trader’s position changedfrom neutral to bullish eventhough no other marketconditionschanged.

Because the deltadependson thevolatility,butvolatility is an unknownfactor,calculationofthedeltacanposeamajorproblemforatrader,especiallyforalargeoption position. Using theimpliedvolatility to calculatethedelta isonlyonepossible

approach.Figure 9-2 shows what

happenstocalldeltasastimepasses. Note the similaritiesto Figure 9-1. Delta valuesmove toward 50 if weincrease either time toexpiration or volatility andmove away from 50 if wereduce either of these inputs.In many situations, time andvolatility will have a similareffectonoptions.More time,

like higher volatility,increases the likelihood oflarge price changes. Lesstime, like lower volatility,reduces the likelihood oflarge price changes. If atrader cannot immediatelydetermine the effect on anoption’s value or sensitivityof changing time, he mightinstead consider the effect ofchanging volatility.Conversely, if he cannotdetermine the effect of

changing volatility, he mightconsider the effect ofchanging time. Both effectsarelikelytobesimilar.

Figure9-2Calldeltavaluesastimepasses.

The effects of volatilityandtimeonputdeltasarethesame as those on call deltas,except that put deltas tendtoward 0 and –100 asvolatility falls or time passesand toward –50 as volatilityrises. This is shown inFigures9-3and9-4.

Figure9-3Putdeltavaluesasvolatilitychanges

Figure9-4Putdeltavaluesastimepasses.

Analternativemethodofdisplaying the effects ofchanging time and volatilityon delta values is shown inFigure9-5. This is similar toFigure 7-6 except that wehave varied time andvolatility. As we lower timeor reduce volatility, deltavalues for calls move veryquickly toward either 0 forout-of-the-money options or100forin-the-moneyoptions.

Figure9-5Calldeltavaluesastimepassesorvolatilitydeclines.

Becausedeltavaluesareaffected by the passage oftime, a position that is deltaneutral today may not bedelta neutral tomorrow, evenif allothermarketconditionsremainunchanged.Ofcourse,withmanymonths remainingto expiration, the passage ofeven several days may havelittle effect on the delta. If,however, expiration isquickly approaching, the

passage of just one day,because it represents a largeportion of the option’sremaining life, can have adramaticeffectonthedelta.

As option traders havebecome more aware of theimportance of riskmanagement, they havebegun to pay closer attentiontochangesinthesensitivitiesthemselves as marketconditions change. In some

cases,theyhavealsobeguntoattach names (although notnecessarily Greek letters) tothese higher-ordersensitivities. The sensitivityof the delta to a change involatility is sometimesreferred to as the option’svanna. The sensitivity of thedeltatothepassageoftimeissometimes referred to as theoption’s delta decay or itscharm.1

Which delta values arethemostsensitive tochangesinvolatility (vanna)and time(charm)?Weknowthatdeltavalueswilltendeithertoward50asweincreasevolatilityortime, or away from 50(toward 0 or 100) as wereduce volatility or time.Logically, delta values thatare already close to 0, 50, or100 are the least likely tochange. At the same time,delta values that are

approximately midwaybetween these numbers aremostlikelytochange.ThisisborneoutbyFigures9-6and9-7, thevannaandcharmforoptions with different deltas.Note that the shapes of thegraphs are identical for callsand puts,with the vanna andcharm approximately 0aroundadeltaof50or–50.2We can also see that vannaand charm are greatest for

call delta values close to 20and 80 and put delta valuescloseto–20and–80.Optionswith these deltas will movethemostquicklytoward50ifwe raise volatility or awayfrom50ifwelowervolatilityorreducetimetoexpiration.

Figure9-6Vannaofanoption.

Figure9-7Charmofanoption.

The three vanna graphsalso show that the vannamoves in the oppositedirection of volatility, fallingas we raise volatility andrisingaswereducevolatility.The graphs of the charmexhibit similar characteristicswithrespecttothepassageoftime, falling with more timeto expiration and rising withlesstimetoexpiration.

In Figures 9-6 and 9-7,

wehaveignoredtheeffectofchanging time on the vannaand the effect of changingvolatilityonthecharm.Fromprevious discussions, wemight expect time andvolatility to have the sameeffect on both these values.However, whereas vannavalues are affected bychangesinvolatility,theyarenot significantly affected bychangesintimetoexpiration.Whereas charm values are

affectedbytimetoexpiration,they are not significantlyaffected by changes involatility.

Theta

The thetaofanoption, therate at which it decays, willvary depending not only onmarketconditionsbutalsoonwhether an option is in themoney, at the money, or out

of themoney. In Figure 9-8,wecanseethatthethetaofanoptionisgreatestwhenitisatthe money. As the optionmoves either into or out ofthemoney, its thetadeclines.Becausethethetaofanoptionisafunctionofitstimevalue,and because very deeply inthe money options and veryfar out of themoney optionshavevery little timevalue, itis logical that such optionshaveaverylowtheta.

Figure9-8Thetaofanoptionastheunderlyingpricechanges.

Note also that when allotherconditionsarethesame,an at-the-money option at ahigherunderlyingpricehasagreaterthetavaluethananat-the-money optionwith lowerunderlying price. Tounderstandwhy,considertwocalls, one with an exerciseprice of 10 and one with anexercisepriceof1,000,wherebothoptionsareatthemoneyandboth callshave the same

amount of time to expirationand the same impliedvolatility. Which option willbe worth more? Clearly, the1,000callwillbeworthmorebecauseitrepresentstherighttobuyamorevaluableasset.3Because both options are atthe money and thereforeconsist solely of time value,the theta of the 1,000 callmustbegreaterthanthethetaofthe10call.

Figure 9-9 shows thetheoreticalvalueofanin-the-money, at-the-money, andout-of-the-money option astime passes. Early in theoption’slife,therateofdecay(the slope of the theoretical-value graph) is similar foreach option. But late in theoption’s life, as expirationapproaches, the rateofdecayslows for in-the-money andout-of-the-money options,whereas it accelerates for an

at-the-money option,approaching infinity at themoment of expiration. Thesecharacteristics, which applyto both calls and puts, areshowninFigure9-10.4

Figure9-9Theoreticalvalueofanoptionastimepasses.

Figure9-10Thetaofanoptionastimepasses.

Theeffectonthethetaofchanging volatility is shownin Figure 9-11. If we ignoreinterest, with a 0 volatility,thethetaofanyoptionwillbe0. As we increase volatility,we increase the timepremium, at the same timeincreasingthetheta.

Figure9-11Thetaofanoptionasvolatilitychanges.

Note that the graph ofthe at-the-money option isessentially a straight line,with the theta being directlyproportional to the volatility.For an at-the-money option,the theta at a volatility of 20percent is exactly double thetheta at a volatility of 10percent. The same is notnecessarily true for higherexercise prices (out-of-the-money calls and in-the-

money puts) or lowerexerciseprices (in-the-moneycalls and out-of-the-moneyputs). The theta tends todecline as volatility declinesbut may become 0 wellbeforethevolatilityis0.

Figure 9-11 wasconstructed with the higherand lower exercise pricesequally far away from thecurrent underlying price.Note that the higher exercise

pricehas agreater theta thanthelowerexerciseprice,withthedifferenceincreasingwithincreasing volatility. Wetouched on the explanationforthisinChapter6.Ifacallandaputarebothequallyoutof the money, under theassumptions of a lognormaldistribution, the out-of-the-money call (the higherexercise price) will carrygreater time premium thantheout-of-the-moneyput(the

lowerexerciseprice).Ifthereis no movement in the priceoftheunderlyingcontract,theoption with more timepremium (thehigher exerciseprice)mustnecessarilydecaymore quickly than the optionwith less time premium (thelowerexerciseprice).

Ifweknow thevalueofan option today, is there anyway to estimate the option’stheta?Thereisnoconvenient

method for estimating thetheta of in-the-money andout-of-the-moneyoptions,butfor an at-the-money option,weknowthatthetaisdirectlyproportional to volatility(Figure9-11).We also knowfromChapter6 thatvolatilityis proportional to the squarerootoftime

The theta of an at-the-

money optionmust thereforebe proportional to the squareroot of time. If TVt is anoption’s theoretical value attimet (indaystoexpiration),thenthetheoreticalvalueonedaylaterTVt–1is

Thethetaistherefore

As time passes, the value

of becomesincreasingly large.Consequently, the thetaofanat-the-moneyoptionwillalsobecome increasingly large(Figure9-7).

For example, consideranat-the-moneyoptionwithatheoretical value of 2.50 and30 days remaining toexpiration.Theoption’sthetawillbeapproximately

Onedaylater,with29daysremaining to expiration, thethetawillbe

Vega

Figure9-12showsthevegaofanoptionaswechangethe

underlying price. Note thatthis figure is almost identicalto Figure 9-8. As with thetheta, the vega is greatestwhen an option is at themoney, and an at-the-moneyoptionwitha higher exerciseprice has a greater vega thananat-the-moneyoptionwithalower exercise price.Moreover, the vega of an at-the-money option isproportional to its exerciseprice.Assumingthatallother

conditions are the same, anat-the-money option with anexercise price of 100 willhaveavegathat is twicethatofanoptionwithanexercisepriceof50.Notethatthetermvanna, which previouslyreferred to the sensitivity ofdeltatoachangeinvolatility,can also refer to thesensitivity of the vega to achange in the underlyingprice.Bothinterpretationsaremathematicallyidentical.

Figure9-12Vegaofanoptionastheunderlyingpricechanges.

Figure 9-13 shows thetheoreticalvalueofanin-the-money, at-the-money, andout-of-the-money option aswe change volatility. Ofparticularnoteisthefactthatthe value of an at-the-moneyoptionisessentiallyastraightline. Because the vega is theslope of the graph, we canconclude that the vega of anat-the-money option isrelatively constant with

respect to changes involatility. Whether volatilityis 20 percent, 30 percent, orsome higher value, the vegaof an at-the-money optionwillbethesame.

Figure9-13Theoreticalvalueofanoptionasvolatilitychanges.

Theeffectonthevegaofchanging volatility is shownin Figure 9-14. While thevega of the at-the-moneyoption is relatively constant,the vega values of in-the-money and out-of-the-moneyoptions tend to rise withhigher volatility.5 This islogicalwhenwerecallthataswe raise volatility, the deltasof in-the-money and out-of-the-money options tend

toward 50, causing theoptionstoactmoreandmoreas if they are at the money.Because at-the-moneyoptionshavethegreatestvega(see Figure 9-12), we wouldexpectthevegavaluestorise.The sensitivity of vega to achange in volatility issometimes referred to aseitherthevolgaorthevomma(both terms are a contractionof volatility and gamma—either volatility gamma or

volatilitygamma).Figure9-14Vegaofanoptionas

volatilitychanges.

Figure9-15showsvolgavaluesforcallsandputswithvarying deltas. We havealready noted that an at-the-moneyoptionwithadeltaofapproximately 50 has arelatively constant vega and,consequently,avolgacloseto0. However, as an optionmoves either into the moneyoroutofthemoney,thevolgabegins to increase, reachingits maximum for calls with

deltas of approximately 10and90andputswithdeltasofapproximately –10 and –90.Additionally, as we increasetime,volgavalues for in-the-money and out-of-the-moneyoptions become moresensitive to the passage oftime, with long-term optionshaving greater volga valuesthanshort-termoptions.

Figure9-15Volga(vomma)ofanoption.

In Figure 9-16, we cansee how vega values changeastimechanges,risingasweincrease time to expirationandfallingaswereducetime.This characteristic, that long-termoptionsarealwaysmoresensitive to changes involatility than short-termoptions, was introduced inChapter 6 (see Figures 6-11and6-12).

Figure9-16Vegaofanoptionastimepasses.

The sensitivity of thevega to changes in time toexpiration, sometimesreferred to as either vegadecay or DvegaDtime, isshown in Figure 9-17. Thevega of options with deltavalues between 10 and 90tendstobethemostsensitiveto the passage of time. Thissensitivity increases as wereduce time to expiration; astime passes, the vega of

short-term options willchangemorequicklythanthevegaoflong-termoptions.

Figure9-17Vegadecayofanoption.

Gamma

The gamma measures thesensitivity of the delta to achange in the underlyingprice.Butthegammaitselfissensitivetochangesinmarketconditions.6

In Figure 9-18, we canseethatthegammaisgreatestwhen an option is at the

money. This is similar tothetaandvega,whicharealsogreatestwhen an option is atthe money, and leads to animportant principle of optiontrading: gamma, theta, andvega are greatest when anoption is at the money.Becauseofthis,at-the-moneyoptions tend to be the mostactivelytradedinmostoptionmarkets. Such options havethecharacteristicsthattradersare lookingforwhen theygo

intoanoptionmarket.Figure9-18Gammaofanoptionas

theunderlyingpricechanges.

Unlike the theta andvegaofat-the-moneyoptions,which increase at higherexerciseprices,thegammaofan at-the-money optiondeclines at higher exerciseprices. To understand why,recall that the gamma is thechange in the delta per one-point change in theunderlying price. Buttheoretical pricing modelsmeasurechangeinpercentage

terms. By this measure, aone-point price change withthe underlying at 50 (a 2percent change) is greaterthanaone-pointpricechangewith theunderlyingat100(a1 percent change). Althoughthe theta and vega of at-the-money options areproportional to their exerciseprices, the gamma isinversely proportional. Thegamma of an option with anexercise price of 50 will be

twice as large as the gammaofanoptionwithanexercisepriceof100.

Because at-the-moneyoptions have the greatestgamma, as the underlyingprice moves toward theexercise price, the gamma ofanoptionwillrise,andastheunderlyingpricemovesawayfrom the exercise price, thegamma will fall. Thesensitivityof thegammatoa

change in the underlyingprice, sometimes referred toas the speed, is shown inFigure 9-19. The speed isgreatest for out-of-the-moneyoptions with deltas close to15 for calls and –15 for putsand for in-the-money optionswith deltas close to 85 forcallsand–85forputs.Asweincreasetimetoexpirationorvolatility, the speed of anoptiondeclines;aswereducetime to expiration or

volatility,thespeedrises.Thegamma is least sensitive tochanges in the underlyingprice for at-the-moneyoptions (a delta close to 50for calls or –50 for puts) orforverydeeply in-the-moneyor very far out-of-the-moneyoptions(deltascloseto0andcloseto100forcallsor–100forputs).

Figure9-19speedofanoption.Putdeltas

Thegammawill alsobesensitivetochangesintimetoexpirationandvolatility.Thisis shown inFigure 9-20.Weknow that gamma is greatestwhen an option is at themoney and declines as theoption moves either into themoney or out of the money.Of particular importance isthefactthatthegammaofanat-the-money option rises astime passes or as we reduce

volatility and falls as weincrease volatility. To seewhy,considera100callwiththemarket at 97.50.Becausethe option is currently out ofthe money, its delta is lessthan50.Wealsoknowthatastime passes or we reducevolatility, delta values moveawayfrom50.Ifweareclosetoexpirationorinaverylow-volatilitymarket, thedeltaoftheoptionwillbewellbelow50, perhaps 25. If the

underlying market shouldthen rise 5 points to 102.50,thedeltaoftheoptionwillbegreater than 50, perhaps 75.With the underlying marketrising from 97.50 to 102.50and the delta rising from 25to 75, the approximategammashouldbe

Figure9-20Gammaofanoptionas

timepassesorvolatilitychanges.

If, however, expiration isfarinthefutureorweareinahigh-volatility market, thedeltaofthe100callwillstayclose to 50. With theunderlying market at 97.50,thedeltaoftheoptionmaybe45.Ifthemarketthenrisesto102.50,thedeltamaybeonly55. The approximate gammaisthen

The effect is just theopposite for in-the-moneyand out-of-the-moneyoptions.Thegammawill fallif we reduce volatility andriseifweincreasevolatility.7Becausegammaandthetaareclosely related, ifwewere tographthegammaofanoptionas time passes, the resultwould be very similar to

Figure9-10,with thegammainsteadof the thetaalong they-axis.

The sensitivity of thegamma to the passage oftime,sometimesreferredtoasits color, is shown in Figure9-21.Thecolorisgreatestforat-the-money calls and puts,withgammavaluesbecomingsmallerasweincreasetimetoexpiration and larger as wereducetime(henceanegative

colorvalue).Callswithdeltascloseto5or95andputswithdeltascloseto–5or–95alsohavelargecolorvalues.Here,however, an increase in timecauses gammavalues to rise,whereas the passage of timecauses gamma values to fall(a positive color). Moreover,reducing time or volatilitywill increase color values,making an option’s gammamore sensitive to changes inthe passage of time.

Increasing time or volatilitywill reduce color values,making an option’s gammaless sensitive to the passageof time. Calls with deltasclose to 15 or 85 and putswithdeltascloseto–15and–85 tend to have color valuescloseto0.Thegammavaluesof such options will berelatively insensitive to thepassageoftime.

Figure9-21Colorofanoption.

The sensitivity of anoption’s gamma to a changein volatility, sometimesreferred to as its zomma, isshown in Figure 9-22.Zomma characteristics aresimilar to colorcharacteristics.Thezommaislarge for at-the-money callsandputs,withgammavaluesbecomingsmallerasvolatilityrises and larger as volatilityfalls (a negative zomma).

Callswithdeltascloseto5or95andputswithdeltascloseto –5 or –95 also have largezomma values. Here,however, an increase involatility causes gammavaluestoriseandadeclineinvolatility causes gammavalues to fall (a positivezomma).Moreover, reducingtimeorvolatilitywillincreasethe zomma, making anoption’s gamma moresensitive to changes in

volatility. Increasing time orvolatility will reduce thezomma, making an option’sgamma less sensitive tochanges in volatility. Callswith deltas close to 15 or 85andputswithdeltascloseto–15 and –85 tend to havezommavaluescloseto0.Thegamma values of suchoptions will be relativelyinsensitive to changes involatility.

Figure9-22Zommaofanoption.

Given the fact that thegamma is greatest for at-the-money options and that thegamma of an at-the-moneyoption increases as timepasses or volatility declines,experiencedtradersknowthatat-the-moneyoptionsclosetoexpiration in a low-volatilityenvironment are among theriskiest options that one cantrade.Although thesegammaoptions initially have delta

valuescloseto50,theirdeltascanchangedramaticallywithonlysmallmovesinthepriceof the underlying contract,moving very quickly toward0or100.

Lambda(Λ)

Thedelta tellsus thepointchange in an option’s valuefor a given point change inthe price of the underlying

contract. But we might alsoask how an option’s valuechanges in percentage termsforagivenpercentagechangeintheunderlyingprice.

Consider a call optionwith a theoretical value of4.00 and a delta of 20, withthe underlying contracttrading at a price of 100. Ifthe underlying contract risesone point to 101, the newdelta of the option (ignoring

the gamma) should beapproximately 4.20.But howmuch are these changes inpercentage terms? Theunderlying changed by 1percent (1/100), whereas theoption changed by 5 percent(0.20/4.00).Theoptionhas alambda,orelasticity,of5. Inpercentage terms, it willchange at five times the rateoftheunderlyingcontract.

We can see that the

lambdaissimplytheoption’sdelta (using the decimalformat) multiplied by theratiooftheunderlyingpriceSto the option’s theoreticalvalue

Λ=Δ×(S/TV)

Inourexample,lambdais

0.20×100/4.00=5

Traders sometimes refer to

the lambda as the option’sleverage value. Althoughlambda is not a widely usedrisk measure, it may still beworth looking at some basiclambda characteristics. Theseare shown in Figures 9-23(calllambdavalues)and9-24(put lambda values).Logically, because thelambdaiscalculatedfromthedelta, calls have positivelambda values and puts havenegative lambda values. We

can see that the lambda isgreatest for out-of-the-moneyoptions—as the underlyingpricerises,calllambdavaluesdecline and put lambdavaluesrise(theytakeonlargenegative values). Lambdavalues are also sensitive tochangesintimeandvolatility.If we increase volatility,lambda values for both callsand puts fall. If we reducevolatility or as time passes,lambda values for both calls

andputsrise.Figure9-23Lambdaofacallastime

passesorvolatilitychanges.

Figure9-24Lambdaofaputastimepassesorvolatilitychanges.

A trader who wants thebiggestpossiblereturnonhisinvestment, in percentageterms, compared with anequal investment in theunderlying contract canmaximize his lambda bytrading out-of-the-moneyoptionsclose toexpiration ina low-volatility environment.Ofcourse,thisistrueonlyintheory. There may be otherconsiderations, such as the

bid-ask spread and liquidityof the option market, thatmight make a large lambdaposition impracticalcompared with a similarposition in the underlyingmarket.

It may seem that wehavegoneintounduedetailinourexaminationoftheoptionriskmeasures.Although it iscertainly true that not everyrisk is important in every

situation, experienced tradershave learned that it is almostimpossible to overemphasizethe importance of riskmanagement in optiontrading. Because options areaffectedbysomanydifferentmarketforces,unlessatraderis aware of and understandsthe many ways in whichoption values change, hecannot hope to successfullymanage the very real risksthatoptiontradingentails.

A summary of theprimary risk characteristicsdiscussed in this chapter isgiven inFigures 9-25 and 9-26.

Figure9-25Traditionalriskmeasures.

Figure9-26Nontraditionalhigher-orderriskmeasures.

1Inmathematics,the“sensitivityofasensitivity”isasecond-ordersensitivity.Thegamma,vanna,andcharmareallsecond-ordersensitivities(thesensitivityofthedeltatoachangeinunderlyingprice,volatility,andtimetoexpiration,respectively).2Thevannaisactually0fordeltavaluesslightlylargerthan50andsmallerthan–50.Thisisduetothenon-symmetricalcharacteristicofthelognormaldistribution.3Infact,thetheoreticalvalueandthetaoftwootherwiseidenticalat-the-moneyoptionsareproportionaltotheirexerciseprices.Inthisexample,the1,000callwillbeworthexactly100

timesmorethanthe10call,anditsthetawillbeexactly100timesgreater.4Thethetavaluesforin-the-moneyandout-of-the-moneyoptionsareactuallyslightlydifferent.However,thevaluesaresoclosethatinFigure9-10weuseonelinetorepresentbothoptions.5Infact,wecanseefromFigure9-14thatthevegaofanat-the-moneyoptiondeclinesveryslightlyasweraisevolatility.ThiswillbediscussedingreaterdetailinChapter18.6Becausethegammaisasecond-ordersensitivity—thesensitivityofthedeltatoachangeintheunderlyingprice—thesensitivityofgammatoachangeinmarketconditionsisathird-order

sensitivity.Foradiscussionofsomeofthehigher-ordersensitivities,seeEspenGaarderHaug,TheCompleteGuidetoOptionPricingFormulas(NewYork:McGraw-Hill,2007);EspenGaarderHaug,“KnowYourWeapon,Part1,”WilmottMagazine,May2003:49–57,alsoavailableathttp://www.wilmott.com/pdfs/050527_haug.pdfandEspenGaarderHaug,“KnowYourWeapon,Part2,”WilmottMagazine,July–August2003:50–56,alsoavailableathttp://www.nuclearphynance.com/Userpercent20Files/2552/0307_haug.pdf.7Thisisageneralrule.Sometimesanoptionthatisonlyslightlyinthemoneyoroutofthemoneywillactlikeanat-

the-moneyoption.Whetheranoption’scharacteristicswillresemblethoseofanat-the-money,in-the-money,orout-of-the-moneyoptionwilldependonavarietyoffactors,includingvolatilityandtimetoexpiration.

10

IntroductiontoSpreading

Inoptionmarkets,asinallmarkets, there are manydifferent approaches totrading.Atonetime,scalpingwasapopularstrategyamongtraders on the floors of

futures exchanges. Byobserving the activity in aparticular market, a scalperwould try to determine anequilibrium price thatreflected a balance betweenbuyers and sellers. Thescalper would then quote abid-ask spread around thisequilibrium price, attemptingtobuyatthebidpriceandsellat the offer price as often aspossiblewithouttakingeithera long or short position for

any extended period of time.The scalpermade no attemptto determine the theoreticalvalue of the contract.Althoughtheprofitfromeachtrade might be small, if atraderwasabletotradeoftenenough, he expected to showa reasonable profit. Scalping,however, requires a highlyliquid market, and optionmarketsarerarelysufficientlyliquid to support this type oftrading.

A different type oftrading strategy involvesspeculating on the directionin which the underlyingcontract will move. Thedirectional position can betakeninavarietyofways—inthecashmarket,inthefuturesmarket, or in the optionmarket. Unfortunately, evenwhen an underlying marketmoves in the expecteddirection, takingadirectionalposition in an option market

will not necessarily beprofitable. Many differentforces, including changes involatility and the passage oftime, can affect an option’sprice. If a trader’s soleconsideration is direction, heis usually better advised totake the position in theunderlyingmarket.Ifhedoesand he is right, he is assuredofmakingaprofit.

Most successful option

traders are spread traders.Because option evaluation isbased on the laws ofprobability and the laws ofprobabilitycanbeexpectedtoeven out only over longperiodsoftime,optiontradersmust oftenholdpositions forextended periods. Over shortperiods of time, while thetraderiswaitingforanoptionposition to move towardtheoreticalvalue,thepositionmay be affected by a variety

of changes in marketconditions that threaten itspotential profit. Indeed, overshortperiodsoftime,thereisno guarantee that an optionposition will react in amanner consistent with atheoretical pricing model.Spreading enables an optiontrader to take advantage oftheoretically mispricedoptions while at the sametime reducing the effects ofshort-term“badluck.”

WhatIsaSpread?

Aspread is a strategy thatinvolves taking opposingpositions in different butrelated instruments. Mostcommonly, a spread willconsistofpositionsthatmoveintheoppositedirectionwithrespect to changes in marketconditions. When marketconditions change, oneposition is likely to gain

value, while the otherposition is likely to losevalue.Ofcourse,ifthevalueschange at the same rate, thevalueofthespreadwillneverchange. A profitablespreading strategy ispredicated on the assumptionthat the values of thepositions will change atdifferentrates.

Many commonspreadingstrategiesarebased

on arbitrage relationships,buying and selling the sameor very closely relatedinstruments in differentmarkets to profit from amispricing. The cash-and-carry strategy common incommodity markets is anexample of this type ofspread. Given the currentcash price, interest rate, andstorageandinsurancecosts,acommodity trader cancalculate the value of a

forwardcontract.Iftheactualmarket price of the forwardcontract is higher than thecalculated value, the traderwill create a spread bypurchasing the commodity,sellingtheoverpricedforwardcontract, and carrying thepositiontomaturity.1

Consider a commoditytrading at a price of $700. Ifinterest rates are 6 percentannually and storage and

insurancecosts combinedare$5permonth,whatshouldbethe value of a two-monthforwardcontract?

If theactualpriceof thetwo-month forward contractis$725, the tradermightbuythe commodity for $700, selltheforwardcontractfor$725,

and carry the position tomaturity.The total cash flowin terms of debits (–) andcredits(+)willbe

The total profit resultingfromthisstrategyis

–$7–$700–$10+$725=

+$8

This is exactly the amountbywhichtheforwardcontractwas mispriced. The resultingprofit will be unaffected byfluctuations in the price ofeitherthecommodityitselforthe forward contract becauseall cash flows weredetermined at the time thestrategy was initiated.Whether thecommodityrisesto $800 or falls to $600, the

profitisstill$8.Another type of

spreading strategy involvesbuying and selling futurescontracts of differentmaturities on the sameunderlyingcommodity.Inthepreceding example, wecalculatedthevalueofatwo-month forward contract on acommodity at $717. We candoa similar calculation for afour-month forward contract.

Here, however, the cost ofborrowing will becompoundedbecausewewillneed to borrow $700 for thefirst two-month period at 6percentandthenborrow$717for the second two-monthperiod,alsoat6percent.2

The value of the four-monthforwardcontractoughttobe

Iftherearetwo-month-andfour-month exchange-tradedfutures contracts on thiscommodity,thereshouldbea$17.17 difference, or spread,betweenthepricesofthetwo

contracts. If the spreadbetween months is actually$20, a trader might buy thetwo-month contract and sellthe four-month contract. Thetrader cannot tell whethereither contract individually isoverpriced or under-priced.But he knows that at a priceof $20, the spread is $2.83tooexpensive.

Assumingthatthetraderhas accurately evaluated the

spread,howwillhemakethis$2.83 profit? One possibilityisthatthepriceofthefuturesspread will return to itsexpected value of $17.17.Having sold the spread (sellthe four-month futurescontract, buy the two-monthfutures contract), the tradercan close out the position bypurchasing the spread (buythe four-month futurescontract, sell the two-monthfuturescontract).

Ifthepriceofthespreaddoesnotreturntoitsexpectedvalue,thetradercancarrytheentire position to maturity.Suppose that the spread wasoriginally created bypurchasing the two-monthforwardcontractatapriceof$717 and selling the four-month forward contract at aprice of $737. If carried tomaturity, the cash flow fromthe entire positionwill be asfollows:

Of course, the tradercouldhaveachievedthesame

result by simply selling thefour-month forward contractand buying the commodity.However, although a tradermay have easy access to afuturesexchange,hemayfindthathisaccesstothephysicalcommoditymarket is limitedbecause such markets aretypically dominated by largecorporations. In such a case,he may find that it is bothsimpler and cheaper toexecute the spread in the

futuresmarket.Spreading strategies are

often done to reduce one ormore risks. In a cash-and-carry strategy, much of thedirectional risk is eliminatedbecausethevalueof thelongcashcontractandthevalueofthe short forward contractwilltendtomoveinoppositedirections. But a spreadingstrategy will not necessarilyeliminate all risks. In our

example,weassumedthatwewereabletoborrowmoneyata fixed rate, therebyeliminating any interest-raterisk. We also assumed thatstorage and insurance costswere fixedwhen the strategywas initiated. If we aredealing only with futurescontracts, changes in interestrates, as well as changes instorage and insurance costs,may affect the pricerelationship between futures

months. If the changes arelarge enough, a seeminglyprofitable spreading strategymay in fact becomeunprofitable.Intheprecedingexample, if interest rates andstorage costs rise after thestrategy has been initiated,the spread between the two-monthandfour-monthfuturescontractwillwiden, resultingin a smaller profit to thetraderorperhapsevenaloss.

Our examples thus farwere both intramarketcommodity spreads, with allcontract values based on thesame underlying commodity.However, if a trader canidentify a price relationshipbetween two differentcommodities or two differentfinancial instruments, hemightconsideranintermarketspread,buying inonemarketand selling in a differentmarket. As with all spreads,

the strategy is based on theassumption that there is anidentifiable relationshipbetween the prices ofdifferent contracts.When thepricespreadbetween the twocontracts appears to violatethis relationship, it representsanopportunityforthetrader.

In the fixed-incomemarkets, a common strategyinvolves buying or sellingshort-term interest-rate

instruments and taking anopposing position in long-terminterest-rateinstruments.The value of the spreaddepends on changes in theyield curve—the relationshipbetweenshort-and long-terminterestrates.

Consider two futurescontracts with the same timeto maturity, a 10-yearTreasury note future tradingatpriceof11614/32anda30-

year Treasury bond futuretrading at a price of 11827/32.3 The spread betweenthetwois

11827/32–11614/32=213/32

The prices of Treasurycontracts move in theopposite direction of interestrates. If interest rates rise,Treasury prices will fall; ifinterest rates fall, Treasury

prices will rise. If a traderbelieves that interest rateswill rise but that long-termrates will rise more quicklythan short-term rates, hemightsellthe10-year/30-yearspread.4 If he is correct, thespreadwillnarrow,perhapsatalaterdatetradingat

11510/32–1137/32=23/32

Ifthetraderoriginallysold

the spread at 213/32 and laterbuysthespreadbackat23/32,hewillshowaprofitof

213/32–23/32=10/32

Asasomewhatdifferentintermarket spread, supposethat a trader observes theprices of two commodities,Commodity A andCommodity B, over anextended period and

concludes thatCommodityBtendstotradeatapricethatisthree times greater than thatofCommodityA.Thatis,

PriceofCommodityB=3×priceofCommodityA

If the price of CommodityA is 50, the price ofCommodity B ought to be150. If the price ofCommodity A is 200, theprice ofCommodityB ought

to be 600. Although pricesmay occasionally deviatefrom this, they eventuallyseem to revert to this 3:1relationship. Given thisrelationship, what will atraderdoif thecurrentpricesofthecommoditiesare

Price ofCommodity A=120Price ofCommodity B

=390

With prices of 120 and390,CommodityB is tradingat a multiple of 3.25 timesCommodity A. Given thehistorical relationship,Commodity B seems to betrading at a price that is toohigh compared withCommodity A. EitherCommodity A ought to betrading at 360 (3 × 120) orCommodity B ought to be

tradingat130(390/3).Ifthetraderbelievesthat

thepricesare likely to returnto their 3:1 historicalrelationship, he mightpurchase three contracts ofCommodity A for 120 eachand sell one contract ofCommodity B at a price of390

If at a later date thecontractprices return to their3:1 relationship, the tradercan close out the position atnocost, leavinghimwiththeexpected profit of 30. Thisprofit will be independent ofthe actual prices of the twocommodities as long as the3:1 relationship ismaintained.

The strategy that wehave just described involves

buying and selling unequalnumbers of contracts,sometimes referred to as aratio strategy. It is commonin markets where there is aperceived relationshipbetweenproductswithsimilarcharacteristics but that tradeat different prices. In theprecious metals market, atrader might spread goldagainst silver, even thoughgold trades at a price manytimes that of silver. In the

agricultural market, a tradermight spread corn againstsoybeans, even thoughsoybean prices are alwaysgreater than corn prices. Inthe stock index market, atrader might spread theStandard and Poor’s (S&P)500 Index against the DowJones Industrial AverageIndex.Allthesespreadsdifferfrom previous strategies inthat they depend on anobserved and perhaps less

well-definedrelationshipthanthat between the cash priceand the futures price orbetween the prices ofdifferent futures months.Because the relationship isless reliable, these types ofspreads carry greateruncertainty and thereforegreaterrisk.Nonetheless, ifatrader believes that hisanalysis of a pricerelationship is accurate, thestrategy may be worth

pursuing.Thus far, all our

spreading examples haveconsisted of two sides, orlegs.Inthefirstexample,oneleg consisted of a physicalcommodity and one legconsisted of a forwardcontract. In the secondexample,thelegsconsistedoftwo different futurescontracts. In the thirdexample,thelegsconsistedof

two different commodities.But spreading strategies mayconsist ofmany legs as longas a price relationshipbetweenthedifferentlegscanbeidentified.

In energy markets, acommon spreading strategyconsists of buying or sellingcrude oil futures and takingan opposing position infutures in products that aremade from crude oil—

gasoline andheatingoil.Thevalue of this crack spreaddepends on the cost ofrefining, or cracking, crudeoil into its derivativeproducts, as well as thedemand for these productsrelative to the cost of crudeoil. If the costs of refiningriseorthedemandforrefinedproducts rises, the value ofthespreadwillwiden.Ifcostsfallordemandfalls,thevalue

ofthespreadwillnarrow.5There are a number of

ratios in which the crackspreadcanbetraded,butonecommonratioisthe3:2:1—3gallonsofcrudeoiltoyield2gallons of gasoline and 1gallonofheatingoil.Becausethe value of the refinedproducts is greater than thatofcrudeoil,atraderissaidtobuy thespreadwhenhebuysthe products and sells crude

oil.

Priceofthe3:2:1crackspread=(2×gasoline)+(1×heatingoil)–(3×crudeoil)

A traderwho believes thatthe demand for refinedproductswill fallcansell thecrack spread. A trader whobelievesthatdemandwillrisecanbuythespread.

In somemarkets, itmaybe necessary to execute each

leg of a spread separatelybecause there may be nocounterparty willing toexecute the entire spread atone time. If the spreadconsists of multiple legs andthe traderhasonlybeenabletoexecuteoneleg,hewillbeat riskuntilhecompletes thespread by executing theremaining legs. If the tradermust execute the spread oneleg at a time, he needs toconsider the risk resulting

from this piecemealexecution. Determining howbest to execute a spread isusually a matter ofexperience. It is often truethat some legs, owing to theliquidity in the respectivemarkets, will be moredifficulttoexecutethanotherlegs.Asaconsequence,mosttraders learn that it isusuallybest to execute the moredifficult leg first. If a traderdoes this, he will find that

execution risk is reducedbecause he will be able tomore easily complete thespread. If, on theotherhand,a trader executes the easierlegfirst,hemaybeleftwithanakedpositionifheisunabletoexecutetheremaininglegsin a timely manner or at areasonableprice.

Fortunately, in manymarkets, spreads are tradedall at one time as if they are

onecontract.Aquote for thespread will typically consistofonebidpriceandoneofferprice,nomatterhowcomplexthespread.Consideraspreadthat consists of buyingContract A and sellingContracts B and C with thefollowingbid-askquotes:

From the bid-ask quotesfor each of the individualcontracts, the current marketforthespreadis

Ifatraderwantstobuythespread, he can immediatelytrade all three contractsindividuallyandpayatotalof16. If he wants to sell thespread,hecandosoatapriceof 9. But a trader may takethepositionthatbecauseheistradingmultiple contracts,heoughttogetsomediscount.Amarket maker in this spreadwill often take the view thatbecausehehaslessriskwhenhe executes all contracts at

one time, he iswilling to doso at a price more favorabletothetrader.Ifthetraderasksfor a market for the entirespread,hewilloftenfindthatthe difference between thebid price and ask price isnarrower than the sumof thebid-ask prices, perhaps 11bid, 14 offer. Executing theentire spread as onetransaction will clearly bebetter than executing thespread as three individual

transactions.Even if a spread is

executed as one trade, manyexchangesrequirethatpartiestrading a spread still reportthe prices of the individualcontracts. If this is the case,what prices should bereported if a trader buys theentirespreadatapriceof14?In fact, the individual pricesreally don’t matter. Whetherthe trader pays 129 for

Contract A and sellsContractsB andC at 48 and68 (129 – 47 – 68 = 14) orpays 131 forContractA andsellsContractsBandCat48and69(131–48–69=14),the total price is still 14.Indeed, the parties coulddecideforwhateverreasontotradeContractAatapriceof200andContractsBandCatprices of 86 and 100 (200 –86 100 = 14). As far as theparties to the trade are

concerned, all thatmatters isthat the individualpricesaddup to the agreed-on spreadpriceof14.6

OptionSpreads

At the beginning of thischapter, we defined a spreadas consisting of opposingpositions in relatedinstruments. Butwhat dowemean by a position? In the

spreadexamples thus far, thepositions were based ondirectional considerations. Ifthevalueofonepositionrisesas a result of a directionalmove in the underlyingmarket, the value of theopposingpositionisexpectedto decline, even thoughultimately the price of thespread is expected toconverge to some projectedvalue. We can also createdirectional spreads in the

option market by takingopposing but unequal deltapositions in differentcontracts. As with our otherspreads, the value of such aspread will depend ondirectional movement in theunderlyingcontract.

While the prices ofoptions are affected bydirectional moves in theunderlying market, they canalso be affected by other

factors. In an option market,we might create a spread bytakingalonggammapositionin one option and a shortgammapositioninadifferentoption, or by taking a longvega position and a shortvegaposition,orevena longand short rho position. Thevalueofeachofthesespreadswill depend on factors otherthandirectionalmoves in theunderlying market. Thegamma spread will be

sensitive to the volatility ofthe underlying market. Thevega spreadwill be sensitiveto changes in impliedvolatility.Andtherhospreadwillbesensitivetochangesininterestrates.

The dynamic hedgingexamples in Chapter 8 aretypical gamma spreads. Weinitiatedthespreadsbyeitherpurchasing or selling optionsand then offsetting the

option’s delta with anopposingdeltapositionintheunderlying contract.However, although anunderlying contract has nogamma, an option does havea gamma. As a result, theentire position had either apositiveoranegativegamma.From this we demonstratedthat the value of the positiondependednotonthedirectionof movement in theunderlyingcontractbutonthe

volatility of the underlyingcontract.

Manyoptionspreadsaredynamic, requiring periodicadjustments.Butaspreadcanalsobestatic.Once initiated,the spread is carried toexpiration withoutadjustments. This is usuallydone only when the riskcharacteristics of the spreadarewelldefinedandlimited.

Perhaps in no other

market are spreadingstrategiesaswidelyemployedastheyareinoptionmarkets.There are a number ofreasonsforthis:

1. A tradermightperceive a relativemispricing betweencontracts. Just as atrader mightcalculate the valueofafuturescontractin relation to the

price of a cashcontract, an optiontrader might try toidentifythevalueofone option contractin relation toanother option.Althoughitmaynotbe possible todetermine the exactvalue of eithercontract, the tradermight be able toestimatetherelative

value of thecontracts. If pricesin the marketplacedeviate from thisrelative value, atrader will try toprofit by eitherbuying or sellingthespread.

Inmanymarkets,traders

express amispricingin termsof howmuch theprice of acontractdiffersfrom itsvalue. Inoptionmarkets,themispricing

is oftenexpressedin termsofvolatility.Considertwooptions,one thathas atheoreticalvalue of7.00 andis trading

at a priceof 8.00andanotherthat has atheoreticalvalue of6.00 andis tradingat a priceof 6.75.Whichoptionrepresents

a greatermispricing?Lookingonly attheoptionprices,the firstoptionappearstobeoverpricedby 1.00,whereasthe

secondoptionappearstobeoverpricedby only0.75. Butsupposethat thevolatilityused tocalculatethetheoretical

value is23percent.Becausebothoptionsareoverpriced,we knowthat theirimpliedvolatilitiesmust begreater

than 23percent.If theimpliedvolatilityof theoptiontrading at8.00is26percent,while theimpliedvolatilityof the

optiontrading at6.75is28percent,an optiontrader islikely toconcludethat involatilityterms, thesecondoption ismore

overpriced.

2. A trader maywant to construct aposition thatreflectsaparticularview of marketconditions. Optionscanbecombinedin

an almost infinitevariety of wayssuch that apositionwill yield a profitwhen marketconditions movefavorably. At thesame time, optionscanbecombinedinways thatwill limitloss whenconditions turnunfavorable. Welooked at some

examples of this inChapter 4. Ofcourse, even if atrader is able toconstruct a positionthat exactly reflectshis view of marketconditions, he willhave to decidewhether the pricesat which the tradescan be executedmake the positionworthwhile.

3. Spreadingstrategies help tocontrolrisk.This isparticularlyimportant forsomeone who ismaking decisionsbased on atheoretical pricingmodel. In Chapter5, we stressed thefact that allcommonly usedpricing models are

probability basedand that outcomespredicated on thelaws of probabilityare only reliable inthe longrun. In theshort run, any oneoutcome candeviate from theexpected outcome.If a traderwants tobe successful inoptions, he mustensure that he

remainsinthegamefor the long run. Ifheisunluckyintheshort run and mustleave thegame, thelong-termprobability theorydoes him no good.Spreading is theprimary method bywhich traders limitthe short-termeffects of “badluck.”

In addition to reducingthe effects of short-term badluck,spreadingstrategiescanalso help protect a traderagainst incorrectly estimatedinputs into the theoreticalpricingmodel.Supposethatatrader estimates that over thelifeofanoption,thevolatilityofanunderlyingcontractwillbe35percent.Basedon this,he determines that a certaincalloption,whichiscurrentlytradingatapriceof4.00,has

a theoreticalvalueof3.50. Ifthecallhasadeltaof25, thetrader might try to capturethismispricingbysellingfourcalls at a price of 4.00 eachand buying one underlyingcontract and dynamicallyhedging thepositionover thelife of the option. The totaltheoretical edge for thepositionis4×0.50=2.00.Ofcourse,ifthetradercanmake2.00 with a 4 × 1 spread, itmayoccurtohimthathecan

make 20.00 if he increasesthesizeofthespreadto40×10. Why stop now? Thetrader canmake 200.00 if heincreases the size to 400 ×100.

Even if the market issufficiently liquid to absorbthe increased size, is this areasonable approach totrading? Should a tradersimply find a theoreticallyprofitable strategy and do it

asmany times as possible inorder to maximize thepotential profit? At somepoint, the intelligent traderwillhavetoconsidernotonlythe potential profit of astrategy but also its risks.After all, the trader’svolatility estimate of 35percent is just that, anestimate.Whatwillhappenifvolatilityactuallyturnsouttobe some higher number,perhaps 40 percent, or 45

percent? If the calls that thetrader sold at 4.00 are worth4.50 at a volatility of 45percent, and volatilityactually turns out to be 45percent, then the hoped-forprofit of 200.00 (assuming asize of 400 × 100) will turnintoalossof200.00.

A trader must alwaysconsider the effects of anincorrect estimate and thendecide how much risk he is

willingtotake.Ifthetraderinthis example decides that hecan survive if volatility goesnohigherthan40percent(a5percentage point margin forerror), he might only bewillingtodothespread40×10.But, if there issomewayto increase his breakevenvolatility to 45 percent (a 10percentage point margin forerror), he might indeed bewillingtodothespread400×100. Option spreading

strategies enable traders toprofitunderawidevarietyofmarket conditions by givingthemanincreasedmarginforerror in estimating the inputsinto a theoretical pricingmodel.Notraderwillsurvivevery long if his livelihooddepends on estimating eachinput with 100 percentaccuracy. But if he hasconstructed an intelligentspreadingstrategythatallowsfor a large margin of error,

the experienced trader cansurvive even when hisestimates of marketconditions turn out to beincorrect.

To see how spreadingstrategies can be used toreduce risk, recall ourexampleinChapter5whereacasinoissellingaroulettebetwithanexpectedvalueof95cents for $1.00. The casinoknowsthatbasedonthelaws

of probability, it has a 5percent theoretical edge.Suppose that a customercomes into the casino andproposestobet$2,000ononenumber at the roulette table.Should the casino allow thebet?Thecasinoownerknowsthat the odds are on his sideand that he will most likelyget to keep the $2,000 bet,but there is always a chancethat the player’s numberwillcomeup.Ifitdoes,thecasino

will lose $70,000 (the$72,000 payoff less the$2,000costofthebet).Ifthecasino is backed by millionsofdollars,thelossof$70,000will not severely interferewith the casino’s continuingoperations. If, however, thecasino is only backed by$50,000, the loss of $70,000will put the casino out ofbusiness. And if the casinogoesoutofbusiness,itcannolonger rely on its 5 percent

edge because this is anexpectation that is onlyreliable in the long run.Andthe long run has just beeneliminated.

Now consider a slightvariation where twocustomers come into thecasino and propose to placebets of $1,000 each at theroulette table, but they alsoagree not to bet on the samenumber. Whichever number

oneplayer chooses, theotherwill choose a differentnumber. As with the firstscenario, where one playermakesasingle$2,000bet,thecasino’s potential reward inthis new scenario is also$2,000. If neither numbercomes up, the casino gets tokeepthetwo$1,000bets.Butwhat is the risk to thecasinonow? In the worst case, thecasinocanonlylose$34,000,the $36,000 payoff if one

player wins less the cost ofthetwo$1,000bets.Thetwobetsaremutuallyexclusive—if one player wins, the othermustlose.

Inreturnforthereducedrisk, it might seem that thecasinomust give up someofits theoretical edge.We tendto assume that there is atradeoff between risk andreward. But the edge to thecasino in both cases is still

the same 5 percent.Regardless of the amountwagered or the number ofindividual bets, the laws ofprobabilityspecifythatinthelong run the casino gets tokeep 5 percent of everythingthatisbetattheroulettetable.Intheshortrun,however,therisk to the casino is greatlyreducedwithtwo$1,000betsbecause the bets have beenspreadaroundthetable.

Casinos do not like tosee an individual playerwager a large amount ofmoney on one outcome,whether at roulette or anyother casino game. This iswhy casinos have bettinglimits. The laws ofprobability are still in thecasino’s favor, but if the betislargeenoughandthebettorhappens to win, the short-termbadluckcanoverwhelmthecasino.Fromthecasino’s

point of view, the idealscenario is for 38 players toplace 38 bets of $1,000 eachon all 38 numbers at theroulettetable.Nowthecasinohasaperfect spreadposition.One player will collect$36,000,butwith$38,000onthe table, the casino has asureprofitof$2,000.

Looking at the situationfrom the player’s point ofview,iftheplayerknowsthat

the odds are against him andhe wants the greatest chanceof showing a profit, his bestcourse is to wager themaximum amount on oneoutcomeandhope that in theshort runhegets lucky. Ifhecontinuestomakebetsoveralongperiodof time, the lawsof probability eventuallywillcatch up with him, and thecasino will end up with theplayer’smoney.

An option trader preferstospreadforthesamereasonthat the casino prefers thebets to be spread around thetable: spreading maintainsprofit potential but reducesshort-term risk. There israrely a perfect spreadposition for an option trader,but an intelligent optiontraderlearnstospreadofftheriskinasmanydifferentwaysas possible to minimize theeffectsofshort-termbadluck.

An important part of anyserious option trader’seducationconsistsoflearninga wide variety of spreadingstrategies.

New traders aresometimes astonished at thesize of the trades anexperienced option trader isprepared to make. How canthe trader afford to do this?His financial resourcescertainly play a role in the

risk he is willing to accept.But equally important is hisability to spread off risk.Anexperiencedtradermayknowmany different ways tospread off the risk, usingother options, futurescontracts, cash contracts, orsome combination of these.While hemay not be able tocompletelyeliminatehisrisk,hemaybeabletoreduceittosuchanextent thathisriskisactually less than that of a

muchsmallertraderwhodoesnot know how to spread orknowsonlyalimitednumberofspreadingstrategies.

1Theoppositetypeofarbitrage,sellingthecommodityandbuyingaforwardcontract,isnotusuallypossibleincommoditymarketsbecausecommodities,unlikefinancialinstruments,cannotbeborrowedandsoldshort.2Forsimplicity,wehaveassumedaconstantinterestrate.Infact,thecostofborrowingmoneyforthesecondtwo-monthperiodmaybedifferentfromthecostofborrowingforthefirsttwo-monthperiod.Wehavealsoignoredthecostofborrowingmoneytopayforstorageandinsurance.Thiswilladdaverysmalladditionalcosttothestrategy.

3Treasurynoteandbondpricesaretypicallyquotedinpointsand32ndsofparvalue.4TradersrefertothisastheNOBspread(notesoverbonds).5Asimilartypeofthree-sidedspreadisavailableinthesoybeanmarket.Thecrushspreadconsistsofbuyingorsellingsoybeanfuturesandtakinganopposingfuturespositionintheproductsthataremadefromsoybeans—soybeanoilandsoybeanmeal.6Inpractice,whenreportingthepriceofaspread,exchangespreferthatthepartiestothetradeusepricesfortheindividualcontractsthatreflectcurrentmarketconditions.Otherwise,itmay

appearthatsomeoneisengaginginunethicalorillegalmarketactivity.Theexchangewillnotbehappyifthepartiesreportpricesof200,86,and100,eventhoughthesepricesstilladduptoatotalspreadpriceof14.

11

VolatilitySpreads

In Chapter 8, we showedthat it is possible, at least intheory, tocaptureanoption’smispricinginthemarketplaceby employing a dynamichedging strategy. The first

step in this process involveshedging the option position,delta neutral, by taking anopposing market position inthe underlying contract. Buttheunderlyingcontract isnottheonlywayinwhichwecanhedgeanoptionposition.Wemight instead take ouropposing delta position withotheroptions.

Consider a call with adeltaof50thatappears tobe

underpriced in themarketplace. If we buy 10calls, resulting in a deltaposition of +500, we mighthedge the position in any ofthefollowingways:

Sellfiveunderlyingcontracts.Buyputswitha

totaldeltaof–500.Sellcalls,

differentfromthose

thatwepurchased,withatotaldeltaof–500.Doa

combinationofanyoftheprecedingsuchthatwecreateatotaldeltaof–500.

There are clearly manydifferentwaysofhedgingour10calls.Regardlessofwhichmethod we choose, each

spread will have certainfeaturesincommon:

Eachspreadwillbeapproximatelydeltaneutral.Eachspreadwill

besensitivetochangesinthepriceoftheunderlyinginstrument.Eachspreadwill

besensitivetochangesinimplied

volatility.Eachspreadwill

besensitivetothepassageoftime.

Spreads with theforegoing characteristics fallunder the general heading ofvolatility spreads. In thischapter, we will look at themost common types ofvolatility spreads, initiallybyexamining their expirationvalues and then by

considering their delta,gamma, theta, vega, and rhocharacteristics.

Straddle

A straddle consists of acall and a put where bothoptions have the sameexercise price and expirationdate. In a straddle, bothoptions are either purchased(a long straddle) or sold (a

short straddle). Examples oflongandshortstraddles,withtheir expiration profit-and-loss(P&L)graphs,areshowninFigures11-1and11-2.

Figure11-1Longstraddleastimepassesorvolatilitydeclines.

Figure11-2Shortstraddleastimepassesorvolatilitydeclines.

At expiration, the valueofastraddlecanbeexpressedas a simpleparitygraph.Butwhat about its value prior toexpiration?Aswithalloptionpositions, some changes inmarket conditions will helpthe strategy and somechanges will hurt. FromFigure11-1,wecanseethatalong straddle becomes morevaluablewhentheunderlyingmarketmovesawayfromthe

exercise price and lessvaluable as time passes if nomovement occurs. At thesame time, any increase involatilitywillhelp,whileanydecline will hurt. Thesecharacteristics are indicatedby the risk measuresassociatedwiththeposition:

+Gamma(desire formovement inthe underlying

contract)–Theta (thevalue of thepositiondeclines astimepasses)+Vega (thevalue of thepositionincreases asimpliedvolatilityrises)

The characteristics of a

short straddle are shown inFigure11-2:

–Gamma(movement inthe underlyingcontract willhurt theposition)+Theta (thevalue of thepositionincreases astimepasses)

–Vega (thevalue of thepositionincreases asimpliedvolatilityfalls)

Straddles aremost oftenexecutedonetoone(onecallfor each put) using at-the-money options.When this isdone, the spread will beapproximately delta neutralbecause the delta values of

thecallandputwillbecloseto50and–50.Astraddlecanalsobedonewithoptionsthatareeitherinthemoneyoroutof the money. For example,with the underlying contracttradingat100,wemightbuytheSeptember 95 straddle. IftheSeptember95calls,whichareinthemoney,haveadeltaof 75 and the September 95puts, which are out of themoney, have a delta of –25,thetotaldeltawillbe75–25

= 50, resulting in a bullstraddle. If we want thestraddle to be delta neutral,we will need to adjust thenumber of contracts bypurchasing three puts foreverycall:

Buy 1September 95call (delta =75).Buy 3September 95

puts (delta=–25).

This spread still qualifiesas a straddle because we arebuying calls and puts at thesame exercise price. But,more specifically, this is aratio straddle because thenumber of long marketcontracts (the calls) and thenumber of short marketcontracts (the puts) areunequal.

Strangle

Like a straddle, a strangleconsists of a long call and alongput(alongstrangle)orashort call and a short put (ashort strangle), where bothoptions expire at the sametime. But in a strangle theoptions have differentexercise prices. Typical longandshortstranglesareshowninFigures11-3and11-4.

Figure11-3Longstrangleastimepassesorvolatilitydeclines.

Figure11-4Shortstrangleastimepassesorvolatilitydeclines.

As with a straddle,stranglesaremostoftendoneone toone (onecall for eachput). In order to ensure thatthe position is delta neutral,exercise prices are usuallychosensothatthecallandputdeltas are approximatelyequal.

Ifastrangleisidentifiedonly by its expiration monthandexerciseprices,theremaybe some confusion as to the

specific options involved. AMarch 90/110 stranglemightconsistofaMarch90putandaMarch110call.Butitmightalso consist of a March 90call and a March 110 put.Both strategies are consistentwith the definition of astrangle.To avoid confusion,a strangle is commonlyassumed toconsistofout-of-the-money options. If theunderlying market iscurrently at 100 and a trader

wants to purchase theMarch90/110 strangle, everyonewill assume that he wants topurchaseaMarch90putanda March 110 call. Althoughboth strangles haveessentially the same P&Lprofile, in-the-money optionstendtobelessactivelytradedthan their out-of-the-moneycounterparts. A strangleconsisting of in-the-moneyoptions issometimesreferredtoasaguts.

Note that the riskcharacteristics of a strangleare similar to those of astraddle:

Long strangle:+gamma/–theta/+vegaShort strangle:+gamma/–theta/+vega

A new option traderoftenfindslongstraddlesand

strangles attractive becausestrategies with limited riskandunlimitedprofitpotentialoffer great appeal, especiallywhen the profit is unlimitedin both directions. However,if the hoped-for movementfails to materialize, a traderwill find that losing money,even a limited amount, canalso be a painful experience.Thisisnotanendorsementofeither longorshortstraddles.Under the right conditions,

either strategy may besensible. But an intelligenttrader needs to consider notonly whether the risk andrewardislimitedorunlimitedbut also the likelihoodof thevarious outcomes. This, ofcourse, is one importantreason forusinga theoreticalpricingmodel.

Butterfly

Thusfarwehavelookedatspreads that involve buyingorsellingtwodifferentoptioncontracts. However, we canalso construct spreadsconsisting of three, four, oreven more different options.A butterfly is a commonthree-sided spread consistingof options with equallyspacedexerciseprices,whereall options are of the sametype (either all calls or allputs) and expire at the same

time. In a long butterfly, theoutside exercise prices arepurchased and the insideexercise price is sold, andvice versa for a shortbutterfly.Moreover, the ratioof a butterfly never varies. Itisalways1×2×1,withtwoof each inside exercise pricetraded for each one of theoutside exercise prices.Typical long and shortbutterflies are shown inFigures11-5and11-6.

Figure11-5Longbutterflyastimepassesorvolatilitydeclines.

Figure11-6Shortbutterflyastimepassesorvolatilitydeclines.

To a new trader, abutterfly may look quitecomplex since it involvesthree different options indifferent quantities. Butbutterflies have very simpleand well-definedcharacteristics that makethem popular tradingstrategies. To understandthese characteristics, let’sconsider the value of a longbutterflyatexpiration:

Iftheunderlyingpriceisbelow90atexpiration,allthecalls will expire worthless,and the value of the position

will be 0. If the underlyingcontract is above 120 atexpiration, the combinedvalueof the90and110callswill equal the value of thetwo 100 calls. Again, thevalue of the butterflywill be0. Now suppose that theunderlying contract isbetween 90 and 110 atexpiration, specifically, rightattheinsideexercisepriceof100.The90callwillbeworth10.00,while the100and110

calls will be worthless. Thepositionwillbeworthexactly10.00. If the underlyingmoves away from 100, thevalue of the butterfly willdecline, but its value cannever fall below 0.Summarizing,atexpiration,abutterfly is worthless if theunderlying contract is aboveorbelowtheoutsideexerciseprices (sometimes referred toasthewingsofthebutterfly).It has its maximum value at

expiration when theunderlyingcontractisrightatthe inside exercise price(sometimes referred toas thebody of the butterfly). Andthemaximumvalueisalwaysequal to the amount betweenexercise prices, in ourexample10.00.

Because a butterfly atexpirationalwayshasavaluebetween 0 and the amountbetween exercise prices, in

our example, a trader shouldbe willing to pay someamount between 0 and 10.00for the position. The exactamount depends on thelikelihood of the underlyingcontractfinishingclosetotheinside price at expiration. Ifthere is a high probability ofthisoccurring, a tradermightbewilling topayasmuchas8.00 for the butterfly since itmightverywellexpandtoitsfull value of 10.00. If,

however, there is a lowprobability of this occurringand, consequently, a highprobability that theunderlying contract willfinish outside the extremeexercise prices, a tradermayonlybewillingtopay1.00or2.00 because he may verywell lose his entireinvestment. This alsoexplains why our exampleposition is a long butterfly.Because the position can

neverbeworth less than0,atraderwillalwaysberequiredto pay some amount for theposition. Otherwise, therewould be a riskless profitopportunity.Whenapositionrequires an outlay of cash, atraderhasbought, or is long,theposition.

A butterfly will tend tobe delta neutral when theinside exercise price isapproximately at the money.

Under these conditions, alongbutterflywilltendtoactlike a short straddle, while ashortbutterflywilltendtoactlike a long straddle. Witheither a long butterfly or ashort straddle, a traderwantsthe underlying market to sitstill (–gamma, +theta) andimplied volatility to fall (–vega). With either a shortbutterflyora longstraddle,atrader wants the underlyingmarket tomakealargemove

(+gamma, –theta) andimplied volatility to rise(+vega). But there is oneimportantdifference.Whileastraddle is open-ended intermsofeitherprofitpotentialor risk, a butterfly is strictlylimited.Itcanneverbeworthless than0normorethantheamount between exerciseprices.Thisisimportantforatraderwhomightwanttosellstraddles but who isuncomfortable with the

possibility of unlimited loss.Of course, there is always arisk-rewardtradeoff.Ifalongbutterfly has reduced riskwhen the trader is wrong, itwillalsohaveincreasedprofitwhen the trader is right. Forthisreason,butterfliestendtobe executed in much largersizes than straddles.A tradermay find that buying 300butterflies (300×600×300)is actually less risky thanselling 100 straddles. In

option trading, size and riskdonotalwayscorrelate.Somestrategies done in large sizescan have a relatively smallrisk, while other strategies,even when done in smallsizes, can have a relativelylarge risk. Risk depends notonly on the size in which astrategy is executed but alsoon the characteristics of thestrategy.

Weknowthatabutterfly

at expiration is worth itsmaximum when theunderlyingcontractisrightatthe inside exercise price. IfweassumethatalloptionsareEuropean,withnopossibilityof early exercise, both a calland a put butterfly with thesame exercise prices and thesame expiration dates desireexactlythesameoutcomeandtherefore have identicalcharacteristics. Both theMarch 90/100/110 call

butterfly and the March90/100/110 put butterfly willbe worth a maximum of10.00 with the underlyingprice exactly at 100 atexpirationandaminimumof0 with the underlying pricebelow 90 or above 110. Ifboth butterflies are nottrading at the same price,there is a sure profitopportunity available bypurchasing the cheaper and

sellingthemoreexpensive.1

Condor

Just as a butterfly can bethought of as a straddlewithlimited risk or reward, acondorcanbethoughtofasastrangle with limited risk orreward.A condor consists offour options, two insideexercise prices (the body ofthe condor) and two outside

exercise prices (the wings ofthe condor).2 The ratio of acondorisalways1×1×1×1. Although the amountbetween the two insideexercisepricescanvary,theremust be an equal amountbetween the two lowestexercise prices and the twohighest exercise prices. Aswith a butterfly, all optionsmust expire at the same timeand be of the same type

(eitherallcallsorallputs).Ina long condor, the twooutside exercise prices arepurchasedand the two insideexercise prices are sold, andviceversaforashortcondor.Typical long and shortcondorsareshowninFigures11-7and11-8.

Figure11-7Longcondorastimepassesorvolatilitydeclines.

Figure11-8Shortcondorastimepassesorvolatilitydeclines.

Thevalueofacondoratexpiration can never be lessthan 0 nor more than theamount between the twohigher or the two lowerexerciseprices.A traderwhobuys a condorwill pay someamountbetweenthesevalues,expecting that the underlyingcontract will finish betweenthe two intermediate exerciseprices,where thecondorwillbe worth its maximum. A

traderwhosellsacondorwilltake in some amount,expecting that the underlyingcontract will finish outsidethe extreme exercise prices,where the condor will beworthless.

A condor will beapproximately delta neutralwhen the underlying contractis midway between the twoinside exercise prices. Whenall options are European, the

valueandcharacteristicsofacall condor and put condorwillbeidentical.

The four volatilityspreadsthatwejustdescribed—straddles, strangles,butterflies, and condors—allhave symmetrical P&Lgraphs.When executed deltaneutral, as is most common,these strategies have nopreferenceas to thedirectionof movement in the

underlying market. Longstraddles and strangles andshort butterflies and condorsprefer movement in theunderlying market and anincrease in implied volatility(+gamma, –theta, +vega).Short straddles and stranglesand long butterflies andcondors prefer no movementin the underlyingmarket andadeclineinimpliedvolatility(–gamma, +theta, –vega).These characteristics are

summarizedinFigure11-9.Figure11-9Symmetricalstrategies.

RatioSpread

In a volatility spread, atrader need not be totallyindifferent to thedirectionofmovement in the underlyingmarket. The trader maybelievethatmovementinonedirection is more likely thanmovement in the otherdirection. Given this, thetradermaywishtoconstructa

spread that either maximizeshis profit or minimizes hisloss when movement occursin one direction rather thantheother. Inorder toachievethis, a trader can construct aratio spread—buying andselling unequal numbers ofoptionswhere all options arethe same type and expire atthe same time.Aswithothervolatilitypositions,thespreadistypicallydeltaneutral.

Consider the followingdelta-neutralpositionwiththeunderlyingcontracttradingat100 (delta values are inparentheses):

Now let’s consider threepossible prices for the

underlying contract atexpiration:

If the underlyingcontract makes a very bigmove in either direction, theposition will show a profit.

Of course, the profit will bemuch larger if the move isupward.Iftheunderlyingsitsat 100 until expiration, theposition will show a loss.Thiscallratiospread, wheremorecallsarepurchasedthansold,wantsmovement in theunderlying contract butclearly prefers upwardmovement, where thepotential profit is unlimited.The P&L diagram for thistype of strategy is shown in

Figure11-10.Figure11-10Callratiospread(buy

morethansell)astimepassesorvolatilitydeclines.

The same type ofpositioncanbecreatedusingputs. A put ratio spread,where more puts arepurchased than sold, alsoprefers movement in theunderlyingcontract.Butnowthere is a preference fordownward movementbecause the profit potentialon the downside will beunlimited. This is shown inFigure11-11.

Figure11-11Putratiospread(buymorethansell)astimepassesorvolatilitydeclines.

A ratio spread wheremore options are purchasedthan sold is sometimesreferred to as backspread.Regardless of whether thespread consists of calls orputs, this type of spreadalways wants movement inthe underlying market(+gamma, –theta) and/or anincrease in implied volatility(+vega).

In a ratio spread where

more options are purchasedthan sold, the spread will beworthless if the underlyingcontract makes a largeenough downward move inthe case of calls or a largeenough upward move in thecaseofputs.Foreitherspreadtoresultinaprofit,itmustbeexecutedinitiallyforacredit,and this is a typicalcharacteristic of these typesof spreads. Indeed, under theassumptions of a traditional

theoretical pricing model, adelta-neutral ratio spreadwhere more options arepurchased than sold shouldalwaysresultinacredit.

Ratio spreads are oftenused to limit the risk in onedirection. If we sell morecalls thanwebuy, the spreadwill act like a short straddle(–gamma, +theta, –vega) butwithlimiteddownsiderisk.Ifwe sell more puts than we

buy, the spread will havelimitedupsiderisk.TheP&Ldiagrams for these types ofspreads are shown Figures11-12and11-13.

Figure11-12Callratiospread(sellmorethanbuy)astimepassesorvolatilitydeclines.

Figure11-13Putratiospread(sellmorethanbuy)astimepassesorvolatilitydeclines.

A ratio spread wheremore options are sold thanpurchased is sometimesreferred to as frontspread.3Using calls, the positionwillbe worthless at expiration ifthe underlying contract isbelow the lower exerciseprice.Usingputs,thepositionwill be worthless atexpiration if the underlyingcontract is above the higherexercise price. The fact that

the value of the positioncannotfallbelow0limitsthedownside risk if more callsare sold than purchased andthe upside risk if more putsaresoldthanpurchased.

When executed as asingle trade, ratiospreadsareusually submitted usingsimple ratios, the mostcommon being 2 to 1.However, other ratios—3 to1,4 to1,or3 to2—arealso

relativelycommon.

ChristmasTree

Ratio spreads tend tomimic straddles,butwith therisk or reward limited in onedirection. We can alsoconstruct strategies thatmimic strangles, but againwithlimitedriskorrewardinone direction. Such spreadsare known as either

Christmastreesorladders.4A call Christmas tree

involves buying (selling) acall at a lower exercise priceand selling (buying) one calleach at two higher exerciseprices. A put Christmas treeinvolves buying (selling) aputat ahigherexercisepriceand selling (buying) one puteach at two lower exerciseprices. All options must bethe same type and expire at

the same time, with exerciseprices most often chosen sothattheentirepositionisdeltaneutral. When one option isbought and two options sold(a long Christmas tree), theposition acts like a shortstrangle butwith limited riskin one direction. When oneoptionissoldandtwooptionsbought (a short Christmastree), the position acts like alongstranglebutwithlimitedprofit potential in one

direction. P&L diagrams fortypical Christmas trees areshown in Figures 11-14through11-17.

Figure11-14LongcallChristmastreeastimepassesorvolatilitydeclines.

Figure11-15ShortcallChristmastreeastimepassesorvolatilitydeclines.

Figure11-16LongputChristmastreeastimepassesorvolatilitydeclines.

Figure11-17ShortputChristmastreeastimepassesorvolatilitydeclines.

Although ratio spreadsand Christmas trees havenonsymmetrical P&L graphs,their volatility characteristicstend to mimic straddles andstrangles. A spread in whichmore options are purchasedthan sold will prefermovement in the underlyingmarket and/or an increase inimpliedvolatility(+gamma,–theta, +vega). A spread inwhich more options are sold

thanpurchasedwillprefernomovement in the underlyingmarket and/or a decline inimplied volatility (–gamma,+theta, –vega). Thecharacteristics ofnonsymmetrical spreads aresummarizedinFigure11-18.

Figure11-18Nonsymmetricalstrategies.

CalendarSpread

If all options in a spreadexpire at the same time, thevalue of the spread atexpiration depends solely onthe underlying price. If,however, the spread consistsof options that expire atdifferent times, the spread’svalue depends not only onwhere the underlying market

iswhentheshort-termoptionexpiresbutalsoonwhatwillhappenbetweenthatdateandthe date on which the long-termoptionexpires.Calendarspreads, sometimes referredto as time spreads orhorizontal spreads,5 consistofoptionpositionsthatexpireindifferentmonths.

The most common typeofcalendarspreadconsistsofopposing positions in two

options of the same type(eitherbothcallsorbothputs)where both options have thesame exercise price. Whenthe long-term option ispurchased and the short-termoptionissold,atraderislongthecalendarspread;whentheshort-term option ispurchased and the long-termoption is sold, the trader isshort the calendar spread.Because a long-term optionwill typically be worth more

than a short-termoption, thisisconsistentwiththepracticeof referring to any strategythatisexecutedatadebitasalong position and any spreadthatisexecutedforacreditasashortposition.

Although calendarspreads are most commonlyexecuted one to one (onecontract purchased for eachcontract sold), a trader mayratio a calendar spread to

reflect a bullish, bearish, orneutralmarketsentiment.Forpurposes of discussion, wewill focus on one-to-onecalendar spreads (one long-term option for each short-term option) that areapproximately delta neutral.Because at-the-moneyoptions have delta valuescloseto50,themostcommoncalendar spreads consist oflong and short at-the-money

options.6The value of a calendar

spread depends not only onmovement in the underlyingmarket but also on themarketplace’s expectationsabout future marketmovement as reflected in theimpliedvolatility.Becauseofthis, a calendar spread hascharacteristics that differfrom the other spreads wehavediscussed.Ifweassume

that the optionsmaking up acalendar spread areapproximately at the money,calendar spreads have twoimportantcharacteristics:

1. A calendarspreadwillincreasein value if timepasses with nomovement in theunderlyingcontract.2. A calendarspreadwillincrease

in value if impliedvolatility rises anddecline in value ifimplied volatilityfalls.

Why should a calendarspreadbecomemorevaluableas time passes? Consider thefollowing spread, where theunderlying contract,which iscurrentlytradingat100,isthesameforbothoptions:

+1 June 100call–1 April 100call

Supposethattherearefourmonths remaining to Juneexpiration and two monthsremainingtoAprilexpiration.If we assume a constantunderlyingpriceof100andaconstant volatility of 20percent, the value of theindividual options as time

passes,aswellasthevalueofthespread,isshowninFigure11-19.

Figure11-19Thevalueofacalendarspreadastimepasses.

The spread is initiallyworth 1.34, but as timepasses, both options begin todecay. However the Apriloption, with less timeremaining to expiration,decaysmore rapidly than theJune option. Over the firstmonth, theApriloption loses0.96, while the June optionloses only 0.61. The spreadhasincreasedto1.69.

Over the next month,

with the underlying contractstill at 100, theApril option,because it is at the money,must give up its entire valueof2.30.TheJuneoptionwillalsocontinuetodecay,andata slightly greater rate, losing0.73.But thecalendarspreadhasstillincreasedto3.26.

The increase invalueofthe calendar spread as timepasses is the result of animportant characteristic of

theta that was noted inChapter 8: as time toexpiration grows shorter, thetheta of an at-the-moneyoption increases. A short-term at-the-money optiondecays more rapidly than along-term at-the-moneyoption.

What will happen if theunderlying contract does notsit still but instead makes alarge upward or downward

move? The value of acalendar spread depends onthelong-termoptionretainingas much time value aspossiblewhile the short-termoption decays. This will betrueifbothoptionsremainatthemoneybecauseanat-the-moneyoptionalwayshas thegreatest amount of timevalue. As an option moveseither into the money or outof the money, its time valuewill disappear. A long-term

option will always havegreater time value than ashort-term option. But, if themovement in the underlyingcontract is large enough andtheoptionmovesverydeeplyintothemoneyorveryfaroutof the money, even a long-term option will eventuallylose almost all or its timevalue. This will cause thecalendar spread to collapse,asshowninFigure11-20.

Figure11-20Thevalueofacalendarspreadastheunderlyingpricechanges.

Now let’s consider the

effect of changing volatilityon a calendar spread. Thevalue of the April/June 100call calendar spread atdifferent volatilities is showninFigure11-21.

Figure11-21Thevalueofacalendarspreadasvolatilitychanges.

As we raise or lowervolatility,bothoptionsriseorfall in value, but the June

option changesmore quicklythan the April option. Wetouched on this characteristicinChapter6,wherewenotedthatachangeinvolatilitywillhave a greater effect on along-term option than on anequivalent short-term option.In other words, long-termoptions have greater vegavalues than short-termoptions. This difference insensitivity to a change involatility causes the calendar

spread to widen if weincrease volatility and tonarrow if we reducevolatility.

A trader who is long acalendar spread wants twoapparently contradictoryconditionsinthemarketplace.First,hewantstheunderlyingcontracttositstillinordertotake advantage of the greatertimedecayfor theshort-termoption. Second, he wants

everyone to think that themarket is going to move sothat implied volatility willrise, causing the long-termoption to rise in price morequickly than the short-termoption.Canthishappen?Canthemarketremainunchangedyeteveryonethinkthatitwillmove? In fact, it happensquite often because eventsthatdonothaveanimmediateeffect on the underlyingcontractmay be perceived to

have a future effect on theunderlying.

The most commonexampleoccurswhennewsispendingthatislikelytoaffectthe underlying contract butwhose exact effect isunknown. Consider acompany that announces thatits CEO will make animportantstatementoneweekfrom today. If no one knowsthe content of the statement,

there is unlikely to be anysignificant change in thecompany’s stock price priorto the statement. But traderswill assume that thestatement, when it is made,willhaveaneffect,perhapsadramatic one, on the stockprice. The possibility offuturemovementinthestockprice will cause impliedvolatility to rise. Thiscombination of conditions—the lack of movement in the

underlying stock togetherwith rising implied volatility—willcausecalendarspreadstowiden.

Of course, theassumption of future stockmovement as a result of theCEO’s statement is just that—an assumption. If thestatement turns out to beirrelevant to the company’sfortunes (theCEOwanted toannouncethatheandhiswife

just became grandparents),any presumption of futurevolatility is removed. Theresult will be a decline inimplied volatility, causingcalendarspreadstonarrow.

The effect of impliedvolatility is whatdistinguishes time spreadsfrom the other types ofspreads we have discussed.Long straddles, longstrangles,andshortbutterflies

all want the volatility of theunderlyingcontractaswellasimplied volatility to rise(+gamma, +vega). Shortstraddles,shortstrangles,andlong butterflies all want thevolatility of the underlyingcontract as well as impliedvolatility to fall (–gamma, –vega). But with calendarspreads underlying volatilityand implied volatility haveopposite effects. A quietmarket or an increase in

implied volatility will help along calendar spread (–gamma, +vega), while a bigmove in the underlyingmarketoradeclineinimpliedvolatility will help a shortcalendar spread (+gamma, –vega).This opposite effect iswhat gives calendar spreadstheiruniquecharacteristics.

Figures11-22and11-23show the value of long andshortcalendarspreadsastime

passes.Figures11-24and11-25 show the value asvolatilitychanges.

Figure11-22Longcalendarspreadastimepasses.

Figure11-23Shortcalendarspreadastimepasses.

Figure11-24Longcalendarspreadasvolatilitydeclines.

Figure11-25Shortcalendarspreadasvolatilitydeclines.

Although the effects oftime and volatility apply tocalendar spreads in allmarkets, there may be otherconsiderations, depending onthe specific underlyingmarket. In the foregoingexamples, we assumed thatthe underlying contract forboththeshort-andlong-termoption was the same. In thestockoptionmarket, thiswillalways be true. The

underlying contract forGeneral Electric (GE)options, regardless of theexpiration month, is alwaysGEstock.AndGEstockcanonly have a single price atanyonetime.Butinafuturesmarket the underlying for afutures option is a specificfuturescontract,anddifferentoption expirations can havedifferent underlying futurescontracts.

Consider a futuresmarket where there are fourfuturesmonths:March, June,September, andDecember. Ifserial months are available,anApril/Junecalendarspreadwillhavethesameunderlyingcontract, June futures. But aMarch/June calendar spreadwill have one underlyingcontract forMarchoptions, aMarch future, and a differentunderlying contract for Juneoptions, a June future.

Although one might expectMarch futures and Junefutures to move together,thereisnoguaranteethattheywill. Particularly incommodity markets, short-term supply and demandconsiderations can causefuturescontractson thesamecommodity to move indifferent directions. Inaddition to volatilityconsiderations, a trader whobuys a June/March call

calendar spread must alsoconsider the possibility thatMarch futures will riserelativetoJunefutures.

Inordertooffsettheriskof futures contracts movingagainst a calendar-spreadposition, it is common incommodity futures marketsfor a trader to offset acalendar spread with anopposing position in thefutures market. In our

example, if a trader buys theMarch/June call calendarspread, he can offset thepositionbypurchasingMarchfutures and selling Junefutures.

How many futuresspreads should the traderexecute? If he wants aposition that issensitiveonlytovolatility,heoughttotradethenumberoffuturesspreadsrequiredtobedeltaneutral.If

both calls are at the money,with deltas of approximately50,atraderwhobuys10callcalendarspreads(buy10Junecalls,sell10Marchcalls)willbe long 500 deltas in Juneand short 500 deltas inMarch. Therefore, he shouldbuy5Marchfuturescontractsand sell 5 June futurescontracts.The entire positionwill be (delta values are inparentheses)

+10 June calls(+500), – Junefutures(–500)–10 Marchcalls (–500),+5 Marchfutures(+500)

This type of balancing isnot necessary—indeed, notpossible—in stock optionsbecausetheunderlyingforallmonthsisidentical.

TimeButterfly

In futures markets, asopposed to optionmarkets, abutterflyisapositioninthreefuturesmonths.A traderwillbuy(sell)oneeachofashort-and long-term futurescontract and sell (buy) twointermediate-term futurescontracts. A similar type ofstrategycanbedoneinoptionmarkets. A traditional option

butterfly consists of optionsat three different exerciseprices but with the sameexpiration date. A timebutterfly (sometimesshortenedtotimefly)consistsof options at the sameexercise price but with threedifferentexpirationdates.Alloptionsmustbethesametype(either all calls or all puts),with approximately the sameamount of time betweenexpirations. The outside

expirationmonthsareusuallyreferred to as the wings andtheinsideexpirationmonthasthe body. Some typical timebutterfliesmightbe

Note that a timebutterfly consists ofsimultaneously buying orselling a long-term calendarspread and taking anopposing position in a short-term calendar spread, whereeachspreadhasonecommonexpiration month. Theexample May/June/July 100call timebutterflyconsistsofbuying theMay100call andselling the June 100 call

(selling the May/Junecalendar spread) andsimultaneously selling theJune100call andbuying theJuly 100 call (buying theJune/Julycalendarspread).

If all options remain atthemoney,astimepasses,thevalue of a calendar spreadwill increase. The short-termspread must therefore beworth more than the long-termspread.Consequently, if

we buy the short-termcalendar spread and sell thelong-term calendar spread(buying the body and sellingthe wings), in total, we willpay more than we receive.Because the entire positionwill result in a debit, we arelong the timebutterfly. Ifwedo the opposite, selling theshort-term calendar spreadand buying the long-termspread (selling the body andbuying the wings), we are

shortthetimebutterfly.7Thiscan be somewhat confusingbecause in a traditionalbutterfly consisting ofdifferent exercise prices, thecombination of buying thewings and selling the bodyresults in a debit. But in atime butterfly consisting ofdifferent expiration months,buying thewings and sellingthebodyresultsinacredit.

Thevalueofalongtime

butterfly as time passes andasvolatility falls is shown inFigures11-26and11-27.Thevalue of the spreadwill tendto collapse as the underlyingcontract moves away fromthe exercise price, implyingthatthespreadhasanegativegamma. Consequently, thespread must also have apositive theta. Finally, thevalue of the spread falls asvolatility declines, implyingthat the spreadhasapositive

vega. In sum, a long timebutterfly has characteristicssimilar to those of a longcalendarspread.

Figure11-26Longtimebutterflyastimepasses.

Figure11-27Longtimebutterflyasvolatilitydeclines.

EffectofChangingInterestRatesandDividends

Thus far we haveconsideredonlytheeffectsofchanges in underlying price,time, and volatility on thevalue of a volatility spread.What about changes ininterest rates and, in the case

ofstocks,dividends?Because there is no

carrying cost associated withthe purchase or sale of afuturescontract, interestrateshaveonly aminor impactonfutures options and,consequently, a relativelyminor effect on the value ofall futures option volatilityspreads.8However,inastockoption market, a change ininterest rates will cause the

forward price of stocks tochange. If all options in aspread expire at the sametime, the change in forwardprice is likely to affect alloptionsequally, causingonlysmallchangesinthevalueofthe spread. However, if wehave a stock option positioninvolving two differentexpiration dates, we mustconsider two differentforwardprices.Andthesetwoforward prices may not be

equally sensitive to a changeininterestrates.

Consider the followingsituation:

Stockprice=100interestrate=8.00%dividend=0

Suppose that a trader buysacallcalendarspread:

+10 June100calls–10 March

100calls

If there are three monthsremaining to Marchexpiration and six monthsremaining to Juneexpiration,the forward prices forMarchstock and June stock are102.00 and 104.00,respectively. If interest ratesriseto10percent,theforwardprice for March will be102.50 and the forwardpricefor June will be 105. With

more time remaining to Juneexpiration, the June forwardprice is more sensitive to achange in interest rates.Assuming that the deltas ofboth options areapproximatelyequal,theJuneoption will be more affectedby the increase in interestrates than the March option,and the calendar spread willexpand. In the same way, ifinterest rates decline, thecalendar spread will narrow

because the June forwardprice will fall more quicklythantheMarchforwardprice.Alongcallcalendarspreadinthe stockoptionmarketmustthereforehaveapositiverho,and a short call calendarspread must have a negativerho.

Changes in interest rateshave the opposite effect onstock option puts. In ourexample, if interest rates rise

from8to10percent,theJuneforward price will rise morethantheMarchforwardprice.Ifwe assume, again, that thedeltas of both options areapproximately equal, andrecalling that puts havenegative deltas, the June putwillshowagreaterdeclineinvaluethantheMarchput.Theput calendar spread willthereforenarrow.Inthesameway, if interest rates decline,the put calendar spread will

expand. A long put calendarspread in the stock optionmarketmust thereforehaveanegative rho, and a short putcalendar spread must have apositiverho.

The degree to whichstockoptioncalendarspreadsare affected by changes ininterest rates dependsprimarily on the amount oftime between expirations. Iftherearesixmonthsbetween

expirations (e.g.,March/September), the effectwill be much greater than ifthere is only one monthbetween expirations (e.g.,March/April).

Changes in dividendscan also affect the value ofstockoptioncalendarspreadsbecause it may change theforward price of the stock.Dividends,however,havetheopposite effect on stock

optionsaschangesininterestrates. An increase in thedividend lowers the forwardpriceof stock,while a cut inthe dividend raises theforward price. If all optionsexpire at the same time, thechange in the forward pricefor the stock will have anequal effect on all options,andthechangeinthevalueofa spread will be negligible.Butinacalendarspread,ifatleastonedividendpaymentis

expected between theexpiration dates, an increaseindividendswillcauseacallcalendarspreadtonarrowandput calendar spread toexpand. A decrease individends will have theoppositeeffect,causingacallcalendarspread towidenanda put calendar spread tonarrow.Even though there isno Greek letter associatedwith dividend risk,wemightsaythatacallcalendarspread

hasnegativedividendrisk(itsvalue falls as dividends rise)andaputcalendarspreadhaspositive dividend risk (itsvaluerisesasdividendsrise).Examples of the effects ofchanging interest rates anddividends on stock optioncalendarspreadsareshowninFigure11-28.

Figure11-28Effectofchanginginterestratesandchangingdividendsonstockoptioncalendarspreads.

InFigure11-28,we canseethatanincreaseininterestrateswill reduce thevalueofa put calendar spread and anincrease in dividends willreduce the value of a callcalendarspread.Indeed,ifweraise interest rates highenough, the put calendarspreadcantakeonanegativevalue,with the long-termputhavingalowervaluethantheshort-termput.Thesamewill

be true for a call calendarspread if we increasedividends enough. If a stockpays no dividends, the valueof a call calendar spreadshould always have somevalue greater than 0. Even ifvolatility is very low, thespreadshouldstillbeworthaminimumofthecostofcarryon the stock betweenexpiration months. This isonlytrue,however,ifatradercan carry a short stock

position between expirationmonths. If a situation ariseswhere no stock can beborrowed, the trader whoowns a call calendar spreadmaybeforcedtoexercisehislong-term option, therebylosing the time valueassociated with the option.This issometimesreferred toasashortsqueeze.

DiagonalSpreads

A diagonal spread issimilar to a calendar spreadexcept that the options havedifferent exercise prices.Although many diagonalspreads are executed one toone(onelong-termoptionforeach short-term option),diagonal spreads can also beratioed, with unequalnumbers of long and short

market contracts. With thelargenumberofvariations indiagonal spreads, it is almostimpossible to generalizeabout their characteristics.Eachdiagonalspreadmustbeanalyzed separately todetermine the risks andrewards associated with thespread.

There is, however, onetypeofdiagonalspreadaboutwhichwecangeneralize.Ifa

diagonal spread is done onetooneandbothoptionsareofthe same type and haveapproximatelythesamedelta,the diagonal spread will actvery much like aconventionalcalendarspread.Examples of this type ofdiagonalspreadareshowninFigure11-29(deltavaluesareinparentheses).

Figure11-29Diagonalspreads.

Even though there are

many different volatilityspreads, traders tend toclassify spreads in terms oftheir basic volatilitycharacteristics. While somevolatility spreads may prefermovement in one directionrather thantheother,a traderwho initiates a volatilityspreadisconcernedprimarilywith the magnitude ofmovement in the underlyingcontractandonlysecondarilywith the direction of

movement. Therefore, allvolatility spreads tend to beapproximately delta neutral.Ifatraderhasalargepositiveor negative delta such thatdirectional considerationsbecome more important thanvolatility considerations, theposition can no longer beconsideredavolatilityspread.

All spreads that arehelped by movement in theunderlying market have a

positive gamma. All spreadsthatarehurtbymovement intheunderlyingmarkethaveanegative gamma. A traderwho has a positive gammaposition is said to be longpremium and is hoping for avolatile market with largemoves in the underlyingcontract. A trader who has anegativegammaissaid tobeshort premium and is hopingfor a quiet market with onlysmall moves in the

underlyingmarket.Because the effect of

market movement and theeffect of time decay alwayswork in opposite directions,any spread with a positivegammawill necessarily havea negative theta, and anyspread with a negativegammawill necessarily havea positive theta. If marketmovement helps, the passageof time hurts, and if market

movement hurts, the passageof time helps. An optiontrader cannot have it bothways.

Finally, spreads that arehelped by rising volatilityhaveapositivevega.Spreadsthat are helped by fallingvolatility have a negativevega. In theory, the vegarefers to the sensitivity of atheoretical value to a changein the volatility of the

underlying contract over thelifeoftheoption.Inpractice,however,tradersassociatethevegawiththesensitivityofanoption’s price to a change inimplied volatility. Spreadswith a positive vega will behelped by any increase inimpliedvolatilityandhurtbyany decline; spreads with anegative vega will be helpedby any decline in impliedvolatility and hurt by anyincrease. The delta, gamma,

theta,andvegacharacteristicsof the primary types ofvolatility spreads aresummarizedinFigure11-30.

Figure11-30Summaryofcommonvolatilityspreads.

Because volatilityspreads tend to be deltaneutral and the theta andgamma are always ofopposite sign, we can placevolatility spreads into one offour categories depending ontheeffectofmovementintheunderlying contract (positiveor negative gamma) and theeffect of changes in impliedvolatility(positiveornegativevega):

Of course, within eachof these categories, somespreads will have larger

gamma or vega values andsome spreads will havesmaller values. Of these,straddlesandstranglestendtohave the largest gamma andvegavaluesandthereforethegreatest risk.Theywill resultinthegreatestprofitwhenthetrader is correct in hisassessment of marketconditions, but they willresult in the greatest losswhen the trader is wrong.Butterfliesandcondorsareat

theotherendofthespectrum.These spreads yield smallerprofits when the trader isrightbutalsoresultinsmallerlosses when the trader iswrong. Ratio spreads andChristmas trees fallsomewhereinbetween.

Volatilityspreadscanbefurther distinguished by theirlimited or unlimited risk-reward characteristics, bothon the upside and on the

downside. Thesecharacteristics are alsosummarizedinFigure11-30.

Figure 11-31 is anevaluation table with thetheoretical value, delta,gamma, theta, vega, and rhoof several different options.Following this table areexamplesofvolatilityspreadsof the typesdiscussed in thischapter,alongwiththeirtotaldelta, gamma, theta, vega,

and rho. (Although theexamples in Figure 11-31assumethat theunderlying isstock, except for the rho, thecharacteristicsofeachtypeofspread will tend to be thesame for options on futures.)The readerwill see that eachspread does indeed have thepositive or negativesensitivities summarized inFigure11-30.Notealsothatavolatility spread need not beexactlydeltaneutral.(Indeed,

as we saw in Chapter 7, notrader can say with absolutecertaintywhetherapositionisreally delta neutral.) Inpractice, a volatility spreadshould have a delta that issmall enough that thedirectional considerations areless important than thevolatilityconsiderations.Thisis often a subjectivejudgment.

Figure11-31Examplesofcommonvolatilityspreads.

Also included in Figure11-31 is the theoreticalvalueofeachspread.Thisissimplythe cash flow that results ifeach spread is executed attheoretical value. Purchasesof options result in a cashdebit (indicated with anegative sign), and salesrepresent a cash credit(indicated with a positivesign). In commonterminology, a trader is said

to be long the spread if itresults in a cash debit andshortthespreadifitresultsinacashcredit.

Note that no price isgiven for any of the optioncontractsinFigure11-31,andtherefore,no theoretical edgecan be calculated for any ofthe spreads. The prices atwhich a spread is executedmaybegoodorbad,resultingin a positive or negative

theoretical edge. But, oncethe spread has beenestablished, the marketconditions that will help orhurt the spread aredetermined by itscharacteristics, not by theinitialprices.Likealltraders,an option tradermust not lethis previous trading activityaffecthiscurrentjudgment.Atrader’s primary concernshouldnotbewhathappenedyesterday but what can be

done today tomake themostof the current situation,whether attempting tomaximizeapotentialprofitorminimizeapotentialloss.

ChoosinganAppropriateStrategy

With so many spreadsavailable,howcanwedecidewhich typeofspread isbest?

First and foremost, we willwant to choose spreads thathave a positive theoreticaledge to ensure that ifwe arerightaboutmarketconditions,we have a reasonableexpectation of showing aprofit. Ideally, we want toconstruct a spread bypurchasing options that areunderpriced (too cheap) andselling options that areoverpriced(tooexpensive).Ifwe can do this, the resulting

spread,whateveritstype,willalways have a positivetheoreticaledge.

More often, however,our opinion about volatilitywill result in all optionsappearingeitheroverpricedorunderpriced. When thishappens,itwillbeimpossibletobothbuyandselloptionsatadvantageous prices. Such amarket can be easilyidentified by comparing our

volatility estimate with theimplied volatility in theoption marketplace. Ifimplied volatility is lowerthan the volatility estimate,options will be underpriced.If implied volatility is higherthan our estimate, optionswillbeoverpriced.Thisleadstothefollowingprinciple:

If implied volatilityis low, such thatoptions generally

appearunderpriced, lookfor spreads with apositive vega. Ifimplied volatility ishigh, such thatoptions generallyappear overpriced,look for spreadswith a negativevega.

The theoretical valuesand deltas in Figure 11-31

have been reproduced inFigures11-32and11-33, butnow prices have beenincluded, reflecting impliedvolatilitiesthatdifferfromthevolatility inputof20percent.The prices in Figure 11-32reflectanimpliedvolatilityof17percent. In this case, onlyspreads with a positive vegawill have a positivetheoreticaledge:

Long straddles

andstranglesShortbutterflies andcondorsRatio spreads—long morethan short(includingshortChristmastrees)Long calendarspreads

Figure11-32

Figure11-33

Theprices inFigure11-33reflectanimpliedvolatilityof23percent.Nowonlyspreadswith a negative vega willhave a positive theoreticaledge:

ShortstraddlesandstranglesLongbutterflies andcondorsRatio spreads

—short morethan long(includinglongChristmastrees)Short calendarspreads

It may seem that if oneencountersamarketwherealloptionsareeitherunderpricedor overpriced, the sensiblestrategies are either longstraddles and strangles or

short straddles and strangles.Such strategies will enable atrader to take apositionwithapositivetheoreticaledgeonboth sides of the spread.Straddles and strangles arecertainly possible strategieswhen all options are toocheap or too expensive. ButwewillseeinChapter13thatstraddlesandstrangles,whileoften having a large positivetheoretical edge, can also beamong the riskiest of all

strategies. For this reason, atrader will often want toconsider other spreads suchas ratio spreads andbutterflies, even if suchspreads entail buying someoverpriced options or sellingsomeunderpricedoptions.

An importantassumption in traditionaltheoretical pricing models isthatvolatilityisconstantoverthe life of an option. The

volatilityinputintothemodelis assumed to be the onevolatility that best describesprice fluctuations in theunderlying instrument overthe life of the option. Whenalloptionsexpireatthesametime, it is this one volatilitythatwill,intheory,determinewhetheraspreadisprofitableor unprofitable. But a tradermayalsobelievethatimpliedvolatilitywillriseorfallovertime.

Because calendarspreads are particularlysensitive to changes inimplied volatility, rising orfalling implied volatility willoften affect the profitabilityof calendar spreads.Consequently, we can addthis corollary to the otherspreadguidelines:

Long calendarspreads are likelyto be profitable

when impliedvolatility is lowbutis expected to rise;short calendarspreads are likelyto be profitablewhen impliedvolatilityishighbutisexpectedtofall.

These are only generalguidelines, and anexperienced trader maydecide to violate them if he

hasreason tobelieve that theimplied volatility will notcorrelatewiththevolatilityofthe underlying contract. Along calendar spread mightstill be desirable in a high-implied-volatilitymarket, butthe trader must make aprediction of how impliedvolatility might change withchanges in realizedvolatility.If the market stagnates, withno movement in theunderlying contract, but the

trader feels that impliedvolatility will remain high, along calendar spread is asensible strategy. The short-termoptionwilldecay,whilethe long-term option willretain its value. In the sameway, a short calendar spreadmight still be desirable in alow-implied-volatility marketif the trader feels that theunderlying contract is likelytomakealargemovewithnocommensurate increase in

impliedvolatility.

Adjustments

Avolatility spreadmaybedelta neutral initially, but thedelta of the position willchange as market conditionschange—as the price of theunderlying contract rises orfalls, as volatility changes,and as timepasses.A spreadthat is delta neutral today is

unlikely to be delta neutraltomorrow. The use of atheoretical pricing modelrequires a trader tocontinuously maintain adelta-neutral positionthroughout the life of thespread. Continuousadjustments are neitherpossible nor practical in therealworldoftrading,sowhena trader initiates a spread, heoughttogivesomethoughtasto how he will adjust the

position.Thereareessentiallyfourpossibilities:

1. Adjust atregularintervals.Intheory, theadjustment processis assumed to becontinuous becausevolatility isassumed to be acontinuousmeasureof the speed of themarket. In practice,

however, volatilityis measured overregular timeintervals, so areasonableapproach is toadjust a position atsimilar regularintervals. If atrader’s volatilityestimateisbasedondailypricechanges,the trader mightadjust daily. If the

estimateisbasedonweekly pricechanges, he mightadjust weekly. Bydoing this, thetraderismakingthebest attempt toemulate theassumptions builtinto the theoreticalpricingmodel.2.Adjustwhentheposition becomes a

predeterminednumber of deltaslongor short.Veryfewtradersinsistonbeing delta neutralall the time. Mosttraders accept thatthisisnotarealisticapproach bothbecause acontinuousadjustment processis physicallyimpossible and

becausenoonecanbe certain that allthe assumptionsand inputs in atheoretical pricingmodel, from whichthe delta iscalculated, arecorrect.Evenifonecould be certainthat all deltacalculations wereaccurate, a tradermight still be

willing to take onsome directionalrisk. But a traderought to know justhow muchdirectional risk heiswillingtoaccept.If he wants topursuedelta-neutralstrategies butbelievesthathecancomfortably livewith a position thatis up to 500 deltas

long or short, thenhe can adjust theposition any timehis delta positionreaches this limit.Unlike the traderwho adjusts atregular intervals, atrader who adjustsbased on a fixednumber of deltascannotbe surehowoften he will needto adjust his

position. In somecases, hemay haveto adjust veryfrequently; in othercases, he may gofor long periods oftime withoutadjusting.

Thenumberof deltas,eitherlong or

short, thatatraderiswilling toacceptwithoutadjustingdependson manyfactors—thetypicalsize ofthetrader’s

positions,hiscapitalization,and histradingexperience.A newindependenttradermay findthat he isuncomfortablewith aposition

that isonly 200deltaslong orshort. Alargetradingfirm mayconsiderapositionthat isseveralthousanddeltas

long orshort asbeingapproximatelydeltaneutral.

3. Adjust by feel.This suggestion isnot madefacetiously. Sometraders have goodmarket feel. Theycan sensewhen the

market is about tomove in onedirection oranother. If a traderhas this ability,there is no reasonwhy he shouldn’tmake use of it.Suppose that theunderlying marketis at 50.00 and atrader is deltaneutral with agamma of –200. If

the market falls to48.00, the tradercanestimatethatheis approximately400 deltas long. If400 deltas is thelimit of the risk heiswillingtoaccept,he might decide toadjust at this point.If, however, he isalso aware that48.00 representsstrong support for

the market, hemightchoosenottoadjust under theassumption that themarket is likely torebound from thesupport level. If heis right, he willhave avoided anunprofitableadjustment. Ofcourse, if he iswrong and themarket continues

downward throughthe support level,he will regret nothaving adjusted.But if the trader isright more oftenthannot,thereisnoreason why heshouldn’t takeadvantage of thisskill.4. Don’tadjustatall.Thisisreallyan

extension of thesecond possibility,adjusting by thenumberofdeltas.Atraderwhodoesnotadjust at all iswilling to accept adirectional riskequal to themaximum numberof deltas that theposition can takeon. If the tradersells five straddles,

the position cantakeonamaximumdelta of ±500. Theappeal of thisapproach is that iteliminates allsubsequenttransaction costs.But, if the positiontakes on a largedelta, thedirectionalconsiderations maybecome more

important than thevolatilityconsiderations. Ifthe position wasinitiated because ofan opinion aboutvolatility, does itmake sense for atrader tosubsequentlychange to anopinion aboutdirection? Usuallynot. If the trader

does not want toadjust the positionbuthealsodoesnotwant directionalconsiderations todominate, the onlychoice left is toclose out theposition. If thetrader decides notto adjust, when heinitiates theposition, he mustdecide under what

conditions he willbe willing to holdthe position andunder whatconditions he willclosetheposition.

SubmittingaSpreadOrder

We noted in Chapter 10that a spread order can often

be executed all at one timeand at one single price. Thisis particularly common inoption markets, wherespreads are quoted with asingle bid price and a singleoffer price regardless of thecomplexity of the spread.Suppose that a trader isinterestedinbuyingastraddleand receives a quote from amarketmakerof6.25/6.75.Ifthe trader wants to sell thestraddle,hewillhavetodoso

at a price of 6.25 (the bidprice); ifhewants tobuy thestraddle, he will have to pay6.75 (the ask price). If thetrader decides that he iswillingtopay6.75,neitherhenor the market maker reallycareswhether the traderpays3.75 for thecalland3.00 forthe put or 2.00 for the calland4.75 for theput or someothercombinationofcallandput prices. The onlyconsideration is that the

prices of the call and puttakentogetheraddupto6.75.

A market maker willalways endeavor to give onebid price and one ask pricefor an entire spread. If thespread is a common type,such as a straddle, strangle,butterfly, or calendar spread,a bid and ask can usually begiven very quickly. Butmarket makers are onlyhuman. If a spread is very

complex, involving severaldifferent options in unusualratios, it may take a marketmaker several minutes tocalculate the value of thespread. Regardless of thecomplexity of a spread,however, the market makerwillmakeanefforttogivehisbest two-sided (bid and ask)market.

Spread orders arecommon in almost all option

markets, whether electronicoropenoutcry.Dependingonthe trading platform, anelectronic exchange willusually allow traders tosubmit bids or offers for themost common types ofspreads—simple call or putspreads, straddles, strangles,and calendar spreads. Morecomplexspreads—butterflies,Christmas trees, and spreadswith unusual ratios—musteither be executed piecemeal

or submitted to a broker forexecution on an open-outcryexchange where an exactdescription of the spread canbe communicated directly tooneormoremarketmakers.

Option spread ordersmay often be submitted withspecific instructions as tohow the spread is to beexecuted.Most commonly, aspread will be submitted aseither a market order (an

order to be filled at thecurrent market price) or alimit order (an order to befilled only at a specifiedprice). But the spread mayalso be submitted as acontingency order withspecialexecutioninstructions.The following contingencyorders, all of which aredefined in Appendix A, areoftenusedinoptionmarkets:

Allornone

FillorkillImmediate orcancelMarket iftouchedMarket oncloseNotheldOne cancelstheotherStop limitorderStoplossorder

A broker executing aspread order is responsiblefor adhering to any specialinstructions that accompanythe order. Unless a trader isfully knowledgeable aboutmarket conditions or has agreat deal of confidence inthe broker who will beexecutingtheorder,itmaybewise to submit specificinstructionswith theorderasto how it is to be executed.Additionally, when one

considers all the informationthat must be communicatedwith a spread order (i.e., thequantity, the expirationmonths, the exercise prices,the type of option, andwhether theorder isabuyorsell), it is easy to see howincorrect information mightinadvertently be transmittedwith the order. For thisreason, it is also wise todouble-check all ordersbefore submitting them for

execution.Optiontradingcanbe difficult enough withoutthe additional problems ofmiscommunication.

1ThisisnotnecessarilytrueforbutterfliesconsistingofAmericanoptions,whereearlyexerciseisapossibility.Asureprofitwouldexistonlyifonecouldbecertainofcarryingthepositiontoexpiration.2Butterfliesandcondorsfallunderthegeneralcategoryofstrategiesknownaswingspreads.3ThetermsbackspreadandfrontspreaddatefromtheearlydaysofoptiontradingintheUnitedStatesbutarenowusedinfrequentlyexceptbysomeoldertraders.Mosttraderssimplyrefertothesestrategiesasratiospreads,specifyingwhethermoreoptionsarepurchasedorsoldandtheratiooflong

toshortoptions.4Thetermladdermayalsorefertoatypeofexoticoption.5Intheearlydaysoffloortradingonoptionexchanges,expirationmonthswerelistedhorizontallyontheexchangedisplayboards—hencethetermhorizontalspreadforstrategiesconsistingofoptionswithdifferentexpirationmonths.6Tobemoreexact,at-the-forwardoptionstendtohavedeltasclosestto50.Forthisreason,atradermightpreferacalendarspreadthatconsistsofat-the-forwardoptions.7Wearemakingtheassumptionherethattheimpliedvolatilityofall

expirationsisthesame.Iftheimpliedvolatilitydiffersacrossexpirationmonths,alongtimebutterflymightinfactresultinacredit.8Interestratescan,ofcourse,affecttherelativevalueofdifferentfuturesmonths.Asnoted,wecanoffsetthisriskbytradingafuturesspreadalongwiththefuturesoptioncalendarspread.

12

BullandBearSpreads

Although delta-neutralvolatility trading is thefoundation of theoreticaloptionpricing,thereisnolawthat requires a trader toinitiate andmaintain a delta-

neutralposition.Manytradersprefer to trade fromabullishor bearish perspective. Thetrader who wishes to take adirectional position has thechoice of doing so in eitherthe underlying instrumentitself, buying or selling afutures contract or stock, orby taking the position in theoption market. If the tradertakesadirectionalpositioninthe option market, he muststillbeawareofthevolatility

implications. Otherwise, hemay be no better off, andperhaps even worse, than ifhe had taken an outrightposition in the underlyingcontract.

NakedPositions

Because the purchase ofcalls or the sale of puts willcreate a positive deltaposition and the sale of calls

orpurchaseofputswillcreatea negative delta position, wecan always take a directionalpositioninamarketbytakinganappropriatenakedpositionin either calls or puts. Ifimplied volatility is high,wecan sell puts to create abullishpositionorsellcallstocreate a bearish position. Ifimplied volatility is low, wecan buy calls to create abullishpositionorbuyputstocreateabearishposition.

The problem with thisapproach is that there isverylittle margin for error. If wepurchaseoptions,wewilllosemoneynotonlyif themarketmovesinthewrongdirectionbutalso if themarket fails tomove fast enough to offsettheoption’stimedecay.Ifweselloptions,timewillworkinour favor, but we face theprospect of unlimited risk ifthe market moves violentlyagainst us. An experienced

trader will prefer a strategythat improves therisk-rewardtradeoff by looking forpositions with the greatestpossible margin for error.This philosophy applies noless to directional strategiesthantovolatilitystrategies.

BullandBearratioSpreads

Consider a situationwherewe believe that impliedvolatility is too high. Onepossible strategy is a ratiospread where more optionsaresoldthanpurchased.Withtheunderlyingmarketat101,ten weeks remaining to Juneexpiration, andavolatilityof30 percent, a June 100 callhas a delta of 56 and a June110callhasadeltaof28.1Adelta-neutral spread might

consistof

Buy 1 June100call(56)Sell 2 June110calls(28)

Because the spread isdelta neutral, it has noparticular preference forupward or downwardmovement in the underlyingmarket.

Now suppose that we

believe that this ratio spreadis a sensible strategy, but atthe same time, we are alsobullish on the market. Thereis no law that requires us todo this spread in a delta-neutral ratio. If we want thespread to reflect a bullishsentiment, we might adjusttheratioslightly

Buy 2 June100calls(56)Sell 3 June

110calls(28)

We have essentially thesame ratio spread,butwith abullish bias. This is reflectedinthetotaldeltaof+28.

There is, however, animportantlimitationifweusea ratio strategy to create abullishorbearishposition.Inour example,we are initiallybullish, but the position isstill a ratio spread with anegative gamma. If the

underlying market moves uptoo quickly, the spread caninvert from a positive to anegative delta. If the marketrises far enough, to 130 or140, eventually all optionswill go deeply into themoney,andthedeltasofboththe June 100 and June 110calls will approach 100. Wewill be left with a deltaposition of –100. Eventhoughwemaybe correct inour bullish sentiment, the

volatility characteristics ofthe position will eventuallyoutweigh any considerationsofmarketdirection.Thedeltavalues of both ratios, 1 × 2and 2 × 3, with respect tochanges in the underlyingprice,areshowninFigure12-1.

Figure12-1Deltaofaratiospreadastheunderlyingpricechanges.80

Thedeltacanalsoinvertin a ratio spread in whichmore options are purchasedthan sold. Unlike a negativegamma position, where theinversion is caused by swiftprice movement in theunderlyingcontract, this typeof ratio spread can invertwhen volatility declines ortime passes. Suppose thatconditionsare the sameas inour preceding example, but

we believe that impliedvolatilityistoolow.Nowwemightdothefollowingdelta-neutralstrategy:

Buy 2 June110calls(28)Sell 1 June100call(56)

However, if we arebullishonthemarket,wecan,as in the preceding example,adjust the ratio to reflect this

sentiment

Buy 3 June110calls(28)Sell 1 June100call(56)

The delta position of +28reflectsthisbullishbias.

We know from Chapter9 that as time passes or asvolatility declines, all deltasmove away from 50. If timepasses with no movement in

the underlying contract, thedeltaoftheJune100callwilltendtorise,whilethedeltaoftheJune110callwill tendtodecline. If, after a period oftime,thedeltaoftheJune100call rises to 70 and the deltaof the June 110 call falls to15, the delta of the positionwillnolongerbe+28butwillinstead be –25. Because thisstrategyisavolatilityspread,the primary consideration, asbefore, is thevolatilityof the

market.Only secondarily arewe concerned with thedirectionofmovement. Ifweoverestimate volatility andthe market moves moreslowly than expected, thespread,whichisinitiallydeltapositive, can instead becomedelta negative. The deltavalues of bothpositionswithrespecttothepassageoftimeareshowninFigure12-2.

Figure12-2Deltaofaratiospreadastimepasses.

BullandBearButterfliesandCalendarSpreads

Butterflies and calendarspreads can also be executedinawaythatreflectsabullishorbearishbias.Aswith ratiospreads, though, their deltacharacteristics can invert asmarketconditionschange.

With high impliedvolatility and the underlyingcontract at 100, we mightcreateadelta-neutralpositionby buying the June95/100/105callbutterfly(buya 95 call, sell two 100 calls,buya105call).Wehopethattheunderlyingwillstaycloseto 100 so that at expirationthebutterflywillwidentoitsmaximum value of 5.00. If,however, we want to buy abutterfly but are also bullish

onthemarket,wecanchooseabutterflyinwhichtheinsideexercise price is above thecurrent price of theunderlying contract. If theunderlyingiscurrentlyat100,we might choose to buy theJune 105/110/115 callbutterfly. Because thispositionwantstheunderlyingcontract to be at the insideexercise price of 110 atexpiration and it is currentlyat 100, the position is a

bullishbutterfly.Thiswillbereflected in the positionhavingapositivedelta.

Unfortunately, if theunderlying market moves uptoo far, say, to 120, thebutterfly will invert from apositive to a negative deltaposition. Now we want themarket to fall back from120to 110. Whenever theunderlying market is below110, the position will be

bullish; whenever theunderlying market is above110, the position will bebearish.

Conversely, if we arebearish,wecanchoosetobuyabutterflyinwhichtheinsideexercise price is below thecurrent price of theunderlyingmarket.Butagain,ifthemarketmovesdowntooquicklyandgoes through theinside exercise price, the

position will invert from anegative to a positive delta.The delta position of abutterfly with respect tochanges in the underlyingpriceisshowninFigure12-3.

Figure12-3Deltaofalongbutterflyastheunderlyingpricechanges.

We can also choose abullish or bearish calendarspread. A long calendarspread always wants theshort-term option to expireexactlyat themoney.A longcalendar spread will beinitiallybullishiftheexerciseprice is above the currentprice of the underlyingcontract.2 With theunderlying at 100, theJune/April 110 calendar

spread (buy the June 110option, sell the April 110optionof the same type)willbe bullish because the traderwillwanttheunderlyingpriceto rise to 110 by Aprilexpiration.TheJune/April90calendarspread(buytheJune90 option, sell the April 90optionof the same type)willbe bearish because the traderwillwanttheunderlyingpriceto fall to 90 by Aprilexpiration. But like a long

butterfly, a long calendarspreadhasanegativegamma.If the underlying contractmoves through the exerciseprice, thedeltawill invert. Ifthemarketmovesfrom100to120, the June/April 110calendar spread, which wasinitially bullish, will becomebearish. If themarketmovesfrom 100 to 80, theJune/April 90 calendarspread, which was bearishinitially,willbecomebullish.

The delta values of longcalendar spreadswith respectto changes in the underlyingpriceareshowninFigure12-4.3

Figure12-4Deltaofalongcalendarspreadastheunderlyingpricechanges.

VerticalSpreads

Although we may take abullishorbearishpositionbychoosing an appropriate ratiospread, butterfly, or calendarspread, in each of thesepositions,volatilityisstilltheprimary concern. We can beright about market direction,but if we are wrong aboutvolatility, thespreadmaynot

retain the directionalcharacteristics that weoriginallyintended.

If we want to focusprimarily on the direction ofthe underlying market, wemight look for a spread inwhich the directionalcharacteristicsaretheprimaryconcern and the volatilitycharacteristics are only ofsecondary importance. Wewouldliketobecertainthatif

the spread is initially bullish(deltapositive),itwillremainbullish under all possiblemarketconditions,andifitisinitially bearish (deltanegative), it will remainbearish under all possiblemarketconditions.

Themost common classof spreads that meet theserequirements are simple calland put spreads. One optionispurchasedandoneoptionis

sold, where both options arethe same type (either bothcallsorbothputs)andexpireatthesametime.Theoptionsare distinguished only bytheirdifferentexerciseprices.Such spreads may also bereferredtoascreditanddebitspreads or vertical spreads.4Typical spreads of this typemightbe

Buy 1 June100call

Sell 1 June105call

or

Buy 1December 105putSell 1December 95put

Simple call and putspreads are initially either

bullish or bearish, and theyremain bullish or bearish nomatterhowmarketconditionschange. Two options thathavedifferent exercisepricesbut that are otherwiseidentical cannot haveidentical deltas. In the firstexample, where the trader islongaJune100callandshortaJune105call, theJune100call will always have a deltagreaterthantheJune105call.If both options are deeply in

themoney or very far out ofthe money, the deltas maytend toward 100 or 0. Buteven then, the June 100 callwill have a delta that isslightly greater than that ofthe June 105 call. In thesecond example, no matterhow market conditionschange, the December 105putwillalwayshaveagreaternegative delta than theDecember95put.

At expiration, a call orputverticalspreadwillhaveaminimum value of 0 if bothoptions areoutof themoneyand amaximumvalue of theamount between exerciseprices if both options are inthemoney. If the underlyingcontract is below 100 atexpiration, the June 100/105call spread will be worthlessbecause both options will beworthless. If the underlyingcontract is above 105, the

spread will be worth 5.00becausetheJune100callwillbe worth exactly five pointsmore than the June 105 call.Similarly, the March 95/105putspreadwillbeworthlessifthe underlying market isabove 105 at expiration, andit will be worth 10.00 if themarketisbelow95.

Because a verticalspread at expiration willalwayshaveavaluebetween

0 and the amount betweenexercise prices, a trader canexpect the price of such aspread to be somewherewithin this range.A 100/105call vertical spreadwill tradefor some amount between 0and 5.00; a 95/105 putvertical spread will trade forsome amount between 0 and10.00. The exact value willdepend on the likelihood ofthe underlying marketfinishing below the lower

exercise price, above thehigher exercise price, orsomewhereinbetween.Ifthemarket is currently at 80 andgives little indication ofrising, the price of the100/105 call vertical spreadwill be close to 0, while theprice of the 95/105 putvertical spread will be closeto 10.00. If the market iscurrently at 120 with littlelikelihoodthatitwillfall,theprice of the 100/105 call

vertical spread will be closeto5.00,whilethepriceofthe95/105 put vertical spreadwillbecloseto0.

If we want to do asimple bull or bear verticalspread, we have essentiallyfour choices. If we arebullish,wecanchooseabullcall spread or a bull putspread; ifwe are bearish,wecanchooseabearcallspreador a bear put spread. For

example,

Ifwearebullish,wecanbuya100callandsella105call,orbuya100putandsell

a105put (inbothcases,buythe lower exercise price andsell the higher). If we arebearish,wecanbuya105calland sell a 100 call, or buy a105put and sell and100put(in both cases, sell the lowerexercise price and buy thehigher). This may seemcounterintuitive because oneexpects spreads that consistofputstohavecharacteristicsthatare theoppositeof thosethat consist of calls. But

regardless of whether aspread consists of calls orputs,whenever a trader buysthe lower exercise price andsells the higher exerciseprice, the position is bullish,and whenever a trader buysthehigherexercisepriceandsellsthelowerexerciseprice,thepositionisbearish.

We can see why this istruebyconsideringeitherthedeltas of the position or the

potential profit and loss(P&L) for the position.Consider the two examplebullspreads:

Both spreads must have apositive delta. The June 100call has a greater positivedelta than the June 105 call.

The June 105 put has agreaternegativedeltathantheJune 100 put. Multiplyingwith a positive sign for apurchase and a negative signfor a sale and adding up thedeltas give a total positivedeltaineachcase.

In terms of potentialprofit or loss, the call spreadwill be done for a debit (theJune 100 call will cost morethan the June 105 call) and

will expand to its maximumvalue of 5.00 if theunderlying contract is above105 at expiration. The putspread will be done for adebit (the June 100 put willcost less than the June 105put) but will collapse to 0 ifthe underlying contract isabove105atexpiration.Eachspread wants the underlyingto rise above 105, so eachspreadmustbebullish.

Not only will the totaldelta be very similar for callandputspreadsthatexpireatthesametimeandthatconsistof the same exercise prices,buttheprofitorlosspotentialfor each spread, whether acallspreadorputspread,willbe approximately the same.5The expiration P&L profilesfor simple bull and bearspreadsare shown inFigures12-5and12-6.

Figure12-5Bullspread.

Figure12-6Bearspread.

Giventhemanydifferentexercisepricesandexpirationmonthsavailable,howcanwechoosethebullorbearspreadthat best reflects ourdirectional expectations andthat gives us the best chanceto profit from thoseexpectations?

Because options havefixed expiration dates, atrader who wants to useoptions to take advantage of

an expected market movemust first determine his timehorizon. Is the movementlikely to occur in the nextmonth? In the next threemonths? In the next ninemonths?IfitiscurrentlyMayand the trader foreseesupward movement butbelievesthatthemovementisunlikely to occur within thenext twomonths, it does notmake much sense to take aposition in June or July

options. If his expectationsarelongterm,hemayhavetotake his position inSeptemberorevenDecemberoptions. Of course, as hemoves farther out in time,market liquiditymaybecomea problem. This is a factorthathewill have to take intoconsideration.

Next, a trader will haveto decide just how bullish orbearish he is. Is he very

confident and thereforewilling to take a very largedirectionalposition?Or isheless certain and willing totake only a limited position?Two factors determine thetotal directionalcharacteristicsoftheposition:

1.Thedeltaoftheselectedspread2. The size inwhich the spread isexecuted

A trader who wants totake a position that is 500deltas long (equivalent topurchasing five underlyingcontracts) can choose aspread that is 50 deltas longand execute it 10 times. Orthe trader can choose adifferent spread that is only25 deltas long but execute it20 times. Both strategiesresult in a position that islong500deltas.

Ingeneral, if all optionsexpire at the same time andareclosetoatthemoney,thegreater the amount betweenexercise prices, the greaterwillbe thedeltavalueof thespread.A 95/110 bull spreadwill be more bullish than a95/105 bull spread, whichwill, in turn, bemore bullishthan a 95/100 bull spread.6Moreover, increasing theamount between exercise

prices will also increase thespread’s maximum potentialprofit or loss. This is showninFigure12-7.

Figure12-7Astheexercisepricesbecomefartherapart,thespreadtakesongreaterbullishorbearishcharacteristics.

Onceatraderdecidesontheoptionexpirationinwhichto take his directionalposition, he must decidewhichspecificspread isbest.Thatis,hemustdecidewhichexercise prices to use.Consider the following tableof theoretical values anddeltas:

Supposethatwewanttodoa bull call spread with theseoptions.Onechoiceistobuythe 95 call and sell the 100

call. A second choice is tobuy the 100 call and sell the105 call. Which spread isbest?

The theoretical valueanddeltaforeachspreadare

In theory, both spreadsseem to be equally bullishbecausetheyarebothlong20deltas. But the 100/105spread,with a value of 1.92,appearstobecheaperthanthe95/100 spread, with a valueof 2.91. From this we mightconclude that the 100/105spread represents the bettervalue. But is the spread’svalue theonlyconsideration?The value of a strategy isonly important if we can

compare it with the price ofthe strategy. But nowherehave we said anything aboutprice.

From an option trader’spointofview,thepriceofanoption or strategy isdetermined by the impliedvolatility in the marketplace.In this example, our bestestimateofvolatilityoverthelifeof theoptionsmaybe25percent, butwhatwill be the

prices of the options if theimplied volatility is eitherhigher or lower than 25percent? Let’s expand ourtabletoincludeoptionvaluesat volatilities of 20 and 30percent (delta values are inparentheses).

If implied volatility in themarketplaceis20percent,theprices of the 95/100 spreadand the 100/105 spread willbe 3.06 and 1.82,

respectively. If our bestvolatility estimate is 25percent, we have a choice.Wecanpay3.06foraspreadthatwebelieve isworth2.91(the 95/100 spread), or wecanpay1.82foraspreadthatwebelieveisworth1.92(the100/105spread).Ifcreatingapositive theoretical edge isourgoal, the100/105spread,withatheoreticaledgeof.10,makes more sense than the95/100 spread with its

negative theoretical edge of–.15.

Now suppose thatimplied volatility in themarketplace is 30 percent.The prices of the 95/100spread and the 100/105spread are 2.81 and 1.98,respectively.Againwehaveachoice.Wecanpay2.81foraspreadthatisworth2.91(the95/100spread),orwecanpay1.98foraspreadthatisworth

1.92 (the 100/105 spread).The 95/100 spread, with itspositive theoretical edge of.10,isnowthebetterchoice.

Even though bothspreads have the same deltavalues, under one volatilityscenario, we seem to preferthe 95/100 spread, whileunderadifferentscenario,weseem to prefer the 100/105spread. The reason becomesclear if we recall one of the

basiccharacteristicsofoptionevaluation introduced inChapter6:

Ifweconsiderthreeoptions—inthemoney,atthemoney,andoutofthemoney—optionthatareidenticalexceptfortheirexerciseprices,theat-the-moneyoptionisalwaysthemost

sensitiveintotalpointstoachangeinvolatility.

If all options appearoverpriced because webelieve that impliedvolatilityistoohigh,intotalpoints,theat-the-money option will bethe most overpriced. If alloptions appear underpricedbecause we believe thatimplied volatility is too low,in total points, the at-the-

money option will be themost underpriced. Thischaracteristic leads to a verysimple rule for choosing bullandbearverticalspreads:

Ifimpliedvolatilityislow,thechoiceofspreadsshouldfocusonpurchasingtheat-the-moneyoption.Ifimpliedvolatilityishigh,thechoice

shouldfocusonsellingtheat-the-moneyoption.

Nowwecanseewhythe100/105callspreadisabettervalue if implied volatility is20 percent, whereas the95/100 spread is a bettervalue if implied volatility is30 percent. If impliedvolatility is low (20percent),we prefer to buy the at-the-money (100) call. Having

done this, we have only onechoice ifwewant tocreateabullspread—wemustselltheout-of-the-money (105) call.Ontheotherhand, if impliedvolatilityishigh(30percent),we want to sell the at-the-money (100) call. Havingdonethis,weagainhaveonlyone choice if we want tocreate a bull spread—wemust buy the in-the-money(95)call.

The same principle isequallytrueforbullandbearput spreads.Wealwayswantto focus on the at-the-moneyoption, buying the at-the-money put when impliedvolatility is low and sellingthe at-the-money put whenimplied volatility is high.This is confirmed in thefollowing table (delta valuesareinparentheses):

Supposethatwewanttodo a bear put spread whenimplied volatility is low. Inthiscase,wewant tobuythe

at-the-money (100) put.Having done this, we areforced to sell the out-of-the-money (95) put to create ourbear spread (buy the higherexerciseprice,sellthelower).We will pay 1.94 for thespread, but the spread isworth2.09.Theresultwillbeadelta positionof –20 and apositive theoretical edge of.15.

Notice that in every

case, whether in a low-volatility or high-volatilityenvironment, the spread thatincludes the in-the-moneyoption always has a higherprice than the spread thatincludes the out-of-the-moneyoption.Tounderstandwhy, consider the result ofchoosing between a 95/100anda100/105bullcallspreadunder three differentscenarios. In scenario 1, themarket rises and is at 110 at

expiration. If this happens,both spreads will show aprofit because they will bothwiden to their maximumvalue of 5.00. In scenario 2,themarketdropsandisat90at expiration. Now bothspreads will show a lossbecause they will bothcollapse to 0. Finally,consider the case where theunderlyingmarketfailstorisebut also does not fall. Itsimply remains at 100 until

expiration. If this happens,the 100/105 spread willcollapse to 0, while the95/100 spread will widen toits maximum value of 5.00.The 95/100 spread is alwaysmore valuable than the100/105 spread because itprofits in more cases. The100/105 spread needs themarket to rise to show aprofit. The 95/100 spreaddoes not need the market torise; it just needs for the

market not to fall. Becausethe 100/105 spread requiresmovement, it has a positivegamma and, consequently, anegative theta. Itwilldeclinein value as time passes. The95/100spreadwillprofitevenifthemarketsitsstill.Ithasapositive theta and,consequently, a negativegamma.

Note also the results ifthemarket doesmove. If the

market rises to 110, bothspreads will show a profit,but the 100/105 spread willshowagreaterprofitbecauseit was purchased at a lowerprice. If the market falls to90, both spreadswill show aloss, but the 100/105 spread,because of its lower price,will show a smaller loss. Ifthere is a greater likelihoodthatthemarketwillmove,wewill always prefer the100/105 spread. We will

maximize our profits whenwe are right, and we willminimizeourlosseswhenwearewrong. The likelihood ofmovementwilldependonourestimate of volatility. If ourestimate is higher than theimplied volatility, we aresaying that there is a greaterlikelihood of movement, sowepreferthe100/105spread.Ifourestimateofvolatilityislower than the impliedvolatility, we are saying that

there isa lower likelihoodofmovement, so we prefer the95/100spread.

Even though we havefocused on the at-the-moneyoption, a trader is notrequired to execute a bull orbearspreadbyfirstbuyingorselling the at-the-moneyoption. Such spreads alwaysinvolve two options, and atrader can choose to eitherexecute the complete spread

inone transactionor leg intothe spread by trading oneoptionat a time. In the lattercase, a trader may decide totradethein-the-moneyorout-of-the-moneyoptionfirstandtradetheat-the-moneyoptionat a later time. This is adecision that a trader mustmake based on practicalconsiderations.Butregardlessof how the spread isexecuted, the trader shouldfocus on the at-the-money

option, either buying itwhenimplied volatility is low orselling it when impliedvolatilityishigh.

Inpractice, it isunlikelythat one option will beexactlyatthemoney.Ifthereis no exactly at-the-moneyoption, a trader can focus onan option that is closer to atthemoney. If the underlyingmarket is at 103, with 95,100, 105, and 100 calls

available,itislogicaltofocuson the 105 call because it isclosest to at the money. Ifimplied volatility is low, atrader will want to buy the105 call; if implied volatilityis high, a traderwillwant tosellthe105call.Hecanthentrade a different option inorder to create a bull or bearverticalspread.

Nordoesatraderhavetoinclude the option that is

closest to the money as partof his spread. A trader whohas a strong directionalopinion can choose a spreadwhere both options are veryfar out of themoney or verydeeply in the money. Thedelta values of such spreadswill be very low, but thetrader can create a highlyleveraged position byexecuting each spread manytimes. For example, with theunderlying market at 100, a

traderwhoisstronglybullishmight buy the 115/120 callspread (assuming that suchexercisepricesareavailable).Thecostofthisspreadwillbevery low because there is ahigh probability that thespread will expire worthless.But the trader will also beable to execute the spreadmanytimesbecauseofitslowcost. If he is right and themarket does rise above 120,the spread will widen to its

maximum value of 5.00,resulting in a very largeprofit. Regardless of theexercise prices chosen, ifimplied volatility is low, thetrader should buy an optionthat is closer to the money,and if implied volatility ishigh,thetradershouldsellanoption that is closer to themoney.

Our choice of bull orbear strategies has focused

thus far on the at-the-moneyoption, typically the optionwhose delta is closest to 50.This does indeed tend to bethe case for options onfutures. In other markets,though, the at-the-moneyoptionmaynotbe theoptionwith a delta closest to 50because, as discussed inChapter 5, the theoreticalvalue of an option dependsnotonthecurrentpriceoftheunderlyingcontractbutonthe

forward price. For thisreason, the choice of bull orbear spreads should reallyfocus on the at-the-for-wardoption.Especiallyinthestockoptionmarket,ifinterestratesare high and there is asignificantamountof time toexpiration, the at-the-forwardoption may have an exerciseprice that is considerablyhigher than the current stockprice. Having noted thisdistinction, for practical

purposes,atraderwillnotgotoofarwrongifhefocusesonthe at-the-money option,buying it when impliedvolatilityislowandsellingitwhen implied volatility ishigh.

Before concluding ourdiscussion of bull and bearspreads, it will be useful tolook at graphs of thetheoretical value, delta,gamma,vega,and theta fora

typicalbullverticalspread,asshown in Figures 12-8through 12-13. The readershouldtakesometimetolookat these graphs not onlybecause they highlight someof the importantcharacteristics of this verycommon class of spreads butalso because they serve asexamplesofsomeofthemoreimportant characteristics ofrisk measurement discussedin Chapter 9. This will be

especially helpful when wetake a closer look at riskanalysisinlaterchapters.

Figure12-8Valueofabullspreadastimepassesorvolatilitydeclines.

For the graphs oftheoretical value, delta,gamma, and vega (Figures12-8 through 12-11), theeffect of time passing orvolatilitydeclining issimilar.For the theta, however, thereare slight differences, soseparate theta graphs fordeclining volatility (Figure12-12) and the passage oftime (Figure 12-13) areshown. Note also that the

maximum gamma, vega, andthetaforverticalspreadstendtooccurwhentheunderlyingprice is either just below thelowest exercise price or justabove the highest exerciseprice.

Figure12-9Deltaofabullspreadastimepassesorvolatilitydeclines.

Figure12-10Gammaofabullspreadastimepassesorvolatilitydeclines.

Figure12-11Vegaofabullspreadastimepassesorvolatilitydeclines.

Figure12-12Thetaofabullspreadasvolatilitydeclines.

Figure12-13Thetaofabullspreadastimepasses.

Finally, we might askwhy a trader with adirectional opinion mightprefer a vertical spread to anoutrightlongorshortpositionin the underlying instrument.For one thing, a verticalspreadismuchlessriskythananoutrightposition.A traderwhowants to take a positionthat is 500 deltas long caneither buy 5 underlyingcontracts or buy 25 vertical

callspreadswithadeltaof20each.The25vertical spreadsmay sound riskier than 5underlyingcontracts,untilweremember that a verticalspread has limited risk,whereas the position in theunderlying has open-endedrisk. Of course, greater riskalsomeansgreater reward.Atrader with a long or shortposition in the underlyingmarketcanreaphugerewardsif the market makes a large

move in his favor. Bycontrast, the verticalspreader’sprofitsare limited,buthewillalsobemuchlessbloodied if themarketmakesan unexpected move in thewrongdirection.

Ignoring interestconsiderations, the only wayto profit from trading theunderlying contract is to beright about direction. If webuy the underlying contract,

the market must rise. If wesell the underlying, themarket must fall. But, withoptions, one need notnecessarily be right aboutmarket direction. Optionsalso offer the additionaldimension of volatility.Depending on the exerciseprices thathavebeenchosen,if the trader has correctlyestimated volatility, a bullspreadcanbeprofitableifthemarketfailstoriseorinsome

cases even if it declines. Abear spread can be profitableevenifthemarketfailstofall.This flexibility is just one ofthefactorsthathasleadtothedramatic growth in optionmarkets.

1Togeneralizethisandsubsequentexamplesandtoeliminatethedifferencesbetweenstockoptionsandfuturesoptions,wewillassumeaninterestrateof0.2Inthefuturesmarket,thesituationmaybecomplicatedbythefactthatdifferentfuturesmonthsmaybetradingatdifferentprices.Insteadofchoosingatraditionalcalendarspread,wherebothoptionshavethesameexerciseprice,thetradermayhavetochooseadiagonalspreadtoensurethatthepositioniseitherbullish(deltapositive)orbearish(deltanegative).3Figures12-3and12-4areverysimilar,andonemightconcludethatthe

characteristicsofbutterfliesandcalendarspreadsaresimilar.Butthisisonlytruewithrespecttochangesintheunderlyingprice,asreflectedinthedelta.Thespreadswillreactquitedifferentlytothepassageoftimeandchangesinimpliedvolatility.4Intheearlydaysoftradingonoptionexchanges,exercisepriceswerelistedverticallyontheexchangedisplayboards—hencethetermverticalspreadforstrategiesconsistingofoptionswithdifferenceexerciseprices.5WeareassumingforthemomentthatalloptionsareEuropean,withnopossibilityofearlyexercise.6Thisisnotnecessarilytrueforvery

deeplyin-the-moneyspreadsorveryfarout-of-the-moneyspreads.Insuchcases,thedeltasofbothoptionsmaybeverycloseto100or0,soseparatingtheexercisepriceswillhavelittleeffectonthetotaldeltaofthespread.

13

RiskConsiderations

When choosing a strategy,a trader must always try tofind a reasonable balancebetween two opposingconsiderations—reward andrisk. Ideally, a trader would

like the greatest possibleprofitat thesmallestpossiblerisk. In the real world,however, high profit usuallygoes hand in handwith highrisk,whilelowriskgoeshandinhandwithlowprofit.Howshould a trader balance thesetwo considerations?Certainly, a strategy shouldhave an expected profit thatmakes itworth executing.Atthe same time, the riskassociated with the strategy

must be kept withinreasonable bounds. Andwhatever the risk, it shouldneverbegreater thanwhat iscommensurate with thepotentialreward.

In option trading, thereward is typically expressedintermsoftheoreticaledge—the average profit resultingfromastrategy,assumingthatthe trader’s assessment ofmarket conditions is correct.

Unfortunately, although thetheoretical edge can beexpressedasonenumber, theriskassociatedwithanoptionposition cannot be expressedin the same way. We knowthat options are subject tomany different risks. If wewanttointelligentlyanalyzeastrategy,wemaybe requiredtoconsideravarietyofrisks.Astrategymaybereasonablewithrespecttosomerisksbutunacceptable with respect to

others.Before proceeding

furtherinourdiscussion,let’ssummarize the basic risksassociated with an optionposition:

Delta (Directional) Risk.The risk that the underlyingmarket will move in onedirection rather than another.When we create a positionthat is delta neutral, we aretrying to ensure that initially

thepositionhasnobiasas tothe direction in which theunderlying instrument willmove. A delta-neutralposition does not necessarilyeliminate all directional risk,but the position is typicallyimmune to directional riskswithinalimitedrange.

Gamma (Curvature) Risk.The risk of a large move inthe underlying contract,regardless of direction. The

gammaposition is ameasureofhowsensitiveapositionisto such moves. A positivegamma position does notreally have gamma riskbecause such a positionwill,in theory, increase in valuewith movement in theunderlying contract. Anegative gamma position,however,canquickly lose itstheoretical edge with a largemove in the underlyingcontract.Theeffectofsucha

move must always be aconsiderationwhenanalyzingtherelativemeritsofdifferentpositions.

Theta (Time Decay) Risk.This is the opposite side ofgamma risk. Positions withpositivegammabecomemorevaluablewith largemoves inthe underlying. But ifmovement helps, the passageof time hurts. A positivegamma always goes hand in

handwith anegative theta; anegative gamma always goeshand in handwith a positivetheta. A trader with anegative theta must considertheriskintermsofhowmuchtime can pass before thespread’s theoretical edgedisappears. The positionwants movement, but if themovementfails tooccuroverthe next day, next week, ornext month, will the spread,intheory,stillbeprofitable?

Vega(Volatility)Risk.Theriskthatthevolatilitythatweinput into the theoreticalpricing model will beincorrect.Ifweusethewrongvolatility,wehave thewrongprobability distribution forthe underlying contract.Because some positions havea positive vega and are hurtby declining volatility andsome positions have anegativevegaandarehurtbyrising volatility, vega

represents a risk to everyposition. A trader mustalways consider how muchthe volatility can moveagainst him before thepotential profit from aposition disappears. Mosttraders prefer to interpretvega as the sensitivity of aposition to a change inimplied volatility. If impliedvolatility rises or falls, howwill thatchange thepricesofoptions that make up a

position? If the changes hurttheposition,willthetraderbeable to maintain the positionin thefaceofadversemarketconditions?

Rho (Interest-Rate) Risk.The risk that the interest ratewill change over the life ofthe option.Apositionwith apositiverhowillbehelpedbyrising interest rates and hurtby declining rates; a positionwith a negative rho has just

the opposite characteristics.1Except for special situations,the interest rate is the leastimportantof theinputs intoatheoretical pricing model.Consequently, rho is usuallyconsideredtheleastimportantoftheriskmeasures.

Let’slookattherelativeimportance of the variousrisks by considering severaldifferentoptionstrategies.

VolatilityRisk

For an option trader,volatility risk comes in twoforms—the risk that he hasincorrectly estimated therealized volatility of theunderlying contract over thelifeofa strategyand the riskthat implied volatility in theoption market will change.Anyspreadthathasanonzerogammaorvegahasvolatility

risk.Consider the prices and

values in the theoreticalevaluationtableinFigure13-1.2 What types of volatilitystrategiesmight be profitableunder these conditions?Whether we compare optionprices with their theoreticalvalues or the impliedvolatilitiesoftheoptionswiththe volatility input of 18percent, we will reach the

same conclusion: all optionsare overpriced. Recalling thegeneralguidelines inChapter11, under these conditions, atrader will want to considerspreadswithanegativevega:

Figure13-1

ShortstraddlesandstranglesCall or putratiospreads—sell more thanbuyLongbutterfliesShort calendarspreads

Which of these categoriesis likely to represent thebest

spreading opportunity? Andwithin each category, whichspecific spread mightrepresentthebestrisk-rewardtradeoff?

For the moment, let’sfocus on May options.Having eliminated thepossibility of calendarspreads, any spread wechoose will necessarily havea negative gamma andnegative vega. But with 12

different May optionsavailable(6callsand6puts),it’s possible to construct anumber of spreads that fallinto this category. How canwe make an intelligentdecision about which spreadmightbebest?

Initially, let’s considerthe three strategies shown inFigure 13-2: a short straddlethat has been done in a 4:3ratiotomakeitclosertodelta

neutral(Spread1),aratiocallspread(Spread2),andalongputbutterfly(Spread3).Eachspread is approximatelydeltaneutral and, as we wouldexpect, has a positivetheoreticaledge.Howcanweevaluatetherelativemeritsofeachspread?

Figure13-2

Initially, it may appearthatSpread1 is best becauseit has the greatest theoreticaledge.Ifthevolatilityestimateof 18 percent turns out to becorrect,Spread1willshowaprofit of 6.65, Spread 2 aprofitof1.80,andSpread3aprofitofonly.60.

But is theoretical edgeour only concern? If this istrue, we can simply do eachspread in larger and larger

size to make the theoreticaledge as big as we want.Instead of doing Spread 2 inour original size of 10 × 20,we can increase the sizefivefoldto50×100.Thiswillalso increase the theoreticaledge fivefold to 9.00. Thisostensibly makes Spread 2 abetterstrategythanSpreads1and 3. Clearly, theoreticaledge cannot be the onlyconsideration.

Theoretical edge is onlyan indication of what weexpect toearn ifweare rightabout market conditions.Becausethereisnoguaranteethatwewillberight,wemustgive at least as muchconsideration to the questionofrisk.Ifwearewrongaboutmarketconditions,howbadlymightwebehurt?

In order to focus on therisk considerations, let’s

change the size of Spreads 2and3sothat their theoreticaledge is approximately equalto that of Spread 1. We canachievethisbyincreasingthesize of Spread 2 to 35 × 70and increasing the size ofSpread3to100×200×100.Thespreadsintheirnewsizeswith their total theoreticaledgeandrisksensitivitiesareshown in Figure 13-3. Withall three spreads having asimilar theoretical edge, we

can now focus on the risksassociatedwitheachspread.

Figure13-3

As with all volatilitypositions, one considerationis the possibility of a largepricemove in the underlyingcontract. Because eachstrategy has a negativegamma, any largemovewillhurt the position. But willeach spread be hurt to thesamedegree?BecauseSpread2 has the smallest negativegamma (–165.5), we mightconclude that it has thesmallestriskwithrespecttoa

large move. But this is trueonly under current marketconditions. As marketconditions change, all riskmeasures, including thegamma,will almost certainlychange. If the underlyingcontract makes a very largemove such that currentmarket conditions no longerapply, it may not be clearwhatwillhappen to the risksassociatedwitheachspread.

It will be easier toanalyze the relative risks ofthe spreads ifwe construct agraphofthetheoreticalprofitor loss with respect tomovement in the underlyingcontract. This has been donein Figure 13-4. We can seethat each spread does indeedlose value as the underlyingprice moves either up ordown.3However,wecanalsoseethatifthereisaverylarge

move, the spreadcharacteristics begin todiverge. On both the upsideanddownside,thelossesfromSpread 1, the short straddle,continuetoincrease,resultingin potentially unlimited riskin either direction. Spread 2,theratiospread,hasunlimitedupsiderisk.Onthedownside,though, it flattens out andeventually results in a verysmall profit. Spread 3, thelongbutterfly, flattensouton

both the upside anddownside, so its risk islimited regardless ofdirection.

Figure13-4

Which spread is best?That depends on what thetraderisworriedabout.Ifthetraderisoblivioustotherisk,it won’tmatterwhich spreadhechooses.Onaverage,eachpositionwillshowaprofitofapproximately 6.00. If,however, the trader is moreworried about a largedownward move in themarket, then perhaps Spread2 isbest.And if the trader is

unwillingtoaccepttheriskofunlimited loss in eitherdirection, then perhapsSpread3isbest.

In addition to thepossibility of a large move,all three positions areexposed to the risk of anincorrect volatility estimate.Because each spread has anegative vega, there will benoproblemif,overthelifeoftheoption,volatilityturnsout

to be lower than 18 percent.In such a case, the spreadswillshowaprofitgreaterthanoriginally expected. On theother hand, if volatility turnsout to be greater than 18percent, this could present aproblem.Whatwillhappenifvolatilityturnsouttobe20or25 percent or some highernumber?Eachspreadwillbehurt because of the negativevega,butwill theybehurt tothesamedegree?

Because Spread 2 hasthesmallestvega(–.875),wemight initially conclude thatit has the smallest volatilityrisk. But the vega, like thegamma, changes as marketconditionschange.Ifweraisevolatility, thevegaofSpread1, the short straddle, willremain essentially unchangedbecausethevegaofanat-the-moneyoptionisconstantwithrespect to changes involatility. But the vega of

Spread 3, the long butterfly,willdeclinebecausethevegaof in-the-money and out-of-the-money options (the May46andMay50puts)willtendto increase as volatility rises.With Spread 2, the vega ofbothoptions,theMay50calland the May 52 call, willbegin to increase, so it’s notimmediately clear what willhappen if we increasevolatility.

We can analyze thevolatility characteristics ofeachspreadbyconstructingagraph of each spread’s valuewith respect to changingvolatility. This is shown inFigure 13-5. With a largechange in volatility, thevalues of the three positionsbegin to diverge. If volatilityrises, the spreads begin tolose value until, at somepoint, the potential profitbecomes a loss. In terms of

volatility risk, we mightlogically ask, how high canvolatilityrisebeforewebeginto lose money? That is, wemight want to determine thebreakeven volatility, orimplied volatility, for eachspread. This is simply anextension of the generaldefinition of impliedvolatility: the volatility overthe life of an option, oroptions,atwhichthepositionwill,intheory,showneithera

profitnoraloss.InFigure13-5, we can see that thebreakeven volatility forSpread 1 (the short straddle)is approximately 21 percent,forSpread2(theratiospread)approximately 23 percent,and for Spread 3 (the longbutterfly) approximately 21.5percent. This seems toconfirm that Spread 2, theratiospread,istheleastriskywithrespecttovolatility.

Figure13-5

However, if volatilityturns out to be higher thanexpected, why should it stopat 23 percent? What willhappen if volatility turns outto be much higher, perhaps30 percent or even 40percent? Eventually, Spread2, the ratio spread, whichinitially seemed to carry theleastvolatilityrisk,willbeginto lose value at almost thesame rate as Spread 1, the

short straddle. On the otherhand, at higher volatilities,the graph of Spread 3, thelong butterfly, begins toflatten out, suggesting thatthereisalimittohowmuchitcanlose.Ofcourse,weknowthis because a butterfly hasboth limited profit potentialandlimitedrisk.

Although we mightworry that volatility willincrease to some value

greater than 18 percent, wemightalsoconsiderwhatwillhappen if volatility turns outtobelessthan18percent.Forthe same reason that risingvolatility will hurt, fallingvolatility should help. InFigure 13-5, we can see thatas volatility falls below 18percent, the profit resultingfromeachspreaddoesindeedincrease. However, asvolatility fallswell below 18percent, the profit from

Spread 2 begins to decline,eventuallyfallingtoalmost0.On the other hand, the profitfrom Spread 3 begins toaccelerate.

TheshapesofthegraphsinFigure13-5area resultofeach position’s volga—thesensitivity of the vega to achange in volatility. (For adiscussion of the volga, seeChapter9,specificallyFigure9-15.) Spread 1 has a volga

close to 0; its vega remainsconstant regardless ofchanges in volatility. Spread2 has a negative volga. Asvolatility rises, the vegabecomes more negative; asvolatility falls, the vegabecomes less negative. Thismeans that as volatility risesor falls, changes in volatilitywork against the position,acceleratingtherateoflossasvolatility rises and reducingthe rateofprofit asvolatility

falls.Incontrast,Spread3haspositive volga. Changes involatilityworkinfavoroftheposition, reducing the rate ofloss as volatility rises andincreasingtherateofprofitasvolatilityfalls.

Although Figure 13-5canbe interpreted as the riskof using an incorrectvolatility over the life of theoptions, it can also beinterpreted as the risk of a

sudden change in impliedvolatility.Intermsofimpliedvolatility risk, Spread 3probably represents the bestvalue. If implied volatilitybegins to rise, Spread 3 willinitially lose money morequickly than Spread 2, but ifimplied volatility risesdramatically, Spread 3 willbegin to outperform bothSpreads 1 and 2 because therateof losswilldecline.Andif implied volatility falls,

Spread 3 will outperformboth Spreads 1 and 2,increasing in value morequicklyatlowervolatilities.

Why are riskconsiderations so important?Everytraderknowsthattherearetimeswhenastrategywillresult in a profit and timeswhen it will result in a loss.No onewins all the time. Inthelongrun,however,agoodtrader’sprofitswillmorethan

offset his losses. Forexample, suppose that atraderchoosesa strategy thatwill show a profit of $7,000halfthetimeandwillshowaloss of $5,000 the other halfof the time. In the long run,the trader will show anaverage profit of $1,000.Suppose,though,thatthefirsttime that the trader executesthestrategy,sheloses$5,000,and the trader only has$3,000? Now the trader will

notbeabletostayinbusinessforallthosetimeswhenheisfortunate enough to show aprofitof$7,000.Everytraderknows that it is only overlongperiodsoftimethatgoodluck and bad luck even out.Hencenotraderwillinitiateastrategywhereshort-termbadluck might end his tradingcareer.

Financial officers atlarge firms know that it is

much easier to manage asteadycashflowthanonethatswings wildly. In a sense,every trader is his ownfinancial officer. He mustsensiblymanage his financesso that he can avoid beingruinedbytheperiodsofbackluck that will inevitablyoccur, no matter howskillfullyhetrades.

PracticalConsiderations

Considering only thegammaandvegarisk,Spread3 probably has the best riskcharacteristics. It has limitedriskifthereisalargemoveineither direction and performsbetter thaneitherSpread1orSpread2ifthereisadramaticchangeinvolatility.Thisdoesnot mean that Spread 3

performs better under allconditions. If the underlyingmarketmakesanydownwardmove or there is a small tomoderate upward move,Spread2outperformsSpreads1and3.Spread2alsohasanadvantage if there is amoderate increase involatility.

Even if we assume thatSpread 3, the long butterfly,offers the best theoretical

risk-reward tradeoff, it mayhave some practicaldrawbacks. Butterflies areactively traded in manymarkets, but Spread 3 is athree-sided spread, asopposed to Spreads 1 and 2,which are two-sided spreads.A three-sided spreadmay bemore difficult to execute inthemarketplaceandalsomaycostmoreintermsofthebid-ask spread. If a trader wantsto execute the complete

spread at one time, he maynot be able to do so at histarget prices. And if he triesto execute one leg at a time,he will be at risk fromadversechangesinthemarketuntil the other legs can beexecuted.

Additionally,thereisthequestion of market liquidity.In order to obtain atheoretical edgecommensuratewithSpreads1

and 2, it was necessary toincrease the size of thebutterflyto100×200×100.If there is insufficientliquidity in the May 46, 48,and 50 puts to support thissize,itmaynotbepossibletoexecute the butterfly in thesize required to meet thetrader’s profit objective.Alternatively, it may bepossibletoexecutepartofthespreadatfavorableprices,butas the size increases, the

prices may become lesssatisfactory. Moreover, for aretail customer, the increasedsize may entail greatertransactioncosts.

If trading considerationsmakeSpread3 impractical, atrader may have to choosebetween Spreads 1 (shortstraddle)and2(ratiospread).If this happens, Spread 2 istheclearwinner.Itallowsfora much greater margin for

errorinbothunderlyingpricechange (gamma risk) andvolatility(vegarisk).Atraderwho is given a choicebetween these two spreadswillstronglypreferSpread2.

In the real world, thechoice of spreads is notalwaysclear.Onespreadmaybe superior with respect toone type of risk, while adifferent spread may besuperior with respect to a

different risk. The ease withwhich a spread can beexecuted, as well as the costofexecution,will alsoplayarole.

Let’sconsiderthreenewspreads—Spread 4 (a shortput calendar spread), Spread5(adiagonalcallspread),andSpread6(aputdiagonalratiospread). In order to againfocusonrisk,thesizeofeachspread has been adjusted so

thatthetheoreticaledgeofallthree spreads is similar. Thetotaltheoreticaledgeandrisksensitivities of each spread(alltakenfromthetheoreticalevaluationtableinFigure13-1)areshowninFigure13-6.

Figure13-6

Becauseeachspreadhas

a negative vega, we willagain want to consider therisk that volatility will turnout to be greater than ourestimate of 18 percent. Thesensitivity of each spread toincreasingvolatility is shownin Figure 13-7. We can seethat Spread 4 has an impliedvolatility of approximately20.5 percent, Spread 5approximately 22 percent,and Spread 6 approximately20percent.Ifrisingvolatility

is our primary concern,Spread 5, the diagonal callspread, seems to entail thelowest risk. However,although Spread 5 loses theleast in a rising-volatilitymarket, it also shows asmaller profit in a falling-volatility market. This mayseem like a reasonabletradeoff, except that withSpread5, thepositive effectsof falling volatility begin todecline very quickly. This is

due to the negative volgaassociated with the position.As volatility falls, the vegabecomes less negative until,at a volatility ofapproximately10percent,thevega falls to0.Spread6, theputdiagonalratiospread,hasanevenlargernegativevolga;its vega turns positive ifvolatility falls below 11percent. In contrast to bothSpreads 4 and 6, Spread 5,theshortcalendarspread,has

avolgaof0.Itsvegaremainsconstant regardless ofwhether volatility rises orfalls. It offers an equaltradeoffbetween losseswhenvolatility rises and profitswhenvolatilityfalls.

Figure13-7

What about the gammariskofeachspread?Herewehaveasituationwherenotallthe spreads have a gammawiththesamesign.Spread6,thediagonal ratiospread,hasa negative gamma, so itshould be hurt by a largemove in the underlying.Spreads 4 and 5, however,have a positive gamma andshould profit from a largemove. The graphs of the

positions with respect tochanges in the underlyingpriceareshowninFigure13-8.

Figure13-8

WecanseeinFigure13-8 that althoughSpread6, thediagonal ratio spread,willbehurtbyamoveinthepriceofthe underlying contract, thedegree to which the movewill hurt depends on thedirection. With an upwardmove,thepotentialprofitwilldecline.Butevenwithaverylarge upward move, thespread will always retainsome profit. On the

downside, however, thespread’s profit rapidlydisappears, turning into apotentially unlimited loss ifthe downward move is largeenough.

Spread 4, the short putcalendar spread, and Spread5, the diagonal call spread,both have positive gammaand will profit from a largemove. Unlike Spread 4,though, which shows

approximately equal profit ineither direction, Spread 5shows a greater profit in anupward move and a smallerprofitinadownwardmove.4

There is, of course, atradeoff between gamma andtheta. If movement in theunderlyingpricewillincreasethevalueofSpreads4 and5(positive gamma), thepassage of time with nomovement will reduce the

value(negativetheta).Itmaybeworthwhiletolookathowmuch time can pass beforeeach spread loses itstheoretical edge. This isshowninFigure13-9.

Figure13-9

InFigure13-9,Spread4exhibits the typical decayprofile for a short calendarspread that is approximatelyatthemoney.Astimepasses,theposition losesvalueatanincreasingly greater rate.Spread 5, the diagonal callspread, also loses value astime passes. But after fiveweeks the decay turnspositive, so that if nothinghappens in the underlying

market the position willeventually show a smallprofit.Spread6, thediagonalratiospread, initiallyshowsasmall increase in value astime passes. Eventually,though, this position is alsosubject to decay.After sevenweeks, its potential profitdisappearscompletely.

As must be obvious bynow, thechoiceof spreads isnever simple. As with all

trading decisions, it is aquestion of risk and reward.Although there are manyrisks with which an optiontradermustdeal,hewilloftenhave to ask himself whichrisk represents the greatestthreat.Sometimes,inordertoavoidonetypeofrisk,hewillbeforcedtoacceptadifferentrisk. Even if the trader iswillingtoacceptsomeriskinacertainarea,hemaydecidethat he will only do so to a

limited degree. Then hemayhavetoacceptincreasedrisksinotherareas.

If given the choicebetween several differentstrategies, a trader can use acomputer to determine therisk characteristics of thestrategies under differentmarket conditions.Unfortunately, it may notalwaysbepossibletoanalyzethe choices in such detail. A

trader may not haveimmediate access to thenecessary computer support,ormarket conditionsmay bechangingsorapidlythatifhefails to make an immediatedecision, opportunity mayquickly pass himby. In suchcases, the trader will oftenhavetorelyonhisinstinctsinchoosingastrategy.Althoughthere is no substitute forexperience, most tradersquickly learn an important

rule: straddles and stranglesaretheriskiestofallspreads.Thisistruewhetheronebuysor sells these strategies.Newtraders sometimes assumethat thepurchaseofstraddlesandstranglesisnotespeciallyrisky because the risk islimited. However, it can bejust aspainful to losemoneydayafterdaywhenonebuysastraddleor strangleand themarketfailstomoveasitistolose the same amount of

money all at once when onesellsastraddleandthemarketmakes a violent move. Ofcourse, a trader who is rightaboutvolatilitycanreaplargerewards from straddles andstrangles.Butanexperiencedtrader knows that suchstrategies offer the leastmargin for error and willtherefore prefer strategieswith more desirable riskcharacteristics.

HowMuchMarginforerror?

What is a reasonablemargin for error in assessingthe risk of a position,particularlywhen itcomes tovolatility risk? There is noclear answer because it willusually depend on thevolatility characteristics of aparticular market, as well asthetrader’sexperienceinthat

market. In some cases, 5percentage points may be anextremely large margin forerror, and the traderwill feelvery confident with anystrategy passing such a test.In other cases, 5 percentagepoints may be almost nomarginforerroratall,andthetrader will find that thestrategy is a constant sourceofworry.

Rather than focusing on

margin for error, a betterapproach might be to focusonthecorrectsizeinwhichtodo a spread given a knownmargin for error. Practicaltrading considerations aside,atradershouldalwayschoosethespreadwith thebest risk-reward characteristics. Butsometimes even the bestspreadwillhaveonlyasmallmargin for error andconsequently will entailsignificant risk. In such a

case, a trader, if hewants tomake a trade, ought to do soin small size. If, however, atrader can execute a spreadwith a very large margin forerror, he ought to be willingto do the spread in a muchlargersize.

Consider a traderwhosebestestimateofvolatilityinacertain market is 25 percent.If implied volatility is lowerthan 25 percent, the trader

will lookforpositionswithapositive vega. If the bestpositive-vega strategy thetradercanfindisa2×1ratiospread with an impliedvolatility of 23 percent (onlya 2-percentage-point marginfor error), he will almostcertainly keep the size of hisstrategy small, perhapsexecuting the spread only 10times (20×10). If,however,the same spread has animplied volatility of 18

percent (a 7-percentage-pointmargin for error) and thetrader believes that such alow volatility is extremelyrare, he may have theconfidence to execute thespread in amuch larger size,perhaps 100 × 50.5 The sizeofa trader’spositions shoulddependontheriskinessofthepositions, and this, in turn,dependsonhowmuchcangowrong before the strategy

turnsagainstthetrader.

DividendsandInterest

In addition to the delta,gamma, theta, andvega risksthatapplytoalltraders,stockoption tradersmay also haveto consider the risk ofchanges in interest rates anddividends.6When all options

expire at the same time, therisk associated with changesininterestratesanddividendstends to be relatively small.Straddles, strangles, ratiospreads, and butterflies maychange slightly because achange in interest rates ordividends will raise or lowerthe forward price. But alloptions are evaluated usingone and the same forwardprice. For calendar spreads,however, where the options

are evaluated using twodifferent forward prices,long-term and short-termoptions can react differentlytochangesintheseinputs.

Consider the evaluationtableforstockoptionsshowninFigure13-10.Withimpliedvolatilitiesbelowtheforecastof29percent, itmakes senseto look for spreads withpositive vegas. Suppose thatwe focuson the four spreads

shown in Figure 13-11.Spreads 7 and 8 are longcalendar spreads, whileSpreads9and10arediagonalspreads.Whataretherelativemeritsofeachspread?

Figure13-10

Figure13-11

Becauseall four spreadsfall into the long calendarspreadcategory,theyallhavethe typical negative-gammaand positive-vegacharacteristicsassociatedwithsuch spreads. This is shownin Figures 13-12 and 13-13.Movementinthepriceof theunderlyingcontractor fallingvolatility will reduce thevalue of the spread. Risingvolatility will increase the

valueof the spread. (Spreads7 and 8 have essentiallyidentical volatilitycharacteristicsandarealmostindistinguishable from eachother in Figure 13-13.)Initially,thechoiceofspreadswill depend on the risk ofmovement in the underlyingcontractaswellastheriskofchangesinimpliedvolatility.

Figure13-12

Figure13-13

Because we are dealingwith stock options, there aretwoadditionalrisks—theriskofchanginginterestratesandthe risk of changingdividends, assuming that atleast one dividend paymentfallsbetweenexpirations.Weknow from Chapter 7 thatstock option calls and putsreactinjusttheoppositewayto changes in interest ratesanddividends.Risinginterest

rates or falling dividendscause calls to rise in valueand puts to fall; fallinginterest rates and risingdividendscausecallstofallinvalue and puts to rise.Moreover, the impact of achange in either of theseinputs will be greater forlong-term options than forshort-term options. We canmeasure the risk of changinginterest rates by determiningthe total rho value for each

spread. Even though there isno Greek for the dividendsensitivity,we can still use acomputer to determine thedividend risk associatedwitheach spread.The sensitivitiesfor the individual options, aswell as the total spreadsensitivities, to changinginterest rates and dividendsareshowninFigure13-14.

Figure13-14Interest-rateanddividendsensitivity.

The call spreads(Spreads 7 and 9) have apositive rho and negativedividend sensitivity. The putspreads (Spreads 8 and 10)have a negative rho andpositive dividend sensitivity.Thevalueofeachspreadwithrespect to changes in theseinputsisshowninFigures13-15and13-16.

Figure13-15Interest-ratesensitivity.

Figure13-16Dividendsensitivity.

The interest-rate anddividend risk associatedwithvolatility spreads is usuallysmall compared with thevolatility (gamma and vega)risk. Nonetheless, a traderought to be aware of theserisks, especially when aposition is large and there issignificantriskofachangeineither interest rates ordividends.

WhatisaGoodspread?

Option traders, beinghuman, would rather talkabout their successes thantheirdisasters.Ifoneweretoeavesdrop on conversationsamong traders, it wouldprobably seem that no oneever made a losing trade.Disasters, when they dooccur, only happen to other

traders.Thefactisthateverysuccessful option trader hashad his share of disasters.What separates successfultraders from theunsuccessfulones is the ability to survivesuchoccurrences.

Consider the traderwhoinitiatesaspreadwithagoodtheoretical edge and a largemargin for error in almostevery risk category. If thetrader still ends up losing

money on the spread, doesthis mean that the trader hasmade a poor choice ofspreads? Maybe a similarspread, but one with lessmargin for error,wouldhaveresulted in an even greaterloss, perhaps a loss fromwhich the trader could notrecover.

It is impossible to takeinto consideration everypossible risk. A spread that

passed every risk test wouldprobably have so littletheoreticaledge that itwouldnot be worth doing. But thetrader who allows himself areasonable margin for errorwill find that even his losseswillnotleadtofinancialruin.A good spread is notnecessarily the one thatshows the greatest profitwhen things go well; it maybe the one that shows theleast loss when things go

badly.Winningtradesalwaystake care of themselves.Losingtradesthatdonotgiveback all the profits from thewinning ones are just asimportant.

EfficiencyOne method that traders

sometimesusetocomparetherelative riskiness of potentialstrategiesfocusesontherisk-

reward ratio,orefficiency, ofthe strategies.Suppose that atrader is considering twopossible spreads, bothwith apositive gamma and anegative theta.The reward isrepresented by the gamma,the potential profit when theunderlying market moves.The risk is the theta, themoney that will be lostthroughthepassageoftimeiftheunderlyingmarketfailstomake sufficiently large

moves.Thetraderwouldlikethereward(thegamma)tobeaslargeaspossiblecomparedwith the risk (the theta).Wemight express thisrelationshipasaratio

gamma/theta

The larger the absolutevalue of this ratio, the moreefficienttheposition.

Inthesameway,atraderwho has a negative gamma

andapositivethetawantstherisk (the gamma) to be assmall as possible comparedwith the reward (the theta).He therefore wants theabsolute value of thegamma/theta ratio to be aslargeaspossible.

For example, we mightgo back and calculate theefficiency of Spreads 1through3inFigure13-3.Theefficienciesare

Because each spread has anegative gamma and positivetheta,wewant the efficiencytobeassmallaspossible.Wecansee thatSpread3 isbest,which is consistent with ourprevious analysis of each

spread.Assuming that all

strategieshaveapproximatelythesametheoreticaledge,theefficiencycanbeareasonablemethodofquicklycomparingstrategies where all optionsexpire at the same time. Insuch cases, the gamma andthetaare theprimary risks tothe position. If a strategyconsistsofoptionsthatexpireat different times, the

efficiency is only oneconsideration, and thesensitivity of the positions tochanges in implied volatility(the vega) may also becomeimportant,astheywereinourother spread examples. Insuch cases, a more detailedrisk analysis will benecessary.

Adjustments

In Chapter 11, weconsidered the question ofwhenatradershouldadjustaposition to remain deltaneutral. In addition todeciding when to adjust, thetraderalsomustconsiderhowbest to adjust because thereare many different ways toadjustthetotaldeltaposition.An adjustment to a trader’sdeltapositionmayreducehisdirectional risk, but if hesimultaneously increases his

gamma,theta,orvegarisk,hemay inadvertently beexchanging one type of riskforanother.

Adeltaadjustmentmadewith the underlying contractis essentially a risk-neutraladjustment. The gamma,theta, and vega of anunderlying contract are 0, soan adjustmentmadewith theunderlying contract will notchangeanyoftheserisks.Ifa

trader wants to adjust hisdelta position but wants toleavetheothercharacteristicsof thepositionunaffected,hecan do so by purchasing orsellinganappropriatenumberofunderlyingcontracts.

An adjustment madewith optionswill also reducethedeltarisk,butatthesametime, itwill change theotherrisk characteristics. Becauseevery option has not only a

deltabutalsoagamma,theta,and vega, when an option isaddedtoorsubtractedfromaposition, it necessarilychanges the total delta,gamma,theta,andvegaoftheposition. This is somethingthat new traders sometimesforget.

Consider a stock optionmarket where the underlyingcontract is trading at 99.25and all options appear to be

overpriced. Suppose that atrader decides to sell the95/105 strangle (sell the 95put, sell the 105 call), withputandcalldeltasof–32and34, respectively. If the tradersells20strangles,thepositionis initially slightly deltanegativebecause

(–20×+34)+(–20×–32)=–40

Suppose that a week

passes and the underlyingmarket has fallen to 97.00,withnewdeltavalues for the95 put and 105 call of –39and +25. Assuming that noadjustmentshavebeenmade,the trader’s delta position isnow

(–20×–39)+(–20×+25)=+280

If the trader wants tohold the position but also

wants to remainapproximately delta neutral,hehasthreebasicchoices:

1. Sellunderlyingcontracts.2.Sellcalls.3.Buyputs.

Whichmethodisbest?All other considerations

being equal, whenever atrader makes an adjustment,she should do so with the

intention of improving therisk-reward characteristics ofthe position. If the traderdecides to adjust his deltaposition by purchasing puts,healsoreduceshisotherrisksbecause the gamma, theta,and vega associated with theput purchase are opposite insigntothegamma,theta,andvega associated with theexisting short strangleposition.

Unfortunately, all otherconsiderations may not beequal. Because impliedvolatility can remain high orlow for longperiodsof time,it is quite likely that if alloptionswereoverpricedwhenthe trader initiated hisposition, they will still beoverpriced when he goesbackintothemarket tomakehis adjustment. Even thoughthe purchase of puts tobecomedeltaneutralwillalso

reducehisotherrisks,suchanadjustment will have theeffect of reducing thetheoreticaledge.Ontheotherhand, if all options areoverpriced and the traderdecidestoselladditionalcallstoreducethedelta,thesaleoftheoverpricedcallswillhavethe effect of increasing thetheoretical edge. If the traderdecides that adding to histheoreticaledgeisofprimaryimportance,hemaydecideto

sell 11 additional 105 calls,leaving him approximatelydeltaneutralbecause

(–20×–39)+(–31×+25)=–5

Now suppose thatanother week passes and themarket has rebounded to101.00,withnewdeltavaluesforthe95putand105callof–24 and +37. The positiondeltaisnow

(–20×–24)+(–31×+37)=–667

Again, if the traderwants to adjust, he has threebasic choices—buyunderlying contracts, buycalls, or sell puts. Assumingthat all options are stilloverpricedandthatthetraderwants tocontinue to increasehis theoretical edge, he maydecidetosellanadditional28of the95puts.Thenewtotal

deltapositionis

(–48×–24)+(–31×+37)=+5

It should be clear whatwill result from theseadjustments. If all optionsremain overpriced and thetrader focuses solely onincreasing his theoreticaledge, he will continue tomake whatever adjustmentsare necessary by selling

overpriced options. Thismethod of adjusting mayindeed result in the greatestprofit to the trader, but thestrangle,whichthetraderwasinitially prepared to sell 20times, now has increased insizeto48×31.Ifthemarketnowmakesaviolentmoveineither direction, the adverseconsequences will be greatlymagnified. The new trader,overlyconcernedwithalwaysincreasing his theoretical

edge, often finds himself injust such a position. If themarket makes a very swiftmove, the trader may notsurvive. For this reason, anew trader is usually welladvised to avoid makingadjustments that increase thesizeofaposition.

No trader can afford toignore the effect thatadjustments will have on thetotal risk to a position. If he

hasapositivegammaorvegaposition, buying anyadditional options willincrease his gamma or vegarisk; if he has a negativegamma or vega position,sellinganyadditionaloptionswill likewise increase hisgammaorvegarisk.Atradercannot afford to selloverpriced options or buyunderpriced options adinfinitum.At somepoint, thesizeofthespreadwillsimply

become too large, and anyadditional theoretical edgewill have to take a back seatto risk considerations. Whenthis happens, there are onlytwochoices:

1. Decrease thesizeofthespread.2. Adjust in theunderlyingmarket.

A disciplined traderknows that sometimes,

because of riskconsiderations, the bestcourseistoreducethesizeofthe spread, even if it meansgiving up some theoreticaledge. When open-outcrymarketswereflourishing,thiscould be particularly hard onatrader’segoifthetraderhadtopersonallygobackintothemarket and either buy backoptions, that he originallysold, at a lower price or selloutoptions,thatheoriginally

purchased, at a higher price.However, if a trader isunwilling to swallow hispride from time to time, histradingcareerislikelytobeashortone.

Ifa traderfindsthatanydeltaadjustmentintheoptionmarket that reduces his riskwill also reduce histheoretical edge and he isunwilling to give up anytheoretical edge, his only

recourse is to makeadjustmentsintheunderlyingmarket. An underlyingcontracthasnogamma,theta,or vega, so the risks of theposition will remainessentiallythesame.

AQuestionofStyleBecause most option

pricing models assume thatmovement in the underlying

contract is random,anoptiontraderwhotradespurelyfromthe theoretical valuesgenerated by amodel shouldnot have any prior opinionabout market direction. Inpractice, however, manyoption traders begin theirtrading careers by takingpositions in the underlyingmarket,wheredirectionistheprimary consideration. Manytraders therefore develop astyle of trading based on

presumed directional movesin the underlying market. Atradermight, forexample,bea trend follower, adhering tothephilosophythat“thetrendis your friend.” Or he mightbe a contrarian, preferring to“buyweakness,sellstrength.”

Traders often try toincorporate their personaltradingstylesintotheiroptionstrategies.Onewaytodothisis to consider beforehand the

adjustments that will berequiredforacertainstrategyif the underlying marketbeginstomove.Atraderwhosells straddles knows thatsuch spreads have negativegamma.Asthemarketmoveshigher, his delta position isbecomingnegative,andasthemarketmoveslower,hisdeltapositionisbecomingpositive.If this trader likes to tradeagainst the trend, he willavoidadjustmentsasmuchas

possible because his positionis automatically tradingagainst the trend. Whicheverway the market moves, theposition always wants aretracement of thismovement. On the otherhand, a trader who sells thesame straddles but prefers totrade with the trend willadjustateveryopportunity.Inorder to remaindeltaneutral,he will be forced to buyunderlying contracts as the

market rises and sellunderlying contracts as themarketfalls.

Theoppositeistrueforatraderwhobuysstraddles.Hehas a positive-gammaposition.As themarket rises,hisdeltapositionisbecomingpositive, and as the marketfalls, his delta position isbecoming negative. If thistrader likes to trade with thetrend, he will adjust as little

as possible in the belief thatthe market is likely tocontinue in the samedirection. If, however, heprefers to trade against thetrend, hewill adjust as oftenaspossible.Everyadjustmentwill represent a profitopportunity if the marketdoes in fact reverse itsdirection.

Atraderwithanegativegamma is always adjusting

with the trend of theunderlying market. A traderwith a positive gamma isalways adjusting against thetrend of the underlyingmarket. If a trader prefers totradewiththetrendoragainstthetrend,heshouldchooseastrategy and an adjustmentprocessthatareappropriatetohis preference. A trader whopreferstotradewiththetrendcan choose a strategy with apositivegammatogetherwith

lessfrequentadjustmentsorastrategy with a negativegamma with more frequentadjustments. A trader whoprefers to trade against thetrend can choose a strategywith a negative gammatogether with less frequentadjustmentsorastrategywitha positive gamma with morefrequent adjustments. Thepurely theoretical trader willnot have to worry about thisbecause for him there is no

such thing as a trend.However, for many traders,old habits, such as tradingwith or against the trend, arehardtobreak.

LiquidityEveryopenoptionposition

entailsrisk.Eveniftheriskislimitedtothecurrentvalueofthe options, by leaving theposition open, the trader is

risking the lossof thatvalue.If the trader wants toeliminate the risk, he willhavetotakesomeactionthatwill, in effect, close out theposition. Sometimes this canbe done through earlyexercise or by takingadvantage of an opposingposition to create anarbitrage. More often,however,inordertocloseoutan open position, a tradermustgo into themarketplace

and buy in any short optionsandselloutanylongoptions.

An importantconsideration in decidingwhether to enter into a tradeis often the ease with whichthe trader can reverse thetrade. Liquid optionmarkets,where there aremany buyersand sellers, are much lessrisky than illiquid markets,where there are few buyersandsellers. In the sameway,

aspread thatconsistsofveryliquid options is much lessrisky than a spread thatconsists of one or moreilliquid options. If a trader isconsidering entering into aspread where the options areilliquid, he ought to askhimselfwhetherhe iswillingtolivewiththatpositionuntilexpiration. If the market isvery illiquid, thismaybe theonlytimethathewillbeableto get out of the position at

anything resembling a fairprice.Ifthespreadconsistsoflong-term options, the tradermay find himself married tothe position for better orworse, in sickness and inhealth, for what may seemlike an eternity. If he isunwilling to commit hiscapital for such a lengthyperiod, perhaps he shouldavoid the position. Becausethereisgreaterriskassociatedwith a long-term investment

than with a short-terminvestment,atraderwhodoesdecide to take a position inlong-term options ought toexpectgreaterpotentialprofitin the form of largertheoreticaledge.7

New traders are oftenadvised to begin trading inliquid markets. If a newtrader makes an errorresulting in a losing trade, ina liquid market, he will be

able to keep his loss to aminimum because hewill beable to exit the trade withrelative ease. On the otherhand, an experienced trader,especially a market maker,will often prefer to deal inless liquid markets. Theremaybelesstradingactivityinsuchmarkets,but thebid-askspread is much wider,resultingingreatertheoreticaledge each time a trade ismade.Ofcourse,anymistake

canbeaproblemwithwhichthetraderwillhavetolivefora long time. However, anexperienced trader isexpectedtokeephismistakestoaminimum.

The most liquid optionsin any market are usuallythose that are short term andthat are either at or slightlyout of the money. Suchoptions always have thenarrowestbid-askspread,and

thereareusuallymanytraderswilling to buy or sell thesecontracts. As a trader movesto longer-term options or tooptions that are more deeplyin the money, he finds thatthe bid-ask spread begins towiden, and fewer and fewertradersare interested in thesecontracts. Although there isconstant activity in at-the-money short-term options,deeply in-the-money long-term options may not trade

forweeksatatime.In addition to the

liquidityofanoptionmarket,a trader should also givesome thought to the liquidityof the underlying market. Inan illiquid option market, atradermay find it difficult toadjust the position usingoptions. If, however, theunderlying market is liquid,he will at least be able tomake his adjustment in that

marketwithrelativeease.Themost dangerous markets inwhich to trade are thosewhere both the options andthe underlying contract areinactively traded. Only themost experienced andknowledgeabletradersshouldentersuchmarkets.

Figure13-17showsend-of-day bid-ask spreads andvolume figures for Standardand Poor’s (S&P) 500 Index

optionstradedat theChicagoBoard Options Exchange onMarch 1, 2010.8 In general,the volumes are lower andbid-ask spreadsarewider forback-month options oroptions thataredeeply in themoney compared with front-monthoptionsoroptionsthatareatthemoneyoroutofthemoney.

Figure13-17SPXindexoptions:Bid-askspreadsandtradingvolumesforMarch1,2010.

1Weareconsideringonlytheinterest-rateriskasitappliestotheevaluationofoptions.Changesininterestratescanalsoaffecttheevaluationofanunderlyingcontract,suchasabond,oreventhesharesinacompany.Butthatisaseparatematter.2Inordertofocusonlyonvolatility,wehaveassumedaninterestrateof0.3Spreads1and2,withtheirslightlypositivedelta,initiallyshowasmallgainasthemarketrises.Spread3,withitsslightnegativedelta,initiallyshowsasmallgainasthemarketfalls.4ItmayappearfromFigure13-8thatSpread5hasunlimitedupsideprofitpotential.Inreality,theprofitislimited

bythefactthatthespreadbetweenthevalueoftheMay52callandtheJuly54callcanneverbegreaterthan2.00.Thiswilloccurifbothoptionsgoverydeeplyintothemoney.5Size,ofcourse,isrelative.Toawell-capitalized,experiencedtrader,even100×50maybeasmalltrade.6Dependingonthesettlementprocedure,changesininterestratescanalsoaffectfuturesoptions.Buttheeffect,asdiscussedinChapter7,isusuallyquitesmall.Changesininterestratescanalsoaffectfuturesoptionsbecausetheymaychangethepriceoftheunderlyingfuturescontract.Butthiscanbeassessedastheriskofachangeintheunderlyingprice,notachangein

interestrates.7Thisisthesamereasonthatlong-terminterestratestendtobehigherthanshort-termrates.Ifoneiswillingtocommitcapitalforalongerperiod,thepotentialrewardshouldalsobegreater.8Figure13-17representsonlyapartiallistingofS&P500Indexoptions.Moreexercisepricesandexpirationmonthswereavailablethancouldconvenientlybedisplayedhere.

14

Synthetics

One importantcharacteristic of options isthat they can be combinedwith other options, or withunderlying contracts, tocreate positions withcharacteristics which arealmostidenticaltosomeother

contract or combination ofcontracts. This type ofreplication enables us to domost option strategies in avariety ofways, and leads tomany useful relationshipsbetween options and theunderlyingcontract.

SyntheticUnderlying

Consider the followingpositionwherealloptionsare

European (no early exercisepermitted):

long a June100callshort a June100put

What will happen to thispositionatexpiration?Itmayseem that one cannot answerthequestionwithoutknowingwheretheunderlyingcontractwill be at expiration.

Surprisingly, the price of theunderlying contract does notaffect the outcome. If theunderlying contract is above100, the put will expireworthless, but the traderwillexercise the100call, therebybuying the underlyingcontract at 100. Conversely,if the underlying contract isbelow 100, the call willexpire worthless, but thetraderwillbeassignedonthe100 put, also buying the

underlyingcontractat100.Ignoringforthemoment

the unique case when theunderlying price is exactly100, at June expiration theabove position will alwaysresultinthetraderbuyingtheunderlying contract at theexercise price of 100, eitherby choice (the underlyingcontract is above 100 and heexercises the 100 call) or byforce(theunderlyingcontract

is below 100 and he isassignedonthe100put).Thisposition, a synthetic longunderlying, has the samecharacteristics as a longunderlying contract, butwon’t actually become anunderlying contract untilexpiration.1

If the trader takes theopposite position, selling aJune 100 call and buying aJune 100 put, he has a

synthetic short underlyingposition. At June expirationhe will always sell theunderlying contract at theexercise price of 100, eitherby choice (the underlyingcontract is below100andheexercises the 100 put) or byforce(theunderlyingcontractis above 100 and he isassignedonthe100call).

We can express theforegoing relationships as

follows:

syntheticlongunderlying≈longcall+shortput

syntheticshortunderlying≈shortcall+longput

wherealloptionsexpireatthe same time and have thesameexerciseprice.

In our examples wecreated a synthetic positionusing the 100 exercise price.Butwecancreateasynthetic

using any available exerciseprice. A long June 110 calltogetherwithashortJune110put is still a synthetic longunderlying contract. Thedifference is that at Juneexpiration the underlyingcontractwill be purchased at110. A short June 95 calltogetherwith a long June 95put is a synthetic shortunderlying contract. At Juneexpiration the underlyingcontractwillbesoldat95.

We can also see why acall and put with the sameexercise price and expirationdate make up a syntheticunderlying by constructingparity graphs of the options.This is shown in Figures14-1aand14-1b.

Figure14-1a

Figure14-1b

While not exactlyidentical(hencetheuseofanequivalentsignratherthananequal sign) a syntheticposition acts very much likeits real equivalent. For eachpoint the underlyinginstrument rises, a syntheticlong position will gainapproximately one point invalue and a synthetic shortposition will loseapproximately one point in

value. This leads us toconclude, correctly, that thedelta of a syntheticunderlying position must beapproximately 100. If thedelta of the June 100 call is75, the delta of the June 100put will be approximately –25. If the delta of the June100 put is –60, the delta ofthe June 100 call will beapproximately 40. Theabsolute value of a call andput delta will always add up

to approximately 100. Wewill see later that thesettlement procedure andinterest rates, as well as thepossibility of early exercise,can cause the delta of asynthetic underlying positionto be slightly more or lessthan 100. But for mostpractical purposes this is areasonableestimate.

SyntheticOptions

By rearranging thecomponents of a syntheticunderlying position we cancreate four additionalsyntheticcontracts:

syntheticlongcall≈longanunderlyingcontract+long

putsyntheticshortcall≈shortanunderlyingcontract+short

put

syntheticlongput≈shortanunderlyingcontract+long

callsyntheticshortput≈longanunderlyingcontract+short

call

Again, all options mustexpire at the same time andhave thesameexerciseprice.Eachsyntheticpositionhasadelta approximately equal to

its real equivalent and willthereforegainorlosevalueatapproximately the same rateas its real equivalent. Theparity graphs for a syntheticlongcallareshowninFigures14-2a and14-2b. The graphsfor a synthetic long put areshown in Figures 14-3a and14-3b.

Figure14-2a

Figure14-2b

Figure14-3a

Figure14-3b

A new trader mayinitially find it difficult torememberwhichcombinationis equivalent to whichsynthetic option. Thissuggestion may help: If wetrade a single option andhedge it with an underlyingcontract, we have the sameposition, synthetically in thecompanion option (thecompanion option being theopposite type,eitheracallor

put, at the same exerciseprice).

Ifwebuyacallandhedgeitbysellingtheunderlyingcontract,wehavesyntheticallyboughtaput.Ifwesellacall

andhedgeitbybuyinganunderlyingcontract,

wehavesyntheticallysoldaput.Ifwebuyaput

andhedgeitbybuyingtheunderlyingcontract,wehavesyntheticallyboughtacall.Ifwesellaput

andhedgeitbysellingan

underlyingcontract,wehavesyntheticallysoldacall.

Thus far we have madeno mention of the prices atwhichanyofthecontractsaretraded. The prices will ofcourse be important whendeciding whether to create asynthetic position, and wewill eventually address thisquestion. But for the present

we are considering only thecharacteristics of a syntheticposition, and these areindependent of the prices atwhich the contracts aretraded. In Figures 14-2a and14-3a theunderlyingpositionwas takenatapricedifferentthan theexerciseprice.Whatgives the position itscharacteristics is not thepricesofthecontracts,buttheslopes of the contracts. Andthe combined slopes are

equivalent to a long call(Figure14-2b)andalongput(Figure14-3b).

Summarizing, there aresix basic synthetic contracts—long and short anunderlyingcontract, longandshort a call, and long andshort a put. If all optionsexpire in June,using the100exercisepricewehave:

syntheticlongunderlying=longJune100call+short

June100putsyntheticshortunderlying=shortJune100call+long

June100put

syntheticlongJune100call=longunderlying+longJune

100putsyntheticshortJune100call=shortunderlying+short

June100put

syntheticlongJune100put=shortunderlying+longJune

100callsyntheticshortJune100put=longunderlying+shortJune

100call

We know from thesynthetic relationship that theabsolutevalueofthedeltasofcalls and puts with the sameexercise price and expirationdateaddup toapproximately100. We can also usesynthetics to identify otherimportantriskrelationships.

We know that thegamma and vega of anunderlying contract is zero.Since a long call and shortput with the same exerciseprice and expiration date canbecombined tocreatea longunderlying contract, thegamma and vega of thesecombinations must also addup to zero. This means thatthe gamma and vega of acompanion call and putmustbe identical. If the June call

has a gamma of 5, so mustthe June100put. If the June105puthasavegaof .20,somust the June 105 call. (Toconfirmthis,itmaybeusefulto go back and compare thecompaniondelta,gamma,andvega values in Figures 7-13,13-1,and13-10.)

Because the gamma andvega of companion calls andputs is identical, optiontraders who focus on

volatilitymakenodistinctionbetween calls and puts withthe same exercise price andexpirationdate.Bothhavethesame gamma and vega, andtherefore the same volatilitycharacteristics. If a traderownsacallandwouldpreferinstead toownaputheneedonly sell the underlyingcontract.Ifheownsaputandwouldprefertoownacall,heneedonlybuytheunderlyingcontract.Thevolatilityriskof

a position depends not onwhetherthecontractsarecallsor puts, but on the exerciseprices and expiration dateswhichmakeuptheposition.

Why isn’t the theta, likethe gamma and vega, ofcompanion options identical?Depending on the underlyingcontract and the settlementprocedure, in some cases thethetavalueswillbethesame.But in other cases the theta

values in a syntheticwill notadduptozerobecauseofthecost of carry associated witheither theunderlyingcontractortheoptioncontracts.

As an example, if wepurchase stock and the stockprice remains unchanged arewe making money or losingmoney? Itmayseem that thepositionisjustbreakingeven.Butifweconsiderthecostofborrowing cash in order to

buy the stock, then theposition is losing moneybecause of the interest cost.This will be reflected in thesynthetic equivalent having anonzerotheta.

Unlikestock, thereisnocostofcarryassociatedwithafutures contract. But ifoptionsonfuturesaresubjecttostock-typesettlement therewill be a cost of carryassociatedwiththeoptions.If

companion options aretrading at different pricesthere will be a different costofcarry,andthiswillresultinthe synthetic underlyingposition having a nonzerotheta.

Finally,ifwearedealingwith options on futures, andthe options are subject tofutures-type settlement, thereisnocostofcarryassociatedwith either the underlying

contractortheoptions.Inthiscase thecompanioncallsandputs will indeed have thesametheta.

Synthetics can explainsome relationships that werepreviously discussed. In ourdiscussionofvertical spreadswe noted that a bull spreadconsists of buying the lowerexercisepriceand selling thehigher exercise price,regardless of whether the

spreadconsistedofallcallsorall puts.Using syntheticswecanseewhythisistrue:

In the synthetic equivalentthelongandshortunderlyingcontractscanceloutleavingabullputspread

+1June100

put–1 June 100put

The call spread and putspread have similarcharacteristics,buttheydifferin terms of cash flow. Thecallspreadisdoneforadebit,while the put spread is donefor a credit. Since the spreadhasamaximumvalueof5.00,in the absence of interestconsiderations, the value of

the two spreads at expirationmust add up to 5.00. If thecallspreadistradingfor3.00,theputspreadmustbetradingfor 2.00. If interest rates arenonzero, and the options aresubject to stock-typesettlement, theirvalues todaymust add up to the presentvalueof5.00.

UsingSyntheticsina

SpreadingStrategy

Since a synthetic hasessentially the samecharacteristics as its realequivalent, any strategy canbe done using a synthetic.This means that there canoften be several differentways to create the samestrategy.

Consider the followingposition:

+ 2 June 100calls–1 underlyingcontract

This combinationdoesn’t seem to fit anypreviouslydiscussedstrategy.But suppose we write theJune100callsseparately:

+1 June 100call+1 June 100

call–1 underlyingcontract

Weknowthatalongcalland shortunderlyingcontractis a synthetic long put.Therefore, the position isreally

+1 June 100call+1 June 100put

which is easilyrecognizable as a longstraddle.

Similarly, suppose wehave

+2 June 100puts+1 underlyingcontract

We can write the June100putsseparately

+1 June 100put+1 June 100put+1 underlyingcontract

A long put and a longunderlying contract is asyntheticlongcall.Theentireposition is again a longstraddle:

+1 June 100

put+1 June 100call

From the foregoingexamples, we can see thattherearethreewaystocreatealongstraddle:

1.buythecallandbuytheput2. buy the call,and buy the putsynthetically

3.buytheput,andbuy the callsynthetically

The latter two methodsare synthetic long straddles.The best way to buy astraddle will depend on theprices of the syntheticscompared to their realequivalents.Weshalladdressthe question of pricingsyntheticsinthenextchapter.

IronButterfliesandIronCondors

Consider these twopositions:

1.+1June95put/+1June105call2.–1June100call/–1June100put

The first strategy is along strangle; the second

strategy is a short straddle.What will happen if wecombine the two strategies?We can answer the questionby rewriting the positionusingonlycallsoronlyputs.If we choose to express allcontracts as calls we canrewrite each put as asynthetic:

Replacing the puts withtheir synthetic equivalents,and canceling out the longand short underlyingcontracts, we are left with alongbutterfly

+1 June 95call–2 June 100calls +1 June105call

If, instead of calls, weexpress all contracts as putswe will also end up with along butterfly. This confirmsthe fact that a call and putbutterfly are essentially thesame. One is simply asyntheticversionoftheother.

An iron butterfly is aposition which combines astrangleandstraddle,withthestraddle centered exactly inthemiddle of the strangle. Ithas the same characteristicsas a traditional butterfly.Butunlike a long butterfly (buythe outside exercise prices /sell the insideexerciseprice)which is done for a debit(hence the term long), theequivalent ironbutterfly (buythestrangle/sellthestraddle)

is done for a credit. Thestraddlewhichwearesellingisalwaysmorevaluable thanthe strangle which we arebuying. Ifwe receivemoneywhenwe put on the positionthen we are short the ironbutterfly.Buyingatraditionalbutterfly is equivalent tosellinganironbutterfly.

Whatisanironbutterflyworth?We know that a longbutterflywill have avalue at

expiration between zero andthe amount between exerciseprices.IfwebuytheJune95/100 / 105 butterfly we willpay some amount betweenzero and 5.00. We hope theunderlying contract willfinish at 100, in which casethebutterflywillbeworthitsmaximumof 5.00. Ifwe sellthe June 95 / 100 / 105 ironbutterflywewilltakeinsomeamount between zero and5.00. We also hope that the

underlyingwill finishat100,inwhich case all the optionswillbeworthlessandwewillprofit by the amount of theoriginalsale.

At expiration the valueof a butterfly and an ironbutterfly must add up to theamount between exerciseprices. Taking interest intoconsideration, the valuestoday must add up to thepresent valueof this amount.

If we assume that interestratesarezero,andtheJune95/100/105butterflyistradingfrom1.75,theJune95/100/105 iron butterfly should betrading for3.25.Whetherwebuy the butterfly for 1.75, orselltheironbutterflyfor3.25,we want the same thing tohappen, themarket to remainclose to the inside exerciseprice of 100. Both spreadswill have the same profit orlosspotential.

We can also create acondor synthetically bycombining long and shortstrangles.

1.+1June90put/+1June110call2.–1June95put/–1June105call

The firstposition isa longJune 90 / 110 strangle; thesecond is a short June 95 /105 strangle. If we express

theentirepositionintermsofcallswecanrewriteeachputasasynthetic:

Replacing the puts withtheir synthetic equivalents,

and canceling out the longand short underlyingcontracts, we are left with alongcondor

+1 June 90call–1 June 95

call–1 June105

call+1June110

call

If we instead express allcontractsasputswewillalsoend up with a long condor.This confirms that a call andputcondorareessentiallythesame. One is simply asyntheticversionoftheother.

An iron condor is aposition which combines along strangle with a shortstrangle, with one stranglecentered in themiddleof theother strangle. While a long

condor (buy the outsideexercise prices / sell theinsideexerciseprice) isdonefor a debit, the iron condorequivalent (sell the outsidestrangle / buy the insidestrangle) is done for a credit.Theinsidestranglewhichweare selling is always morevaluable than the outsidestranglewhichwearebuying.Ifwereceivemoneywhenweput on the position then weare short the iron condor.

Buyingatraditionalcondorisequivalent to selling an ironcondor.

At expiration the valueof a condor and an ironcondor must add up to theamount between the insideandoutsideexerciseprices,inour example 5.00. Takinginterest into consideration,thevaluesmustadduptothepresent valueof this amount.If we assume that interest

ratesarezero,andtheJune90/ 95 / 105 / 110 condor istradingfor3.75,theJune90/95 / 105 / 110 iron condorshould be trading for 1.25.Whether we buy the condorfor 3.75, or sell the ironbutterfly for 1.25, we wantthesamethingtohappen,themarket to remain within theexercise prices of the insidestrangle. Both spreads willhave the same profit or losspotential.

The characteristics ofsome volatility spreads canoften be more easilyrecognized when written insynthetic form. For example,in Chapter 11 we looked atspreads commonly known asChristmas trees. A typicallongChristmastreemightbe

+1 June 95call / –1 June100 call / –1June105call

The characteristics of thisposition may not have beenimmediately apparent. Butsupposewe use synthetics torewrite the June 95 and 100callsasputs

Replacing the June 95 and100 callswith their syntheticequivalents, and cancelingout the long and shortunderlying contracts, we areleftwith

+1 June 95put–1 June100

put–1 June105

call

If we focus first on theJune 100 put and June 105call,thepositionconsistsofashort strangle (the June100 /105 strangle) combined withalongputatalowerexerciseprice(theJune95put).Ifwefocuson theJune95putandtheJune100put,thepositionconsists of a bull put spread(the June100 / 105 putspread) combined with ashortcallatahigherexerciseprice (the June 105 call). In

both cases, we have aposition with limiteddownside risk and unlimitedupsiderisk.

1Becausethepositionwillnotturnintoanunderlyingcontractuntilexpiration,itissometimesreferredtoasasyntheticforwardcontract,whichisperhapsamoreaccuratetheoreticaldescription.Wewillseelaterthatpricingofthiscombinationdependsonthevalueofaforwardcontract.

15

OptionArbitrage

Suppose that we want totake a short position in anunderlying contract that iscurrently trading at 102.00.We can simply sell theunderlyingcontractat102.00.

However, we have anadditional choice—we cantake a short positionsyntheticallybysellingacalland buying a put with thesame expiration date andexerciseprice.Whichofthesestrategies is best? Supposethat we sell the December100callfor5.00andbuytheDecember 100 put for 3.00,for a total credit of 2.00. Ifthe options are European,with no possibility of early

exercise, at expiration, wewill always sell theunderlyingcontractat100.00,eitherbyexercisingtheputorbybeingassignedonthecall.Because we have a credit of2.00 from the option trades,we are in effect selling theunderlying contract at itscurrent price of 102.00. Ifthere are no interest ordividend considerations, theprofit or loss resulting fromoursyntheticpositionwillbe

identical to the profit or lossresulting from the saleof theunderlyingcontractat102.00.Indeed, regardless of theindividual prices of theDecember 100 call and put,as long as the price of theDecember100call is exactly2.00greater than thepriceofthe December 100 put, theprofitorlosswillbethesamefor both positions. This isshowninFigure15-1.

Figure15-1

Now let’s assume thatwe already have a syntheticshortposition:

–1 December100call+1 December100put

Ifwewanttogetoutoftheposition,whatcanwedo?Wecan, of course, close out oursynthetic by buying back the

December 100 call andsellingouttheDecember100put. However, we can alsooffset the synthetic shortposition by buying theunderlyingcontract.

–1 December100call+1 December100put+1 underlyingcontract

This position, usuallyreferred to as a conversion,1is the most common type ofoption arbitrage. In a classicarbitrage strategy, a traderwill try to buy and sell thesame or very closely relatedcontracts indifferentmarketstoprofitfromamispricing.Ina conversion, the trader isbuying the underlyingcontract in the underlyingmarket and selling the

underlying contract,synthetically, in the optionmarket. Taken together, thetradesmakeupanarbitrage.

A trader can also takethe opposite position,executing a reverseconversion (or reversal), byselling the underlyingcontract and buying itsynthetically:

+1 December100call

–1 December100put–1 underlyingcontract

Summarizing,

where the call and putalways have the sameexercise price and expirationdate.

Whether a trader willwant to take either of thesepositions depends on theprices of the contracts. If thesynthetic portion (the longcall and short put) is tooexpensive comparedwith theunderlying contract, a traderwillwant todoaconversion.If thesyntheticportion is toocheap, a trader will want todoareverseconversion.Howcanwedeterminewhetherthesyntheticismispriced?

Let’sbeginbyassumingthattheunderlyingcontractisstock. In a December 100conversion,wewill

Sell aDecember 100callBuy aDecember 100putBuystock

If we do all these trades

and carry the position toexpiration, what are theresultingcreditsanddebits?

First, the credits. Whenwe sell the call, we willreceive the call priceC. Wecan invest this amount overthelifeoftheoptionandearninterestC×r×t.Becauseweown the stock, we willreceive any dividendsD thatare paid prior to theDecemberexpiration.Finally,

at expiration, we will eitherexercise the put or beassignedonthecall.Ineithercase, we will sell the stockandreceivetheexercisepriceX.Thetotalcreditsare

CallpriceCInterest earnedonthecallC×r×tDividends, ifany,DExercise price

X

Next,thedebits.Wewillhave to pay the put price PandthestockpriceS.Inbothcases,wewillhavetoborrowthe money, so there is theadditional interest costP × r× t and S × r × t. The totaldebitsare

PutpricePInterestcost tobuy the put P

×r×tStockpriceSInterestcost tobuythestockS×r×t

In an arbitrage-freemarket, all credits and debitsmustbeequal:

C+C×r×t+D+X=P+P×r×t+S+S×r×t

Traders sometimes refer

to the synthetic portion of aconversion or reversal as acombo, eithera longcall andshort put or a short call andlong put. We can determinewhether there is a relativemispricingand,consequently,an arbitrage opportunity bysolving for the combo valueC – P in terms of all othercomponents.

First, we group the calland put components on the

left side and everything elseontherightside

C+C×r×t–P+P×r×t=S+S×r×t–D–X

Next, we separate theinterest-ratecomponent

C×(1+r×t)–P×(1+r×t)=S×(1+r×t)–D–X

andthenisolateC–P

(C–P)×(1+r×t)=S×(1+r×t)–D–X

At this point, we mightrecognize part of theexpressionontheright:S×(1+ r × t) – D. This is theforward price for the stock.To simplify our notation,wecanreplaceS×(1+r×t)–DwithF

(C–P)×(1+r×t)=F–X

Finally, we divide bothsides by the interestcomponent1+r×t

Simply stated, thedifference between the callprice and put price forEuropean options with thesame exercise price andexpirationdatemustbeequalto the present value of the

difference between theforward price and exerciseprice. This relationship, oneof the most important inoption pricing, goes byvarious names. In textbooks,it iscommonly referred toasput-call parity. Traders mayalso refer to it as the combovalue, the syntheticrelationship, or theconversionmarket.

The exact calculation of

put-callparitydependsontheunderlying market and thesettlement procedures for theoptionsmarket. Let’s look atseveraldifferentcases.

OptionsonFutures

The simplest calculationoccurswhentheunderlyingisa futures contract and theoptionsaresubjecttofutures-type settlement. In this case,

theeffective interest rate is0because no money changeshands when either theunderlyingfuturescontractorthe options are traded.Moreover, futures contractspay no dividends, so we canexpress put-call parity in itssimplestformas

C–P=F–X

With a December 100call trading at 5.25 and a

December 100 put trading at1.50,whatshouldbethepriceof the underlying Decemberfuturescontract?

Because

C–P=5.25–1.50=3.75

F–Xalsomustequal3.75.

The futures price must be103.75.

Whatwillhappeninourexample if the underlyingfuturescontractisnottradingat 103.75 but instead istradingat104.00.Wecanseethat

5.25–1.50≠104.00–100

and

3.75≠4.00

Everyone will want toexecute a reverse conversionby buying the less expensivesynthetic (buy the call, selltheput) and selling themoreexpensive underlying (thefutures contract). Ignoringtransactioncosts, if all tradesactually canbedone at theseprices,thestrategywillresultin an arbitrage profit of .25,theamountofthemispricing.

What will be the result

of everyone attempting to doa reverse conversion?Because everyone wants tobuy the call, there will beupward pressure on the callprice.Ifthecallpricerisesto5.50 while all other pricesremain unchanged, put-callparityismaintainedbecause

5.50–1.50=104.00–100

Alternatively,aspartofthereverse conversion, everyone

wantstoselltheput.Thiswillputdownwardpressureontheputprice.Iftheputpricefallsto 1.25, put-call parity isagainmaintainedbecause

5.25–1.25=104.00–100

Finally, everyonewants tosell the futures contract,putting downward pressureon the futures price. If thefutures contract falls to103.75, put-call parity is

againmaintainedbecause

5.25–1.50=103.75–100

Whether the call pricerises, the put price falls, thefutures price falls, or somecombination of all three, thefinalresultmustbe

C–P=F–X

This application of put-callparity,whereallcontracts

are subject to futures-typesettlement, is typically usedfor options traded on futuresexchanges outside NorthAmerica.When an exchangesettlesoptionpricesattheendof the tradingday, theremaybe inconsistencies having todowiththevolatilityvalueofan option. But the exchangewill always try to assignsettlement prices that areconsistent with put-callparity. A table of settlement

prices for Euro-bund optionstraded on Eurex is shown inFigure 15-2.2 Note that inevery case, put-call parity ismaintained.

Figure15-2Settlementpricesforeuro-bundoptionsonMay25,2010

Put-call paritycalculations become slightlymore complicated whenoptionsonfuturesaresubjectto stock-type settlement, asthey are on most futuresexchanges inNorthAmerica.Nowwemustdiscountbytheinterest-ratecomponent

With six monthsremaining to expiration andanannualinterestrateof6.00percent,aDecember100callis trading for 4.90. Whatshould be the price of theDecember 100 put if theunderlying December futurescontract is trading at 97.25?Weknowthat

Thedifferencebetweenthecall price and put pricemustbe 2.67, with the negativesign indicating that the putprice is greater than the callprice

C–P=–2.67P=C–(–2.67)=4.90+2.67

=7.57

Theputmustbetradingfor7.57.

LockedFuturesMarkets

Manyfuturestradersprefernot to become involved inoptions markets because ofthe apparent complexity ofoptions. There is, however,one situation in which afutures trader ought tobecome familiar with basicoption characteristics. If afuturestraderwantstomakea

trade but is prevented fromdoing so because the futuresmarket has reached its dailylimit,hemaybeabletotradefuturessyntheticallybyusingoptions. The price at whichthe synthetic futures contactis trading can be determinedthroughput-callparity.

Consider a futuresmarket thathas adailyupordown limit of 5.00. Thefutures contract closed the

previousdayat126.75but isnow up limit at 131.75. Nofurther futures trading cantake place unless someone iswilling to sell at a price of131.75 or less. If, however,the options market is stillopen,atradercanbuyorsellfutures synthetically, even ifthisprice isbeyond thedailylimit.Hecaneitherbuyacalland sell a put (buying thefuturescontract)orsellacalland buy a put (selling the

futures contract) with thesame exercise price andexpiration date. The price ofthecallandput,togetherwiththe exercise price, willdetermine the price at whichthe synthetic futures contractistrading.

Below is a hypotheticaltable of call and put pricestogether with the resultingsynthetic futures price. Forsimplicity, we assume that

there are no interestconsiderations. Because C –P =F –X, we can calculatetheequivalentfuturespriceF=C–P+X.

There is some variationin the equivalent syntheticprices, possibly because theprices do not reflect the bid-askspreadorperhapsbecausethe option prices have notbeen quotedcontemporaneously.However, one can see that ifthefuturescontractwerestillopen for trading, its pricewould probably besomewhere in the range of133.20to133.30.Ifafutures

trader wants to buy or sellfutures synthetically in theoptionmarket, he can expectto tradeatapricewithin thisrange.

OptionsonStock

Calculating put-call parityfor stock options entails anadditional step because wemust first calculate theforward price for the stock.

Withsixmonthsremainingtoexpiration and an annualinterestrateof4.00percent,aDecember 65 call is tradingfor 8.00. If the underlyingstock is trading at 68.50 andtotal dividends of .45 areexpected prior to expiration,what should be the price ofthe65put?

We begin with theforwardprice

F=68.50×(1+0.04×6/12)

–0.45=69.42

Then

Theputpricemustbe

8.00–4.33=3.67

AnApproximationforStockOptions

When exchanges firstbegan trading options, allactivity took place in anopen-outcry environment.Traders often had to makepricingdecisionsquicklyandwithouttheaidofcomputers.Asaresult,theyoftensoughtshortcutsbywhichtheycouldmore easily approximateprices. Even if the shortcutresulted in small errors, thevalue of being able to makefaster decisions more than

offset the small loss inaccuracy.

Let’s go back to basicput-call parity for stockoptions and replace theforward price F with theactual forward price for thestock

How might we simplifythiscalculation?

Note that we aremultiplyingthestockpricebythe interest-rate componentand then dividing the stockprice, the dividend, and theexercise price by the sameinterest-rate component. Weend up with the stock priceitself less the discountedvalues of the dividend andexerciseprice

Dividends are typicallysmall compared with thestockpriceandexerciseprice,so a reasonableapproximation for thediscounted value of thedividend is simply thedividendD itself. We mightapproximate the discountedvalue of the exercise priceand eliminate the need to doany division by subtractingthe interest on the exerciseprice from the exercise price

itself

Substituting ourapproximations into the put-callparityequation,wehave

C–P≈S–(X–X×r×t)–D=S–X+X×r×t–D

Thedifferencebetweenthecall price and put price isapproximately equal to thestockpriceminustheexerciseprice plus interest on theexercisepriceminusexpecteddividends.

How good anapproximation is this?Clearly, if interest rates arevery high, the dividend isvery large, orwe are dealingwith long-term options, the

errors will begin to increase.Butforshort-termoptionsourapproximation oftenrepresents a reasonabletradeoff between speed andaccuracy.

Let’s go back to ourprevious stock optionexample:

Stock price =68.50Time toexpiration = 6

monthsInterest rate =4.00percentExpecteddividends =.45

Wecalculated thevalueofthe65combo(C–P)as4.33.How will our approximationcompare?

Our approximation differsby .02 from the true value.Depending on marketconditions, this might be anacceptablemarginoferror inreturnforbeingabletomakeafastertradingdecision.

All experienced tradersare familiar with put-callparity, so any price

imbalances are likely to beveryshort-lived.Ifthecombois overpriced compared withtheunderlying,alltraderswillwant toexecuteaconversion(i.e., buy the underlying, sellthe call, buy the put). If thecombo is underpriced, alltraderswillwanttoexecuteareversal (i.e., sell theunderlying, buy the call, selltheput).Suchactivity,whereeveryone is attempting to dothe same thing, will quickly

force prices back intoequilibrium. Indeed, priceimbalances in the syntheticrelationshipareusually smallandrarelylastformorethanafew seconds. Whenimbalances do occur, anoption trader is usuallywilling to executeconversions or reversals inverylargesizebecauseofthelowriskassociatedwithsuchstrategies.

Put-call parity specifiesthepricerelationshipbetweenthreecontracts—acall,aput,andanunderlyingcontract.Ifthepriceofanytwocontractsis known, it should bepossibletocalculatethepriceof the third contract. If theprices in the marketplace donotseemtobeconsistentwiththis relationship, what mightatraderinfer?

Consider this stock

optionsituation:

90call=7.2090put=1.40Time toexpiration = 3monthsInterest rate =8.00percentExpecteddividends =.47

Whatshouldbethepriceof

theunderlyingstock?Using our stock option

approximation for put-callparity,weknowthat

C–P≈S–X+X×r×t–D

Therefore,

Suppose,however, that thestock is actually trading at

94.30. Does this mean thatthere is an arbitrageopportunity?

The stock pricecalculation depended onassumptions about interestand dividends. Are we surethat those assumptions arecorrect? One possibility isthat the interest rate we areusing,8percent,istoolow.Ifwe assume that the contractprices and dividends are

correct, we can calculate theimpliedinterestrate

Another possibility isthat the dividend we areusing, .47, is too high. Ifweassume that the contract

pricesandinterestarecorrect,we can calculate the implieddividend

The marketplace seems tobe expecting a dividend ofonly .30. If our originalcalculation was based on anestimate of the expecteddivided,weoughttoconsider

the possibility that thecompany will cut thedividendpriortoexpiration.

ArbitrageRisk

New traders who arelearning to trade optionsprofessionally are oftenencouraged to focus onconversions and reversalsbecause, they are told, thesestrategies, once executed, are

essentiallyriskless.Awordofwarning: very few strategiesare truly riskless. Somestrategies entail greater risk,whileothersentaillesserrisk.Rarely, however, does astrategy entail no risk. Therisksofdoingconversionsorreversals may not beimmediately apparent, buttheyexistnonetheless.

ExecutionRiskBecause no one wants to

give awaymoney, a trader isunlikely to be offered aprofitable conversion orreversal all at one time.Consequently, a trader whofocuses on these strategieswill have to begin byexecutingoneortwolegsandthenhopetoexecutethefinalleg(s)atalatertime.Hemay,for example, initially

purchase puts together withunderlyingcontractsandhopeto later complete theconversion by selling calls.However, if call prices beginto fall, hemayneverbeableto profitably complete theconversion. Even aprofessional trader on anexchange,whowouldseemtobeinagoodpositiontoknowthe prices of all threecontracts, can be mistaken.He may create a long

synthetic underlying positionby purchasing a call andselling a put at what hebelieves are favorable prices.However, when he tries toselltheunderlyingcontracttocompletethereversal,hemayfind that the price is lowerthanheexpected.Wheneverastrategyisexecutedonelegata time, there is always therisk of an adverse change inpricesbeforethestrategycanbecompleted.

PinRiskWhen we introduced the

concept of a syntheticposition, we assumed that atexpiration, the underlyingmarketwouldbeeitherabovethe exercise price, in whichcase the call would beexercised, or below theexercise price, inwhich casethe put would be exercised.But what will happen if theunderlying market is exactly

equal to, or pinned to, theexercise price at themomentofexpiration?

Supposethatatraderhasexecuted a June 100conversion:heisshortaJune100call,longaJune100put,and long the underlyingcontract. If the underlyingcontract is above or below100 at expiration, there is noproblem. Either he will beassignedonthecallorhewill

exercise the put. In eithercase, the long underlyingpositionwillbeoffset,andhewill have nomarket positionon the day followingexpiration.

But suppose that at themoment of expiration, theunderlying market is right at100.Thetraderwouldliketobe rid of his underlyingposition.Ifheisnotassignedon the call, he can exercise

his put; if he is assigned onthe call, he can let the putexpire. In order to make adecision, he must knowwhether the call will beexercised.Buthewon’tknowthis until the day afterexpiration, when he eitherdoes or does not receive anassignmentnotice.Ifhefindsout that he was not assignedonthecall, itwillbe toolatetoexercise theputbecause itwillhaveexpired.

It may seem that anoption that is exactly at themoney at expiration willnever be exercised because,in theory, it has no value. Infact, many at-the-moneyoptions are exercised. Eventhough the option has notheoreticalvalue,itdoeshavesome practical value. Forexample, suppose that theownerofacallthatisexactlyat the money at expirationwants to take a longposition

intheunderlyingcontract.Hehas two choices. He caneitherexercisethecallorbuythe underlying contract.Because an exchange-tradedoption typically includes theright of exercise in theoriginal transaction cost, it isalmost always cheaper toexercisethecall.Evenifthereis a small transaction cost toexercise, an option, it willalmostalwaysbelessthanthecostoftradingtheunderlying

contract. Anyone owning anat-the-money option andchoosing to take a long orshort position at expirationwill find that it is cheaper toexercise the option than tobuy or sell the underlyingcontract.

Clearly, a trader who isshort an at-the-money optionat expiration has a problem.What can he do? Onepossibility is to make an

educatedguessas towhetherthe at-the-money option willbe exercised. If the marketappears to be strong on thelast trading day, the tradermight assume that it willcontinue higher followingexpiration.Iftheholderofthecall sees the situationsimilarly, it is logical toassume that the call will beexercised. Hence the traderwillchoosenottoexercisehisput. Unfortunately, if the

trader is wrong and he doesnot get assigned on the call,he will find himself with along underlying position thathe would rather not have.Conversely, if the marketappearstobeweakonthelasttrading day, the tradermightmake the assumption that hewill not be assigned on thecall.Hewillthereforechooseto exercise the put. But,again,ifheiswronganddoesget an assignment notice, he

will find himself with anunwanted short underlyingpositiononthedayfollowingexpiration.

The risk of a wrongguess can be furthercompounded by the fact thatconversions and reversals,becauseof their lowrisk,areoftendoneinlargesize.Ifthetraderguesseswrong,hemayfind that on the day afterexpiration, he is naked long

or short not one or two butmanyunderlyingcontracts.

There can be no certainsolutiontotheproblemofpinrisk. With many, perhapsthousands, of open contractsoutstanding, some at-the-money options will beexercised and somewon’t. Ifthetrader lets thepositiongoto expiration and relies onluck,heisatthemercyofthefates, and this is a position

that an intelligent optiontrader prefers to avoid. Thepractical solution is to avoidcarryingashortat-the-moneyoption position to expirationwhen there is a realpossibility of expiration rightat the exercise price. If thetrader has a large number ofJune 100 conversions orreversals and expiration isapproaching with theunderlying market close to100, the sensible course is to

reduce the pin risk byreducing the size of theposition. If the traderdoesn’treduce the size, he may findthat he is under increasingpressure togetoutof a largenumber of risky contracts asexpirationapproaches.

Sometimes even acarefultraderwillfindthathestillhassomeoutstandingat-the-money conversions orreversals as expiration

approaches. If he is veryconcerned with the potentialpin risk, he might simplyliquidate the position at theprevailing market prices.Unfortunately,thisislikelytoresult in a loss because thetraderwill be forced to tradeeach contract at anunfavorable price, eitherbuying at the offer or sellingat the bid. Fortunately, it isoftenpossible to tradeoutofsuchapositionallatonceata

fairprice.Becauseconversionsand

reversals are commonstrategies,atraderwhohasanat-the-money conversion andisworried about pin risk canbefairlycertainthattherearealsotradersinthemarketwhohave at-the-money reversalsand are worried about thesame pin risk. If the traderwith the conversion couldfind a trader with a reversal

andcrosspositionswithhim,both traders would eliminatethe pin risk associated withtheir positions. This is whyon option exchanges oneoftenfindstraderslookingforother traders who want totradeconversionsorreversalsat even money. This simplymeans that a trader wants totrade out of his position at aprice that is fair to everyoneinvolvedsothateveryonecanavoidtheproblemofpinrisk.

Whatever profit a traderexpected to make from theconversion or reversalpresumably resulted from theopening trade, not from theclosingtrade.

Pin risk only occurs inoption markets whereexercise results in a long orshort position in theunderlying contract. In somemarkets, such as stockindexes,optionsaresettledat

expiration incashrather thanwith the delivery of anunderlying contract. Whenthe option expires, there is acash payment equal to thedifference between theexercisepriceandunderlyingprice, but no underlyingposition results.Consequently, there isnopinrisk associatedwith this typeofsettlement.

SettlementRiskLet’s go back to our

December 100 conversionexample. But now let’sassumethat theunderlying isaDecemberfuturescontract

–1 December100call+1 December100put+1 Decemberfutures

contract

If the December futurescontract is trading at 102.00,there are three monthsremaining to Decemberexpiration, interest rates are8.00 percent, and all optionsare subject to stock-type(cash)settlement,thevalueofthe December 100 syntheticcombination (the differencebetween the December 100call and the December 100

put)shouldbe

Suppose that a trader isable to sell a December 100callfor5.00,buyaDecember100 put for 3.00, and sell aDecemberfuturescontractfor102.00. At expiration, thetrader should realize a profitof .04 because he has donetheDecember100conversion

at.04betterthanitsvalue.Shortly after the trader

executes the conversion, theunderlying December futurescontract falls to 98.00.Whatwill be the cash flow? Thesyntheticpositionwillshowaprofit of approximately 4.00;the short call and long puttogether, because they makeup a short underlyingposition, will appreciate by4.00.Butbecausetheoptions

are settled like stock, theprofit on the syntheticpositionwill be unrealized—therewillbenocashcreditedto the trader’s account. Onthe other hand, the trader isalsolongaDecemberfuturescontract, and this contract,because it is subject tofutures-type settlement, willresult in an immediate debitof 4.00 when the marketdrops to98.00.Tocover thisdebit, the trader must either

borrowthemoneyortakethemoney out of an interest-bearing account. In eithercase, there will be a loss ininterest, and this interest losswill not be offset by theunrealized profit from theoptionposition. If the loss ininterest is great enough, itmay more than offset theprofit of .04 that the traderoriginally expected from theposition. In themostextremecase, where the trader does

not have access to the fundsrequired to cover thevariation on the futuresposition,hemaybeforcedtoliquidate the position.Needless to say, forcedliquidations are neverprofitable.

Of course, this worksbothways.Ariseinthepriceof the underlying futurescontract to 106.00will resultin a loss of 4.00 on the

synthetic option position; theshort call and long puttogetherwilldeclineby4.00.But this loss is unrealized—no money will actually bedebited from the trader’saccount.3 On the other hand,theriseinthefuturescontractwill result in an immediatecash credit on which thetrader can earn interest. Thisinterest will increase thepotential profit beyond the

expectedamountof.04.Option traders tend to

assume that conversions andreversals are delta-neutralstrategies. But this is notalwaystrue.Anexactlydelta-neutral position has nopreferenceas to thedirectionof movement in theunderlying contract. In ourexample,we can see that thetrader prefers upwardmovement because he can

earn interest on the variationcredited to his account.Withthe underlying futurescontract at 102, the deltas inourexamplemightbe

Thetwoextradeltasreflectthefactthatthetraderprefersthemarket to rise rather thanfallsothatcashwillflowintohis account from the futuresposition. The interest fromthiscashflowcanresultinanunexpected profit. A declinein the futurespricewill havethe opposite effect and canresultinanunexpectedloss.

Under normalcircumstances, few traders

will concern themselveswiththe risk of being two deltaslongorshort.Butconversionsand reversals, because theyare low-risk strategies, areoftendoneinverylargesize.Atraderwhoexecutes300ofoursampleconversionshasadeltariskof300×+2=+600.Thisisthesameasbeinglonganextrasixfuturescontracts.The risk comes from theinterestthatcanbeearnedonany cash credit or that must

be paid on any cash debitresulting from movement inthe underlying futurescontract.

The amount by whichthedeltaofasyntheticfuturesposition will differ from 100depends on the interest riskassociated with the position.This,inturn,dependsontwofactors—the general level ofinterest rates and the amountof time remaining to

expiration. The higher theinterest rate and the moretime remaining to expiration,thegreatertherisk.Thelowerthe interest rate and the lesstime remaining to expiration,thelesstherisk.A10percentinterestratewithninemonthsremaining to expirationrepresentsamuchgreaterriskthan a 4 percent interest ratewithonemonth remaining toexpiration.Intheformercase,the deltas of a synthetic

position may add up to 93,while in the latter case thedeltas may add up to 99. Ingeneral, the total delta for asynthetic futures contract,where theoptionsaresubjecttostock-typesettlement,is

wherer is the interest rateand t is the time to maturityoftheoptions.

This type of settlementrisk occurs only when theoptions and the underlyingcontract are subject todifferent settlementprocedures.4 There is nosettlement risk when bothcontracts are subject to thesamesettlementprocedure.Ifall contracts are subject tostock-typesettlement,astheyare in a typical stock optionmarket, no cash flow results

fromfluctuationsinthepricesof the contracts prior toexpiration.Ifallcontractsaresubject to futures-typesettlement, as they are onmost futures exchangesoutsidetheUnitedStates,anycash flow resulting fromchanges in the price of theunderlying futures contractwill exactly offset the cashflow resulting from changesin prices of the optioncontracts.

InterestandDividendRiskLet’s againgoback toour

December 100 conversion,butnowlet’sassumethat theunderlyingcontractisstock.

–1 December100call+1 December100put+1 stock

contract

What are the risks ofholdingthisposition?

The stock price willalways be greater than theoption prices, so the entireposition will be done for adebit approximately equal tothe option’s exercise price.Because the trader will haveto borrow this amount, therewill be an interest costassociated with the position.

If interest rates rise over thelife of the position, theinterest costs will also rise,increasingthecostofholdingthe position and,consequently, reducing thepotential profit. If interestrates fall, the potential profitwill increase because thecostsofcarrying thepositionwilldecline.5

Theoppositeistrueofareverseconversion:

+1 December100call–1 December100put–1 stockcontract

Because the trader willreceive cash from the saleofthe stock, the position willearn interest over time. Ifinterest rates rise, the interestearnings will also rise,increasing the value of the

position. If interest rates fall,the interestearningswill fall,reducing the value of theposition.

Clearly,conversionsandreverse conversions aresensitive to changes ininterestrates.Thisisreflectedin their rho values. In thestock option market, aconversion has a negativerho, indicating a desire forinterestratestofall.Areverse

conversionhasapositiverho,indicatingadesireforinterestrates to rise. This is logicalwhen we recall that in thestock option market callshave positive rho values andputshavenegativerhovalues.In a conversion or reverseconversion, the signs of thecalland theput rhopositionswill be the same, either bothpositive or both negative,because we are buying oneoptionandsellingtheother.

The fact that aconversion or reverseconversion includes a stockpositionalsomeansthatthereis the riskof risingor fallingdividends. In a conversion,we are long stock, so anyincrease in dividends willincrease the value of theposition, and any cut individends will reduce thevalue. In a reverseconversion, the opposite istrue.

Even though there is noGreekletterusedtorepresentdividend risk, we might saythataconversionhaspositivedividend risk and a reverseconversion has negativedividendrisk.Theformerwillbe helped by any increase individends, while the latterwillbehurt.

Wecanseetheeffectofchanging interest anddividends by recalling our

earlierexample:

Stock price =68.50Time toexpiration = 6monthsInterest rate =4.00percentExpecteddividend=.45

We calculated theapproximate value of the

combo(C–P)as4.35

Ifinterestratesriseto5.00percent,thevaluewillnowbe

68.50–65+65×.05×6/12–.45≈4.68

If, on the other hand, thedividend is increased to .65,

thevaluewillbe

68.50–65+65×.04×6/12–.35=4.15

Aconversionorreversalentails risk because thesestrategiescombineasyntheticunderlying position,which iscomposedofoptions,withanactual position in theunderlying contract. The riskarises because a syntheticposition and the actual

position, while very similar,can still have differentcharacteristics,eitherintermsofsettlementprocedure,asinthe futures optionmarket, orin terms of interest ordividends, as in the stockoption market. Is there anywaytoeliminatethisrisk?

One way to eliminatethis risk is to eliminate theposition in the underlyingcontract. Consider a

conversion:

ShortacallLongaputLong anunderlyingcontract

Ifwewanttomaintainthisposition but would also liketo eliminate the risk ofholding an underlyingposition,wemightreplacethelongunderlyingpositionwith

something that acts like anunderlying contract but thatisn’t an underlying contract.One possibility is to replacethe long underlying positionwith a deeply in-the-moneycall:

ShortacallLongaputLong a deeplyin-the-moneycall

If the deeply in-the-moneycall has a delta of 100 andtherefore acts like a longunderlying contract, theposition will have the samecharacteristics as theconversion.

Inthesameway,insteadof buying a deeply in-the-money call, we can sell adeeplyin-the-moneyput:

ShortacallLongaput

Short a deeplyin-the-moneyput

This type of position,where the underlyinginstrumentinaconversionorreversal is replaced with adeeply in-the-money option,is known as a three-way.Although it eliminates somerisks, a three-way is notwithoutitsownproblems.Ifatrader sells a deeply in-the-

money option to complete athree-way,hestillhastheriskof the market going throughthe exercise price. Indeed, asthe underlyingmarketmovescloser and closer to theexercise price of the deeplyin-the-money option, thatoption will act less and lesslike an underlying contract,and the entire position willact less and less like a trueconversionorreversal.

BoxesWhat else acts like an

underlying contract but isn’tan underlying contract?Another possibility is toreplace the underlyingposition with a syntheticposition, but a syntheticwithadifferentexerciseprice.Forexample, suppose that wehaveaJune100conversion:

–1 June100

call+1June100

put+1

underlyingcontract

At the same time,we alsoexecute a June 90 reversal.Thecombinedpositionis

The long and shortunderlying contracts cancelout,leaving

We have a synthetic longunderlying position at the 90exercisepriceandasyntheticshort underlying position atthe 100 exercise price. Thisposition, known as a box, issimilar to a conversion or

reversal except that we haveeliminated theriskassociatedwithholdingapositionintheunderlying contract. A traderis long the box when he issynthetically long at thelower exercise price andsynthetically short at thehigher exercise price. He isshort the box when he issynthetically short at thelower exercise price andsynthetically long at thehigher exercise price. The

example position is long aJune90/100box.

Like a conversion orreversal,aboxisanarbitrage—we are buying and sellingthe same contract but indifferent markets. In ourexample, we are buying theunderlying contract in the 90exercise price market andselling the same underlyingcontract in the 100 exercisepricemarket.

How much is a boxworth? Ignoring pin risk, atexpiration,atraderwhohasabox will simultaneously buytheunderlyingcontractatoneexercise price and sell theunderlying contract at theother exercise price. Thevalueoftheboxatexpirationwill be exactly the amountbetween exercise prices. Inour example, at expiration,the90/100boxwillbeworthexactly 10.00 because the

trader will simultaneouslybuytheunderlyingcontractat90(exercise the90callorbeassigned on the 90 put) andselltheunderlyingcontractat100 (exercise the 100 put orbe assigned on the 100 call).If the box is worth 10.00 atexpiration, how much is itworth today? If the optionsare subject to futures-typesettlement, thevalue today isthe same as the value atexpiration. If, however, the

options are subject to stock-type settlement, the value ofthe box today will be thepresent value of the amountbetween exercise prices. Ifour 90/100 box expires inthree months with interestrates at 8 percent, the valuetodayis

Because a box

eliminates the risk associatedwithcarryingapositionintheunderlying contract, boxesare even less risky thanconversions and reversals,which are themselves low-risk strategies. When alloptionsareEuropean(thereisnoriskofearlyexercise)andtheoptionsaresettledincashrather than through deliveryof the underlying contract(there is no pin risk), thepurchase or sale of a box is

identical to lending orborrowingfundsoverthelifeof the options. In ourexample, a trader who sellsthe 90/100 box for 9.80 hasessentially borrowed fundsfromthebuyeroftheboxforthree months at an interestrate of 8 percent. Selling thebox at a lower price isequivalenttoborrowingfundsatahigherinterestrate.Ifthetrader sells the three-monthboxatapriceof9.70,hehas,

ineffect,agreedtoborrowatan annual interest rate of 12percent.

When no other methodis available, a trading firmmay be able to raise neededshort-term cash by sellingboxes. Because the firmwillprobably have to sell theboxes at a price lower thanthetheoreticalvalue,thiswillincreasethefirm’sborrowingcosts. Moreover, there will

stillmargin requirements andtransaction costs associatedwith this strategy, increasingtheborrowingcostsfurther.

Weoriginallyintroduceda box as a conversion at oneexercise price and a reversalat a different exercise price.With the long and shortunderlying positionscanceling out, we are leftwithtwosyntheticunderlyingpositions:

Theleftsideoftheboxisasynthetic longposition at 90,and the right side is asynthetic short position at100. Instead of dividing theboxintoarightsideandaleftside,supposethatwedivideitinto upper portion and alowerportion:

Thestrategyonthetopisabull vertical call spread (i.e.,long June 90 call, short June100 call), whereas thestrategy on the bottom is abear vertical put spread (i.e.,longJune100put,shortJune90 put). Because a box is acombination of two verticalspreads, the combined pricesof the vertical spreads must

equalthevalueofthebox.With three months

remaining to expiration andinterestratesat8percent,thevalueofourJune90/100boxis9.80.Suppose thata traderknows that the June 90/100callspreadistradingfor6.00.The trader can estimate thefairmarketpricefor theJune90/100putspreadbecauseheknows that the90/100box isworth9.80andthatthevalue

ofacallandputspreadmustadd up to the value of thebox. The price of the putspreadmustthereforebe

9.80–6.00=3.80

If the trader believes thathe can either buy or sell thecallspreadfor6.00andheisasked foramarket in theputspread, he will make hismarket around an assumedvalue of 3.80. He might, for

example, make a market of3.70bid/3.90ask.Ifheisabletobuytheputspreadfor3.70,hecanthentrytobuythecallspread for 6.00. If he issuccessful, hewill have paidatotalof9.70foraboxwitha theoretical value of 9.80.Conversely, if he is able tosell the put spread for 3.90,hecanthentrytosellthecallvertical for 6.00. If he issuccessful, he will have soldaboxwithatheoreticalvalue

of9.80forapriceof9.90.

RollsIn a box, the risk of

holding the underlyingcontract is offset bycombining a conversion andreversal in the same monthbut at different exerciseprices:

Suppose that we insteadcombine a conversion andreversal, not at differentexercise prices, but indifferentexpirationmonths:

If the long and shortunderlying positions cancelout,weareleftwitharoll:

We have a synthetic longunderlying position in Juneand a synthetic shortunderlying position inAugust,wherebothpositionshavethesameexerciseprice.

Although it is always

possible to combine aconversioninonemonthwitha reversal in a differentmonth, in a roll, theunderlying positions mustcancelout.Forexample, inafutures option market, theunderlyingforJunemaybeaJune futures contract and theunderlyingforAugustmaybean August futures contract.Because they are differentcontracts, the long and shortunderlying positions will not

offset each other. Hence thepositionisnotatrueroll.

Rolls are done mostcommonly in a stock optionmarket,where theunderlyingcontract for all expirationmonths is the sameunderlying stock. The longstock position in oneexpirationmonthwill alwaysoffsettheshortstockpositionintheotherexpirationmonth.

What should be the

value of a roll in the stockoption market? The value oftherollmustbethedifferenceinthevaluesofthecombos

(Cl–Pl)–(Cs–Ps)

where Cl and Pl are thelong-termcallandput,andCsandPsare theshort-termcallandput.

For the moment, let’sassumethatthestockpaysno

dividends. We know thevalueofacombo

Thevalueoftherollshouldthereforebe

Excluding dividends, thevalue of the roll is the

difference between thediscounted values of theexercise price. Note that thevalue of the roll depends ontwo different interest rates—rs,theinteresttoshort-termexpiration,andrl,theinterestto long-term expiration.These rates are usually verysimilar,whichmeansthat thepreceding expression isalmost always a positivenumber because the

discountingon theshort-termexercisepriceis less thanthediscounting on the long-termexerciseprice.

If the stock pays adividend D betweenexpirations, the value of theroll should also include thisamount. Ignoring interest ondividends,therollvalueis

Consider ourJune/August90roll,withtwomonthstoJuneexpirationandfour months to Augustexpiration. If we assume aconstant interest rate of 6percent, and the stock isexpectedtopayadividendof.40 between expirations, thevalueofthe90rollis

A trader who needs tomake calculations withoutcomputer supportmight, aswith conversions andreversals,mightbewilling togive up some accuracy inreturnforgreaterspeed.Howmight a trader simplify thecalculationofaroll?A traderwho is short a roll (i.e., longthe short-term synthetic andshortthelong-termsynthetic)will buy stock at the short-termexpirationandsellstock

at the long-term expiration,with both transactions takingplace at the same exerciseprice. Additionally, becausethe traderwill own the stockover the life of the roll, hewill receive any dividendspaidoutoverthisperiod.Thevalue of the roll should beapproximately the cost ofcarrying the exercise pricefrom one expiration to theother less any dividends thataccrue

X×r×t–D

wheretisthetimebetweenexpirations. In our example,wehave

90×.06×2/12–.40=.90–.40=.50

Depending on the tradingenvironment and the trader’sultimategoal,thiserrorof.03may or may not beacceptable.

Instead of writing a rollasacombinationofsyntheticlong and short underlyingpositions, we can also writethe roll as a combination ofcalendarspreads:

–1 June 90call/+1August90call+1 June 90

put/–1 August90put

Thestrategyonthetopisalongcallcalendarspread;thestrategy on the bottom is ashort put calendar spread. Ifwe buy the call calendarspread and sell the putcalendar spread, we have aroll. The value of the rollshould therefore be equal tothe difference between thetwocalendarspreads.6

Because the interestcomponent is almost always

greaterthandividends,alongroll (i.e., buy the long-termsynthetic, sell the short-termsynthetic)will typically tradeforapositivevalue,requiringan outlay of cash.Consequently, the callcalendar spreadwill bemorevaluablethantheputcalendarspread.However,ifdividendsaregreaterthaninterest,arollcan have a negative value.7Then the normal relationship

will be inverted: the putcalendar spreadwill bemorevaluable than the callcalendarspread.

In our earlier example,wecalculatedthevalueoftheJune/August 90 roll as .47.SupposethattheJune/August90 call calendar spread istradingfor2.25.Whatshouldbe the value of theJune/August 90 put calendarspread? We know that the

difference between thespreads must be .47. Thevalueof theputspreadoughttobe

2.25–.47=1.78

Inthesameway,iftheputspreadistradingfor1.50,thecall spread ought to betradingfor

1.50+.47=1.97

Because dividends arediscrete amounts that applyequally to all rolls with thesame expiration dates, thevaluesof rollswith the sameexpiration date but differentexercise prices should differby approximately the intereston exercise prices. In ourexample, the value of theJune/August 90 roll was .47.ThevalueoftheJune/August80rollshoulddifferfromthevalue of the 90 roll by the

interest on the differencebetween80and90

0.47–(90–80)×.06×2/12=.47–.10=0.37

Whiles a trader mayexecute a roll with theintention of eliminating theriskofholdingtheunderlyingcontract, this risk is onlyeliminated up to the short-termexpiration.At that time,the trader will either buy or

sell the underlying stock atthe exercise price. Theposition is therefore sensitiveto changes in interest ratesanddividends.Rollsfluctuatein value as interest rates riseor fall and as dividends areraised or lowered. The moretimebetweenexpirations, themore sensitive a roll will betothesechanges.

TimeboxesA box or roll consists of

long and short syntheticpositions, either in the samemonth but at differentexercise prices (a box) or indifferent months but at thesame exercise price (a roll).We can also combine thesestrategies by taking syntheticpositionsatdifferentexerciseprices and in differentmonths:

This position is usuallyreferred to as either a timeboxordiagonalroll.

We can calculate thevalue of a time box in thesame way we calculated thevalueofaroll—bytakingthedifference between thediscounted exercise priceslessexpecteddividends

wherethesubscriptssandlrefer to short-term optionsandlong-termoptions.

What should be thevalue of the June 90/August100timeboxiftherearetwomonthstoJuneexpirationandfour months to Augustexpiration,interestratesareaconstant 6 percent, and thestock is expected to pay a

dividend of .40 over thisperiod?

Thenegativesignindicatesthatifatraderwantstoputonthis position, hewill have topay 9.33. This is logicalbecause the position consistsofbuying the lower-exercise-price synthetic (i.e., buy theunderlying at 90 at June

expiration) and selling thehigher-exercise-pricesynthetic (i.e., sell theunderlying at 100 at Augustexpiration).

In the same way thatboxesaremadeupofbullandbear spreads and rolls aremadeupofcalendar spreads,time boxes are made up ofdiagonal spreads. We canwrite our time box as twodiagonalspreads:

+1 June 90call/–1 August100call–1 June 90put/+1 August100put

Are we paying orreceiving money for each ofthesespreads?Weareclearlypaying for the put spreadbecause the August 100 putwillalwaysbemorevaluablethantheJune90put.But it’s

not clear what the cash flowis for the call spread. Thelowerexercisepriceseemstoimply that the June call willbe more valuable, but thegreateramountoftimemightin fact make the August callmorevaluable.Thevaluesofthe call options will dependon both the underlying priceandvolatility. In somecases,we may pay for the callspread; in other cases, wemay be paid. Regardless of

the prices of the individualspreads, though, the totaldebitmustbe9.33.Ifthecallspreadistradingfor3.50,theputspreadoughttobetradingfor9.33–3.50=5.83. If theputspreadistradingfor7.75,the call spread ought to betradingfor9.33–7.75=1.58.

Because a timebox is acombination of a box and aroll, if we can value a boxand a roll, we ought to be

able to value a time box.SupposethatwebuytheJune90/100box

and at the same time selltheJune/August100roll

The long and short June

100 synthetics cancel out,leaving the June 90/August100timebox:

The time box mustthereforebeacombinationofbuying the June 90/100 boxand selling the June/August100roll.

Similarly, suppose thatwe buy the August 90/100

box

and at the same time selltheJune/August90roll

The longandshortAugust90 synthetics cancel out,again leaving the June

90/August100timebox:

Inthiscase,thetimeboxisa combination of buying theAugust 90/100 box andselling the June/August 90roll.

From the foregoingexamples, we can see that ifwe buy a long-term box andsell a lower-exercise-price

roll or buy a short-term boxand sell a higher-exercise-price roll, both combinationsresult in the same time box.We can confirm this bycalculating the value of theJune and August 90/100boxes as well as theJune/August90and100rolls

Ifwe buy the June 90/100box and sell the June/August100roll,thetotalvalueis

–9.90+.57=–9.33

If we buy the August90/100 box and sell theJune/August90 roll, the totalvalueis

–9.80+.47=–9.33

The total in both cases isequaltothevalueofthetimebox.

UsingSyntheticsinVolatilitySpreadsAny mispriced arbitrage

relationship will be quicklyrecognized by almost alltraders. Consequently, therearefewopportunitiestoprofitfrom a mispriced conversionor reversal. When amispricing does arise, it islikely to be small and veryshort-lived. Only a

professional trader, who haslow transaction costs andimmediate access tomarkets,is likely to be able to profitfrom such a situation. Buteven if a trader does notintendtoexecuteanarbitrage,he may be able to use aknowledge of arbitragepricing relationships toexecute a strategy at morefavorableprices.

InChapter14,wenoted

that because there is asyntheticequivalentforeverycontract, therearethreewaystobuyastraddle:

1. Buyacall,buyaput.2. Buyacall,buya put synthetically(buy two calls, sellan underlyingcontract)3. Buy a call

synthetically, but aput (buy two puts,buy an underlyingcontract)

Suppose that we havethefollowingpricesforacall,a put, and an underlyingstock:

If there are three monthsremaining to expiration,interestratesare4.00percent,and we expect the stock topayadividendof.25priorto

expiration, what is the bestwaytobuythe50straddle?

Assuming that we mustsellatthebidpriceandbuyattheofferprice, ifwebuy thestraddleoutright,wewillpayatotalof4.20+2.40=6.60.Suppose, however, that webuytheputsynthetically(i.e.,buy the call, sell theunderlying). How much arewe actually paying for theput?

Recall theapproximation for put-callparityforstockoptions

Callprice–putprice=stockprice–exerciseprice+

interestonexerciseprice–expecteddividends

If we buy the putsynthetically,wewillhavetopay4.20 for thecall and sellthestockat51.45.Therefore,

4.20–??=51.45–50+50×.04×3/12–.25=1.70

Thecostofbuying theputsynthetically must be 2.50.This ishigher than theactualprice of 2.40, so this is aworsechoicethanbuyingthestraddleoutright.

What about buying thecallsynthetically(i.e.,buytheput,buytheunderlying)?Wewillpay2.40 for theput and51.50 for the stock. This

givesus

??–2.40=51.50–50+50×.04×3/12–.25=1.75

The synthetic call price is4.15. This is in fact betterthan the actual call price of4.20. If we buy the straddlesynthetically, buying twocallsandbuyingstock,wearepaying a total of 4.15+2.40= 6.55, or .05 better thanbuyingthestraddleoutright.

How important is asavingsof.05?Thatprobablydepends on several factors—the size in which the spreadwill be done, the liquidity ofthe market, and executioncosts and brokerage fees. Aprofessional trader, who hasvery low transaction costsand tends to trade in largevolumes, ought to be veryhappy to save .05. On theother hand, a retail customermay find that the outright

straddle, because it involvesonlytwocontractsratherthanthree, entails lowertransaction costs and can beexecuted more easily in themarketplace. It might be abetterpracticalchoice,evenifitmeansgivingupapotentialsavingsof.05.

Itmightseemthatwhenweareabletotradeacontractsynthetically at a better pricethan the actual price, there

mustanarbitrageopportunityavailable.Butinourexampleno arbitrage opportunityexists. Ifwe do a conversion(i.e., sell call, buy put, buystock), the put-call paritycalculationis

We will be selling stock,synthetically, at 1.70 andbuyingat1.75.

If we instead do areverse conversion (i.e., buycall, sell put, sell stock), thecalculationis

Nowwe are buying stock,synthetically, at 1.85 andselling at 1.70. Because wemustbuyatthebidandsellatthe offer, no arbitrage isavailable in either case. Our

goal, however, was avolatility spread, not anarbitrage. And the bid-askspreads were such that wewereable tobuy the straddlesynthetically at a savings of.05.

Let’sexpandthenumberof options and consider adifferentexample:

If,asbefore,therearethreemonths remaining toexpiration and interest ratesare4percent,whatisthebestway to buy the 45/50/55butterfly?

We might begin bycomparing the prices of thecall and put butterflies. Weknow that these areequivalent strategies andought to have the sameprices.

Buying the put butterfly isslightlybetterthanbuyingthecallbutterfly.

Inadditiontobuyingthecallorputbutterfly,wehaveathirdchoice—wecansellaniron butterfly. InChapter 14,we noted that selling an ironbutterfly (i.e., buy a strangleand sell a straddle) isequivalent to buying abutterfly.Moreover,thevalueof the iron butterfly and thevalue of an actual butterflymust add up to the presentvalue of the amount betweenexercise prices. In our

example,thevaluesmustaddupto

5.00/(1+.04×3/12)=4.95

Therefore, paying 1.30 forthe put butterfly is the sameas selling the iron butterflyfor 4.95 – 1.30 = 3.65. Atwhat price can we sell theironbutterfly?

If buying the put butterflyfor 1.30 is equivalent toselling the iron butterfly for3.65, then selling the ironbutterfly at a price of 3.70must be .05 better. This, intheory, seems to be the best

way to buy the 45/50/55butterfly.

Even though selling theiron butterfly is best intheory, other factors, such asease of execution andtransaction costs,may play arole. Everything else beingequal,though,sellingtheironbutterfly for 3.70 is the bestway to execute our butterflystrategy.

The relationship

between the prices of callbutterflies,putbutterflies,andiron butterflies is based onsynthetic relationships—theabilitytoexpressanycontractasasyntheticequivalent.Thereader may wish to confirmthatnoarbitrageopportunitiesexist, either in the form ofconversions, reverseconversions, or boxes. Ourgoal, however, was not totakeadvantageofanarbitrageopportunitybut rather to find

thebestpriceatwhichtobuyabutterfly.Ourknowledgeofsyntheticpricingrelationshipsenabledustodothis.

Figure 15-3 is asummary of basic arbitragepricing relationships.Whenever a trader isconsidering a strategy, heought to always ask whetherhecandobetterbyexecutingsome part of his strategysynthetically. Usually this

will not be possible becausesyntheticrelationshipstendtobe very efficient.Occasionally, though, thetrader will find that thesynthetic position is slightlymore favorable. And over acareer of trading, even smallsavingscanaddup.

Figure15-3Summaryofarbitragerelationshipsforeuropeanoptions.

1Sometradersrefertoaconversionasaforwardconversionbecausethesyntheticportionofthestrategyisreallyasyntheticforwardcontract.Itwillnotturnintoanunderlyingcontractuntilexpiration2TheoptionsinFigure15-2areinfactAmericanandthereforeentailthepossibilityofearlyexercise.However,whenoptionsonfuturesaresubjecttofutures-typesettlement,astheyareonEurex,wewillseeinChapter16thatthereiseffectivelynodifferencebetweenaEuropeanandanAmericanoption.3Theremaybeamarginrequirementassociatedwithchangesintheoption

prices.But,asdiscussedinChapter1,margindeposits,intheory,belongtothetraderandthereforeentailnolossofinterest.4Asimilartypeofsettlementriskoccurswhenafuturescontractisusedtohedgeaphysicalcommodityorsecurityposition.Whenthevalueofthephysicalcommodityorsecurityrisesorfalls,anyprofitorlossisunrealized.Buttheprofitorlossonthefuturespositionisimmediatelyrealizedintheformofvariation.Thecorrecthedgeisthereforenotonetoonebutisdeterminedbytheinterestonthevariationfromthefuturesposition.Hedgerssometimerefertothisriskastailing.

5Intheory,atradercanborrowmoneyatafixedrate,eliminatinganyinterest-raterisk.Inpractice,however,tradersusuallyfinancetheirtradingactivitiesthroughtheirbrokerorclearingfirmatavariablerate.Thecostofborrowingorlendingchangesdailyasinterestratesriseorfall.6Notethatthevalueofaboxisequaltothesumoftwospreads,abullspreadandabearspread,whilethevalueofarollisequaltothedifferencebetweentwospreads,acallcalendarspreadandaputcalendarspread.7Tradersneedtobecarefulaboutwhattheymeanbybuyingandselling.Usually,buyingmeanspayingsome

amount(acashdebit),whilesellingmeansreceivingsomeamount(acashcredit).Withsomestrategies,however,itmaynotbeclearwhetherthetraderispayingorreceiving.Rollsareanexampleofthis.

16

EarlyExerciseofAmericanOptions

Thus farwehaveassumedthat all option strategiesinvolve holding a position toexpiration. Because many

exchange-traded options areAmerican,carryingwiththemthe right of early exercise, itwill be worthwhile toconsider some of thecharacteristics of Americanoptions. Specifically,wewillwant to answer threequestions:

1. Under whatcircumstancesmight a traderconsider exercising

anAmericanoptionpriortoexpiration?2.Ifearlyexerciseis deemeddesirable, is therean optimal time todoso?3. How muchmore should atrader bewilling topay for anAmerican optionover an equivalent

Europeanoption?

In order for earlyexercisetobedesirable,theremust be some advantage toholding a position in theunderlying contract ratherthan a position in the optioncontract. This advantage cancomeintheformofdividendsthat the owner of stock willreceive or in the form of apositive cash flow on whichinterestcanbeearned.Ifthere

are no dividend or interestconsiderations, there is novaluetoearlyexercise.Inthatcase,

valueofanAmericanoption=valueofaEuropeanoption

This is generally true foroptions on futures traded onexchangesoutside theUnitedStates,where the options aresubject to futures-typesettlement. Futures contracts

donotpaydividends,andnocash flow takes place wheneither the underlying futurescontract or options on thatcontract are traded. Eventhough the options may beAmerican,thereiseffectivelyno early exercise valueassociatedwithsuchoptions.

ArbitrageBoundaries

When evaluating acontract,atradermighttrytodetermine an arbitrageboundary for that contract—the lowest price (the lowerarbitrage boundary) orhighest price (the upperarbitrage boundary) at whichthecontractcantradewithoutthere being some arbitrageopportunity. Identifying thearbitrage boundaries forEuropean and Americanoptions can help us

understand the early exercisecriteriaforAmericanoptions.

Considertheseprices:

If the June 90 call isAmerican, everyone willwant tobuythecallfor9.90,sell the underlying contract

for 100.00, and immediatelyexercise the option. Theresultingcashflowwillbe

Thereisanarbitrageprofitof0.10.

Now consider theseprices:

If the June 70 put isAmerican, everyone willwant tobuy theput for4.80,buy the underlying contractfor 65.00, and immediatelyexercise the option. Theresultingcashflowwillbe

Thereisanarbitrageprofitof0.20.

We can conclude fromthese examples that anAmerican option shouldnever trade for less thanintrinsic value. If it does,everyonewillbuytheoption,hedge the position with the

underlying contract, andexercise the option, all ofwhich will result in animmediate arbitrage profit.We can express the lowerarbitrage boundary for anAmericanoptionas

where X is the exerciseprice,andSisthepriceoftheunderlyingcontract.

We have included thequalifieratleastforbothcallsand puts because, aswewillsee, the lower arbitrageboundary for an Americanoptionmay in factbegreaterthan intrinsic value. For thepresent, we will simply saythat it cannot be less thanintrinsicvalue.

To determine the lowerarbitrage boundary for aEuropean option,we can use

put-callparity

The lowest possible pricefor a put is 0, so the lowerarbitrage boundary for aEuropeancallmustbe

whereF is either thepriceof an underlying futures

contract or the forward priceforanunderlyingstock.

Fora futuresoption, thelower arbitrage boundary fora call is the present value ofthe intrinsic value. Thismeans that if Europeanoptionsonfuturesaresubjectto stock-type settlement, thelowerarbitrageboundarywillalways be less than intrinsicvalue because the presentvalue must be less than

intrinsicvalue.Forexample,

Futuresprice=1,167.00Time toexpiration = 6monthsInterest rate =4.00percent

If options are subject tostock-type settlement, thelower arbitrage boundary for

the1,100callis

Even though the intrinsicvalue is 67.00, the lowerarbitrageboundaryis65.69.

For stock options, ifwereplace F with the forwardprice for the stock and weignore interest on dividends,the lower arbitrage boundaryforaEuropeancallis

A stock option call cannottrade for less than the stockprice minus the discountedvalue of the exercise priceless dividends. This meansthat the lower arbitrageboundary for an out-of-the-money stock option call canbe greater than 0. Forexample,

A50call,eventhoughitisoutofthemoney,hasalowerarbitrageboundaryof

Ifthecallistradingforlessthan 0.48, say, 0.40, we can

buy the call, sell the stock,and exercise the call atexpiration. The cash flowswillbe

This is exactly thedifference between the call

price of 0.40 and the lowerarbitrageboundaryof0.48.

In this example,we canbe certain of an arbitrageprofitofatleast0.08becausewe know that we can closeout the position at expirationbyexercisingthecall,therebypurchasingthestockbackataprice no higher than 50.Suppose, however, that thestock price at expiration isless than 50. Instead of

exercising the call, we canpurchase the stock at itsmarket price.Thiswill resultinanevengreaterprofit than0.08. The lower arbitrageboundary tells us the pricebelow which there is anarbitrage opportunity and atthesame timedetermines theminimum amount that can bemade.Themaximumamountcan be even greater if atexpirationthestockistradingat a price below the exercise

price.What is the lower

arbitrageboundaryforthe50call if it is an Americanoption? We might assumethat itmust be0because theoption is out of the money,and no one would everexercise an out-of-the-moneyoption.Butearlyexerciseisaright, not an obligation. Wecan convert an AmericanoptionintoaEuropeanoption

simply by choosing not toexercise it early. The lowerarbitrage boundary for anAmerican option is thereforeat least intrinsic value. If thelower arbitrage boundary foran equivalent Europeanoptionisgreaterthanintrinsicvalue,asitisinthisexample,then this number also servesas the lower arbitrageboundary for the Americanoption:

Americancall≥maximum[0,S–X,(F–X)/(1+r×t)]

In this example, the lowerarbitrageboundaryforthe50call is 0.48 regardless ofwhether the option isEuropeanorAmerican.

Let’s change ourexampleslightly:

Stock price =49.50Time to

expiration = 6monthsInterest rate =4.00percentDividend =0.65 payableevery threemonths (totaldividend of1.30)

Whatisthelowerarbitrageboundary for a European 45call?

If the call isAmerican, itsintrinsic value (49.50 – 45 =4.50) is greater than theEuropean value of 4.08.Therefore,thelowerarbitrageboundaryforanAmerican45callis4.50.

By reversing F and X,we can use put-call parity todetermine the lowerarbitrageboundaryforaEuropeanput

As with a futures optioncall, the lower arbitrageboundary for a European putis the present value of theintrinsicvalue.

For stock options, wecan replaceF with the stockforward price, giving us thelower arbitrage boundary foraput

A stock option put cannottrade for less than thediscounted value of theexercisepriceminusthestockpriceplusdividends.

Stock price =49.50Time toexpiration = 6months

Interest rate =4.00percentDividend=0

The lower arbitrageboundary for a European 50putmustbe0because

If, however, the option isAmerican,thelowerarbitrageboundarywillbetheoption’s

intrinsicvalueof0.50.Because we can always

turn an American put into aEuropean put simply bychoosing not to exercise it,the lower arbitrage boundaryforanAmericanputis

Americanput≥maximum[0,X–S,(X–F)/(1+r×t)]

Because the lowerarbitrage boundary forEuropean options is a

function of time, interestrates, and, in the case ofstock, dividends, as timepasses, the boundary isconstantly changing. Forfutures options that aresubject to stock-typesettlement, the boundary isalways rising because thepresentvalueisalwaysrisingtoward intrinsic value. Forstock options, however, theboundary may rise or falldepending on whether the

forward price is greater thanor less than thecashprice. Ifthe forward price is greaterthanthecashprice(interestisgreater than dividends), thelowerarbitrageboundarywillriseforcallsandfallforputs.If the forward price is lessthanthecashprice(interestisless than dividends), theboundary will fall for callsand rise for puts. A graphicrepresentation of thesechanges is shown in Figures

16-1through16-4.Figure16-1Lowerarbitrage

boundaryforaEuropeancallonfutures(stock-typesettlement).

Figure16-2LowerarbitrageboundaryforaEuropeanputonfutures(stock-typesettlement).

Figure16-3LowerarbitrageboundaryforaEuropeancallonstock.

Figure16-4LowerarbitrageboundaryforaEuropeanputonstock.

If the lower arbitrageboundary for a Europeanoption is less than intrinsicvalue,aEuropeanoptioncan,in some cases, be worth lessthan intrinsic value. Whenthis occurs, as time passes,the value of the option willrisetowardintrinsicvalue.Asa consequence, the optionwill have a positive theta.This was discussed inChapter 7 and shown

graphicallyinFigure7-9.Although traders are

primarily interested in thelower exercise boundary foran option, for completeness,we might also want todetermine theupperarbitrageboundary for an option.Because the underlyingcontract cannot fall below 0,the upper arbitrage boundaryforanAmericanput,whetheron futures or stock, must be

the exercise price. For aEuropean put, which issubject to stock-typesettlement, the upperboundaryis thepresentvalueoftheexerciseprice

Americanput≤XEuropeanput≤X/(1+r×

t)

To determine the upperarbitrageboundary foracall,wecanuseput-callparity

We know that themaximum value for aEuropeanputisX/(1+r×t).Therefore, the maximumvalueforaEuropeancallis

A European call on afutures contract has amaximum value equal to thefutures price discounted byinterest.Ifoptionsaresubjecttofutures-typesettlement,themaximumvalueissimplytheprice of the underlyingfuturescontract.

For a European call onstock,we can replaceF with

the forward price for stockS×(1+r×t)–D

Ignoring interest ondividendsgivesus

A European call on stockhas a maximum value equalto the stock price less

dividends. An American callhas a maximum value equaltothestockprice.Asummaryof arbitrage boundaries isshowninFigure16-5.

Figure16-5Summaryofarbitrageboundaries.

EarlyExerciseofCallOptionsonstock

Under what conditionsmight we choose to exercisean American call option onstock prior to expiration? Toanswer this question, let’sthink about the componentsthat make up the value of acall.

Clearly, if we areconsidering exercising anoption, it must be in themoney. Therefore, onecomponent must be intrinsicvalue.Acallalsoofferssomeprotective value over a stockposition because the call’slossislimitedbytheexerciseprice.The likelihood that thestock will fall below theexercisepricedependsonthevolatility, so we might referto this protective value as

volatility value. As volatilityrises, we are willing to paymore for the call. The callalso includes some interest-rate value. As interest ratesrise, thecallbecomesamoredesirable substitute forholding a stock position.Finally, there is dividendvalue. But unlike volatilityvalueandinterestvalue,bothofwhichincreasethevalueofthecall, thedividendreducesthe value of the call.

Therefore,

Callvalue=intrinsicvalue+volatilityvalue+interestvalue–dividendvalue

Supposethatweareabledetermine the value of eachofthesecomponentsandfindthat the dividend value isgreater than the combinedvolatility value and interestvalue

Dividendvalue>volatilityvalue+interestvalue

In this case, the value ofthe call will be less thanintrinsic value. And, indeed,European options can, insome cases, trade for lessthan intrinsic value. But, ifthe call is American, itbecomes an early exercisecandidate because we cancollecttheintrinsicvaluenowby simultaneously exercising

thecallandsellingthestock.Howcanweestimatethe

value of the volatility,interest-rate, and dividendcomponents? The dividendcomponentissimplythetotaldividendthestockisexpectedto pay over the life of theoption. The interest valuemust be the interest that wewouldhavetopayifwewereto sell the call and buy thestock and carry this position

to expiration. If the call isdeeplyinthemoney,itsvaluewillbeveryclosetointrinsicvalue,andthetotalcashflowwill be approximately equaltotheexerciseprice

Intrinsicvalue=stockprice–exerciseprice

We might reach the sameconclusion by observing thatif we exercise the call, wewill have to pay the exercise

price.Theinterestvaluemustbe the approximate cost ofcarrying theexerciseprice toexpiration.

The volatilitycomponentissomewhatmoredifficulttodetermine.Butweknowthat thevolatilityvaluedepends on the likelihood ofthe stock price falling belowthe exercise price. The valueofthecompanionput(theputwith the same exercise price

and expiration date as thecall)mustbeagoodestimateof thisvalue.Weknow fromput-call parity that the vegasof calls and puts with thesame exercise price andexpiration date are the same—they have the samesensitivity to changes involatility. Therefore, theirvolatility values ought to bethesame.1

For example, consider

thefollowing:

Stock price =100Time toexpiration = 1monthInterest rate =6.00percentDividend =0.75, payablein15days

Is the 90 call an early

exercisecandidateifthepriceofthe90putis0.20?

We know the dividendvalue(0.75)andthevolatilityvalue (0.20), so the onlycomponent we need tocalculate is the cost ofcarrying theexerciseprice toexpiration

90×.06×1/12=.45

The early exercise criteriaaresatisfiedbecause

Dividendvalue>volatilityvalue+interestvalue0.75>

0.20+0.45=0.65

In Figure 16-6 we cansee why the 90 call hasbecome an early exercisecandidate—its Europeanvalue has fallen belowintrinsic value.2 If given thechoice between exercisingnow or carrying the optionpositiontoexpiration,wewillcomeoutaheadby0.10ifwe

exercisenow.Figure16-6

But are those our onlytwochoices—exercisenowornot at all? An Americanoptioncanbeexercisedatanytime prior to expiration.Instead of exercising today,what about exercising theoption tomorrow?Or thedayafterthat?

Supposethatweexercisetoday instead of exercisingtomorrow.Whatwillwegain,and what will we lose? We

will lose one day’s worth ofvolatility value.Wewill alsolose one day’s worth ofintereston theexerciseprice.Inreturn,weget...nothing.We are exercising to get thedividend. But the dividendwill not be paid for 15 days.Because we always give upsome volatility value andsome interest valuewhenweexercise an American calloption on stock prior toexpiration, the only time we

will consider exercising theoptionearlyisthedaybeforethe stock pays the dividend.On no other day will earlyexercisebeoptimal.

For an American calloptiononstocktobeanearlyexercise candidate, the earlyexercise criteria must holdtrueovertheentirelifeoftheoption

Dividendvalue>volatilityvalue+interestvalue

But,foranoptiontobeanimmediate early exercisecandidate,thisconditionmustalso hold true over the nextday. For a call option onstock, theonlydayonwhicha trader need consider earlyexerciseis thedaybeforethestock pays a dividend.Indeed, if a stock pays nodividend over the life of theoption, there is never anyreason to exercise the callpriortoexpiration.

EarlyExerciseofPutOptionsonstock

Under what conditionsmight we choose to exercisean American put option onstockpriortoexpiration?Justasweseparatedthevalueofastock option call into itscomponents, we can do thesamewithastockoptionput.Again, we begin with theintrinsic value. To this, we

canaddthevolatilityvalue—the protective value affordedby the put in the event thatthestockpricerisesabovetheexerciseprice.Therewillalsobe interest value—if weexercise the put, we willcollectinterestontheexerciseprice. Finally, there will besomedividendvalue.

Put value = intrinsic value+ volatility value – interestvalue+dividendvalue

Notethatvolatilityvalueand dividend value increasethevalueoftheput,whiletheinterest value reduces theput’s value. Suppose thatweare able to determine thevalue of each of thesecomponentsand find that theinterest value is greater thanthe combined volatility valueanddividendvalue

Interestvalue>volatilityvalue+dividendvalue

If this is true, the value ofthe option will be less thanintrinsic value. But, if theoption is American, itbecomes an early exercisecandidate because we cancollect the intrinsic valueright now by exercising theput.

We can estimate thevalueof thesecomponents inthe same way we estimatedthem for a call. The interest

valueistheamountofinterestwe will earn on the exerciseprice to expiration if weexercisetheput.Thedividendvalueisthetotaldividendthestock isexpected topayoverthe life of the option. Thevolatility value isapproximatelythepriceofthecompanion out-of-the-moneycall.

Considerthissituation:

Stock price =

100Time toexpiration = 2monthsInterest rate =6.00percentDividend =0.40

Is the 120 put an earlyexercisecandidateifthepriceofthe120callis0.55?

We know the volatility

value (0.55) and dividendvalue (0.40). The interest onthe exercise price toexpirationis

120×.06×1/6=1.20

The early exercise criteriaaresatisfiedbecause

Interestvalue>volatilityvalue+dividendvalue1.20>

.55+.40=.95

WecanseeinFigure16-7thatatastockpriceof100,thevalueoftheEuropean120put falls below intrinsicvalue, making the put anearly exercise candidate. Ifgiven the choice betweenexercising now or carryingthe option position toexpiration,wewill come outahead by 0.25 ifwe exercisenow.3

Figure16-7

Aswithacall, foraputto be an immediate earlyexercise candidate, the earlyexercise criteria must holdtrue not only over the entirelife of the option but alsoover the next day. We willexercise today only if weexpect to gainmore over thenext day through earlyexercise than we lose. Willthisbetrueforour120put?

Suppose that the

dividendof0.40willbepaidtomorrow. If we exercisetoday instead of tomorrow,wewillgainoneday’sworthofinterest

120×0.06/365=0.02

Inreturn,wearegivingupone day’s worth of volatilityvalue aswell as the value ofthe dividend. Even if weassume that the volatilityvalue is negligible, the

dividendof0.40 thatwewilllose is far greater than theinterest of 0.02 that we willearn.Clearly,weshouldwaitonedaybeforeexercisingtheoption, foregoing one day’sworthofinterestbutretainingthevalueofthedividend.

Suppose that thedividend will be paid twodays from now. If weexercise today instead ofwaiting until the dividend is

paid,wewill earn two days’worthofinterest,0.04,butwewill still lose thedividendof0.40. Waiting two days toexercise is still a betterstrategy.

When should weexerciseaputearly?Becauseatraderwillnotwant togiveup the value of the dividend,the most common day onwhich to exercise a stockoptionputearlyisthedayon

which the stock pays thedividend. But unlike stockoption calls, where the onlyday on which the optionoughttobeexercisedearlyisthedaybeforethestockpaysthe dividend, a stock optionput might be exercised anytimepriortoexpiration.Earlyexercisewillbeoptimaliftheinterest that can be earned isgreater than the combinedvolatilityanddividendvalue.

Ignoring the volatilityvalue, we can see that notrader will exercise a putearly if the total interest thatcanbeearnedislessthanthedividend. In our example,whereweexpecttoearn0.02in interest per day, earlyexercisecanneverbeoptimalif the dividend will be paidwithin the next 20 daysbecause

0.40/.02=20

Withfewerthan20daystothedividendpayment,wecanneverearnenough interest tooffset the loss of thedividend.Forputoptions,thisblackoutperiodcanbeeasilycalculated by dividing thedividendby thedaily interestthat can be earned on theexercise price. During thisperiod, no knowledgeabletrader will exercise a putbecause the loss of thedividendwill be greater than

thetotalinterestearned.

This does notmean that

a put should never beexercised prior to thedividend payment. In ourexample, if the dividendwillbe paid in 30 days and weexercisenow,wewillearn30days’ worth of interest, thatis, 30 × 0.02 = 0.60. This isgreaterthanthe0.40valueofthe dividend. As long as thevolatility value over the next30 days is less than 0.20,immediate early exercise is asensiblechoice.

ImpactofShortStockonEarlyExercise

Interest rates are animportant factor in decidingwhether to exercise a stockoption early. If we reduceinterest rates, calls are morelikely to be exercised early(early exercise results in asmaller interest loss), andputs are less likely to be

exercised early (earlyexercise results in smallerinterest earnings). Because ashort stock position entails alowerinterestrate(therateisreduced by the borrowingcosts), a trader who has ashort stock position is morelikelytoexerciseacalloptionearly. At the same time, atrader who does not alreadyown stockwill be less likelytoexerciseaputoptionearly.This is consistent with the

generalrulethatweproposedinChapter7:

Wheneverpossible, atrader shouldavoid a shortstockposition.

Ifatraderiscarryingashortstockposition,exerciseofacallwillreduceoreliminatethisposition.Ifatraderiscarryingnostock

position,exerciseofaputwillresultinashortstockposition.Intheformercase,acallismorelikelytobeexercisedearly;inthelattercase,aputislesslikelytobeexercisedearly.

EarlyExerciseofOptionsonFutures

What happens when we

exercise a futures option?Exercise of a call optionenables us to buy theunderlyingfuturescontractattheexerciseprice.Exerciseofaputoptionenablesustosellthe underlying futurescontractat theexerciseprice.Because the futures contractis subject to futures-typesettlement, there will be avariation credit equal to theoption’s intrinsic value, thedifference between the

exercisepriceandthepriceofthe futures contract. If theoption is subject to futures-type settlement, exercisewillcausetheoptiontodisappear,andwewillbedebitedbyanamount equal to the option’svalue. Assuming that thepriceoftheoptionisequaltoits intrinsic value, the creditand debit will cancel out,resulting in no cash flow.Because there is no cashflow, there can be no

advantage to early exercise.If, however, the option issubject to stock-typesettlement, as is the practiceon futures exchanges in theUnitedStates,thereisnocashflowwhentheoptionpositiondisappears. The only cashflowisthevariationcreditonthe futures position, a crediton which we can earninterest.

For a futures option to

be an early exercisecandidate, theoptionmustbesubject to stock-typesettlement, and the interestthat can be earned on theintrinsic value must begreater than the volatilityvaluethatwearegivingup

Interestvalue>volatilityvalue

Theinterestontheoption’sintrinsicvalueiseither

(F–X)×r×t

forcallsor

(X–F)×r×t

forputs.As with stock options,

wecanestimatethevolatilityvalueofanoptionbylookingatthepriceofthecompanionout-of-the-money option.Suppose that we have thefollowing:

Futuresprice=100Time toexpiration = 3monthsInterest rate =8.00percent

Is the 80 call an earlyexercisecandidateifthepriceofthe80putis0.15?

Theinterestwecanearnthroughearlyexerciseis

(100–80)×0.08×3/12=0.40

Becausethisisgreaterthanthevolatilityvalueof.15,theoption is an early exercisecandidate.Ifgiventhechoicebetween exercising now andholding the position toexpiration,wewill come outahead by 0.25 ifwe exercisenow.For theoption to be animmediate early exercisecandidate, it must also meet

theearlyexercisecriteriaoverthenextday.Oneday’sworthof interest must be greaterthan one day’s worth ofvolatilityvalue.

We can easily calculateoneday’sworthofinterest

(100–80)×.08/365=0.0044

How canwe calculate oneday’s worth of volatilityvalue? We know that theprice of the companion

option, in this case, the 80put, is almost all volatilityvalue.Aseachdaypasses,thevalue of the option will fallby one day’s worth ofvolatility value. This dailyloss in value is simply theoption’s theta. Bydetermining the theta of thecompanion out-of-the-moneyoption, we can estimate oneday’s worth of volatilityvalue. Unlike the othercalculations, thiswill require

the use of a theoreticalpricingmodel.

Using theBlack-Scholesmodel, we find that theimplied volatility of the 80put is 24.68 percent. At thisimplied volatility, theoption’s theta is –0.0046,slightly greater (in absolutevalue) than thedaily interest.If we exercise the 80 calltoday instead of tomorrow,we will gain 0.0044 in

interest, but we will lose0.0046 in volatility value.Because we will lose morethanwegain,theoptionisnotan immediate early exercisecandidate.

When should weexercise the 80 call?Assuming that the earlyexercise criteria aremet overthe entire life of the option,we will want to exercisewhen the daily volatility

value is less than the dailyinterest. In our example, wewill want to exercise whentheoption’s theta is less than.0044. Using the Black-Scholes model, we canestimatethatthiswilloccurinfour days, at which time thetheta of the 80 putwill be –0.0043.4

Notexercisinganoptionto retain the theta valuemayseem counterintuitive. If we

do not exercise the 80 calland the price of the futurescontract does not move, wenotonlyloseoneday’sworthof interest, but we also loseoneday’sworthoftheta.Butthisistrueonlyifthepriceofthe futures contract does notmove. If the futures contractdoes move, the fact that wehave a positive gammaposition will work in ourfavor. If the movement islarge enough, we will prefer

to hold the option positionratherthanafuturesposition.In an extreme case, if thefutures contract were to fallbelow 80, we would clearlyprefer the option positionbecause of the protectivevalue offered by the 80 call.How likely is it thatwewillget sufficient movement inthefuturespriceoverthenextday to justify holding the 80call rather than exercising it?This is one day’s worth of

volatility value—the theta ofthe80put.

For an American optionthat might be an earlyexercise candidate, we haveconsidered two choices—hold the option or exercisethe option. There is also athird choice—sell the optionandreplace itwithapositionin the underlying contract.The result is equivalent toexercising theoptionbecause

both strategies result in theoption position beingreplaced by an underlyingposition.

When does selling anoption rather than exercisingmakesense?Whenwedecideto exercise an Americanoptionpriortoexpiration,wehave,ineffect,concludedthatthe value of the option isequal to its intrinsicvalue. Iftheprice of theoption in the

marketplace is exactlyintrinsic value, there is nodifferencebetweenexercisingthe option or selling theoption and replacing it withan underlying position. If,however,theoptionistradingat a price greater thanintrinsic value, andtransaction costs are not afactor, the best choice willalways be to sell the optionandreplace itwithapositionintheunderlyingcontract.As

a practical matter, however,selling an option that is anearly exercise candidate willusually not be a viablealternative. If the option isdeeply enough in the moneyto justify early exercise, themarket for theoptionwill berelatively illiquid. Underthese conditions, the bid-askspreadislikelytobesowidethat any sale will almostcertainlyhavetobedoneataprice that is no greater than

intrinsicvalue.

ProtectiveValueandEarlyExercise

When we exercise anoptionpriortoexpiration,weare giving up the protectivevalueaffordedbytheoption’sexerciseprice. If thepriceofthe underlying contract wereto fall through the exercise

price in the case of a call orrisethroughtheexercisepriceinthecaseofaput,wewouldalways prefer the optionposition to an underlyingposition.Tobetterunderstandthe consequences of givingup this protectivevalue, let’sgo back to an earlier stockoption example, butwith thedividendpayabletomorrow:

Stock price =100

Time toexpiration = 1monthInterest rate =6.00percentDividend =0.75, payabletomorrow

Ifthe90putistradingat0.20,weknowthatthe90callisanimmediateearlyexercisecandidatebecause

Dividendvalue>volatilityvalue+interestvalue.75>

0.20+.45=.65

If we exercise the 90 call,theresultisthatwewillhaveno option position, but wewill have a long position inthe underlying stock. This isthe same position thatwouldresulthadwesold theoptionand bought the stock.However,ifwesellacallandbuy the underlying, this is

synthetically equivalent toselling a put. In a sense,exercising the 90 call is thesame as selling the 90 put.What will cause us to regretselling the 90 put? Whetherwesellthe90putorexercisethe 90 call early, in bothcases, we will regret ourchoice if the stock price isbelow90atexpiration.

If exercising the 90 callis the same as selling the 90

put, we might ask, if weexercise the 90 call, at whatprice are we selling the 90put?Because

.75>0.20+0.45=0.65

we can see that we willgain .10byexercising the90call.Thismustmean thatwehavesoldthe90putatapricethat is .10 better than itsmarket price of 0.20.Therefore, exercising the 90

call is equivalent to sellingthe90putatapriceof0.30.

How can a trader whobelievesthatearlyexerciseisindicated protect himselffrom the possibility of theunderlying contract goingthrough the exercise price?Thesolution issimple:at thesametimethetraderexercisesan option, he can purchasethe companion out-of-the-money option. In our

example, if the traderexercises the 90 call andsimultaneously buys the 90putatapriceof0.20,hewillhave the same protectionaffordedbythe90call,butata cost that is 0.10 lower.Whether the trader actuallychooses to purchase the 90put is a decision that hewillhave to make based on hisassessment of marketconditions. If the traderbelieves that implied

volatility is low, a price of0.20willseemcheap,andheoughttobehappytopurchasethe 90 put. If impliedvolatility is high, a price of0.20willseemexpensive,andthe traderwill look for someother way of controlling hisdownsiderisk.

PricingofAmericanOptions

OurdiscussionthusfarhasfocusedonwhyandwhenanAmerican option might beexercised prior to expiration.Butwealsowant toconsiderthe question of pricing.Howmuch is an American optionworth? Unless interest ratesare 0 and there are nodividend considerations, anAmerican option shouldalwaysbeworthmorethananequivalent European option.Buthowmuchmore?

The Black-Scholesmodel makes no attempt toevaluate American optionsbecause it is a Europeanpricing model. When theChicago Board OptionsExchangeopenedin1973,thefirstlistedstockoptionswereAmerican. In spite of this,traders continued to use theBlack-Scholes model forseveral years because nomodel of equal simplicityexistedforAmericanoptions.

Traders tried to approximateAmerican values by makingadjustments to Black-Scholes–generatedvalues.

For example, when astock is expected to pay adividend, an American callvaluecanbeapproximatedbycomparing theBlack-Scholesvalueofthecalloptionundertwocircumstances:

1.Thecallexpiresthe day before the

stock goes ex-dividend.2.Thecallexpireson its customarydate, but theunderlying stockprice used toevaluate the call isthe current priceless the expecteddividend.

Whichevervalueisgreateristhepseudo-Americancall

value.Inthecaseofoptionson

futures or put options onstock, traders used Black-Scholes–generatedvalues butraised any option with atheoretical value less thanparity to exactly parity.Unfortunately, neither ofthese methods resulted in atruly accurate value for anAmericanoption.

The first widely used

model to evaluate Americanoptions was introduced in1979 by John Cox of theMassachusetts Institute ofTechnology,StephenRossofYale University, and MarkRubinstein of the Universityof California at Berkeley.5Unlike the Black-Scholesmodel, which is closed formand therefore returnsasingleoption value, the Cox-Ross-Rubinstein, or binomial,

modelisanalgorithmorloop.The more times the modelpasses through the loop, thecloser it comes to the truevalueofanAmericanoption.The Cox-Ross-Rubinsteinmodel is relatively easy tounderstand, both intuitivelyand mathematically, and isthemostcommonmethodbywhichstudentsareintroducedtooptionpricing theory. (Wewill take a closer look atbinomial option pricing in

Chapter 19.) However, interms of computation, themodelmay requirenumerouspasses through the loop togenerate anacceptablevalue.In an effort to reduce thecomputational time requiredby the Cox-Ross-Rubinsteinmodel, in 1987, GiovanniBarone-Adesi of theUniversity of Alberta andRobert Whaley of DukeUniversity introduced analternative model for pricing

American options.6 Althoughthe Barone-Adesi-Whaley, orquadratic, model is morecomplex mathematically, itconverges to an acceptablevalue for American optionsmuch more quickly than theCox-Ross-Rubinstein model.The Barone-Adesi-Whaleymodel has the limitation oftreating all cash flows as ifthey were interest paymentsthat accumulate at a constant

rate.Dividends,however,arepaidallinonelumpsum,andforthisreason,theCox-Ross-Rubinstein model is moreoftenusedtoevaluateoptionsondividend-payingstocks.

Inadditiontogeneratingvalues forAmerican options,both the Cox-Ross-Rubinstein and Barone-Adesi-Whaleymodelsspecifywhen early exercise of anAmerican option is optimal.

Althoughwewere somewhatsubjectiveonthispointinourearlier discussion, using atrueAmericanpricingmodel,an option is optimallyexercised early when itstheoreticalvalueisexactlytoparity and itsdelta is exactly100.

The extent to whichAmerican and Europeanoption values differ dependson many factors, including

time to expiration, volatility,interestrates,and,inthecaseof stock options, the amountof the dividend. Thelikelihood of early exercisewill increase, andwith it thedifferencebetweenAmericanand European values, as theoptiongoesmoredeeply intothemoney.WecanseethisinFigure16-8,thevalueofa90callonstockwhere

Figure16-8Theoreticalvalueofa

90call.

Time toexpiration = 7weeksInterest rate =6.00percentDividend =1.00, payablein4weeksVolatility=25percent

As the underlying stockpricerisesfrom90to110,the

call moves from out of themoney, with a very smalllikelihood of early exercise,to in themoney,with a veryhigh likelihood. Figure 16-9shows the net difference invaluesnotonlyatavolatilityof 25 percent but also atvolatilities of 15 and 35percent.Atahighervolatility,the difference in values issmallerbecausetheAmericancall is less likely to beexercised early. At a lower

volatility, the difference isgreater because the call ismore likely to be exercisedearly. In all cases, as theoptiongoesmoredeeply intothe money, the differenceapproaches 0.67, the amountof the dividend less theinterestcostofpurchasingthestockat90thedaybeforethedividend ispaidandcarryingthepositiontoexpiration

Figure16-9DifferencebetweenthetheoreticalvalueofanAmericanand

European90call(AmericanvaluelessEuropeanvalue).

1.00–(90×0.06×22/365)≈0.67

Now consider the valueof a 110 put under the sameconditions.Aswithacall,themoredeeplytheputgoesintothe money, the greater thedifference between theAmerican value and theEuropean value. This can beseeninFigure16-10.Thenetdifference under three

volatility assumptions isshown in Figure 16-11. At ahigher volatility, thedifferenceinvaluesissmallerbecause the American put isless likely to be exercisedearly. At a lower volatility,the difference is greaterbecausetheputismorelikelyto be exercised early. In allcases, as the option goesmore deeply into themoney,the difference approaches0.38, the amount of interest

that can be earned on theexercise price for the threeweeks remaining toexpirationfollowingpaymentofthedividend

Figure16-10Theoreticalvalueofa110put.

Figure16-11DifferencebetweenthetheoreticalvalueofanAmericanandEuropean110put(AmericanvaluelessEuropeanvalue).

110×0.06×21/365≈0.38

In our discussion ofsynthetics, we noted that forEuropean stock options, thedeltavaluesofacallandputwith the same exercise priceand expiration date alwaysadd up to 100. But, forAmerican options, the deltascanadduptomorethan100.Thisisbecausethedeltaofanin-the-money American

option goes to 100 morequickly than an equivalentEuropeanoption.Atthesametime, the companion out-of-the-moneyoption still retainssomedeltavalue.Asaresult,ifwecalculatethedeltaofthesynthetic underlying (longcall and short put) by addingthe American call delta andsubtracting theAmerican putdelta, we find that the deltasadd up to more than 100.Figure 16-12 shows the

Americandeltavaluesforthe100syntheticunder thesameconditionsasinourprecedingexample:

Figure16-12Deltaofthe100synthetic(100calldelta–100putdelta)ifalloptionsareAmerican.

Time toexpiration = 7weeksInterest rate =6.00percentDividend =1.00, payablein4weeksVolatility=25percent

Higher volatility tends toreduce the differences

between American andEuropeanoptions,sothedeltaof the synthetic will remaincloserto100.

Becausedeltavaluesareaffected by the likelihood ofearly exercise, arbitragestrategiessuchasconversionsand reversals, boxes, androlls, which may be deltaneutral if all options areEuropean, may not be deltaneutral if the options are

American. Although thesestrategies may deviate fromdelta neutral only by a smallamount,thefactthattheyareoften done in large sizes canresultinadditionalriskthatatradershouldnotignore.

An American pricingmodel is necessary toevaluate individualAmericanoptions, but it may still bepossibletoestimatethevalueofsomestrategieswithoutthe

use of a pricing model. Forexample, suppose we knowthefollowing:

Time toexpiration =24daysInterest rate =6.00percentDividend =0.60, payablein9days

Whatshouldbethevalue

ofa100/110boxifalloptionsareAmerican?Toanswerthisquestion,wecanfirstevaluateanequivalentEuropeanbox.Thenwecanadjusttheboxvaluedependingonwhichoptionsmightbeexercisedearly.

The value of theEuropean box is simply thepresent value of the amountbetweenexerciseprices

Now we can consider thevarious possibilities for earlyexercise:

Case1:Boththe100and110putareexercisedearly.Theputswillbeexercisedthedaythedividendispaid.Theboxvalue

willincreasebytheinterestearnedon10.00for15days

9.96+(10×0.06×15/365)=9.96+0.025=9.985

Case2:Boththe100and110callareexercisedearly.Thecallswillbeexercisedthedaybeforethedividendispaid.Thebox

valuewillincreasebytheinterestearnedon10.00for16days

9.96+(10×0.06×16/365)=9.96+0.026=9.986

Case3:Onlythe110putisexercisedearly.Theboxvaluewillincreasebytheinterestearnedon110for

15days

9.96+(110×0.06×15/365)=9.96+0.271=10.231

Case4:Onlythe100callisexercisedearly.Theboxvaluewillincreasebytheamountofthedividendlesstheinterestcoston100for16days

9.96+0.60–(100×0.06×16/365)=9.96+0.60–0.263

=10.297

Case5:Boththe100calland110putareexercisedearly.Theboxvaluewillincreasebytheamountofthedividendplustheinterestearnedon110for15dayslesstheinterestcost

on100for16days

9.96+0.60+(110×0.06×15/365)–(100×0.06×

16/365)=9.96+0.60+0.271–0.263=10.568

Atverylowstockprices,where both puts are earlyexercise candidates, and atveryhighstockprices,whereboth calls are early exercisecandidates, theboxwillhavea value close to 9.99. If one

option, either the 100 call or110 put, is an early exercisecandidate, the value of thebox will be somewherebetween 10.23 and 10.30.Finally, the boxwill have itsmaximum value ofapproximately 10.57 if boththe 100 call and the 110 putare earlyexercise candidates.This will occur if bothoptions are in the money,most likely with the stockprice close to 105. Volatility

mustalsobelowbecauseinahigh-volatilitymarket,noonewill want to give up anoption’s volatility value byexercisingearly.Thevalueofthe 100/110 box at differentstock prices and under threedifferent volatilityassumptions is shown inFigure16-13.

Figure16-13Valueofa100/110boxifalloptionsareAmerican.

The difference betweenEuropean and Americanvalues is usually greatest foroptions on dividend-payingstocks. But even futuresoptions, if the options aresubject to stock-typesettlement, have someadditional early exercisevalue. We can see this inFigure 16-14, the value of a90 call on a futures contract,where

Figure16-14Theoreticalvalueofa90callonafuturescontractwheretheoptionissubjecttostock-typesettlement.

Time toexpiration = 3monthsInterest rate =8.00percentVolatility=25percent

ThedifferencebetweentheEuropean and Americanoption values is shown inFigure 16-15.Unlike a stockoption, where there is a

maximum difference, thedifference for options onfutures continues to increaseastheoptiongoesfurtherintothe money. This is becausethe early exercise valuedepends on the interest thatcanbeearnedontheoption’sintrinsicvalue.And themoredeeply in the money, thegreater the intrinsic value. Inour example, with theunderlying futures contracttrading at 110, the additional

earlyexercisevalueforthe90callwillapproachtheinterestthat can be earned on theintrinsicvalue

Figure16-15DifferencebetweenthetheoreticalvalueofanAmericanandEuropean90callonafuturescontractwheretheoptionsaresubjecttostock-typesettlement(Americanvalue–Europeanvalue).

(110–90)×0.08×3/12=0.40

Regardless of themodelatraderchooses,theaccuracyof model-generated valueswill depend at least asmuchon the inputs into the modelasonthetheoreticalaccuracyofthemodelitself.IfatraderevaluatesanAmericanoptionusing an incorrect volatility,an incorrect interest rate, or

anincorrectunderlyingprice,the fact that he derives hisvalues from an AmericanratherthanaEuropeanmodelis likely to make littledifference. Both models willgenerate incorrect valuesbecause the inputs areincorrect. The Americanmodel may produce lesserror, but that will be smallconsolation if the incorrectinputs lead to a large tradingloss.

The importance of earlyexercise is greatest whenthere is a significantdifferencebetweenthecostofcarrying an option positionand the cost of carrying aposition in the underlyingcontract. This difference canberelativelylargeinthestockoptionmarket,wherethecashoutlay required to buy stockismuchgreater thanthecashoutlay required to buyoptions. Moreover, dividend

considerationswillalsoaffectthe cost of carrying a stockposition compared with thecost of carrying an optionposition. A trader in a stockoption market will usuallyfind that the additionalaccuracy afforded by anAmerican model will indeedbeworthwhile.

In futures optionsmarkets, where the optionsare subject to futures-type

settlement,thereisnocostofcarry associated with eitheroptions or the underlyingfutures contract. In this case,a European pricing modelwill suffice because there isno difference betweenEuropean and Americanoptionvalues.Evenifoptionson futures are subject tostock-typesettlement,thereisa relatively small costassociated with carrying anoption position because the

price of the option is smallcompared with the price ofthe underlying futurescontract.Theadditionalvalueforearlyexerciseis thereforesmallandisonlylikelytobea consideration for verydeeply in-the-money options.Practicalconsiderations,suchastheaccuracyofthetrader’svolatility estimate, his abilityto anticipate directionaltrends in the underlyingmarket, and his ability to

control risk through effectivespreading strategies, will faroutweigh any smalladvantagegainedbyusinganAmerican rather than aEuropeanmodel.7

EarlyExerciseStrategies

Earlyexerciseofanoptionis a right rather than an

obligation, and there arestrategies that depend onsomeonemakinganerrorandnotexercisinganoptionearlywhenitoughttobeexercised.For example, consider thissituation:

Stock price =98.75Time toexpiration = 5daysDividend =

1.00, payabletomorrow

Suppose that there is a 90call that is American andoughttobeexercisedtodayinordernottolosethedividendof 1.00. If this is true, theoption ought to be worthapproximatelyparity,or8.75.Suppose that a trader is ableto sell a 90 call for 8.75 andat the same time buy 100shares of stock for 98.75.

Because the 90 call ought tobeexercisedtoday,thetraderprobably will be assigned,requiringhimtosellthestockat 90. If this occurs,excluding transaction costs,thetraderwillbreakeven:

Butsupposethat thetrader

isnotassignedonthe90call.Ifthestockopensunchanged,its new price will be 97.75(the stockpriceof98.75 lessthe dividend of 1.00).Because the call is tradingatparity, it will open atapproximately 7.75. Thetraderwillshowalossof1.00on the stock and a profit of1.00 on the 90 call. But thetrader, because he owns thestock, will also receive thedividend. Excluding

transaction costs, the profitfor theentirepositionwillbeequaltothedividendof1.00.

Inadividendplay,astheex-dividend day approaches,atraderwilltrytoselldeeplyin-the-money calls andsimultaneously buy an equalamountofstock.Ifthetraderisassignedonthecalls,asheshouldbe, hewill essentiallybreak even. But, if he is notassigned, he will show a

profit approximately equal tothe amount of the dividend.What is the likelihoodof thetrader being assigned?Because assignment formostexchange-traded options israndom, one determinant istheamountofopeninterestinthe call that was sold. Themore outstanding calloptions, the lower thelikelihood of assignment. Asecond determinant is therelative sophistication of the

market—whether mostmarket participants arefamiliar with the criteria forearly exercise. Dividendplays were much morecommon in the early days ofoption trading when themarketwas lesssophisticatedandmanyoptionsthatshouldhavebeenexercisedwerenot.As markets have becomemore efficient, only aprofessional trader with verylow transaction costswill try

to take advantage of such apossibility. Even then, hemay find that he is assignedon thegreatmajorityofcallshehassold.

A trader also mightattempttoexecuteaninterestplay by selling stock andsimultaneouslysellingdeeplyin-the-money American putsthat ought to be exercisedearly. If the puts are notexercised, the trader will

profit by the amount of theinterest he can earn on theexercise price (the proceedsof the stock sale and the putsale combined). This profitwill continue to accrue aslong as the puts remainunexercised. If the puts areexercised, the trader does noworse than break even.Again, only a professionaltrader, with low transactioncosts,islikelytoattemptsuchastrategy.

If options are subject tostock-type settlement, aninterestplaycanalsobedoneina futuresoptionmarketbyeither purchasing a futurescontract and simultaneouslysellingadeeplyin-the-moneycall or selling a futurescontract and simultaneouslysellingadeeplyin-the-moneyput. If the option is deeplyenoughinthemoney,itoughtto be exercised early. But, ifthe option remains

unexercised, the trader willcontinue to earn interest onthe proceeds from the optionsale. Because the amount onwhich the trader will earninterest is approximately theintrinsicvalue(thedifferencebetween the exercise priceand futures price), this willnot be as profitable as asimilar strategy in the stockoption market where thetraderwillearninterestontheexercise price. Still, if the

transaction costs are lowenough, it may beworthwhile.

Instead of entering intoan early exercise strategy byselling options and tradingthe underlying contract, atrader may also be able toexecute the strategy bytrading deeply in-the-moneycall or put spreads. In ourdividend-play example, thetrader sold 90 calls and

bought stock. Suppose thatboth the 85 call and the 90call ought to be exercised toavoid losing the dividend. Ifthis is true, the 85/90 callspread ought to be worth5.00, exactly the differencebetween exercise prices.Onemight assume that ifrequested, a market makerwillquoteabidpriceforthisspread below 5.00, perhaps4.90,andanaskpriceforthespread above 5.00, perhaps

5.10. In fact, amarketmakermight quote an identical bidand ask price of 5.00. Thismay seem illogical, quotingthe same bid and ask price,butconsiderwhatwillhappenifthemarketmakerisabletoeitherbuyorsellthespreadatapriceof5.00.

Ifthemarketmakerbuysthe spread (i.e., buy the 85call, sell the 90 call), hewillimmediately exercise the 85

call, thereby purchasingstock. He has effectivelyentered into the samedividend play that weoriginally described (i.e.,shortcall,longstock).Ifheisnotassignedonthe90call,hewill again profit by theamount of the dividend. If,instead, the market makersells the spread (i.e., sell the85 call, buy the 90 call), hewill immediatelyexercise the90call.Nowhehasexecuted

the dividend play bypurchasing stock and sellingthe 85 call. If he is notassigned on the 85 call, hewill again profit by theamount of the dividend. Themarket maker is willing togive up the edge on the bid-ask spread in return for thepotentialprofitthatwillresultif the short options gounexercised.

EarlyExerciseRisk

How concerned should atrader be that an option thathehas soldwill be exercisedearly?“WhatwillhappenifIamsuddenlyassigned?”Earlyassignment can sometimesresult ina loss.But therearemanyfactorsthatcancauseatrader to lose money; earlyexercise is only one suchfactor. A trader should be

prepared to deal with thepossibility of early exercise,justasheshouldbepreparedtodealwiththepossibilityofmovement in thepriceof theunderlying contract or thepossibility of changes inimplied volatility. Marginrequirements established bythe clearinghouses oftenrequire a trader to keepsufficientfundsinhisaccountto cover the possibility ofearly assignment. But this is

not always true. If the traderis short deeply in-the-moneyoptions, an early assignmentnotice may cause a cashsqueeze. If this happens, hewillneedsufficientcapital tocover the situation.Otherwise, hemay be forcedtoliquidatesomeoralloftheremaining position. Andforced liquidations areinvariably losingpropositions.

In spite of the risk ofearly assignment, it shouldrarely come as a surprise. Atrader needonly askhimself,“If I owned this option,would I logically exercise itnow?” If the answer is yesthen the trader ought to beprepared for assignment. Ifthe answer is no and thetrader is still assigned, it isprobably good for the trader.It means that someone hasmistakenly abandoned the

option’s interest or volatilityvalue.Whenthathappens,thetrader who is assigned willfindthatheistherecipientofanunexpectedgift.

1Althoughthecompanionputalsohassomeinterestanddividendvalue,thesecomponentswilltendtobesmall.Changinginterestratesordividendswillcausetheforwardpricetochange,whichissimilartochangingtheunderlyingprice.Buttheput,withitssmalldelta,willberelativelyinsensitivetothesechanges.Consequently,theout-of-the-moneyputhasonlyasmallinterest-rateanddividendvalue.Thereisnosensitivitymeasurefordividends,butwecanconfirmthattheputisrelativelyinsensitivetochangesininterestratesbynotingthatanout-of-the-moneyoptionhasasmallrhovaluecomparedwithanin-the-moneyoption.

2Figure16-6isclearlynotdrawntoscale.ThepointatwhichtheEuropeanlowerarbitrageboundarygraphbendsappearstobehalfwaybetween90and100.TheactualpointisX/(1+r×t)+D=90/(1+0.06/12)+0.75=90.30.3Figure16-7,likeFigure16-6,isnotdrawntoscale.ThepointatwhichtheEuropeanlowerarbitrageboundarygraphbendsisX/(1+r×t)+D=120/(1+0.06/6)+0.40=119.21.4Thetermfugitissometimesusedtorefertothenumberofdaysremaininguntilanoptionbecomesanimmediateearlyexercisecandidate.5JohnC.Cox,StephenA.Ross,andMarkRubinstein,“OptionPricing:A

SimplifiedApproach,”JournalofFinancialEconomics7:229–263,1979.6GiovanniBaron-AdesiandRobertWhaley,“EfficientAnalyticApproximationofAmericanOptionValues,”JournalofFinance42(2):301–320,1987.7Earlyexerciseconsiderationsmayalsobeimportantinaforeign-exchangemarketiftheinterestratesassociatedwiththedomesticcurrency(thecurrencyinwhichtheoptionissettled)andforeigncurrency(thecurrencytobedeliveredintheeventofexercise)aresignificantlydifferent.

17

HedgingwithOptions

Futures and options wereoriginally introduced asinsurance contracts, enablingmarketparticipantstotransferthe riskofholdingapositionin the underlying instrument

from one party to another.Butunlikea futurescontract,whichessentiallytransfersallthe risk, an option transfersonly part of the risk. In thisrespect, an option acts muchmore like a traditionalinsurance policy than does afuturescontract.

Even though optionswere originally intended tofunction as insurancepolicies,optionmarketshave

evolvedtothepointwhere,inmostmarkets, hedgers (thosewantingtoprotectanexistingposition) make up only asmall portion of marketparticipants. Other traders,including arbitrageurs,speculators, and spreaders,typically outnumber truehedgers. Nevertheless,hedgers still represent animportant force in themarketplace, and any activemarketparticipantoughttobe

aware of the strategies ahedgermightusetoprotectaposition.

Many hedgers come tothe marketplace as eithernatural longs or naturalshorts.Throughthecourseofnormalbusinessactivity,theywill profit from either a riseor fall in the price of someunderlying instrument. Theproducerofacommodityisanatural long; if the price of

the commodity rises, theproducer will receive morewhen he sells in themarketplace. The user of acommodityisanaturalshort;ifthepriceofthecommodityfalls,theuserwillhavetopaylessforitwhenhebuysinthemarketplace. In the sameway, lenders and borrowersarenaturallongsandshortsintermsof interest rates.A risein interest rates will helplendersandhurtborrowers.A

decline in interest rates willhavetheoppositeeffect.

Other potential hedgerscome to the marketplacebecausetheyhavevoluntarilychosentotakealongorshortpositionandnowwish to layoff part or all of the risk ofthat position.A speculator inacommoditymayhavetakena long or short position butwishes to temporarily reducethe risk associated with an

outright long or shortposition. A fund managermayholdaportfolioofstocksbutbelieves that thevalueofthe portfolio may decline intheshortterm.Ifso,itmaybeless expensive to temporarilyhedgethestockswithoptionsor futures than to sell thestocksandbuythembackatalaterdate.

Aswith insurance, thereisacosttohedging.Thecost

maybeimmediatelyapparentin the form of a cash outlay.But the cost may also bemoresubtle,eitherintermsoflost profit opportunity or intermsofadditionalriskundersome circumstances. Everyhedging decision is atradeoff: what is the hedgerwilling to give up under oneset of market conditions inreturn for protection under adifferent set of marketconditions. A hedger with a

long position who wants toprotect his downside willalmost certainly have to giveupsomethingontheupside;ahedger with a short positionwho wants to protect hisupside will have to give upsomethingonthedownside.

ProtectiveCallsandPuts

Thesimplestwaytohedgean underlying position usingoptionsistopurchaseeitheraput toprotecta longpositionor a call to protect a shortposition. In each case, if themarket moves adversely, thehedger is insulated from anyloss beyond the exerciseprice.Thedifferencebetweenthe exercise price and thecurrent price of theunderlying is similar to thedeductible portion of an

insurancepolicy.Thepriceofthe option is similar to thepremium that one has to payfortheinsurancepolicy.

Consider an Americanfirm that expects to takedelivery of €1 million worthof German goods in sixmonths. If the contractrequires payment in euros atthe time of delivery, theAmericanfirmhasacquiredashortpositionineurosagainst

U.S. dollars. If over the nextsix months the euro risesagainst the dollar, the goodswill cost more in dollars; ifthe euro falls, the goodswillcost less. If the euro iscurrently trading at 1.35($1.35per euro) and remainsthereforthenextsixmonths,thecosttotheAmericanfirmwill be $1,350,000. If,however,atdelivery theeurohas risen to 1.45 ($1.45 pereuro), the cost to the

American firm will be$1,450,000.

The American firm canoffset therisk ithasacquiredbypurchasingacalloptiononeuros, for example, a 1.40call. For a complete hedge,the underlying contract willbe€1million,and theoptionwill have an expiration datecorresponding to the date onwhichpaymentisrequired.Ifthe value of the euro begins

toriseagainsttheU.S.dollar,the firm will have to pay ahigher price than expectedwhen it takes delivery of thegoods in sixmonths.But theprice it will have to pay foreuros can never be greaterthan 1.40. If the price isgreater than 1.40 atexpiration, the firm willsimply exercise its call,effectively purchasing eurosat1.40.Ifthepriceofeurosisless than 1.40 at expiration,

the firm will let the optionexpire worthless because itwill be cheaper to purchaseeurosintheopenmarket.

When used to hedgeinterest-rate risk, protectiveoptions are sometimesreferredtoascapsandfloors.A firm that borrows funds ata variable interest rate has ashort interest-rate position—falling interest rates willreduce its cost of borrowing,

whilerisinginterestrateswillincrease its costs.Tocap theupside risk, the firm canpurchasean interest-rate call,thereby establishing amaximumamountitwillhaveto pay for borrowed funds.No matter how high interestrates rise, the borrower willnever have to paymore thanthecap’sexerciseprice.

An institution that lendsfunds at a variable interest

rate has a long interest-rateposition—rising interest rateswillincreaseitsreturns,whilefalling interest rates willreduce its returns. To set aflooronitsdownsiderisk,theinstitution can purchase aninterest-rate put, therebyestablishing a minimumamount it will receive forloaned funds.Nomatterhowlow interest rates fall, thelenderwillnever receive lessthan the floor’s exercise

price.A hedger who chooses

topurchaseacalltoprotectashort position or a put toprotect a long position hasrisk limited by the exerciseprice of the option. At thesame time, the hedger stillmaintains open-ended profitpotential. If the underlyingmarketmovesinthehedger’sfavor, he can let the optionexpire and take advantage of

the position in the openmarket. If, in our example,the euro falls to 1.25 at thetimeofdelivery,thefirmwillsimplyletthe1.40callexpireunexercised. At the sametime, the firm will purchase€1 million for $1,250,000,resulting in a windfall of$100,000.

There is a cost involvedin buying insurance in theform of a protective call or

put, namely, the price of theoption. The cost of theinsurance is commensuratewiththeamountofprotectionaffordedby theoption. If thepriceofasix-month1.40callis 0.02, the firm will pay anextra $20,000 (0.02 × 1million) no matter whathappens.Acalloptionwithahigherexercisepricewillcostless, but it also offers lessprotection in the form of anadditionaldeductibleamount.

If the firm chooses topurchasea1.45calltradingat.01,thecostforthisinsurancewill only be $10,000 (0.01×1 million), but the firm willhave tobearany lossup toaeuro price of 1.45. Onlyabove 1.45 is the firm fullyprotected.Inthesameway,alower-exercise-price call willofferadditionalprotectionbutat ahigherprice.A1.35callwill protect the firm againstanyriseabove1.35,butifthe

price of the call is 0.04, thepurchase of this protectionwill add an additional$40,000(0.04×1million) tothefinalcost.

Thecostofpurchasingaprotective option and theinsurance afforded by thestrategyareshowninFigures17-1(protectiveput)and17-2(protective call). Becauseeach strategy combines anunderlying position with a

long option position, itfollowsfromChapter14 thatthe resulting protectedposition is a synthetic longoption

Figure17-1Longanunderlyingpositionandlongaprotectiveput.

Figure17-2Shortanunderlyingpositionandlongaprotectivecall.

Shortunderlying+longcall≈syntheticlongput

Longunderlying+longput≈syntheticlongcall

Ahedgerwhobuysaputtoprotect a long underlyingposition has effectivelycreatedalongcallpositionatthe same exercise price. Ahedger who buys a call toprotect a short underlyingposition has effectively

created long put position. Inour example, if the firmpurchases a 1.40 call toprotect a short euro position,the combined position (i.e.,shortunderlying,longcall)isequivalent to owning a 1.40put.

Which protective optionshould a hedger buy? Thisdependsontheamountofriskthe hedger iswilling to bear,something that each hedger

must determine individually.One thing is certain: therewill always be a costassociated with the purchaseof a protective option. If theinsurance afforded by theoption enables the hedger toprotect his financial position,thecostmaybeworthwhile.

CoveredWrites

If a hedger is averse to

payingforprotectiveoptions,whichofferlimitedandwell-defined risk, the hedgermayinstead consider selling, orwriting, an option against anunderlying position. Thiscovered write (sometimesreferred to as an overwrite)doesnotofferthelimitedriskaffordedbythepurchaseofaprotective option but doeshave the obvious advantageofcreatinganimmediatecashcredit. This credit offers

limited protection against anadverse move in theunderlyingmarket.

Consider an investorwhoownsstockbutwants toprotect against a short-termdeclineinthestockprice.Hecan, of course, buy aprotective put. But if hebelieves that any decline islikelytobeonlymoderate,hemight instead sell a calloption against the long stock

position. The amount ofprotection the investor isseeking, as well as thepotential upside appreciation,will determine which call hesells, whether in the money,at the money, or out of themoney. Selling an in-the-money call offers a highdegree of protection but willeliminate most of the upsideprofit potential. Selling anout-of-the-money call offersless protection but leaves

room for additional upsideprofit.

Supposethataninvestorownsastockthatiscurrentlytradingat100.Ifhesellsa95callatapriceof6.50,thesaleof the call will offer a highdegreeofprotectionagainstadecline in the price of thestock. As long as the stockdeclinesbynomorethan6.50to93.50, the investorwilldono worse than break even.

Unfortunately, if the stockbeginstorise,therewillbenoopportunity to participate intherisingstockpricebecausethestockwillbecalledawaywhentheinvestorisassignedon the 95 call. Still, even ifthe stock rises, the investorwillatleastprofitbythetimepremium of 1.50 that hereceived from the sale of the95call.

Ontheotherhand,ifthe

investor wants to participatein upside movement in thestock and is also willing toaccept less protection on thedownside,hemightsella105call. If the105call is tradingatapriceof2.00, thesaleofthis option will only protectthe investor down to a stockprice of 98.But, if the stockprice rises, the investor willparticipate up to a price of105. Above 105, he canexpect the stock to be called

away,eliminatinganyfurtherprofit.

Whichoptionshouldtheinvestor sell? This is asubjective decision based onhowmuchrisktheinvestoriswilling to accept, as well asthe amount of upsideappreciation in which hewants to participate. Manycoveredwritesinvolvesellingat-the-money options. Suchoptions offer less protection

than in-the-money calls andless profit potential than out-of-the-moneyoptions.Butanat-the-money option has thegreatest amount of timepremium. If the marketremains close to its currentprice, a position that ishedged by selling at-the-moneyoptionswill show thegreatest amount ofappreciation.

The characteristics of a

covered write and theprotection afforded by thestrategyareshowninFigures17-3 (covered call) and 17-4(covered put). Because eachstrategy combines anunderlying position with ashort option position, itfollowsfromChapter14 thatthe resulting protectedposition is a synthetic shortoption:

Figure17-3Longanunderlyingpositionandshortacoveredcall.

Figure17-4Shortanunderlyingpositionandshortacoveredput.

Longunderlying+shortcall≈syntheticshortput

Shortunderlying+shortput≈syntheticshortcall

A hedger who sells a callagainst a long underlyingposition has effectivelycreatedashortputpositionatthe same exercise price. Ahedger who sells a put toprotect a short underlyingposition has effectively

createdshortcallputposition.Inourexample, if thehedgersells a 105 call to protect along stock position, thecombined position (i.e., longunderlying, short call) isequivalent to selling a 105put.

Selling a covered callagainst a long stock positionis one of the most popularhedging strategies in equityoption markets. When

executed all at one time—buying stock andsimultaneously selling a callon the stock—the strategy isreferredtoasabuy/write.TheDecember 105 buy/writeconsists of buying one stockcontract (usually 100 shares)and simultaneously selling aDecember 105 call. As withany spread, it can be quotedas a single price (the stockprice – the call price) andexecuted with a single

counterparty. With a stocktrading at 100 and theDecember105call tradingat2.00, the December 105buy/write is trading at 98.00.Thepricequotedbyamarketmaker might be 97.90 –98.10. In total, the marketmaker is willing to buy thestock and sell the call for97.90.Heiswillingtosellthestock and buy the call for98.10.

Buy/writes are suchcommon strategies that someexchanges publish indexesreflecting the performance ofthestrategy,usuallyagainstamajor stock index. TheChicago Board OptionsExchange BuyWrite Index(BXM) reflects theperformance of a strategyconsisting of buying aStandard and Poor’s (S&P)500 Index (SPX) portfolioand each month selling a

slightly out-of-the-moneyone-month S&P 500 Indexcalloption.1

Acoveredwritecanalsobe used to set a target priceforeitherbuyingorsellinganunderlying instrument. Aninvestor who owns a stockmay decide that if the stockreaches a certain price, hewill be willing to sell. Bysellingacallwithanexercisepriceequaltothetargetprice,

the investor has effectivelylockedinthesaleifthestockreaches the exercise price. Ifthe stock does not reach theexercise price, the investorstillgetstokeepthepremiumreceived from the sale of thecall.

Similarly, an investorwhoiswillingtobuystockifthe price declines by somegiven amount can sell a putwith an exercise price equal

tothetargetpurchaseprice.Ifthe stock falls below theexercise price, the investorwill be assigned on the put,forcing him to purchase thestock. But that was hisoriginalintention.Ifthestockfailstofallbelowtheexerciseprice, the investor gets tokeep the premium receivedfromthesaleoftheput.Thisstrategy of selling puts totrigger the purchase of stockis often used by companies

that want to initiate a buy-backprogramfor their stock.By selling putswith exercisepricesequaltothetargetbuy-back price, the companyeitherbuysbackitsownstockor profits by the amount oftheputpremium.

The primary differencebetweensellingacalltosetasalepriceandsellingaputtoset a purchase price is theway in which the trade is

secured.The sale of a call issecuredwithownershipofthestock.But thesaleof theputmustbesecuredwithenoughcash to support the purchaseofthestockshouldtheputbeexercised.Thesaleofacash-secured put requires theinvestor to keep on depositcash equal to the exerciseprice of the put. If the put isEuropean with no possibilityofearlyexercise,theinvestorcan keep on deposit cash

equal to the present value oftheexerciseprice

The purchase of aprotectiveoptionandthesaleof a covered option are thetwo most common hedgingstrategies involving options.If given a choice betweenthese strategies, which oneshould a hedger choose? In

theory, the hedger ought tobasehisdecisiononthesamecriteriausedbyatrader:priceversusvalue. If optionpricesseem low, the purchase of aprotective option makessense. If option prices seemhigh, the sale of a coveredoption makes sense. From atrader’spointofview, loworhigh is typicallyexpressed interms of implied volatility.By comparing impliedvolatility with the expected

volatility over the life of theoption, a hedger ought to beable to make a sensibledetermination as to whetherhe wants to buy or selloptions.Of course, he is stillleft with the question ofwhich exercise price tochoose. This will depend onthe amount of adverse orfavorable movement thehedger foresees, as well astheriskheiswillingtoacceptifheiswrong.

While theoreticalconsiderations often play arole in a hedger’s decision,these may be less importantthan practical considerations.If a hedger knows that amove in the underlyingcontract beyond a certainpricewillrepresentathreattohis business, then thepurchase of a protectiveoptionatthatexercisemaybethe most sensible strategyregardless of whether the

option is theoreticallyoverpriced.2

Many hedgers seem tohave an aversion to buyingprotective options. “Whyshould I pay for an optionwhenIwillprobablylosethepremium?” This is, indeed,true. Most protective optionsdo expire out of the money.The reasoning, however,seems illogical when oneconsiders that most people

willingly purchase insuranceto protect their personalproperty. And the greatmajorityofinsurancepoliciesexpire without claims everbeing made against them:houses do not burn down;people do not die; and carsare not stolen. This is thereason insurance companiesmake a profit. But mostpeople do not buy insurancetomake a profit. They do soforthepeaceofmindthatthe

insurancepolicyaffords.Thesame philosophy ought toapply to the purchase ofoptions. If a hedger needswell-defined protection, thepurchaseofanoptionmaybethe best choice regardless ofthe fact that the option willmostoftenexpireworthless.

Collars

A hedger may want the

limited risk afforded by thepurchase of a protectiveoption but may also bereluctant to pay the premiumassociated with such astrategy.What can he do?Acollar involvessimultaneously purchasing aprotective option and sellinga covered option against aposition in an underlyingcontract.3Collarsarepopularhedging tools because they

offer known protection at alow cost. At the same time,they still allow a hedger toparticipate, at least partially,in favorable marketmovement. With anunderlying stock trading at100, a hedger with a longpositionmightchoose tobuya95putandatthesametimesella105call.Thehedger isinsulated from any fall inprice below 95 because hecan then exercise his put.At

the same time, he canparticipate in any upwardmoveupto105.

The terms long andshort, when applied tocollars, typically refer to theunderlying position. A longunderlying position togetherwith a protective put andcovered call is a long collar.A short underlying positiontogetherwithaprotectivecalland covered put is a short

collar.Thecharacteristicsofacollar are shown in Figures17-5and17-6.Becauseeverycontractcanbeexpressedasasynthetic equivalent, we cansee that a longcollar (Figure17-5)issimplyabullverticalspread, while a short collar(Figure17-6)issimplyabearvertical spread. Bothstrategies have limited riskandlimitedreward.

Figure17-5Longcollar(longanunderlyingcontract,longaprotective

put,shortacoveredcall).

Figure17-6Shortcollar(shortanunderlyingcontract,longaprotectivecall,shortacoveredput).

Because a collar is avertical spread, it will havethe risk characteristicsdescribed in Chapter 12. Alongcollarwillalwayshaveapositive delta; a short collarwill always have a negativedelta.Thegamma, theta, andvega will be determined bythe choiceof exercise prices.If the underlying price iscloser to the protectiveoption, the position will

usually have a positivegamma, negative theta, andpositive vega. If theunderlying price is closer tothe covered option, theposition will usually have anegative gamma, positivetheta, and negative vega.Unless one option is muchfurtheroutofthemoneythantheother,theseriskmeasuresare likely to be similar,resulting in only a smallgamma, theta, and vega

position.Ahedgermightalsochoose exercise prices suchthat the collar will beapproximately neutral withrespect to the gamma, theta,orvega.

Collars are also popularbecause the sale of thecovered option may offsetsomeorallof thecostof theprotective option. When thepriceof theprotectiveoptionisgreaterthanthepriceofthe

covered option, as it is inFigure 17-5, the midsectionofthecombinedpositionwillfall below theprofit and loss(P&L) graph for theunderlying position. Whenthe price of the protectiveoptionislessthanthepriceofthecoveredoption,as it is inFigure 17-6, the midsectionofthecombinedpositionwillbe above the P&L graph fortheunderlyingposition.Ifthepriceof theprotectiveoption

andthecoveredoptionarethesame, the strategybecomesazero-cost collar. A summaryof basic hedging strategies isgiveninFigure17-7.

Figure17-7Summaryofbasichedgingstrategies.

ComplexHedgingStrategies

Because most hedgers arenot professional optiontraders and have neither thetime nor the desire tocarefully analyze optionprices, simple hedgingstrategies involving thepurchase or sale of single

options are the most widelyused. However, if one iswilling todoamoredetailedanalysis of options, it ispossible to construct a widevariety of hedging strategiesthat involve both volatilityand directionalconsiderations. To do this, ahedgermust be familiarwithvolatility and its impact onoption values, as well as thedelta as a measure ofdirectional risk. The hedger

can then combine hisknowledge of options withthepracticalconsiderationsofhedging.

As a first step inchoosinga strategy, ahedgermightconsiderthefollowing:

1. Doesthehedgeneed to offerprotection against aworst-casescenario?2. How much of

the currentdirectional riskshould the hedgeeliminate?3.Whatadditionalrisks is the hedgerwillingtoaccept?

A hedger who needsdisaster insurance to protectagainst aworst-case scenarioonly has a choice of whichoption(s) tobuy.Evenso,hestill needs to decide which

exercisepricetopurchaseandhow many options. With alongpositioninanunderlyingcontract currently trading at100, a hedger decides to buya put because he needs tolimit the downside risk tosome known and fixedamount.Whichputshouldhebuy?

If the hedger hasdetermined that options aregenerally overpriced (i.e.,

implied volatility seemshigh), any option purchasewillclearlybetothehedger’sdisadvantage. If his solepurpose is to hedge hisdownside riskwithout regardto upside profit potential, heought to avoid options andhedge his position in thefuturesorforwardmarket.If,however,hestillwantsupsideprofit potential, he must askhimself howmuch of a longpositionhewantstoretain.If

he is willing to retain 50percent of his current longposition,heoughttopurchaseputswithatotaldeltaof–50.Hecandothisbypurchasingone at-the-money put with adeltaof–50orseveralout-of-the-money puts whose deltasadd up to –50. In a high-implied-volatility market,however, it is usually best tobuy as few options aspossible and sell as manyoptions as possible. (This is

analogous to constructing aratio spread.) Hence,purchasing one put with adelta of –50 will be lesscostly, theoretically, thanpurchasing several puts witha total delta of –50. If thehedger wants to eliminateeven more of the directionalrisk, say, 75 percent, underthese circumstances, he willbe better off purchasing oneputwithadeltaof–75.

Allotherfactorsbeingequal,inahigh-implied-volatilitymarket,ahedgershouldbuyasfewoptionsaspossibleand/orsellasmanyoptionsaspossible.Conversely,inalow-implied-volatilitymarket,ahedgershouldbuyasmanyoptionsas

possibleand/orsellasfewoptionsaspossible.

This means that if alloptions are overpriced (i.e.,impliedvolatilityseemshigh)andthehedgerdecidesthatheis willing to accept theunlimited downside risk thatgoes with the sale of acovered call, in theory, heoughttosellasmanycallsaspossible to reachhishedging

objectives. If he is trying tohedge 50 percent of his longunderlying position, he cando a ratio write by sellingseveral out-of-the-moneycalls with a total delta of 50ratherthansellingasingleat-the-moneycallwithadeltaof50.

There is an obviousdisadvantage if one sellsmultiplecallsagainstasinglelong underlying position.

Nowthehedgernotonlyhasthe unlimited downside riskthat goeswith a covered callposition, but he also hasunlimitedupsideriskbecausehehassoldmorecallsthanhecan cover with theunderlying. If the marketmovesup enough,hewill beassignedonallthecalls.Mosthedgers want to restrict theirunlimited risk to onedirection, usually thedirection of their natural

position. A hedger with alongunderlyingpositionmaybewillingtoacceptunlimiteddownside risk, but he isprobably unwilling to acceptunlimited upside risk. Ahedger with a shortunderlying position may bewilling to accept unlimitedupsiderisk,butheisprobablyunwilling to acceptunlimiteddownsiderisk.Ahedgerwhoconstructs a position withunlimited risk in either

direction is presumablytaking a volatility position.There is nothing wrong withthisbecausevolatility tradingcan be highly profitable. Buta true hedger ought not losesight of what his ultimategoalis—toprotectanexistingposition and to keep the costof this protection as low aspossible.

A hedger can alsoprotect a position by

constructing one-to-onevolatility spreads with deltasthatyield thedesiredamountof protection. A hedger whowantstoprotect50percentofa short underlying positioncan buy or sell calendarspreads or butterflies with atotal delta of +50. Suchspreads offer partialprotectionwithinarange.Theentire position still hasunlimitedupsideriskbutalsoretains unlimited downside

profit potential. Suchvolatility spreads also givethe hedger the choice ofbuyingorsellingvolatility.Ifimpliedvolatility isgenerallylow, with the underlyingmarket currently at 100, thehedger might protect a shortunderlying position bypurchasing a 110 callcalendar spread (i.e.,purchase a long-term 110call, sell a short-term 110call). This spread has a

positive delta and is alsotheoretically attractivebecause the low impliedvolatility makes a longcalendar spread relativelyinexpensive. If the 110 callcalendarspreadhasadeltaof+25, to hedge 50 percent ofhis directional risk, thehedger can buy two spreadsfor each short underlyingposition. Conversely, ifimpliedvolatility is high, thehedger can consider selling

calendarspreads.Nowhewillhave to choose a lowerexercise price to achieve apositive delta. If he sells the90 call calendar spread (i.e.,purchaseashort-term90call,sell a long-term 90 call), hewill have a position with apositive delta and a positivetheoretical edge. If he wantsto protect 75 percent of hispositionand the spreadhasadelta of +25, he can sell thespread three times for each

underlying position. (SeeChapter11forcharacteristicsof calendar spreads andbutterflies.)

Ahedgercanalsobuyorsell vertical spreads toachieve a desired amount ofprotection. Depending onwhetheroptionsaregenerallyunderpriced or overpriced(i.e., implied volatility isexcessively low or high), thehedger will work around the

at-the-money option. Withthe underlying marketcurrently at 100, the hedgerwho wants to protect a longposition can execute a bearvertical spread (i.e., sell thelower exercise price, buy thehigher exercise price). Ifimplied volatility is high, hewill prefer to sell an at-the-money option and buy anoption at a higher exerciseprice. If implied volatility islow,hewill prefer tobuyan

in-the-money option and sellanoption at a lower exerciseprice.Eachspreadwillhaveanegative delta but will alsohave a positive theoreticaledge because the at-the-money option is the mostsensitive to changes involatility.(SeeChapter12forcharacteristics of verticalspreads.)

As is obvious, usingoptions to hedge a position

can be just as complex asusing options to constructtrading strategies. Manyfactors go into the decision-making process. When apotentialhedgerisconfrontedfor the first time with themultitude of possiblestrategies, he canunderstandably feeloverwhelmed, to the pointwhere he decides to abandonoptionscompletely.Perhapsabetterapproachistoconsider

alimitednumberofstrategies(perhaps four or five) thatmake sense and compare thevarious risk-rewardcharacteristics of thestrategies.Giventhehedger’sgeneral market outlook andhis willingness orunwillingness to acceptcertainrisks,itshouldthenbepossibletomakeaninformeddecision.

HedgingtoReduceVolatility

In addition to protecting aposition against an adversemove in the underlyingcontract, hedging strategieshave an additional importantadvantage—they tend toreduce the volatility of aposition. To understand whythis may be important,consider a portfolio manager

who generates the followingannual returns over a periodoffiveyears:

+19%–14%+27%–9%+22%

His average annual returnis

(19%–14%+27%–9%+22%)/5=+9%

Now consider a second

portfolio manager whogenerates these annualreturns:

+25%–20%–23%+44+24

His average annual returnis

(25%–20%–23%+44%+24%)/5=+10%

Finally, a third portfoliomanager generates these

returns:

+35%+15–35+65%–20%

His average annual returnis

(35%+15%–35%+65%–20%)/5=+12%

Portfolio Manager 3trumpets his average annualreturn of 12 percentcompared with Portfolio

Managers 1 and 2, withreturns of only 9 and 10percent.Clearly,weought toinvest our money withPortfolio Manager 3. Orshould we? Perhaps weshouldconsidernotonlywhatis happening each year butalso how each portfolioperformsovertheentirefive-year period. We can do thisby taking the product of allthe annual changes for eachportfolio:

Portfolio 1:1.19 × 0.86 ×1.27 × 0.91 ×1.22 = 1.4429(up44.29%)Portfolio 2:1.25 × 0.80 ×0.77 × 1.44 ×1.24 = 1.3749(up37.49%)Portfolio 3:1.35 × 1.15 ×0.65 × 1.65 ×

0.80 = 1.3320(up33.20%)

EventhoughPortfolioManager3hadthebestaverageannualreturn,hisportfoliofaredtheworst.PortfolioManager1,withthelowestannualreturn,faredthebest,making11percentmoreoverthefive-yearperiodthanPortfolioManager3.

The explanation for thisseemingly unexpected resulthas to do with the volatility,or standard deviation, of thereturns. The returns forPortfolio Manager 3fluctuatedwildlyfromahighof +65 percent to a low of –35 percent. The returns forPortfolioManger1fluctuatedmuch less, between +27percentand–14percent.Thegreater volatility seemed toreducethetotalreturn.

The results for eachportfolio manager aresummarized in Figure 17-8.We have also added a veryboring Portfolio Manager 4,whoplodsalongwithareturnofexactly8percenteachyearfor the five-year period. Inspite of having the lowestaverage return, his portfolioperformed the best, gaining46.93 percent over the entireperiod.

Figure17-8Thegreaterthevolatility,thelowerthetotalreturn.

Our example does notmean that high volatility is

unacceptable. A portfoliomanager with highly volatilereturnsmaystillbepreferableif his average return is alsocommensurably higher. Thistradeoff between returns andvolatility is often expressedby the Sharpe ratio,originally suggested byWilliamSharpein19664

Averagereturn/standarddeviationofreturns

The greater the Sharperatio, the more favorable thetradeoff between risk(volatility) and reward(returns). The standarddeviationandtheSharperatiofor all four portfoliomanagers are also given inFigure17-8.

PortfolioInsurance

Imagine that we hold a

longpositioninanunderlyingasset such as stock and thatwewould like to protect ourposition against a possibledecline in price over someperiod of time. One possiblestrategy is to purchase aprotectiveput.Unfortunately,when we go into the marketto purchase the put, we findthat no market exists foroptions on our stock. Whatcanwedo?

Ifwewerereallyabletopurchase a put, our positionwouldbe

Longstock+longput

But we know that a longunderlying position togetherwith a long put is equivalenttoalongcall.Whatwereallywant is a long call positionwith the same exercise priceandexpirationdateastheputthat we wanted but were

unabletobuy.What would be the

characteristics of this call?We can determine this byusing a theoretical pricingmodel. To do this, we needthe basic inputs into thetheoreticalpricingmodel:

ExercisepriceTime toexpirationUnderlyingstockprice

InterestrateVolatility

Becausewearenotdependentonlistedexercisepricesandexpirationdates(becausenoneexist),theexercisepriceandexpirationdatecanbeofourownchoosing.Wecandeterminethestockpriceandinterestratefromcurrentmarketconditions.Onlythevolatilitycannotbedirectlyobservedin

themarketplace.But,ifwehaveadatabaseofhistoricalpricechangesforthestock,wemaybeabletomakeareasonableestimateofthestock’svolatility.

Supposethatwefeedallthe inputs into a theoreticalpricing model and determinethat our intended call has adelta of 75. To replicate thecallposition,weneedtoown75 percent of the underlying

contract.Wecanachievethisby selling off 25 percent ofour holdings in the stock. Ifwe originally owned 1,000shares, we need to sell 250shares,leavinguswithalongpositionof750shares.

Now suppose that atsomelaterdatewelookatthenew market conditions,recalculate the delta of thecall, and find that it is now60. To achieve the desired

delta position, we must nowsell off an additional 15percent of our originalholdings, or 150 shares. Weare now long 600 shares ofstock.

Suppose that wecontinue this process ofperiodically calculating thedelta from current marketconditions and buying orselling some percentage ofour original holding in the

underlyingstocktoachieveaposition with the same deltaas the presumed call option.Finally, suppose that at thetargetexpirationdatewebuyback a sufficient amount ofthestocksothatwehave100percent of our originalholding. What should be theresultofthisentireprocess?

Weareessentiallygoingthrough thedynamichedgingprocess described in Chapter

8.Whereas in Chapter 8 weused dynamic hedging tocapture the differencebetween an option’s price inthe marketplace and itstheoretical value, in ourcurrent example, we cannotprofit from a mispricedoption because no optionexists. But we can replicatethe characteristics of theoption to achieve a desiredoptionposition.

In Chapter 8 wepresented a stock optionexampleanda futuresoptionexample. In the stock optionexample,we bought a call ata price thatwas less than itstheoretical value and thensold the call, through thedynamichedgingprocess,ataprice that was equal to itstheoretical value. In thefutures option example, wesoldaputataprice thatwasgreater than its theoretical

value and then bought theput, through the dynamichedging process, at a pricethat was equal to itstheoretical value. In bothexamples,weendedupwithaprofit equal to the differencebetween the option’s priceanditstheoreticalvalue.

Portfolio insurance, oroption replication, is amethod by which thedynamic hedging process is

usedtocreateapositionwiththesamecharacteristicsasanoption. In theory, themethodshould achieve the sameresultsasbuyingaprotectiveoption but without actuallypurchasing the option.Portfolio insurance can beused by a fund manager toinsure the value of thesecurities in a portfolioagainst a drop in value. If amanager has a portfolio ofsecurities currently valued at

$100 million and wants toinsure the value of theportfolio against a drop invalue below $90 million, hecan either buy a $90millionput or replicate thecharacteristics of a $90millioncall.Ifheisunabletofind someone willing to sellhima$90millionput,hecanevaluatethecharacteristicsofthe $90 million call andcontinuously buy or sell aportion of his portfolio

required to replicate the callposition. In effect, he hascreatedhisownput.

Portfolio insurancestrategies were widely usedbyfundmanagerspriortothestock market crash of 1987,especiallybymanagerswithaportfolio that tended to trackamajorindex.Iftheportfoliomanager wanted to buyprotective puts but alsobelieved that the prices of

puts were inflated, he couldcreate theputshimself at the“correct” theoretical valuethrough thedynamichedgingprocess. Insteadof buyingorselling a portion of theportfolio, which could beexpensive in terms oftransaction costs, theportfolio manager couldmimic the delta adjustmentsby buying or selling indexfutures to increase or reducethe total value of the

portfolio. In return for a fee,firms that marketed portfolioinsurance strategies assumedthe responsibility ofdetermining thecharacteristics of the optionthat the portfolio managerwanted to purchase byestimating the correctvolatility and choosing themost appropriate optionpricing model.5 Someportfolio insurance firms

generated additional fees byacting as a broker andexecuting the necessaryadjustments in the indexfuturesmarket.

Unfortunately, followingthe crash of 1987,practitioners came to realizethatportfolioinsurancewouldonly achieve the desiredresults if the inputs into themodel were correct and themodel itself was based on

realistic assumptions.6 Noone foresaw the dramaticincreaseinvolatilityresultingfrom the crash, so thevolatilityinputthatwasbeingusedwasclearlyincorrect.Atthe same time, many of themodel assumptions aboutdynamic hedging seemed tobeviolated in the realworld.The upshotwas that the costof replicating an optionthrough thedynamichedging

process became much moreexpensive than anyone hadanticipated. As a result,portfolio insurance strategiesfell out of favor with mostfundmanagers.

1AcompletedescriptionoftheCBOEBuy/WriteIndex,aswellasitshistoricalperformance,canbefoundathttp://www.cboe.com/micro/bxm/.2Ofcourse,ifoptionsseemwildlyoverpriced,ahedgermaybereluctanttobuyaprotectiveoption.Butthisisanunlikelyscenario.Ifoptionpricesarehigh,thereisusuallyavalidreason.3Thecollarstrategygoesbyawidevarietyofnames,includingfence,tunnel,cylinder,rangeforward,orsplit-strikeconversion.4ThereturnsusedtocalculatetheSharperatioaresometimesexpressedasthereturnsinexcessofsomebenchmark,suchasarisk-freeTreasury

instrument.5Thefirmmostcloselyassociatedwithportfolioinsurancepriortothecrashof1987wasLosAngeles–basedLeland,O’Brien,Rubinstein(withprincipalsHayneLeland,JohnO’Brien,andMarkRubinstein).6SomestudieshavesuggestedthatthedynamichedgingrequiredtoimplementportfolioinsuranceexacerbatedthestockmarketcrashofOctober19,1987.Becauseofthedramaticdropinthestockmarket,portfolioinsurerswererequiredtoselleverlargernumbersofindexfuturescontracts,creatingacascadingeffectinthemarket.

18

TheBlack-ScholesModel

Because of its importanceas a foundation of optionpricing theory, as well as itswidespread use by traders, itwill be worthwhile to take acloser look at the Black-

Scholes model.1 Thediscussion in this chapter isnotmeant tobearigorousordetailed derivation of themodel,which is better suitedtoauniversity textbookor toa class in financialengineering.Rather,wehopeto present a more intuitivediscussionoftheworkingsofthe model, as well as someobservations on the valuesgeneratedbythemodel.

Initially, rather thancalculating the theoreticalvalueofanoption,BlackandScholes tried to answer thisquestion: if the stock pricemoves randomly over time,but in a manner that isconsistent with a constantinterest rate and volatility,whatmustbetheoptionpriceafter each moment in timesuch that an option positionthat is correctly hedged willjust break even? The answer

to this question resulted inrather intimidating-lookingequation

Although this equationmight look mysterious tomany readers, it is just amathematician’s way ofexpressing how changes inone set of variables—stockpriceSandtimet—affect the

value of something else, acallC.Todeterminetheexacteffect caused by changes inthevariables, onemust solvetheequation.

Note that we did notrefer to the volatility σ andinterestraterasvariables. Inthe Black-Scholes equation,only thestockpriceand timeare changing. As inputs intothe model, the volatility andinterest rate will affect the

valueoftheoption.Butoncethey have been chosen, theyare assumed to remainconstant over the life of theoption.Thisisconsistentwiththe dynamic hedgingexamples in Chapter 8.Overthe life of an option, weassumed that only theunderlying price and timewere changing. Everythingelseremainedconstant.

We have already

encountered several of thecomponents of the Black-Scholes equation in slightlydifferentform.ThetermsC

are the more formalmathematicalnotationfor theoption’s delta (Δ), gamma(Γ),andtheta(Θ).TheBlack-Scholes equation states thatchanges in an option’s value

depend on the sensitivity ofthe option to changes in thestock price (the delta), thesensitivity of the option’sdelta to changes in the stockprice (the gamma), and thesensitivityoftheoptiontothepassageoftime(thetheta).

Of course, the equationalso includes volatility andinterest-ratecomponents.Theinterest-rate component playstwo roles. First, because the

Black-Scholes model valuesoptions from the forwardprice, the interest rate takesus from the spot price to theforward price (assuming thatthe stock pays no dividend).This spot-to-forwardrelationship appears in theequationas

rS

Second, the Black-Scholes equation initially

givesustheexpectedvalueofthe option as time passes. Ifwe want to determine theoption’stheoreticalvalue,wemust discount the expectedvalue backwards to get itspresentvalue.This expected-value-to-present-valuerelationship appears in theequationas

rC

Finally, there is a

volatility component. Therate at which the deltachangesdependsnot onlyonthe gamma but on the speedat which the stock price ischanging. The speed isexpressed as a volatility orstandard deviation σ. Thevolatility component and itseffect on the gamma appearintheBlack-Scholesequationas

We will not go into theformal derivation of theBlack-Scholes equation inthis text because it can bemathematically complex. Butwe might note that there issome similarity between theBlack-Scholes equation andthemethodusedinChapter7to estimate the change in anoption’s value as the

underlying price changesfromS1toS2.Toapproximatethis change, we used theaverage delta over the pricerange

(S1–S2)×Δ+(S1–S2)2×Γ/2=(S1–S2)×Δ+1/2(S1–

S2)2×Γ

Recallingthat

represent the delta andgamma,wecanseethatthereis a similarity between thisrelationship and the first twoterms of the Black-Scholesequation.

The primary differencesare the interest-ratecomponent attached toS (thestock price must move fromspot to forward) and thevolatilitycomponentattachedto the gamma. Although weassumed a discrete pricechange from S1 to S2, theBlack-Scholes equationassumes an infinitesimallysmall, or instantaneous, pricechange.

This is, admittedly, avery simplistic attempt toexplain the roles played bythevariouscomponentsintheBlack-Scholes equation.However, even for someonewho fully understands themodel,beingabletowriteoutthe equation does notnecessarilyyieldavalue.Thereal goal is to solve theequationso that it ispossibletocalculatetheexactvalueofanoption.

The solution to theBlack-Scholes equationyields thewell-knownBlack-Scholesmodel:if

C=theoreticalvalue of aEuropeancallS = the priceof a non-dividend-payingstockX = exerciseprice

t = time toexpiration, inyearsσ = annualstandarddeviation(volatility) ofthestockprice,inpercentr = annualinterestrateln=thenaturallogarithm

e = theexponentialfunctionN = thecumulativenormaldistributionfunction

It may not be

immediately apparent whatthe values in the Black-Scholes model represent, butone starting point is put-callparity, discussed in Chapter15

If the underlying contractis a non-dividend-payingstock,theforwardpriceis

F=S×(1+r×t)

Substituting this into theput-call parity relationshipgivesus

Inourexamplesthusfar,wehaveusedsimpleinterest.If,instead,weusecontinuousinterest, rather than dividingby1+r×t,wecanmultiply

bye–rt.Thisgivesus

C–P=S–Xe–rt

Becauseaputcanneverbeworth less than 0, we knowfrom Chapter 16 that thelower arbitrage boundary fora European call option onstockisthegreaterofeither0or

S–Xe–rt

This expression lookssimilar to the Black-Scholesvalue for a call option, butwithout the terms N(d1) andN(d2)attachedtoSandXe–rt,respectively. What do N(d1)andN(d2)represent?

In Chapter 5, weproposed a very simplemethodforevaluatingoptionsby considering a series ofunderlying prices atexpiration and assigning

probabilities to each of thoseprices. Using this approach,the expected value for a calloption is the sum of theintrinsic valuesmultiplied bythe probability associatedwitheachunderlyingprice

To determine the option’sintrinsic value, we combinedthe underlying price and

exercise price into oneexpression(Si–X).

The Black-Scholesmodel takes a slightlydifferent approach byseparating the underlyingprice and exercise price intotwo distinct components andthenaskingtwoquestions:

1. If held toexpiration, what istheaveragevalueof

all the stock abovetheexerciseprice?2. If held toexpiration, what isthe likelihood thatthe owner of anoption will end uppaying the exerciseprice?

If we can answer thesequestions, the differencebetween theaveragevalueofthe stock above the exercise

price and the likelihood ofpaying the exercise priceshould equal the option’sexpectedvalue.

To help explain theapproach takenbyBlackandScholes, let’s consider adiscrete distribution of stockprices at expiration, but onethatmorecloselyresemblesalognormal distribution withanextendedrighttail.Suchadistribution, resulting from a

total of 153 occurrences, isshown in Figure 18-1.Usingthis distribution, how mightweevaluateacalloptionwithanexercisepriceof12½?

Figure18-1

First,wemustdeterminethe value of all stock above12½, that is, the valueresultingfromalloccurrencesthat fall into troughs 13through 27. The number ofoccurrences and the value ofthe occurrences in eachtroughareasfollows:

The average value of allstockabovetheexercisepriceof12½isthetotalvalue,987,divided by the total numberofoccurrences,153

987/153=6.45

Next, we need todetermine the likelihood thatwewillpaytheexercisepriceof 12½. There are 60occurrenceswhere theoption

is in the money (the stockpriceisabove12½),butthereareatotalof153occurrences.The likelihood that we willpaytheexercisepriceis

60/153=0.392

The average payoutresultingfromexerciseoftheoption0.392×12½=4.90.

In the Black-Scholesmodel, the average value ofall stock above the exercise

price is given by SertN(d1),where Sert is the forwardprice of the stock. Theaverageamountwewillhaveto pay is given by XN(d2).Theexpectedvalueforacalloption is the differencebetweenthesetwonumbers

SertN(d1)–XN(d2)=6.45–4.90=1.55

These terms are slightly

different from the terms thatappear in the model, SN(d1)and Xe–rtN(d2), but we willshow shortly how SertN(d1)becomes SN(d1) and howXN(d2)becomesXe–rtN(d2).

We can confirm that1.55isthecorrectvalue(withslight rounding error) byreturning to our originalapproach of adding up theintrinsic valuesmultiplied by

their probabilities (thenumber of occurrencesdividedby153).

This is essentially theapproach taken by the Blackand Scholes. The primarydifference is that the Black-Scholes model, rather thanusing discrete outcomes aswedid,assumesacontinuouslognormaldistribution.

n(x)andN(x)

Before continuing, it will

be useful to define twoimportant probabilityfunctions—n(x) and N(x). Inthis chapter and in previousdiscussions of volatility, wehave often referred to theconcept of a bell-shaped, ornormal, distribution.Depending on the mean andstandard deviation, there canbe many different normaldistributions, but n(x), thestandardnormaldistribution,isperhaps themostcommon.

It has a mean of 0 and astandard deviation of 1. Thestandard normal distribution,shown in Figure 18-2, alsohas one very usefulcharacteristic: the total areaunder the curve adds up toexactly 1. That is, the curverepresents 100 percent of alloccurrences that form a truenormaldistribution.

Figure18-2n(x)—thestandardnormaldistributioncurvewithmean=0andstandarddeviation=1.

Although the standardnormal distribution takes in100 percent of alloccurrences,wemaywant toknow what percent of theoccurrences fall within aspecific portion of thestandard normal distribution.This is given by N(x), thestandard cumulative normaldistribution function. If x issome number of standarddeviations, N(x) returns the

probability of getting anoccurrence less than x bycalculatingtheareaunderthestandard normal distributioncurvebetweenthevaluesof–∞ and x, as shown in Figure18-3. That is, N(x) tells uswhat percentage of allpossible occurrences fallbetween–∞andx.Obviously,N(+∞)must be 1.00 because100percentofalloccurrencesmustfallbetween–∞and+∞.AndN(–∞)mustbe0because

there can be no occurrencestotheleftof–∞.Becausethenormal distribution curve issymmetrical,with 50 percentof the occurrences falling tothe left of 0 and 50 percentfallingtotheright,N(0)mustequal 0.50. It also followsthat the area under the curvebetween –∞ and x must beequal to the area under thecurve between –x and +∞,resulting in this usefulrelationship

Figure18-3N(x)—theareaunderthestandardnormaldistributioncurvebetween–∞andx.

N(x)=1–N(–x)

The Black-Scholesmodel makes all calculationsusing the probabilitiesassociated with a normaldistribution. This may seeminconsistent with ourassumption that the prices ofan underlying contract arelognormally distributedbecauseanormaldistributionand a lognormal distribution

are clearly not the same.However, by making someadjustmentstothevalueofx,we can useN(x) to generateprobabilitiesassociatedwithalognormaldistribution.

It will also be useful todefine three numbers used todescribe many commondistributions:

Mode. The peak of thedistribution. The point atwhich thegreatestnumberof

occurrencestakeplace.

Mean. The balance pointof the distribution. The pointatwhichhalfthevalueoftheoccurrences fall to the leftandhalftotheright.

Median. The point atwhich half the occurrencesfall to the leftandhalf to theright.

In a perfect normal

distribution, all these pointsfallinthesameplace,exactlyin the middle of thedistribution.But consider thedistribution in Figure 18-1.Themode,mean,andmedianof this distribution all fall atdifferent points, as shown infigureFigure18-4.Themodeis approximately 9.3, themean is approximately 12.7,and the median isapproximately10.5.Tomaketheappropriateadjustmentsto

a lognormal distribution sothat we can use theprobabilitiesassociatedwithanormal distribution, we mustlocatethesenumbers.

Figure18-4

The Black-Scholesmodelbeginsbydefining therelationship between theexercise price and theunderlyingprice. Inanormaldistribution,thisissimplyS–X, but in a lognormaldistribution, the relationshipis

If S > X, this value ispositive,andthecallisinthemoney; ifS<X, thevalue isnegative,andthecallisoutofthemoney.

Next, because optionsare valued off the forwardpriceandtheforwardpriceisa function of interest rates,we must adjust thisrelationship by the interestcomponentoverthelifeoftheoptionrt.Thisgivesus2

The number of standarddeviations associatedwith anoccurrence depends on howfartheoccurrenceisfromthemeanof thedistribution.Inanormaldistribution,themean,like the mode, is located inthe exact center of thedistribution.ButinFigure18-4, which approximates a

lognormal distribution, withitselongatedrighttail,wecansee that the mean must besomewheretotherightofthemode. How far to the right?Thisdependson thestandarddeviation of the lognormaldistribution. The higher thestandarddeviation,thelongerthe right tail, andconsequently, the further tothe right we must shift themean. Mathematically, the

shiftisequaltoσ2t/2.Addingthisadjustmentgivesus

Combiningtheinterest-rateand volatility componentsgivesusthenumeratorford1

Finally,wemustconvertthisvaluetosomenumberofstandard deviations. If weknow the value of onestandard deviation, we candivide by this value todeterminethetotalnumberofstandard deviations. In fact,we know that over any timeperiod t, one standarddeviation isequal to . Ifwe divide by this value, theresult,d1,tellsus,instandard

deviations, how far theexercise price is from themean when adjusted for alognormaldistribution

IntheequationshowninFigure 18-5, the calculationof d1 may seem somewhatcomplicated, but it is reallyjust a collection ofadjustments to the exerciseprice and underlying pricethat enable us to use acumulative normal

distribution function tocalculateprobabilities.

Figure18-5

Once we have determinedthe value of d1, multiplyingtheforwardpriceofthestockbyN(d1)givesustheaveragevalue of all stock above theexercisepriceatexpiration.

Having calculated theaverage value of all stockabove the exercise price, westill need to determine thelikelihoodthattheoptionwillbe exercised. To do this, we

need the median of thedistribution, the point thatexactly bisects the totalnumber of occurrences. InFigure 18-4, we can see thatthe median in a lognormaldistribution falls somewhereto the left of themean.Howfar to the left? In fact, themedian falls to the left by

ThevalueN(d2)usesthemedian to calculate theprobability of the optionbeing in the money atexpirationandthereforebeingexercised. Multiplying thisprobability by the exerciseprice gives us the averageamount we will pay atexpiration if we own theoption

XN(d2)

Takingtheaveragevalueofthestockwewillreceiveatexpirationandsubtractingtheaverage amount we will payat expiration gives us theexpectedvalueforthecall

SertN(d1)–XN(d2)

There is still one finalstep in calculating thetheoretical value of a calloption, and this stepexplains

how the terms S–rtN(d1) andXN(d2) become SN(d1) andXe–rtN(d2), which is the waythey appear in the Black-Scholes model. TheexpressionSertN(d1)–XN(d2)represents theexpectedvalueof theoptionatexpiration. Ifwe must pay for the optiontoday,thetheoreticalvalueisthe present value of theexpected value. Multiplyingthe expected value by e–rt

yieldsthefamiliarformoftheBlack-Scholesmodel

C=[SertN(d1)–XN(d2)]e–rt=SN(d1)–Xe–rtN(d2)

In the original Black-Scholes model, theunderlying contract wasassumed to be a non-dividend-paying stock.However, since itsintroduction, the model has

been extended to evaluateoptions on other types ofunderlying instruments. Thisis most commonly done byincluding an adjustmentfactorbthatvariesdependingon the type of underlyinginstrumentand thesettlementprocedurefortheoptions.Ifris the domestic interest rateand rf is the foreign interestrate,then

The complete Black-Scholes model, withvariationsandsensitivities, isgiveninFigure18-6.

Figure18-6TheBlack-Scholesmodel.

AUsefulApproximation

A trader might wonderwhether it is possible tocalculate a Black-Scholesvalue without using acomputer. In general, theanswer is no; thecomputations are just toocomplex. However, there is

one type of approximationthatmany traders are able tomake without too muchdifficulty.

Supposethatanoptionisexactlyat themoney (X =S)and that there is one year toexpiration (t = 1). Supposealso that the interest rate is0(r=0)andthatvolatilityis1percent (σ = 0.01). Thismeans that ln(S/X) = 0 andthat =0.01.Calculating

d1andd2,weget

If we calculate N(d1) andN(d2),wefindthat

N(d1)=0.501995andN(d2)=0.498005

Because the interest rate is0,thevalueofthecalloptionmustbe

(S×0.501995)–(X×0.498005)

IfX = S, the value of thecallis

X×(0.501995–0.498005)=X×0.003990

What does this numbertell us? For a one-yearEuropean option that isexactly at the forward (i.e.,the forward price is equal to

the exercise price), for eachpercentagepointofvolatility,the expected value for theoptionisequaltotheexerciseprice multiplied by 0.00399.If the exercise price is 100,theexpectedvalueis0.00399× 100 = 0.399 for eachpercentagepointinvolatility.

Why doesn’t this valuechange as we increasevolatility? Although the firstpercentage point of volatility

may be worth 0.00399,perhaps the secondpercentage point is wortheither more or less than0.00399. But recall fromChapter9 that thevegaofanat-the-money option isrelatively constant withrespect to changes involatility. Therefore, at avolatility of 20 percent, thevalueofa100callshouldbe

20×100×0.00399=7.98

At a volatility of 35percent,thevalueshouldbe

35×100×0.00399=13.965

We also know that thetheoreticalvalueofanat-the-forwardoptionisproportionalto its exercise price. If thevalue of a one-year 100 callatavolatilityof20percentis7.98, under the sameconditions,thevalueofanat-the-forward50callshouldbe

20×50×0.00399=3.99

andthevalueofa125callshouldbe

20×125×0.00399=9.975

We can further refineourapproximation ifwenotethatanat-the-moneyoptionismade up entirely of timevalueand that the timevalueofanoptionisproportionaltothe square root of time. If a

one-year 100 call is worth7.98 at a volatility of 20percent, the same call withsixmonths to expiration (t =0.5)mustbeworth

Lastly, this is anapproximation for theexpectedvalue.Todeterminethetheoreticalvalue,wemustdiscountbyinteresttogetthepresent value. Putting

everything together, for anexactly at-the-forwardEuropean option, theexpected value at expirationisapproximately3

andthetheoreticalvalueis4

Thisapproximationapplies

tobothcallsandputsbecauseunder put-call parity, anexactly at-the-forwardEuropean call and put musthavethesamevalue.

Forexample,ifvolatilityis 18 percent, what is theexpected value of a three-month(t =¼) at-the-forwardoptionwith an exercise priceof65?

If interest rates are 4percent, the option’stheoretical value isapproximately

Although this is acommonly usedapproximation, it is only anapproximation. As weincrease time and volatility,the approximation will

actually be slightly greaterthan the true Black-Scholesvalue. This is because thevega of an at-the-moneyoptiondeclinesslightlyasweincrease volatility, and thisdecline is magnified withgreater time to expiration.ThiscanbeseeninFigure9-14: the vega of an at-the-money option, althoughrelatively constant withrespect to changes involatility,doesinfactdecline

slightly with increasingvolatility. If, in our example,we raise the volatility to 40percentand increase the timetoexpirationtotwoyears,theapproximation for theexpectedvalueis

while the actual Black-Scholes expected value is14.48.

The reader who is

familiar with thecharacteristics of a standardnormal distribution mayalready have recognized thesignificance of the value0.00399. Referring to Figure18-2, for a standard normaldistributionwithameanof0and standard deviation of 1,the peak of the distributionhas a value of approximately0.399 (more exactly,0.398942). Because avolatility of 1 percent

represents1/100ofastandarddeviation, the value from themodel is 0.399/100 =0.00399.

TheDelta

In the Black-Scholesmodel, thedeltaof anoptionis equal to N(d1). When wedefined the delta in Chapter7,wesuggestedthatthedelta

is approximately theprobabilitythatanoptionwillfinish in the money. But wenow know that the trueprobabilitythatanoptionwillfinish in the money is equaltoN(d2).AlthoughN(d1)andN(d2) are often very close invalue, especially for short-termoptions,N(d1)(thedelta)isalwayslargerthanN(d2).

For a call option that isat the forward, the deltawill

be greater than 50, even ifonly slightly. Because weknowthat

Putdelta=calldelta–100

the delta of a put will beless than –50 in absolutevalue.Thismeans that an at-the-forwardstraddlewillhavea positive delta. If a call andput have the same exerciseprice, at what forward pricewill the delta of the call and

put be identical? This willoccurwhend1isexactly0.Astraddle will therefore beexactlydeltaneutralwhen

Solving,forS,weget

S=Xe–[r+(σ2/2)]t

Forastraddletobeexactlydelta neutral, the forward

price will be less than theexercisepricebyafactorof

e–[r+σ2/2)]t

As time or volatilityincreases,theforwardpriceatwhich the straddle is deltaneutral drops further andfurther below the exerciseprice—the call goes furtheroutofthemoney,andtheputgoes further into the money.With a 0 interest rate, the

underlying price at which a100 straddle will be exactlydelta neutral is shown inFigure 18-7. At very lowvolatilities, the delta-neutralprice is close to 100. But, atveryhighvolatilitiesandwithincreasing time to expiration,thedelta-neutralprice iswellbelow100.

Figure18-7Theunderlyingpriceatwhichastraddleisexactlydelta-neutral.

TheTheta

Of all the sensitivitiesderived from the Black-Scholes model, the formulaforthetaisprobablythemostcomplex. Depending on theunderlyinginstrumentandtheoption settlement procedure,the passage of time affectsoption values in threedifferentways. First, there is

a decay in the option’svolatility value—as timepasses, the distribution ofpossible prices at expirationbecomesmorerestricted.Thisis represented by the firstterminthethetaformula

Second, for an underlyingcontract such as stock, thespotpriceisassumedtomove

toward the forward price astime passes. This isrepresented by the secondterminthethetaformula

(b–r)Se(b–r)tN(d1

Finally, the present valueoftheoption’sexpectedvalueat expiration is changing astime passes. This appears intheformulaas

rXe–rtN(d2)

We know from put-callparitythatthevolatilityvaluefor a call and put withidentical contractspecifications must be thesame. The sign of the firstcomponent, the decay involatility value, mustthereforebethesameforcallsandputs.Theothertwothetacomponents depend on the

effects of interest rates andmay be either positive ornegative depending on thesettlement procedure andwhethertheoptionisacalloraput.

The decay in volatilityvalue is almost always moreimportant than interestconsiderations and will tendto dominate the thetacalculation. If interest ratesare0or ifoptionson futures

are subject to futures-typesettlement, the second andthirdcomponents in the thetaformula will be 0, leavingonly the volatility decaycomponent. In this case, thevolatility decay component,sometimes referred to as thedriftlesstheta,willbethesolefactorthatdetermineshowanoption’s theoretical valuechangesastimepasses.

MaximumGamma,Theta,andVega

InChapter7,wesuggestedthat an option has itsmaximum gamma, theta, andvegawhenitisexactlyatthemoney. But, just as we tendto assign a delta of 50 to anat-the-money option, this isonly an approximation.Where does the maximumgamma,theta,andvegareally

occur?Without going into the

mathematical derivation, wecan summarize the criticalunderlying prices S asfollows:

If b = 0, the maximum

gammaandthetawilloccuratan underlying price that ishigherthantheexerciseprice,and the maximum vega willoccur at an underlying pricethatislowerthantheexerciseprice. Moreover, themaximum gamma and thetawill occur at the sameunderlying price. This isshown in Figure 18-8 for aone-year option with anexercise price of 100. If weraiseinterestrates(b>0),the

underlyingpriceatwhichthemaximum gamma and vegaoccur will fall, and theunderlying price where themaximum theta occurs willrise. This is shown in Figure18-9.

Figure18-8Ataninterestrateofzero,theunderlyingpriceatwhichthemaximumgamma,theta,andvegaoccur.*

*We can also relate thecritical underlying prices tothe higher-order riskmeasures. If we ignore theextremes,wheretheoptioniseither very deeply in themoney or very far out of themoney,themaximumgammawill occur when the option’sspeed is 0. The maximumtheta will occur when theoption’s charm is 0. Themaximum vega will occur

whentheoption’svannais0.

Figure18-9Ataninterestrateof4percent,theunderlyingpriceatwhichthemaximumgamma,theta,andvegawilloccur.

We might also considerwhatwillhappen to thevegaof an option as we changetime. The answer may seemobvious because wepreviously made theassumption that the vegaalways increaseswith time—long-term options are moresensitive to a change involatility than short-termoptions. But this is true onlyif the underlying price is

equaltotheforwardprice,asit is assumed to be whenevaluatingoptionsonfutures.Ifweevaluateastockoption,theforwardprice forstock isa function of both time andinterest rates. If interest ratesare greater than 0, andassumingnodividends,asweincrease time, the forwardprice will increase, causingthe option to become eithermore or less at the forward.Because an at-the-forward

option tends to have thehighest vega, changing timecan cause the vega of anoption to either rise or fall.This means that under someconditions, it is possible forthevegaof a stockoption todecline if we increase toexpiration. We can see thiseffectinFigure18-10.

Figure18-10Vegaastimeandinterestchange.

With an underlyingstockpriceof100,avolatilityof 20 percent, and interestrate of 0, the vega of a 100call always increases as weincrease time to expiration.Butasweraiseinterestrates,thereissomepointintimeatwhich the opposite occurs—the option’s vega begins todeclineasweincreasetimetoexpiration.Atan interest rateof 10 percent, this occurs if

there are more than 33months remaining toexpiration.Atan interest rateof20percent,thecriticaltimeis 10 months remaining toexpiration.

We can also see wherethese critical points are bylookingatagraphofthevegadecay,asshowninFigure18-11. At an interest rate of 0,the vega decay is alwayspositive.Ataninterestrateof

10percent, thevegadecay ispositive with less than 33months to expiration butnegative with more than 33months. And at an interestrate of 20 percent, the vegadecay is positive with lessthan 10months to expirationand negative with more than10months.

Figure18-11Vegadecayastimeandinterestchange.

1TheBlack-ScholesmodelissometimesreferredtoastheBlack-Scholes-MertonmodelbecauseRobertMerton,originallyassociatedwiththeMassachusettsInstituteofTechnology,contributedsignificantlytothetheoryofoptionpricing.MertonandScholeswerejointlyawardedtheNobelPrizeinEconomicsin1997fortheirworkonoptionpricing.FischerBlack,sadly,diedin1995.2.WecouldinfactdroprtandatthesametimereplaceSwithitsforwardpriceSert.Thevaluesarethesame:ln(S/X)+rt=ln(Sert/X).3Tofurthersimplifythisapproximation,manytradersround

.00399to.004.Thisleadstowhatissometimesreferredtoasthe40%rule:theexpectedvalueofanat-the-forwardoptionisequaltoapproximately40%ofonestandarddeviation,whereonestandarddeviationisequaltoF×σ√t.4Foramoreexactcalculation,1+r×tcanbereplacedbyert.

19

BinomialOptionPricing

The Black-Scholes modelisthemostwidelyusedofalltheoretical option pricingmodels. Unfortunately, a fullunderstanding of the modelrequiressomefamiliaritywith

advancedmathematics.Inthelate 1970s, three professors,John Cox of theMassachusetts Institute ofTechnology,StephenRossofYale University, and MarkRubinstein of the Universityof California at Berkeley,were trying to develop amethod of explaining basicoption pricing theory to theirstudents without usingadvanced mathematics. Themethod they proposed,

binomial option pricing,1 isnot only relatively easy tounderstand, but the binomialmodel (also known as theCox-Ross-Rubinstein model)that resulted from thisapproachcanbeusedtopricesome options (primarilyAmerican options) thatcannot be priced using theBlack-Scholesmodel.

ARisk-NeutralWorld

Consider a security that iscurrently trading at 100 andthat, on some day in thefuture,cantakeononeoftwoprices,120and90.Assumingthat there are no interest ordividend considerations,would you rather buy or sellthis security at today’s priceof100?

Instinctively, it seemsthatonewouldratherbelongthissecurityatapriceof100than short the security at thesame price. After all, thesecurity can go up 20 butdownonly10.

The decision to go longis probably based on theassumptionthatthelikelihoodofthepricerisingandfallingis the same, 50 percent. Butwhy should the probabilitiesbe the same? Perhaps theprobability of movement inone direction is greater thanthe probability of movementintheotherdirection.Indeed,there should be someprobability of upward

movement p and downwardmovement1–p such thataninvestorwillbeindifferentasto whether he buys or sellsthe security. For an investorto be indifferent, the totalexpectedvaluemustbeequaltothecurrentpriceof100

p×120+(1–p)×90=100

Solvingforp,weget

120p+90–90p=100>>

30p=10>>p=⅓

Wecanconfirmthatthisiscorrect by doing thearithmetic

⅓×120+⅔×90=40+60=100

If S is the currentsecurity price, we cangeneralize this approach bydefining u and d asmultipliers that represent the

magnitudes of the upwardand downward moves. Thisresults in a one-periodbinomialtree:

Inarisk-neutralworld,

pSu+(1–p)Sd=S

Solvingforp,

p(Su)+(1–p)Sd=S>>pu+d–pd=1>>p=(1–d)/(u–

d)

In our original example, uand d were 1.20 and 0.90,respectively,withpequalto

Whatshouldpand1–pbe for a non-dividend-payingstock? For an investor to beindifferent to buying orselling, the risk neutralprobabilities must yield avalue that is equal to theforward price for the stockS(1+r×t).Therefore,

ValuinganOption

Suppose that we want tovalue an option using a one-period binomial tree. Weknow at expiration that anoption is worth exactly itsintrinsicvalue, themaximumof[S–X,0]foracallandthemaximumof [X –S, 0] for a

put. Inaone-periodbinomialtree, the expected value of acallis

p×max[Su–X,0]+(1–p)×max[Sd–X,0]

Thetheoreticalvalueofthecallisthepresentvalueoftheexpectedvalue

Using the same reasoning,the theoretical value of theputis

Suppose thatwe expandour binomial tree to twoperiodseachoflengtht/2andalsomaketheassumptionthatu and d are multiplicativeinverses.Then

d=1/u>>u=1/d>>ud=du=1

This means that an upmove followed by a downmove or a down movefollowed by an up moveresults in the same price. Ifthemagnitudesof theupanddownmovesu andd are thesame at every branch in ourtree, then in a risk-neutralworld, the probability of anupwardmovewillalwaysbe

and the probability of adownmovewill alwaysbe1–p.

There are now threepossible prices for theunderlying at expiration—Suu,Sud,andSdd.Thereisonlyonepaththatwillleadtoeither Suu or Sdd. But thereare two possible paths to themiddle price Sud. Theunderlying can go up andthen downor down and thenup.Thetheoreticalvalueofacall in the two-periodexampleis

Thevalueofaputis

Using this approach,wecanexpandourbinomial treetoanynumberofperiods.

If

n = number ofperiods in thebinomialtreet = time toexpiration inyearsr = annualinterestrate

the possible terminalunderlyingpricesare

Sujd(n–j)forj=0,1,2,…,n

The number of paths thatwill lead to each terminalpriceisgivenbythebinomialexpansion2

The values of a Europeancallandputare

AThree-PeriodExampleSupposethat

n=3

S=100t = 9 months(0.75year)r = 4 percent(0.04)u=1.05d = 1/u ≈0.9524

Thenthevaluesofpand1–pare

The complete three-periodbinomial tree is shown inFigure19-1.3

Figure19-1Athree-periodbinomialtree.

Using the three-periodbinomialtree,whatshouldbethevalueof a100 call and a100put?

Thevalueofthe100callis

Thevalueofthe100putis

If thevalues for the100call and put are correct, theyshouldbeconsistentwithput-callparity

We can check this by first

calculating the forward priceforthestock.Becausewearecompounding interest overthree time periods, theforwardpriceis

F=100×(1+0.75×0.04/3)3=100×1.0303=

103.03

Then

whichis indeedequal toC–P

5.22–2.28=2.94

BinomialNotationWhen constructing a

binomial tree, it iscustomaryto denote each price in thetreeasSi, j,where i, j=0,1,2, . . . , n. The value of ilocates S along the tree

movingfromlefttoright.Thevalue of j locates S movingfrom bottom to top. A five-period binomial tree usingthis notation is shown inFigure19-2.

Figure19-2Binomialnotationforafive-periodbinomialtree.

Instead of filling in abinomial tree with theunderlyingpricesSi, jateachnode, we can instead fill inthe tree with option values,eitherCi,jforcallsorPi,j forputs. Figure 19-3 shows thevalue of a 100 call at eachnode along the binomial treein Figure 19-1. The terminalvalues C3,j are simply themaximumof eitherS3j – 100

or 0. For S3,3 = 115.76, thevalueofthecallC3,3 isequalto 115.76 – 100= 15.76; forS3,2=105.00,thevalueofthecallC3,2 is equal to105.00–100= 5.00. ForS3,1 = 95.24andS3,0=86.38,the100callis out of themoney, so bothC3,1andC3,0are0.

Figure19-3Acallvalueatanypointalongthebinomialtree.

It’s obvious what thevalue of the 100 call is atexpiration, either intrinsicvalue or 0. But what shouldbe the value of the call atother nodes along the tree?To determine these values,wecanworkbackwardsfromthe terminal values using theprobabilities of upward anddownward moves anddiscounting by interest todetermine the present value.

For example, what is thevalueofC2,2?Weknow thatthere is a 59 percent chancethat at S2,2 the stock willmove up in price, in whichcase theoptionwillbeworth15.76. We also know thatthere is a 41 percent chancethatthestockwillmovedownin price, in which case theoption will be worth 5.00.The expected value of theoption at C2,2 is therefore

know that there is a 41percent chance that the stockwill move down in price, inwhichcasetheoptionwillbeworth 5.00. The expectedvalue of the option atC2,2 istherefore

(0.59×15.76)+(0.41×5.00)=11.35

Thetheoreticalvalueoftheoption at C2,2 is the present

valueof11.35

Using the same reasoning,the theoretical value of theoptionatC2,1is

The value of the option atC2,0mustbe0becauseeither

an upward or downwardmove results in a value of 0.Wecanexpressthevalueofacall at any point along thebinomialtreeas

Working backwardsalong the tree, we comefinally to C0,0, the option’sinitial theoretical value. Ofcourse, we already know

fromourpreviouscalculationthatthisvalueis5.22,sowhygo through the process ofcalculating the call value atevery point along thebinomialtree?Thereasonforcalculatingtheseintermediatevalues is that they not onlyenable us to determine someof the risk sensitivitiesassociated with the option,butalso,aswewillsee later,theyenableustocalculatethevalueofanAmericanoption.

TheDelta

We know the initial valueof the 100 call, 5.22. Butwhat is the option’s delta atC0,0?Thedelta is thechangein the option’s value withrespect to movement in theprice of the underlyingcontract.Wecanexpress thisasafraction

As we move fromC0,0 toeitherC1,1orC1,0, theoptionwillgoupinvalueto7.75ordowninvalueto1.71.Atthesame time, the stock willmoveupinpriceto105.00ordown in price to 95.24. Thedeltaistherefore

Using the whole-numberformat,theinitialdeltaofthe100callis62.

We can calculate thedeltaateverypointalongthebinomial treebydividing thechange in the option’s valueby the change in theunderlyingprice

Figure 19-4 shows thestock price, the value of the100call, and thedeltaof thecall at every node along thebinomialtree.

Figure19-4Deltaofanoptionusingabinomialtree.

InChapter8,weshowedthat the dynamic hedgingprocessenablesus tocapturethe difference between anoption’s value and its price.We can see this sameprincipal at work in thebinomialmodel.ReturningtoFigure19-4, suppose thatwebuy the 100 call at itstheoretical value of 5.22 andcreate a delta-neutral hedge(Δ=62)byselling62percent

of an underlying stockcontract at a price of 100.Whatwillbe the result ifweholdthepositionforonetimeperiod?

If the stockpricemovesup to 105.00, the optionwillbe worth 7.75, resulting in aprofitontheoptionof7.75–5.22 = 2.53. At the sametime,wewilllose0.62×(100– 105) = –3.10 on the stockposition, giving us a loss on

thehedgeof

+2.53–3.10=–0.57

If the stock price movesdownto95.24,theoptionwillbe worth 1.71, resulting in aloss on the option of 1.71 –5.22 = –3.51. At the sametime, we will make 0.62 ×(100 – 95.24) = 2.95 on thestock position, giving us alossonthehedgeof

–3.51+2.95=–0.56

It seems that we will losemoney, either 0.56 or 0.57,regardless of whether thestock moves up or down inprice. In fact, both numbersare the same, the differencebeingduetoaroundingerrorin our calculations (the truedelta is 61.88). But this stillleaves us with a loss whenoptionpricingtheorysaysweoughttobreakeven.

Recall that when webought the option and soldstock, the cash flow was acredittoouraccountof

–5.22+0.62×100=+56.78

Ataninterestrateoverthistime period of 1.00 percent,weareabletoearninterestonthiscreditof

0.01×56.78≈+0.57

Including this in ourcalculations, we do in factjustbreakeven.

If we go through thedelta-neutral rehedgingprocess at every node in thetree,takingintoconsiderationthevalueofthehedgeaswellasanyinterestconsiderations,regardless of the path thestock follows, at expiration,wewillbreakexactlyeven.Ittherefore follows that if we

areabletobuyanoptionataprice less than theoreticalvalue or sell an option at aprice greater than theoreticalvalue, we will show a profitat expiration equal to thedifference between the priceatwhichwetradedtheoptionanditstheoreticalvalue.Thisis the principle of dynamichedgingdescribed inChapter8.

TheGamma

Thegammaofanoptionisthe change in the option’sdelta with respect tomovement in thepriceof theunderlying contract. As wedid with the delta, we canexpress the gamma as afraction

InFigure19-4,wecanseethataswemovefromC0,0 toeither C1,1 or C1,0, theoption’s delta will either goupto81ordownto31.Atthesame time, the stock willeither move up to 105.00 ormove down to 95.24. Thegammaistherefore

The initial gamma of the100callis5.1.

We can calculate thegammaatanypointalongthebinomial treebydividing thechange in the option’s deltaby the change in theunderlyingprice

TheTheta

The theta is the change inan option’s value as timepasses, assuming everythingelse,includingtheunderlyingprice, remains unchanged. Ina binomial model, at eachtime period, the underlyingprice is assumed to moveeither up or down. Theunderlying price remainsunchanged only after two

time periods, when theunderlying price either goesupanddownordownandup.Toapproximate the theta,wemust therefore consider thechange in the option’s valueovertwotimeperiods.

In Figure 19-4, we canseethataswemovefromC0,0toC2,1, the value of the 100call drops from 5.22 to 2.92,fora loss invalueof2.30. Ifwewanttoestimatethedaily

theta, we can divide by thenumber of days during thistwo-periodtime

We can approximate thedailythetaatanypointalongthetreeas

VegaandRho

It would be convenient ifwecouldusethesamesimplearithmetic to calculate thevegaand rho thatweused tocalculate the delta, gamma,and theta. Unfortunately,there isnosimplesolution tothevolatilityand interest-ratesensitivities.Todeterminethevega, we must change thevolatility input—we will see

shortly how we determinethis input—and thenseehowthe option’s value changes.To determine the rho, wemust change the interest-rateinput.

TheValuesofuandd

We have chosen theupward move u anddownward move d so thatthey form a recombining

binomial tree. The terminalprice for the security isindependent of the order inwhichthepricemovesoccur.Whether the security movesup first and then down ordown first and then up, theresultisthesame

u×d=d×u

If the upward anddownward moves were notrecombining, the number of

calculationswouldbegreatlyincreased because each nodeon the binomial tree wouldyieldacompletelynewsetofupward and downwardvalues.

We have also chosen uand d so that they are themultiplicativeinverseofeachother

u×d=d×u=1.00

This ensures that if the

security makes an upwardmove followed by adownward move or adownwardmove followedbyan upward move, theresulting underlying pricewillbesamepriceatwhichitbegan. If u and d were notinverses, there would be adrift in the underlying price.If,forexample,uanddwerechosen to be 1.25 and 0.75,then there would be adownwarddriftbecause

u×d=1.25×0.75=0.9375

In order to calculate thetheta, as we did previously,weneedtoeliminatethedriftin the underlying price. Thiswill be true if u and d aremultiplicativeinverses.

Other than therestrictions that u and d areinverses and result in adriftlessunderlyingprice,wehave not specified exactlywhat the values of u and d

shouldbe.Itwillnotcomeasa surprise that u and d mustbederivedfromthevolatilityinput. If we want binomialvalues to approximateBlack-Scholesvalues,uanddmustbechosen in suchaway thatthe terminal pricesapproximate a lognormaldistribution. We can achievethisbydefiningu andd as aone standard deviation pricechangeovereachtimeperiodinourbinomialtree

In our three-periodexample,whatvolatilitydoesu = 1.05 represent? Todetermine this, we can workbackwards to solve for thevolatilityσ

Taking the naturallogarithmofeachside,weget

ln(1.05)=ln(e0.5σ)>>0.0488=0.5σ>>σ=0.0976(9.76%)

In our three-periodexample,weusedavolatilityof9.76percent.

GammaRent

In theory, every volatilityposition in the optionmarketrepresents a tradeoffbetweenthe cash flow created by the

dynamichedgingprocessandthe decay in the option’svalue as time passes. Apositive gamma, negativetheta position will makemoney through dynamichedging but lose moneythrough time decay. Anegative gamma, positivetheta position will performjust the opposite, losingmoney through dynamichedging but making moneythrough time decay. Traders

sometimes refer to volatilitytradingasrentingthegamma,with the rental costs beingequaltothetheta.

Over a given timeperiod,howmuchmovementis required in the underlyingcontract to offset the effectsof time decay?We can givean approximate answer bygoing back to our binomialtree. We know that a delta-neutral position taken at

theoretical value will justbreak even if the underlyingcontractmoveseitherupbyuor down by d. Themagnitudesoftheuanddareequalto

But these values are equalto a one standard deviationprice change over the timeinterval t/n. Therefore, overany interval of time, the

amount of price movementneeded in the underlyingcontract to just break evenmustbeequaltoonestandarddeviation.

The reason that this isonlyanapproximationis thatwhile u and d re-mainconstant, theta changes,sometime very rapidly, astime passes. For very shorttime intervalsorwithagreatdeal of time remaining to

expiration,thisapproximationwill be reasonably accurate.However, over longer timeintervals or with very littletime remaining to expiration,the changes in the theta willcausetheapproximationtobelessaccurate.

AmericanOptions

Let’sgobacktoour three-periodbinomialtreeinFigure

19-1. But instead ofcalculatingthevalueofa100call,aswedidinFigure19-4,let’s work backwards fromthe terminal prices tocalculate the value of a 100put. The underlying prices,theoretical values, and deltaandgammavaluesforthe100put are all shown in Figure19-5.Thereadermaywishtoconfirm that the call and putvalues in Figures 19-4 and19-5areconsistentwithbasic

principles of option pricing:at every node, put-call parityis maintained; the absolutevalues of call and put deltasalwaysaddupto100;andthecall and put gammas areidentical.

Figure19-5Thevalueofa100putatanypointalongthebinomialtree.

If we assume that the100 put is European andcannotbeexercisedearly,theonly reason to calculate theintermediate values is todetermine the delta andgamma.But suppose that the100 put is American. Mightthere be any reason toexercise the option prior toexpiration?

Lookcloselyatthevalue

of the 100 put at P2,0 inFigure 19-5. The theoreticalvalue of the put is 8.31. Butwith an underlying price of90.70 theputhasan intrinsicvalue of 9.30. If the put isAmerican, anyone holdingtheputunderthoseconditionswill choose to exercise itearly. If we are using abinomial tree to evaluate anAmerican option, we mightcompare the value of the

European option with theintrinsic value at each node.Iftheintrinsicvalueisgreaterthan the European value, wecan replace the value at thatnode with the option’sintrinsic value and thencontinue to work backwardsto determine the option’svalueateachprecedingnode.IfwereplacethevalueatP2,0with 9.30, the put value atP1,0willbe

We need to replace theEuropeanvalueof4.50atP1,0with the American value of4.90.

Finally, the initialvalue,P0,0,is

Because the delta and

gamma are calculated fromoption values at every node,these new values will affectthe calculation of the deltaand gamma for an Americanoption.Theinitialdeltaofthe100putifitisAmericanis

The delta of the European100putwas–38,butthedeltaof American 100 put is –42.

The values for an American100 put at every node areshown in Figure 19-6.Because the delta is affectedby the possibility of earlyexercise,thegammawillalsobe affected. The gamma forthe100putisnow

Figure19-6ThevalueofanAmerican100put.

ratherthanagammaof5.1fortheEuropeanoption.

Dividends

Howdoesthepossibilityofearlyexerciseaffectthevalueof a call? If we look at thecall value at every node in

Figure 19-4, we find that atno point is it less thanintrinsic value. This meansthat the European andAmericanvaluesmustbe thesame.And, indeed,weknowfrom Chapter 16 that if astockdoesnotpayadividendover the life of the option,there is never any reason toexercise an American stockoptioncallearly.

But what if the stock

doespayadividend?Supposethat the stock in Figure 19-1willpayadividendof2.00atsome point during the lasttime period. When a stockpays a dividend, its pricetypicallydropsbytheamountof the dividend.Consequently, each terminalpriceinourbinomialtreewillbe reduced by the amount ofthe dividend,4 as shown inFigure 19-7. (The terminal

values if there isnodividendare shown in parentheses.) Ifwewanttocalculatethevalueof the 100 call, we can usethese new terminal prices.Then, as before, we can usethe probabilitiesp and 1 – pto calculate the theoreticalvalueanddeltaofthe100callat each node of the binomialtree. These values are showninFigure19-8.

Figure19-7Abinomialtreewithdividendpayment

Figure19-8ThevalueofaEuropeancallonadividend-payingstock.

The value for the 100call in Figure 19-8 is aEuropean value because wenever considered thepossibility of early exercise.But lookmore closely at thevalue of the call one timeperiod prior to expirationwiththestockpriceat110.25.The theoretical value of the100 call is 9.26. But, with astockpriceof110.25,thecallhas an intrinsic value of

10.25.IfthecallisAmerican,anyoneholdingthecallunderthese conditions will choosetoexerciseitearly.AswedidwithanAmericanput,ateachnode, we can compare theEuropean value of the callwith its intrinsicvalue. If theintrinsicvalue isgreater thanthe European value, we canreplacethevalueatthatnodewith the option’s intrinsicvalue and then continue toworkbackwardstodetermine

the option’s value at eachpreceding node. The initialvalue of the call, C0,0, willthen be the value of anAmerican call. The completebinomial tree for theAmerican 100 call is showninFigure19-9.

Figure19-9ThevalueofanAmericancallonadividend-payingstock.

Ifwewanttoconstructabinomial tree for a dividend-paying stock, it might seemthatwecansimplyreduceallstock prices following thedividend payment by theamount of the dividend. InFigure 19-7, where thedividend was paid over thelast timeperiod, this reducedthe terminal prices by 2.00.Butsupposethatthedividendispaidduringthenext-to-last

time period, as shown inFigure 19-10. The stockprices at the following nodesarereducedby2.00.Butlookat what happens when wecontinue to calculate stockpricesusingu=1.05andd=0.9524.Thesubsequentstockprices do not recombine.Each node begins a newbinomial tree. In our three-periodbinomialtree,thismaynot seem like a significantproblem. We can still

calculate the value of anoption using the terminalstock prices (now there aresix terminal prices instead offour) and then workbackward to determine theoption’s theoretical value.The value for the 100 callusing our new binomial treeisshowninFigure19-11.

Figure19-10ThevalueofanAmericancallonadividend-payingstock.

Figure19-11Abinomialtreewithanearlydividendpayment.

What if there aremultiple dividend paymentsover the life of the option?Andwhatifourbinomialtreeconsists of many timeperiods? Because eachdividendpaymentgeneratesanew set of binomial prices,the number of calculationsrequired to value an optionwill be greatly increased,perhaps to thepointofbeingunwieldy. This presents a

problem towhich there isnoideal solution. Perhaps thesimplest way to handledividend payments is tocreate a complete binomialtree without dividends andthenreducethestockpriceateachnodebythetotalamountof dividends. An example ofthisisshowninFigure19-12,which represents anapproximation of the calloption value generated inFigure 19-11. Instead of

generating new binomialprices after the dividendpayment, we have simplyreducedallsubsequentvaluesby the 2.00 amount of thedividend.Wecanseethatthisisonlyanapproximation.Thecall values in Figure 19-12tendtobeslightlylargerthanthevaluesinFigure19-11.

Figure19-12ThevalueofanAmericancallonadividend-payingstock.

One final commentaboutthevaluesforpand1–p. We typically expect aprobability to fall between 0to 1.00, that is, somewherebetween “no chance” and“absolute certainty.”However, this is notnecessarilytrueforpand1–p.Consider the conditions inFigure19-11:

StockpriceS=

100Time toexpiration t =9monthsNumber ofperiodsn=3Interest rate r=4percentu=1.05d = 1/u =0.9524

Thevaluesforpand1–p

resulting from these valuesare 0.59 and 0.41,respectively.Butsupposethatwe are in a high inflationaryclimate and that instead ofsetting r equal to 4 percent,we set r equal to 40 percent.Thenewvaluesofpand1–pwillbe

Thuspand1–pnolongerlook like traditionalprobabilities:p exceeds 1.00,and1–pisnegative.Infact,p and 1 – p can fall outsidethe range for a typicalprobability. For this reason,they are sometimes referredtoaspseudoprobabilities.

What is the implicationof p being greater than 1.00and 1 –p being less than 0?Thismeans that the potential

for movement in theunderlying stock is notsufficientlylargetooffsettheinterest loss should we buythe stock. In our examplewith u = 1.05, if the stockprice always rises over eachtime period, we will show aprofit of 5 percent. But withaninterestrateof30percent,wewouldalwaysbebetteroffleaving our money in thebank and earning interestover each three-month time

periodof

0.30/4=7.5%

Of course, if we increasethe stock volatility byincreasingthevalueofu,thenthe potential profit frominvesting in thestockwillgoup. If we choose a largeenoughvalueforu,thevaluesfor p and 1 – p will indeedfall between 0 and 1.00.Because thevalueforumust

be greater than 1 + r × t/n,with an interest rate of 30percent, u must be greaterthan

1+0.3×0.75/3=1.075

As we did with theBlack-Scholesmodel,wecanuse the binomial model toevaluate options on differentunderlying instruments. Thebinomial model and itsvariations are shown in

Figure19-13.Figure19-13

How close are optionvalues generated by abinomial model to thosegenerated by the Black-Scholesmodel?Thisquestiononly makes sense forEuropeanoptionsbecausetheBlack-Scholes model cannotbeusedtoevaluateAmericanoptions. In our three-periodbinomial tree, the value of aEuropean 100 call is 5.22,and thevalueof a100put is

2.28. Using the Black-Scholesmodel,thevaluesare5.01and2.05.Bothbinomialvalues are greater than thetrue Black-Scholes values.Wecanincreasetheaccuracyof the binomial model byincreasingthenumberoftimeperiods. In a four-periodbinomial tree, the values are4.79 and 1.84. Figure 19-14showsthedifferencebetweenthe Black-Scholes andbinomial values for the 100

call as we increase thenumber of time periods from1 to 10.We can see that theerror oscillates betweenpositive and negative, withtheabsolutevalueoftheerrorbecoming smaller andsmaller.Indeed,ifwebuildatree with an infinite numberoftimeperiods,theerrorwillconverge to 0. The binomialandBlack-Scholesvalueswillbeidentical.

Figure19-14Asweincreasethenumberofperiods,thebinomialvalueconvergestotheBlack-Scholesvalue.

How many periodsshould we use in a binomialmodel?Aswedividethetimetoexpirationintosmallerandsmaller increments, weincrease the accuracy. But agreater number of periodsalso increases the number ofcalculations, and thisnumberincreases exponentially.Given the tradeoff betweenaccuracy and speed, acommon choice is often

somewhere between 50 and100periods.

The accuracy of abinomial calculation can befurther increased by takingthe average value generatedby two periods, sometimesreferred to as half-steps. Forexample, the 9-period treeovervalues the 100 call byabout 0.07 (Black-Scholesvalue – binomial value = –0.07), whereas the 10-period

tree undervalues the call byabout 0.09. If we take theaverage of the 9- and 10-period values (a 9½-periodvalue), the option isundervalued by only 0.01.The results of this averagingprocedure can be seen inFigure19-14.

1JohnC.Cox,StephenA.Ross,andMarkRubinstein,“OptionPricing:ASimplifiedApproach,”JournalofFinancialEconomics7(3):229–263,1979.2Thebinomialexpansionissometimes

writtenas3Forsimplicity,binomialtreesareoftendrawnsymmetricallyfromtoptobottom.However,thiscanbesomewhatmisleading.Ifdrawntoscale,thebranchestypicallybecomenarroweraswemovefromtoptobottom.WecanseethisinFigure19-1:115.76–105.00=10.76(thetoptwobranches);105.00

–95.24=9.76(themiddletwobranches);95.24–86.38=8.86(thebottomtwobranches).Because10.76>9.76>8.86,thebranchesmustbegettingnarrower.4Forsimplicity,weignoretheinterestthatcanbeearnedonthedividendpayment.Amoreaccuratebinomialtreeshouldalsoincludethisamount.

20

VolatilityRevisited

When a trader enters avolatility into a theoreticalpricingmodel,whatexactlyishe feeding into the model?We know the mathematicaldefinition of volatility—one

standarddeviation,inpercentterms,overaone-yearperiod.Beyondthis,westillhavethequestion of interpretation.Does the number represent arealized volatility or animplied volatility? Are wetalking about historicalvolatility or future volatility?Longtermorshortterm?Thevolatility a trader choosesmay vary depending on theanswerstothesequestions.

Considerthissituation:

Underlyingprice=100.00Time toexpiration = 8weeksInterest rate =0Impliedvolatility = 20percent

Suppose that we buy the

100 straddle at a price equalto its impliedvolatilityof20percent,inthiscase6.25.Theposition should beapproximately delta neutralbecauseboththe100callandthe100putareatthemoney.After we buy the straddle,implied volatility rises to 22percent.Howarewedoing?

We might instinctivelyassume that the position willshow a profit because the

increase in implied volatilityshould be a reflection ofrisingoptionprices.Indeed,ifthereisanimmediateincreasein implied volatility and allother conditions remainunchanged, the price of the100straddlewillriseto6.87,resultinginaprofitof

6.87–6.25=+0.62

Butsupposethatimpliedvolatility slowly rises to 22

percentoveraperiodofthreeweeks. Even though theincrease in implied volatilitywill work in our favor, thepassageoftimewillcausetheoptionstodecay.Infact,withtheunderlyingcontractstillat100.00, the straddle will beworthonly5.43,resultinginalossof

5.43–6.25=–0.82

The benefits of rising

implied volatility wereoverwhelmed by the costs oftimedecay.

Now suppose thatinstead of rising, impliedvolatility falls to 18 percent.How will this affect ourposition? If there is animmediate decline with nochanges in any other marketconditions, the price of the100straddlewill fall to5.62,leavinguswithalossof

5.62–6.25=–0.63

But suppose that asimplied volatility falls to 18percent, the underlying priceisalsochanging.Wearenowbenefiting from a positivegamma. If the underlyingprice rises immediately to105.00, the 100 straddle willbe worth 7.09, resulting in aprofitof

7.09–6.25=+0.84

If the underlying pricemoves in the other directionand falls immediately to95.00, the straddle will beworth6.87,nowresultinginaprofitof

6.87–6.25=+0.62

The disadvantages offallingimpliedvolatilityweremore than offset by thebenefits of movement in theunderlyingstockprice.

This example illustratean important principle ofoptiontrading:

Thelongeranoptionpositionisheld,themoreimportantistherealizedvolatilityoftheunderlyingcontractandthelessimportantistheimpliedvolatility.Ifapositionisheldto

expiration,realizedvolatilityistheonlyconsideration.

We saw this principle atwork in Chapter 8 ondynamic hedging. The delta-neutral adjustment processeventually determinedwhether a position wouldshow a profit or loss,irrespectiveofanychangesinimpliedvolatility.This isnotto say that implied volatility

is unimportant; prices arealways important becausethey will often determineinterimcashflowsandcapitalrequirements.But,inordertomake sensible tradingdecisions, we need to knowvalueaswell asprice. In thefinalanalysis,thevalueofanoption position will bedetermined by the volatilityoftheunderlyingcontract.

Determining the right

volatility input can be adifficult and frustratingexercise, even for anexperienced option trader.Theforecastingofdirectionalprice movements, eitherthrough fundamental ortechnical analysis, is acommonly studied area intrading, and there are manysourcestowhichatradercanturn for information on thesesubjects. Unfortunately,volatility is a much newer

concept, and there is less toguideatrader.Inspiteofthisdifficulty, an option tradermust make some effort tocome up with a reasonablevolatility input if he intendsto use a theoretical pricingmodel to make tradingdecisionsandmanagerisk.

HistoricalVolatility

Because the realized

volatility over the life of anoption will eventuallydominate any changes inimplied volatility, we willcertainly want to give somethought to how we mightpredict future realizedvolatility. Such a predictionwilloftenbeginbylookingathistoricalvolatilitydata.Howshouldwecalculatehistoricalvolatility?

We know that volatility

represents a standarddeviation. Two methods arecommonlyusedtocalculateastandarddeviation,either

Ineachcase,xiarethedatapoints, μ is the mean of alldatapoints,andn is the totalnumber of data points. Theonly difference between the

two methods is thedenominator, either n or n –1.

If we want to know thestandard deviation of anentire population of datapoints, we can use the firstmethod,dividingbyn.Thisisknown as the populationstandard deviation. Suppose,however, that we have asamplesetofdatapointsfroma larger population, and we

want to use this sample toestimate the standarddeviation of the entirepopulation. Because oursample is limited, we arelikely to miss some of themore extreme data points inthelargerpopulation.Forthisreason, our estimate of thestandard deviation for theentire population is likely tobe too low. To improve ourestimate,weoughttoincreasethe standard deviation

calculation. This iscommonly done by reducingthe size of the denominatorfromnton–1,resultinginasample standarddeviation ofthe larger population.Becausehistoricalvolatilityismostoftenusedtoestimateafuture volatility, historicalvolatility calculations aremost often made using thesample standard deviation,thatis,dividingbyn–1.

The data points xi in avolatility calculation are theprice returns, either thepercent change in theunderlying price from onetimeperiodtothenext

or, more commonly, thelogarithmicchange

Time periods may be anylength, but for exchange-traded contracts, returns areusually based on the pricechange from one day’ssettlementtothenext.

Inthestandarddeviationcalculation, μ (the Greeklettermu)istheaverageofallprice returns. Because the

volatility is the deviationfrom average, if a contractgoes up 1 percent each dayfor 10 consecutive days, itsvolatility over the 10-dayperiod is 0; the price changenever deviated from itsaverage.Tomosttraders,thisfeels wrong. The upwardmoves of 1 percent ought torepresent some volatilityother than 0. In fact, mosthistorical volatilitycalculations use a zero-mean

assumption: μ is alwaysassumedtobe0regardlessoftheactualmean.

When calculatinghistorical volatility, traderstypically exclude weekendsand holidays, resulting in atrading year of between 250and260days.Butonemightalso calculate volatility usingall 365 days, assigning a 0price change to nontradingdays. This method might be

appropriate when trying tocompare the volatilities ofproducts traded on twodifferent exchanges withdifferent trading calendars.The two methods willobviously yield slightlydifferent historicalvolatilities. But, if historicalvolatility isusedasageneralguideline to future realizedvolatility, the differences areunlikely to be significant.This can be seen in Figure

20-1,which shows the three-month volatility of theStandard and Poor’s (S&P)500 Index calculated usingonly trading days(approximately 252 days peryear)andusingall365days.1The graphs are almostindistinguishable.

Figure20-1S&P500Indexthree-monthhistoricalvolatility:2001–2010.

Although daily pricereturnsaremostoftenusedtocalculate historical volatility,wemight instead useweeklyprice returns. How will thisaffect the historical volatilitycalculation? Figure 20-2shows the three-monthvolatility of gold from 2001through2010calculatedusingboth daily and weekly pricereturns.Ingeneral,thegraphsshow similar characteristics,

althoughfluctuationsseemtobe slightly greater usingweekly returns. This isprobably due to the smallernumber of data points (13weeklydatapointsratherthan91 daily data points). Thegreaternumberofdatapointswilltendtohaveasmoothingeffect. Because the graphsshow similar characteristics,we can conclude that if acontract is volatile from dayto day, it will be equally

volatilefromweektoweekormonth to month. Dailyreturnsareusedmostofteninorder to increase the numberofdatapointsinthevolatilitycalculation and thereforeyield a more accuratevolatility.

Figure20-2Goldthree-monthhistoricalvolatility:2001–2010.

Supposethatthepriceofa contract fluctuates wildlyduring a tradingday,makingdramaticupanddownmoves,yet finishes the dayunchanged. If this is acommon occurrence, thenusing only settlement pricesto calculate the historicalvolatility may result in anincomplete picture of acontract’s true volatility. Totake into consideration

intraday price movement,several alternative methodshave been proposed tocalculatehistoricalvolatility.

The extreme-valuemethod,proposedbyMichaelParkinson,2usesthehighandlow values during a 24-hourperiod.Thismethodnotonlygivesamorecompletepictureof volatility but may also beuseful when no definitivesettlement prices are

available.Using theextreme-valuemethod, theannualizedhistorical volatility is givenby

wheren=numberofpricereturns, hi = highest priceduring the chosen time

interval, li = lowest priceduring the chosen timeinterval, ln = naturallogarithm, and t = the lengthofeachtimeintervalinyears.

An alternative approachproposed by Mark Garmanand Michael Klass3 expandstheParkinsonmethodbyalsoincluding the opening andclosing prices for anunderlying contract. Usingthis method, the annualized

historical volatility is givenby

whereoi=openingpriceatthebeginningof trading, andci=closingpriceattheendoftrading.

As with the traditionalclose-to-close estimator, both

the Parkinson and Garman-Klass estimators areannualizedbydividingbythesquare root of t, the timebetweenpriceintervals.(Thisisthesameasmultiplyingbythesquarerootofthenumberoftimeintervalsinayear.)

Figure 20-3 shows thethree-month volatility of theEuroStoxx50Index,awidelyfollowed index of largeEuropean companies. The

volatility has been calculatedusingthreemethods:close-to-close, high-low (Parkinson),and open-high, low-close(Garman-Klass).Thelasttwomethods seem to yield aconsistently lower volatilitythan the first method. Theexplanation probably has todo with the fact thatParkinson and Garman-Klassare used only when marketsare open and trading iscontinuous. But the

EuroStoxx 50 Index is notcalculated continuously. It iscalculated during a period ofjust under 10 hours, fromapproximately 9:00 a.m. to6:50 p.m. European time.During the remaining hoursof the day, the volatility ofthe index is unobservable.Becauseof this, forcontractsthat tradeonlyduringpartofthe day, Garman and Klassrecommend giving someweight to the close-to-close

estimate. One approach is togive the observable volatility(eitherParkinsonorGarman-Klass)weightproportional tothefractionofthedayduringwhichthemarketisopenandgive the remainingweight tothe close-to-close volatility.This usually means givinggreaterweighttotheclose-to-close volatility estimatebecause many markets areclosed more hours than theyare open. But the Parkinson

and Garman-Klass methodsaregenerallyconsideredmoreaccurate estimates, at leastwhen a market is tradingcontinuously. Thus, it mightmakemore sense to increasethe weightings for theseestimates and reduce theweightings for the close-to-close estimate. Garman andKlass propose a preciseformula for weighting theestimates, but a practicalsolution might be to simply

weighttheestimatesequally.Figure20-3EuroStoxx50Index

three-monthhistoricalvolatility:2001–2010.

Because we have goneinto the calculation ofhistorical volatility in somedetail, the reader may havebeen leftwith the impressionthat the method chosen willbe an important determinantofwhetheranoptionstrategyissuccessfulornot.Formosttraders, though, historicalvolatility is simply aguidelinetowhatthetraderisreally interested in—the

future realized volatility.Because the results of eachmethod are unlikely to differsignificantly, in practice, itprobablydoesnotmakemuchdifference which method ischosen. It is far moreimportant to be able tointerpret historical volatilitydata rather than to worryabouttheexactmethodused.

SomeVolatilityCharacteristicsIn Chapter 6, we used the

analogy that the volatility inits different interpretations—historical, future, implied—issimilar to the weather. Thevolatility-weather analogycanalsohelpusidentifysomebasic volatilitycharacteristics.

Suppose that we are

tryingtoestimatetomorrow’shigh temperature, and wehave only one piece ofinformation, today’s hightemperature.Whatisourbestestimate? Becausetemperatures do not usuallychangedramaticallyfromoneday to the next, our bestestimate of tomorrow’s hightemperature is probably thesame as today’s hightemperature. Temperaturereadings are said tobe serial

correlated. In the absence ofother information, the bestguessaboutwhatwillhappenover the next time period iswhat happened over the lasttime period.Volatility seemsto exhibit this serialcorrelation characteristic.What will happen in thefutureoftendependsonwhathappenedinthepast.

Now suppose that weknow not only today’s high

temperaturebutwealsoknowthe average high temperatureatthistimeofyear.Iftoday’shigh temperature is higherthan the average, anintelligent estimate fortomorrow’s high probablywill be lower than today’shigh. If today’s hightemperature is lower than theaverage, an intelligentestimate for tomorrow’s highwill be higher than today’shigh. We know that

temperaturestendtobemeanreverting. Volatility alsoseems to exhibit thischaracteristic. There is agreater likelihood thatvolatility, like temperature,will move toward the meanratherthanawayfromit.

We can see the mean-reverting characteristic ofvolatility if we compareFigure 20-2, the three-monthvolatilityofgold,withFigure

20-4, the price of gold overthesameperiod.4Bothpricesand volatility sometimes riseand sometimes fall. Butunlike the price of anunderlying contract, whichcanmoveinonedirectionforlong periods of time, thereseems to be an equilibriumnumber to which volatilitytends to return.Over the 10-year period in question, thepriceofgoldrosefromunder

$300 per ounce to over$1,400 per ounce. Althoughprices fluctuated, they neveragain reached the lows of2001.Ontheotherhand,goldvolatility,inspiteofdramaticfluctuationsbetweenalowof9 percent and a high of over40percent,alwaysseemedtoreturneventually to the10 to20percentrange.

Figure20-4Goldfuturesprices:2001–2010.

WemightconcludefromFigure20-2thatgoldtendstoexhibit a long-term averageor mean volatility. Whenvolatility rises above themean, one can be fairlycertain that it will eventuallyfall back to its mean. Whenvolatility falls below themean, one can be fairlycertain that it will eventuallyrise to its mean. There is aconstant gyration back and

forththroughthismean.Mean reversion is a

common volatilitycharacteristic of almost alltraded underlying contracts.Figures 20-1 and 20-5 showthe three-month historicalvolatility,usingdaily returns,for the S&P 500 Index andBund futures from 2001 to2010.Inspiteofthedramaticfluctuations, both the S&P500 Index and Bund futures

tend to exhibit a meanvolatility to which bothcontractstendtoreturn.Inthecase of the S&P 500 Index,this seems to be somewherebetween15and20percent.InthecaseoftheBund,amuchless volatile contract, themean volatility seems to bearound5percent.

Figure20-5Bundfuturesthree-monthhistoricalvolatility:2001–2010.

In Figures 20-6 through20-8,wecanseemoreclearlythe mean-revertingcharacteristic of volatility.These graphs show theminimum, maximum, andaverage realized volatilitiesfor the S&P 500 Index, goldfutures, and Bund futuresfrom2001 to2010over timeperiodsrangingfrom2to300weeks. For example, inFigure 20-6, if we consider

every possible two-weekperiodfrom2001to2010,wecan see that the minimumtwo-week volatility for theS&P 500 Index wasapproximately 5 percent,while the maximum two-week volatilitywas just over100 percent. The averagetwo-week volatility wasapproximately 18 percent.For every possible 300-weekperiod, the minimumvolatility for the S&P 500

Index was approximately 14percent, the maximumvolatilitywas24percent,andthe average volatility wasapproximately 19 percent.The graphs for the goldfutures (Figure 20-7) and forBund futures (Figure 20-8)show the same generalcharacteristics. As weincrease the length of timeover which the volatility iscalculated, the results tend toconverge to an average or

meanvolatility.Figure20-6S&P500Index

historicalrealizedvolatilitybytimeperiod:2001–2010.

Figure20-7Goldfutureshistoricalrealizedvolatilitybytimeperiod:2001–2010.

Figure20-8Bundfutureshistoricalrealizedvolatilitybytimeperiod:2001–2010.

Graphs similar to thoseinFigures20-6 through 20-8areoftenusedtoillustratetheterm structure of volatility—the likelihood of volatilityfalling within a given rangeover a specified period oftime. The term-structuregraph typically has a conicshape,withgreatervariationsovershortperiodsoftimeandsmaller variations over longperiods.5Becauseoftheterm

structure of volatility, it isoften easier to predict long-termvolatilitythanshort-termvolatility. This may seemcounterintuitive because wetend to expect greatervariability over long periodsof time than over shortperiods. However, volatilitycan be thought of as anaveragevariability.Overlongperiodsoftime,thelargeandsmall price fluctuations tendtooffseteachother, resulting

inmorestableresults.Because long-term

volatility tends to be morestable than short-termvolatility, one might assumethatitiseasiertovaluelong-term options than short-termoptions.Thiswouldbetrueifall options were equallysensitive to changes involatility. But we know thatlong-term options havegreater vega values than

short-term options—they aremore sensitive to changes involatility. This means thatany volatility error will begreatly magnified whenevaluating a long-termoption. Depending on thetime to expiration, the effectof a two or three percentagepoint volatility error on along-term option may begreater than a five or sixpercentage point error on ashort-termoption.

What else can we sayabout volatility? Lookingagain at Figure 20-2, wemight surmise that volatilityhas some trendingcharacteristics. From early2004 through the middle of2005, there was a persistentdownward trend in goldvolatility. This was followedby a more dramatic upwardtrendfromthemiddleof2005to the middle of 2006. Andfromearly2007throughmost

of2008,thereseemedtobeastepping-stone increase involatilitytoahighofover40percent. Within these majortrends, therewerealsominortrends as volatility rose andfell for shortperiodsof time.In this respect, volatilitycharts seem to display someof thesamecharacteristicsaspricecharts,anditwouldnotbe unreasonable to applysome of the same principlesused in technical analysis to

volatility analysis. It isimportant to remember,however, that although pricechanges and volatility arerelated,theyarenotthesamething.Ifatradertriestoapplyexactly the same rules oftechnicalanalysistovolatilityanalysis, he is likely to findthat in some cases the ruleshavenorelevanceandthatinother cases the rulesmustbemodifiedtotakeintoaccountthe unique characteristics of

volatility.

VolatilityForecasting

How canwe use historicalvolatility data, together withthe characteristics ofvolatility, to predict futurerealized volatility? Supposethat we have the followinghistorical volatility data foranunderlyingcontract:

Wemightprefer tolookatmore volatility data, but ifthese are the only dataavailable, how should we goabout making a volatilityforecast?

Onepossibleapproachisto simply average all the

availabledata:

(28%+22%+19%+18%)/4=21.75%

Using this method, eachpiece of historical data isgiven equal weight. But isthis reasonable? Perhapssomedataaremoreimportantthan other data. A tradermight assume, for example,thatthemorecurrentthedata,the greater their importance.

Because the 28 percentvolatility over the last sixweeks is more current thanthe other volatility data,perhaps 28 percent shouldplay a greater role in ourvolatility forecast.Wemight,for example, give twice theweight,40percent,tothesix-week volatility but only 20percentweight toeachof theothertimeperiods:

(40%×28%)+(20%×22%)

+(20%×19%)+(20%×18%)=23.0%

Our volatility forecast hasincreased slightly because oftheadditionalweightgiventothe six-week historicalvolatility.

Of course, if it is truethatthemorerecentvolatilityoverthelast6weeksismoreimportantthantheotherdata,it follows that the volatilityover the last 12weeks ought

tobemoreimportantthanthevolatilityoverthelast26and52weeks.Italsofollowsthatthevolatilityover the last 26weeks must be moreimportant that the volatilityover the last 52 weeks. Wecan factor this into ourforecastbyusingaregressiveweighting, giving moredistant volatility dataprogressively less weight inourforecast.Forexample,wemightcalculate

(40%×28%)+(30%×22%)+(20%×19%)+(10%×

18%)=23.4%

Herewehavegiven the6-week volatility 40 percent ofthe weight, the 12-weekvolatility 30 percent of theweight,the26-weekvolatility20percentoftheweight,andthe 52-week volatility 10percentoftheweight.

We have made theassumption that the more

recent the data, the greatertheir importance. Is thisalways true? If we areinterestedinevaluatingshort-term options, it may be truethat data that cover shortperiods of time are the mostimportant. But suppose thatwe are interested inevaluating very long-termoptions.Overlongperiodsoftime, the mean-revertingcharacteristic of volatility islikely to reduce the

importance of any short-termfluctuations in volatility. Infact, over very long periodsof time, the most reasonablevolatility forecast is simplythe long-termmean volatilityof the instrument. Therefore,therelativeweightwegivetothe different volatility datawilldependontheamountoftime remaining to expirationfor the options in which weareinterested.

In a sense, all thehistoricalvolatilitieswehaveat our disposal are current;they simply cover differentperiods of time. How do weknowwhichdataarethemostimportant? In addition to themean-reverting characteristic,we know that volatility alsotends to be serial correlated.Thevolatilityoveranygivenperiodislikelytodependon,or correlate with, thevolatility over the previous

period, assuming that bothperiods cover the sameamount of time. If thevolatility of a contract overthe last four weeks was 15percent,thevolatilityoverthenext four weeks is morelikely to be close to 15percentthanfarawayfrom15percent.Oncewerealizethis,wemight logically choose togivethegreatestweighttothevolatilitydatacoveringatimeperiod closest to the life of

the options in which we areinterested. That is, if we aretrading very long-termoptions, the long-term datashould be given the mostweight.Ifwearetradingveryshort-termoptions, the short-termdatashouldbegiventhemost weight. And if we aretrading intermediate-termoptions,theintermediate-termdatashouldbegiventhemostweight.

Given the serialcorrelation characteristic ofvolatility, what volatilityshould we assign to optionsthat expire in five months ifwe have only our fourhistoricalvolatilities:6-week,12-week, 26-week, and 52-week volatilities? Because 5months is closest to 26weeks, we can give the 26-week volatility the greatestweight and give other datacorrespondinglylesserweight

(15%×28%)+(25%×22%)+(35%×19%)+(25%×

18%)=20.85%

Alternatively, if we areinterested in evaluating 3-month options, we can givethegreatestweight tothe12-weekhistoricalvolatility

(25%×28%)+(35%×22%)+(25%×19%)+(15%×

18%)=22.15%

In the foregoingexamples,we used only fourhistorical volatilities. But themore volatility data that areavailable, the more accurateany volatility forecast islikely to be. Not only willmore data, covering differentperiodsof time,give abetteroverview of the volatilitycharacteristics of anunderlying instrument, theywill also enable a trader tomorecloselymatchhistorical

volatilities to options withdifferent periods of time toexpiration. In our examples,we used historical volatilitiesoverthelast12and26weeksasapproximations to forecastvolatilities over the next sixandthreemonths.Ideally,wewould like historical datacovering exactly six- andthree-monthperiods.

This approach toforecasting volatility is one

that many traders useintuitively. It depends onidentifying the typicalcharacteristics of volatilityand then projecting avolatility over some futureperiod.

The analysis of a dataseries in order to predictfuturevaluesfallsintoanareaofstudyusuallyreferredtoastime-series analysis. Wemight wish to apply time-

series models to volatilityforecasting, but to do so, weneed a series of data pointswhere each point isindependent of every otherpoint. In our examples, thevolatilities we used to makeour prediction do not form atrue time series because thevolatilities overlap and, assuch, are not reallyindependent of each other.The 52-week volatilityoverlaps the 26-, 12-, and 6-

week volatilities. The 26-week volatility overlaps the12- and 6-week volatilities.And the 12-week volatilityoverlaps the 6-weekvolatility. But suppose thatinstead of using as our datapoints, the historicalvolatilities, we use theunderlying returns. Thesereturns create a true timeseries to which we might beable to apply a time-seriesmodel.

One time-series modeloften used to estimate futurevolatility is the exponentiallyweighted moving average(EWMA) model. In thismodel, greater weight isalways given to more recentreturns, with older returnsgiven progressively smallerweightings. If α is theweighting assigned to eachreturn r, then the estimatedvariance (the square of the

standard deviation) σ2 overthe next period of time isgivenby

wherernisthemostrecentreturn. The constraints arethat all the weightings mustaddupto1.00

and that the more recentthe return, the greater is theweighting

αn>αn–1

By choosing a variable λbetween 0 and 1.00, theconstraintswillbemetif

Aswe reduce the value of

λ, more recent returns areassigned progressivelygreater weight—the varianceestimatetendstodiscounttheeffectofolderreturns.Asweincrease the value of λ, theestimate makes less and lessdistinctionbetween returns—older returns become just asimportant as newer returns.As λ approaches 1.00 (it cannever be exactly 1.00), theweight for all returnsconverges to a single value,

1.00/n.Acommonchoiceforλ in many risk-managementprograms is something closeto0.94.

The EWMA model isrelatively simple method forpredicting volatility. Twofactors that it ignores are thelikely correlation betweensuccessive returns and themean-reversion characteristicof volatility. The time-seriesmodels most often used to

forecast volatility were anoutgrowth of theautoregressive conditionalheteroskedasticity (ARCH)model first proposed byRobert Engle in 1982.6 Thetechniques used in ARCHmodels have subsequentlybeen refined and extendedinto what is now commonlyreferred toas thegeneralizedautoregressive conditionalheteroskedasticity (GARCH)

family of volatilityforecasting models. GARCHmodels consist of threecomponents: a volatilityestimate, such as EWMA; acorrelation componentreflecting the fact that themagnitude of successivereturns tends tobecorrelated(i.e., large returns tend to befollowedbylargereturns,andsmall returns tend to befollowed by small returns);and a mean-reversion

component specifying howfast volatility tends to revertto its mean. An in-depthdiscussion of GARCHmodelsisbeyondthescopeofthis text, but furtherinformation on these modelsisavailableinmostadvancedtextsontime-seriesanalysis.

ImpliedVolatilityasaPredictorofFuture

Volatility

If,asmanytradersbelieve,prices in the marketplacereflect all availableinformation affecting thevalue of a contract,7 the bestpredictor of the futurerealizedvolatilityoughttobethe implied volatility. Justhow good a predictor offuture volatility is impliedvolatility?Althoughitmaybe

impossible to answer thisquestiondefinitively,becausethat would require a detailedstudy of many markets overlongperiodsof time,we stillmight gain some insight bylookingatsampledata.

Figure 20-9 shows thethree-month realizedvolatility (approximately 63tradingdays)fortheS&P500Index and a rolling impliedvolatility for three-month at-

the-money options8 on theindex from 2002 through2010. However, the valuesfor the three-month realizedvolatility have been shiftedforward so that each datapoint represents the futurerealizedvolatilityoftheindexoverthenextthreemonths.Ifimplied volatility is a perfectpredictor of future volatility,both graphs would beidentical, but obviously, this

isnotthecase.Ingeneral,thevolatility of the S&P 500Index tends to lead theimpliedvolatility.Iftheindexbecomes more volatile,impliedvolatility rises; if theindex becomes less volatile,implied volatility falls. Themarketplaceseemstoreacttothe volatility of the index.Thiswas particularly evidentduring 2008, when impliedvolatility rose following thedramaticincreaseinvolatility

of the index, and in 2009,whenimpliedvolatilityfellasthe index itself became lessvolatile.

Figure20-9S&P500Indexthree-monthfuturevolatilityversusthethree-monthimpliedvolatility.

We can do the samecomparisonusinga12-monthperiod. Figure 20-10 showsthe 12-month future realizedvolatility of the S&P 500Index (approximately 252tradingdays)versusa rolling12-month at-the-moneyimplied volatility over thesame time period. Here thelag is evenmore evidentduetothelongertimeframe.

Figure20-10S&P500Index12-monthfuturevolatilityversusthe12-monthimpliedvolatility.

Clearly, the impliedvolatilityinourexamplesdidnot accurately predict futurevolatility. But, even if theimplied volatility was not atotally accurate predictor,perhaps we can draw someconclusionsbylookingat thedifference between theimplied volatility and thefuturerealizedvolatility.Thisis shown inFigure20-11 forboth 3- and 12-month

options. A positive valueindicatesanimpliedvolatilitythat was too low (the futurerealized volatility turned outto be higher), while anegative value indicates animpliedvolatilitythatwastoohigh (the future realizedvolatility turned out to belower).

Figure20-11DifferencebetweenfuturevolatilityandimpliedvolatilityfortheS&P500Index.

WecanseeinFigure20-11thatformuchoftheperiodinquestion,impliedvolatilityseemed to predict a futurevolatilitythatwastoohighbyup to 10 percentage points.But there are some dramaticexceptions. During 2008, thethree-monthimpliedvolatilityatonepointpredictedafuturevolatilitythatwastoolowbyalmost 50 percentage pointsandatanotherpointpredicted

a volatility thatwas too highby 20 percentage points.Admittedly, 2008was a yearof extremes, but even duringotheryears,adifferenceof10percentage points betweenimplied volatility and futurevolatilitywasnotuncommon.

Implied volatility is atbestanimperfectpredictoroffuture volatility. What elsemight we conclude fromthese graphs? Under normal

conditions, implied volatilityseems to be too high—optionstendtobeoverpriced.Buyers of options may bewilling to pay this extrapremiuminreturnforthefewoccasions when impliedvolatility is dramatically toolowandthereisasubsequentvolatility explosion. This isanalogous to insurance. Arationalbuyerof insurance isaware that the price of aninsurance contract is almost

certainly higher than itsvalue. Otherwise, theinsurance company wouldhave no profit expectation.But buyers of insurance arewilling to pay this extrapremium for those rareoccasions when anunforeseen event occurs andthe insurance becomesabsolutelynecessary.

There are, of course,other reasons why options

tendtobeoverpriced.Forthesellerof anoption, suchas amarketmaker,theremaybeacost to replicating the optionthrough thedynamichedgingprocess,acostthatthemarketmaker is likely to pass on tothecustomer.Moreover,theremay be weaknesses in thetheoretical pricing modelfromwhich impliedvolatilityis derived. Taken together,these factors may in factjustify the seemingly inflated

prices of options in themarketplace.

TheTermstructureofImpliedVolatilityIf held to expiration, the

soledeterminantofanoptionposition’svalue is, in theory,the realized volatility of theunderlying contract.However,atradermaydecideforavarietyofreasonsthata

position should be closedprior to expiration. Theposition may have achievedits expected profit potentialprior to expiration. Or theposition, even if it hasn’tachieved its expected profit,may have become too risky.Or holding the position mayrequire a large amount ofcapital, capital that could beput to better use. Regardlessof why a trader decides toclose out a position prior to

expiration, there is usuallyone primary cause: changesin implied volatility.Although we haveemphasizedtheimportanceofrealized volatility, in the realworld of option trading,changes in implied volatilitycan often make or break astrategy. For this reason, asensibletraderwillgivesomethought to how changes inimpliedvolatilitywillaffectaposition.

It may seem thatdetermining the sensitivityofa position to changes inimpliedvolatilityisrelativelysimple. We need onlydetermine the total positionvega, which we can do byadding up all the individualvega values. Unfortunately,determining the true impliedvolatility risk can besignificantly more complex.We know that vega valueschangewithchangingmarket

conditions, so today’s vegamaynotbetomorrow’svega.Moreover, the vega valuesacross different exerciseprices and expirationmonthsmay not be a true reflectionofimplied-volatilityrisk.

Consideramarketwherethere are three expirationmonths, all in the samecalendar year—March, June,andSeptember.Let’s assumethatthemeanvolatilityinthis

market is 25 percent, andalthough this almost neverhappens, let’s also assumethat the current impliedvolatility for every month isthesame,25percent.

Suppose that thevolatility of the underlyingcontract begins to rise.What

will happen to impliedvolatility? Implied volatilitywillalmostcertainlyrise,butwillitriseatthesamerateforeach month? If the impliedvolatility for March rises to30 percent, will the impliedvolatility of June andSeptember also rise to 30percent? Traders know thatvolatility is mean reverting,and there is a greaterlikelihood that volatility willrevert to its mean over long

periods of time than overshort periods. Therefore, aswe move to more distantexpirations, impliedvolatilityis likely to remain closer toits mean, in this case, 25percent. The new impliedvolatilitiesmightbe

Meanreversionwillalso

affect falling impliedvolatility. If the underlyingmarket becomes less volatileand implied volatility inMarchfallsto20percent,thenewimpliedvolatilitiesmightbe

Even if there is a largechange in the implied

volatility of short-termoptions,theimpliedvolatilityoflong-termoptionswilltendtochange lessbecauseof themean-reversioncharacteristics of volatility.Figure 20-12 shows thetypical term structure ofimpliedvolatility.

Figure20-12Thetermstructureofimpliedvolatility.

The fact that impliedvolatilities across differentexpiration months change atdifferent rates can haveimportant implications forrisk analysis. Consider anoption position consisting offour different expirationmonths with the followingvegavaluesforeachmonth:

What is the implied-volatilityriskoftheposition?Wemightbeginbyaddingupallthevegas

+15.00–36.00–21.00+42.00=0

With a total vega of 0, itmight appear that there is noimplied-volatility risk. This,however, assumes thatimpliedvolatilitywillchangeat the same rate across allmonths. But we know thatthis is unlikely. The impliedvolatility of short-termoptions will tend to changemorequicklythantheimpliedvolatility of long-termoptions. Given this, howshouldwedetermineourtotal

implied-volatilityrisk?Suppose that the mean

volatility in thismarket is25percent and that we believethat the term structure ofimpliedvolatilityissimilartothatshowninFigure20-13.IfApril implied volatility risesto 28 percent, what will bethe profit or loss to theposition? If there are twomonths remaining to Aprilexpiration and implied

volatility inApril rises to 28percent, we expect Juneimplied volatility to rise toonly 27 percent, Augustimpliedvolatilitytoonly26.5percent, and October to only26.1 percent. Adjusting forthe different rates of change,theresultisalossbecause

Figure20-13RelativechangesinimpliedvolatilityforApril,June,August,andOctoberoptions.

(3×15.00)–(2×36.00)–(1.5×21.00)+(1.1×42.00)

=–12.30

And if April impliedvolatility falls to 22 percent,theresultwillbereversed;wewill show a profit of 12.30.Clearly, the position is notvega neutral. We wouldmuchpreferimpliedvolatilitytofallthanrise.

In order to form amore

accurate picture of theimplied-volatility risk, wemust adjust the vega valuesfor each month. We knowthatforeachpercentagepointchange in April impliedvolatility, June impliedvolatilitywillchangeby

2/3=0.67

For each percentage pointchange in April impliedvolatility, August implied

volatilitywillchangeby

1.5/3=0.50

And for each percentagepointchangeinAprilimpliedvolatility, October impliedvolatilitywillchangeby

1.1/3=0.37

If we want to know ourtotal implied-volatility risk interms of changes in April

implied volatility, we canadjust our vega valuesaccordingly

Junevega=–36.00×0.67=–24.12

Augustvega=–21.00×0.5=–10.50

Octobervega=+42.00×0.37=+15.54

Adding everything up, wecan see that we do indeedhave a short vega position.

For each percentage pointchange in April impliedvolatility, the value of thetotalpositionwillchangeby

+15.00–24.12–10.50+15.54=–4.08

In order to accuratelyassess implied-volatility risk,a trader will need somemethod of determining howimplied volatilities are likelyto change across multiple

expirations. This usuallytakestheformofanimplied-volatility term-structuremodel. There is no singlemodel that all traders use.Models are often “homegrown,” with a trader tryingto develop a model that isconsistent with hismathematical sophistication,as well as his experience inthe marketplace. Whateverthe model, it will usuallyrequireatleastthreeinputs:a

primarymonth againstwhichall other months will becompared, a mean volatilityto which implied volatilitytends to revert, and a“whippiness” factor thatspecifies how impliedvolatility changes acrossotherexpirationswithrespectto changes in the primarymonth. The primary monthwilloftenbethefrontmonth,where trading activity tendstobeconcentrated.Butthisis

not always the case. Inagricultural markets, tradingactivity is often concentratedinexpirationmonths that fallclosetoeithertheplantingorharvesting calendar. If this isthecase,oneofthesemonthsmaybeabetterchoiceas theprimarymonth.Additionally,impliedvolatility in the frontmonth can be unstable,especially as expirationapproaches. It often changesin ways that are inconsistent

with the term structure ofotherexpirationmonths.Asaresult, many traders evaluatetheir position in front-monthcontracts separately fromtheir positions in othermonths, with the volatilityterm-structure modelapplyingtoallmonthsexceptthefrontmonth.Theprimarymonth chosen in thisapproach will be somethingotherthanthefrontmonth.

Figure20-14showshowthe term structure of impliedvolatility can evolve overtime.Thevaluesrepresenttheimplied volatilities during2010ofat-the-moneyoptionson the EuroStoxx 50 Indexfor expirations extending out24 months. Values werecalculated at two-monthintervals, on the first Fridayof February, April, June,August, October, andDecember. The reader may

find it useful to compare thechanges in the term-structuregraphs with the 30-dayhistorical volatility of theEuroStoxx 50 Index duringthis period, shown in Figure20-15. In early February, theterm-structure graph wasdownwardsloping: long-termoptionsweretradingat lowerimpliedvolatilitiesthanshort-term options. By April, as aresult of declining indexvolatility, not only had

implied volatility declined,but the term-structure graphhadinvertedandwasupwardsloping: long-term optionswere trading at higherimpliedvolatilitiesthanshort-termoptions.Afteradramaticincrease in index volatility,the June term-structuregraphagain became downwardsloping. Finally, afterdeclining index volatility inthe lasthalfof2010, impliedvolatilities seemed to settle

into a middle area, with arelativelyflattermstructure.

Figure20-14Implied-volatilitytermstructureforeurostoxx50Indexoptionsduring2010.

Figure20-15eurostoxx50Index30-dayhistoricalvolatilityduring2010.

Noteoneotherimportantpoint:thedisconnectbetweenthe front-month impliedvolatility and the remainderoftheterm-structuregraphinDecember. The graph isgenerallyupwardsloping,butthe front-month impliedvolatility is stillmuch higherthanallothermonths.Thisisa common characteristic inmany option markets. Thefront-monthimpliedvolatility

canoften trade in away thatis inconsistent with the termstructureofothermonths.

The term structure inFigure 20-12 is typical ofmarkets where the onlyfactors that tend to affectimplied volatility are therecent volatility of theunderlying contract and themean volatility. However, insomemarkets,theremayalsobeaseasonalvolatilityfactor.

Given the possibility ofextremely hot temperatures,as well as droughts, summerexpiration months inagriculturalmarkets typicallytrade at higher impliedvolatilitiesthanothermonths,regardlessofthetimeofyear.Inenergymarketswherefuelis needed for heating in thewinter and cooling in thesummer, the possibility ofvery cold winters and veryhot summers may result in

some months trading atpersistently higher impliedvolatilitiesthanothermonths.In such markets, it can bedifficult to create a reliableterm-structuremodel.

Figure 20-16 shows thechanging term structure ofimplied volatility for optionson natural gas futures during2009. Although not asobvious as the Eurostox 50IndexinFigure20-14,wecan

still detect the tendency oflong-term implied volatilityto revert to a mean, perhapsaround 40 percent. But inaddition, there is also aseasonal volatility factor.Note the impliedvolatilityofthe October option contract,which has been highlightedwith a circle. Regardless ofthe term structure, Octoberoptions always seem to tradeat an inflated impliedvolatility. This is perhaps

easiertoseeinFigure20-17,which shows the averageimplied volatility of eachexpiration month during2009.October clearly carriesa higher implied volatilitythan any other month. Thereason for this has to doprimarily with the Atlantichurricane season, whichextends from approximatelyearly June to lateNovember,with theheightof the seasonfalling in August and

September. During thisperiod, any major hurricanecan disrupt natural gasoperations, which in theUnited States areconcentrated along thenorthern coast of theGulf ofMexico. October options,which expire toward the endof September, will captureany volatility occurringduring the height of thehurricane season.Consequently, October

options tend to trade atconsistently higher impliedvolatilitiesthanothermonths.

Figure20-16Implied-volatilitytermstructureforoptionsonnaturalgasfuturesduring2009.

Figure20-17Averageimpliedvolatilitybyexpirationmonthofoptionsonnaturalgasfuturesduring2009.

ForwardVolatility

Let’s return to the term-structure graphs of impliedvolatilities across expirationmonths shown in Figure 20-14. Can we identify anytrading opportunities fromthese graphs? We mightsimply decide that impliedvolatilityiseithertoohigh,inwhich case wewill prefer to

sell options, or too low, inwhich case wewill prefer tobuy options. In either case,we can, in theory, capture aperceived mispricing bydynamically hedging theposition with the underlyingcontract. But we might alsoask a different question: areany expiration monthsmispriced with respect toother expiration months?Should we consider sometype of calendar spread,

selling options in one monthand buying options in adifferentmonth?

Let’sfocusononegraphfrom Figure 20-14, the termstructure of Eurostoxx 50IndexoptionsonFebruary5,2010.ThisisshowninFigure20-18. The large dotsrepresent the at-the-moneyimplied volatilities, with thesolid black line representingthe best fit generated by a

term-structuremodel.Wecanseethatsomecontractmonthsseemtodeviatefromthebest-fit line. June 2010 impliedvolatilityfallsbelowtheline,whiles September andDecember2010fallabovetheline. Assuming that eachmonthisinfacttradingattheindicated implied volatility,9do these deviations representatradingopportunity?Shouldwe be buying June options

and selling September orDecemberoptions?

Onemethod that tradersuse to determine themispricing of a calendarspread is to consider thespread’s implied volatility.That is,whatsinglevolatilityapplied to both expirationmonths will cause the valueofthespreadtobeequaltoitsprice in themarketplace? Tobetter understand this, let’s

use the volatilities in Figure20-18 to calculate the pricesof several calendar spreads.For simplicity, we willassume that the underlyingcontract is tradingat100andthat there are no interest-rateconsiderations. The relevantdata is shown in Figure 20-19.

Figure20-18Impliedvolatilityforat-the-moneyeurostoxx50IndexoptionsonFebruary5,2010.

Figure20-19CalendarspreadvaluesusingimpliedvolatilitiesonFebruary5,2010.

Looking at theFebruary/March calendarspread, the impliedvolatilitiesforthetwomonthsare 29.61 percent forFebruary and 28.06 percentforMarch.The values of theat-the-money calls are 2.31and3.80,withaspreadvalueof 1.49. If we evaluate theseoptions using the samevolatility, what singlevolatility will yield a value

equal to the price of 1.49?Logically, this volatility hasto be less than 28.06 percentbecause at this volatility theMarchoptionisfairlypriced,buttheFebruaryoptionistooexpensive. The entire spreadwillbeworthmorethan1.49.We need to reduce thevolatility until we find thesingle volatility that willcause the spread to beworth1.49. Using a computer, wefind that the February/March

calendar spread has animplied volatility of 25.94percent.

We can go through thisprocess for each successivecalendar spread, calculatingthe implied volatility of eachspread. These volatilities areshownatthebottomofFigure20-19. How will thesecalendar spread impliedvolatilitieslookifweoverlaythemonFigure20-18?Thisis

shown in Figure 20-20. Wecan see clearly that the June2010optionsaresignificantlyunderpriced in themarketplace compared withnearby expirations,while theSeptember 2010 options aresignificantly overpriced. Ifgivenachoiceofstrategies,itmightmake sense to buy theApril/June 2010 calendarspread and sell theJune/September 2010calendar spread. Together

thesespreadsmakeupatimebutterfly.

Figure20-20eurostoxx50IndexcalendarspreadimpliedvolatilitiesonFebruary5,2010.

We use these impliedvolatilities not to determinewhether implied volatility inthe entire option complex iseithertoohighortoolowbutrather to determine whetherparticular months aremispriced with respect toother months. The implied-volatility graph acts as amagnifyingglass,enablingusto more easily determinewhichmonths are overpriced

andwhichareunderpriced.When the term-structure

graph is downward sloping,as it is in Figure 20-20, allcalendar spread impliedvolatilitieswillfallbelowtheterm-structure graph.Alternatively, if the term-structure graph is upwardsloping, all calendar spreadimplied volatilities will fallabove the graph. If allimplied volatilities fall

exactly along the best-fitgraph, regardless of whetherthe graph is upward ordownward sloping, theimplied-volatility curve willbe smooth, suggesting thatthere are no obviouslymispricedcalendarspreads.

Determining the exactimplied volatility of acalendar spread usuallyrequires a computerprogrammed with a pricing

model. However, it is oftenpossible to estimate theimpliedvolatilityofanat-the-money calendar spread ifwerecall that the vega of an at-the-moneyoptionisrelativelyconstant with respect tochangesinvolatility.Supposethatweknowboth thepricesO1 and O2 and the vegavaluesV1 andV2 of the twooptions that make up thecalendar spread.Thepriceof

thespreadisO2–O1,andthevegaofthespreadisV2–V1.The implied volatility of thespread, given as a wholenumber, is approximatelyequal to the price of thespreaddividedbyitsvega

This method is not exactbecause there is likely to be

rounding error, and the vegadoes change slightly as wechange volatility. However,thisapproachmaybeusefulifa trader needs to make aquick estimate of whether acalendar spread is overpricedorunderpriced.

The vega values for theindividualoptions, aswell asfor the various calendarspreads, are given in Figure20-20.Thereadermayfindit

worthwhile to estimate theimplied volatility of eachspreadusing thismethodandthen compare the result withthe true implied volatility ofthespread.

Instead of analyzing thevolatility term structure bylooking at the impliedvolatility of successivecalendar spreads, we mighttake a slightly moretheoreticalapproach.Suppose

that we have two optionexpirations, a short-termoption expiring at t1 and along-term option expiring att2. If the impliedvolatilityoftheshort-termexpirationisσ1and the implied volatility ofthelong-termexpirationisσ2,we might ask this question:what forward volatility σf isthe marketplace implyingbetween expiration of theshort-term option and

expiration of the long-termoption?

This is analogous to aforward rate in an interest-

rate market. Given a short-term interest rateanda long-term interest rate, what ratemust apply between the twomaturities such that noarbitrage opportunity exists?Unlike interest rates, whichare directly proportional totime,volatilityisproportionalto the square root of time.Using this, we can calculatetheforwardvolatility10

We can expand thisrelationshiptoanynumberofvolatilities over any numberof consecutive time periods.Given forward volatilities σicoveringthetimefromti–1 toti,thevolatilityovertheentiretimeperiodfromt0totnmustbe

Suppose that wecalculate the forwardvolatilities for the volatilityterm structure in Figure 20-20.Howwould this comparewith the implied volatilitiesof thecalendarspreads?ThisisshowninFigure20-21.Theforward volatility graph hasthe same general structure as

the calendar spread graph.Both graphs serve the samepurpose—to highlight anymispricing of a particularexpirationmonth.

Figure20-21

Every experiencedoption trader knows thatdealingwithvolatilitycanbea difficult task. To facilitatethe decision-making process,we have attempted to makesome generalizations aboutvolatility characteristics.Eventhen,itmaynotbeclearwhat the right strategy is.Moreover, looking at alimited number of examplesmakes the generalizations

even less reliable. Everymarket has its owncharacteristics, andunderstanding the volatilitycharacteristics of a particularmarket,whetherinterestrates,foreign currencies, stocks, orcommodities, is at least asimportant as knowing thetechnical characteristics ofvolatility. And thisknowledge can only comefrom careful study of amarket combinedwith actual

tradingexperience.

1Becausevolatilityisalwaysquotedonanannualizedbasis,whetherwecalculatehistoricalvolatilityusingall365daysoronlytradingdays,thestandarddeviationofpricechangesmustbemultipliedbythesquarerootofthenumberoftradingperiodsinayear.Fora365-daytradingyear,thestandarddeviationmustbemultipliedby

2MichaelParkinson,“TheExtremeValueMethodofEstimatingtheVarianceoftheRateofReturn,”JournalofBusiness53(1):61–64,1980.3MarkB.GarmanandMichaelJ.Klass,“OntheEstimationofSecurity

PriceVolatilitiesfromHistoricalData,”JournalofBusiness53(1):67–78,1980.4HistoricalgoldvolatilityinFigure20-2andBundvolatilityinFigure20-5werecalculatedfromsettlementpricesofthefront-monthfuturescontract.5Foradditionaldiscussionofvolatilitycones,seeGalenBurghardtandMortonLane,“HowtoTellIfOptionsAreCheap,”JournalofPortfolioManagement,Winter:72–78,1990.6RobertF.Engle,“AutoregressiveConditionalHeteroskedsticitywithEstimatesoftheVarianceofUnitedKingdomInflation,”Econometrica50(4):987–1000,1982.Englewasawardedthe2003NobelPrizein

Economics.7Thisisknowninfinanceastheefficient-markethypothesis.8Thethree-monthimpliedvolatilitywascalculatedbyinterpolatingbetweentheimpliedvolatilityofoptionsbracketingthreemonths.9Wemakethisprovisobecauseoptionsettlementpricesdonotnecessarilyreflectactualtradingactivity.Whenthishappens,anyoneusingsettlementpricesasaguidetopotentialtradingstrategiesmaybedisappointedtofindthatthesettlementpriceisnotanaccuratereflectionofwhereanoptioncanactuallybetraded.10Somereadersmayrecognizethatthe

forwardvolatilitycalculationresultsfromthefactthatthesquareofvolatilityorvarianceσ2isdirectlyproportionaltotimeσf2×(t2–t1)=(σ2

2×t2)–(σ12×t1)

21

PositionAnalysis

Investors or speculators inoption markets often have aparticular view of marketconditions in terms of eitherdirection or volatility. Theyattempt to profit from this

viewthrough theselectionofspreading strategies such asthose discussed in Chapters11and12.InChapter13,welooked at the riskcharacteristics of some ofthese strategies underchanging market conditions.Because each spreadconsistedofalimitednumberof contracts, it was arelatively simple matter todetermine the risks that eachspreadentailed.

An active option trader,suchasamarketmaker,maybuildupmuchmorecomplexpositions consisting of manydifferent options across awiderangeofexercisepricesand expiration months.Unlike simple strategies,where the risks are relativelyeasytoidentify,analysisofacomplex position can beparticularly difficult becauseof the many ways in whichrisks can change as market

conditionschange. Ifa tradercannotdetermine the risksofa position, he will beunprepared to take thenecessary action to protecthimself when marketconditions move against himor to take advantage of hisgood fortune when marketconditionsmoveinhisfavor.

Before theoreticalpricing models came intowidespread use, analyzing a

complexpositionmadeupofmany different options wasoften an impossible task.Even if a trader had someidea of how each optionchangedasmarketconditionschanged, combining manydifferentoptionsoftencausedthe entire position to changein unexpected ways. Still, ifhe expected to survive, anintelligent trader needed tomake some effort to analyzetheposition.

In the early days ofoption trading, one commonapproach to analyzing riskwas to use syntheticrelationships to rewrite aposition in a more easilyrecognizable form. If therewritten position conformedto a strategy with which thetraderwasfamiliar,thetradermight then be able todetermine the risks of theposition.

For example, considerthisposition:

+29underlyingcontracts–44 March 65calls+44 March 65puts–7 March 70calls+49 March 70

puts–33 March 75calls–51 March 75puts+30 March 80calls+12 March 80puts

Suppose that theunderlyingcontract is tradingat a price of 71.50. What is

the delta of this position—positive,negative,orneutral?Without a theoretical pricingmodel, thismay look like animpossible question toanswer.And, indeed,withouta model, there is no way ofknowingtheexactdeltaoftheposition. But even if wecannot determine the exactdelta, perhaps we candetermine the direction inwhich we want theunderlyingcontracttomove.

Using syntheticrelationships, positions thatconsistofbothcallsandputscan be rewritten so that theyconsist of a single type ofoption, either all calls or allputs. This can sometimesmake a position easier toanalyze. Let’s take ourpositionandrewriteitsothatit consists only of calls,rewriting each put as itssyntheticequivalent:

Ifwetotalallthecontracts,whatareweleftwith?

We really have thisposition.

+42 March 70calls–84 March 75calls+42 March 65calls

As complex as thepositionfirstappeared,itwassimplyalongbutterfly.Andalong butterfly always wantsthe underlying contract tomove toward the insideexercise price, in this case,

75. With the underlyingcontract currently trading at71.50, the position must bedelta positive. If we hadrewritten the position so thatit consisted only of puts, theresult would have been thesame because a call and putbutterfly have essentially thesamecharacteristics.

The foregoing examplewas admittedly created sothat when the position was

rewrittenintermsofsyntheticequivalents, its riskcharacteristicswererelativelyeasytoidentify.Inreality,therisk characteristics of acomplex position rarely fallneatly intoplace.Analysisofa complex position willalmost always require atheoretical pricing model.Eventhen,themodelmaynottelltheentirestory.

Suppose that we have

the following marketconditions:

Underlyingprice=99.60Time toSeptemberexpiration = 9weeksVolatility=18percentInterestrate1=0

TheSeptember 95 put andSeptember 105 call havetheseriskcharacteristics:

What are the risks if wehavethefollowingposition2:

Long 10September 95putsShort 10September105callsLong 5underlyingcontracts

The total risksensitivities for the positionare

It appears thatwehavenodirectionalrisk(deltais0),norealized volatility risk(gamma is 0), no risk withrespecttothepassageoftime(theta is 0), and no impliedvolatility risk (vega is 0). Ifthepositionwasinitiatedwith

some positive theoreticaledgeandtherisksensitivitiesassociated with the positionare all 0, then the position iscertain to show a profit. Sowhat’stheproblem?

The problem is that thedelta,gamma,theta,andvegaare only measures of theposition’s risk under currentmarket conditions. Buttoday’s market conditionsmay not be—in fact, cannot

be—tomorrow’s conditions.Even if the underlying priceand volatility remainunchanged, time will pass.And we know that thepassageof timecanchangeaposition’s characteristics.Looking at a position’scharacteristics under currentmarketconditions isonly thefirst step in analyzing risk.Weneedtoasknotonlywhatthe risks are right now butalso what the risks might be

under different marketconditions.Whatwillhappenif the underlying contractmoves up or down in price?What will happen if impliedvolatility risesor falls?Whatwillhappenastimepasses?

We can expand ouranalysis by using what wealreadyknowabouthowrisksensitivitieschangeasmarketconditions change. Supposethat the underlying price

beginstofall.Howmightourrisk change? We know thatgamma is greatest for at-the-money options. As theunderlying price begins tofall, it is moving toward thelower exercise price, 95, andaway from the higherexercise price, 105. Thegamma of the September 95putmustbeincreasing,whilethe gamma of the September105 call must be declining.Because we are long the 95

putandshortthe105call,thetotal gamma position isbecomingpositive.Moreover,ifwehaveapositivegamma,as the market falls, ourposition, which was initiallydelta neutral, will becomedeltanegative.

What if the underlyingpricebeginstorise?Nowthemarket ismoving away from95 and toward 105: thegamma of the September 95

put is declining, and thegammaoftheSeptember105call is increasing. The entireposition is now becominggamma negative.Consequently, as the marketrises, our position willbecomedeltanegative.

This seems odd. Theposition becomes deltanegative if the underlyingprice falls or rises. Theexplanation is the changing

gamma:thepositionbecomesgamma positive on the waydownbutgammanegativeonthewayup.

Now let’s considerwhatwillhappenifvolatilityrises.As volatility increases, thedelta of calls moves toward50, and the delta of putsmoves toward–50,while thedelta of the underlyingcontract remains constant at100. Because we are long

puts,nowwithadeltagreater(in absolute value) than –25,and short calls, now with adelta greater than 25, theposition is becoming deltanegative. If the delta of theSeptember95putgoesto–30and the delta of the 105 callgoes to +30, the total deltapositionwillbe

(10×–30)–(10×30)+(5×100)=–100

In the same way,reducing volatility causesdelta values to move awayfrom 50. If the delta of theSeptember95putgoesto–20and the delta of the 105 callgoes to +20, the total deltapositionwouldbe

(10×–20)–(10×20)+(5×100)=+100

Summarizing, if volatilityrises,wewanttheunderlying

market to fall. If volatilityfalls,wewant theunderlyingmarkettorise.

Whatwillhappen to theposition as time passes?Reducing time, like reducingvolatility, causesdeltavaluestomoveaway from50.Withno change in the underlyingprice as time passes, the callandputwillmovefurtheroutof the money. The fiveunderlyingcontractswilltend

to dominate the position,resultinginapositivedelta.

We have initiallyfocused on the delta andgamma,butwecanalsoinferwhatwillhappen to the thetaand vega because thesevalues, like the gamma, aregreatest for at-the-moneyoptions. If the underlyingcontract begins to fall, ourtheta position will becomenegative(thepassageof time

will begin to hurt), and ourvega position will becomepositive (we will wantimpliedvolatilitytoincrease).If the underlying contractbegins to rise, our thetapositionwillbecomepositive(the passage of time willbegin to help), and our vegapositionwillbecomenegative(we will want impliedvolatility to decline). If theunderlying price does notchange, the gamma, theta,

and vega of the position areunlikely to be significantlyaffected by changes in eithertime or volatility. We cansummarize the effect ofchanging market conditionson the risk characteristics ofthepositionasfollows:

Ifapositionisnotoverlycomplex,atradermaybeableto do this type of analysis,firstlookingattheinitialrisksensitivities and thenconsidering how thesensitivities might change asmarket conditions change.However, a trader can get amore complete picture of aposition’sriskbylookingatagraph of the position’s valueover a broad range of

conditions. Let’s do this forthecurrentposition:

Long 10September 95putsShort 10September105callsLong 5underlyingcontracts

Figure 21-1 shows the

value of the position withrespect to movement in theunderlying contract. Thethree graphs represent thevalue at the current volatilityof 18 percent, as well as atvolatilities of 14 and 22percent. From Figure 21-1,we can see the graphicinterpretation of delta andgamma. For a negative deltaposition, the graph extendsfrom the upper left to thelower right—as the

underlying price rises, theposition loses value. For apositive delta position, thegraphextendsfromthelowerlefttotheupperright—astheunderlying price rises, theposition gains value. In ourexample, the position isalways delta negative athighervolatilities.Aroundthecurrent underlying price of99.60, the position is deltaneutral—the graph is exactlyhorizontal. At lower

volatilities, the delta willbecome positive around thecurrentunderlyingprice.

Figure21-1Positionvalueastheunderlyingpriceandvolatilitychange.

For a negative gammaposition, the graph curvesdownward, taking on theshape of a frown; pricemovement in either directiondecreases the value of theposition. For a positivegamma position, the graphcurvesupward, takingon theshape of a smile; pricemovement in either directionincreases the value of theposition. Our position has a

positive gamma below thecurrent price of 99.60 and anegativegammaabove99.60.At lower volatilities, thegammaismagnified(there isgreater curvature), while athighervolatilities,thegammais muted (there is lesscurvature). The currentunderlying price of 99.60 isan inflection point—thegamma is changing frompositive to negative. At thisprice, thegraph is essentially

a straight line. The graphicinterpretations of a positiveand negative delta andgamma are shown in Figure21-2.

Figure21-2Positiveandnegativedeltaandgamma.

Because gamma andtheta are of opposite signs, apositive gamma positionwilllosevalueastimepasseswithno movement in theunderlying contract. Anegative gamma will gainvalue. This is shown inFigures21-3and21-4.

Figure21-3Positivegamma,negativethetapositionastimepasses.

Figure21-4Negativegamma,positivethetapositionastimepasses.

Although gamma andtheta are always of oppositesigns, gamma and vega maybe either the same or theopposite. Regardless ofwhether we have a positivegamma (we want theunderlying contract tomove)or a negative gamma (wewant the underlying contracttositstill),wecanhaveeithera positive vega (we wantimpliedvolatilitytorise)ora

negative vega (we wantimpliedvolatilitytofall).Thegraphic representations ofthese positions are shown inFigures21-5and21-6.

Figure21-5Positivegammapositionasvolatilitychanges.

Figure21-6Negativegammapositionasvolatilitychanges.

Itmay also be useful tolook at graphs of the risksensitivities as marketconditions change. In Figure21-7,wecanseethechangingdelta as the underlying priceand volatility change. Closeto the current underlyingprice of 99.60, raisingvolatility causes the delta tobecome negative, whileloweringvolatility causes thedelta to become positive. As

we have already seen, if theunderlying contract eitherrises or falls, the deltabecomes negative. In Figure21-8,wecanseethechanginggamma as the underlyingprice and volatility change.Close to the currentunderlyingpriceof99.60,thegamma is unaffected bychanges in volatility. Thegamma becomes positive ifthe underlying price falls ornegative if the underlying

contractrises.Figure21-7Positiondeltaasthe

underlyingpriceandvolatilitychange.

Figure21-8Positiongammaastheunderlyingpriceandvolatilitychange.

In addition toconsidering the risksensitivities—delta, gamma,theta,andvega—andthewayinwhich thesevalueschangeasmarket conditions change,traders are well advised tolook at the net contractposition. If themarketmakesa dramatic downward movesuch that all calls move farout of the money while allputs go deeply into the

money,orthemarketmakesadramatic upward move suchthat all putsmove far out ofthe money while all calls godeeply into the money, whatwill be the result? In otherwords,ifthemarketfallsandallputsbegintoactlikeshortunderlying contracts, or themarket rises and all callsbegin to act like longunderlying contracts, what isthe trader left with? In ourposition, the downside

contractpositionisshortfive.At very low underlyingprices,thelong10September95 puts together combinedwith the long 5 underlyingcontracts will act like aposition that is short 5underlying contracts. Theupside contract position isalso short five. At very highunderlying prices, the short10 September 105 callscombined with the long 5underlying contracts will act

like a position that is alsoshort 5 underlying contracts.ThisisapparentinFigure21-7; the delta approaches –500ineitherdirection.

Thenetcontractpositionmay sometimes seemirrelevant, particularly if aposition consists of very farout-of-the-money options.Afterall,howlikelyisit thatthey will go so deeply intothe money that they will act

likeunderlyingcontracts?Buttraders have learned,sometimes through painfulexperience, that big movesoccur more often in the realworld thanonemightexpect.Extraordinary andunpredictable events—political and economicupheavals, scientificbreakthroughs, naturaldisasters, corporate takeovers—can sometimes causemarkets to move

dramatically. When thisoccurs,atradermayfindthatoptions that “couldn’tpossibly go into the money”havedonejustthat.

A trader who is shortvery far out-of-the-moneyoptionsmaybelievethatthereis so little chance that theoptions will go into themoney that there is no pointin buying them back. Thismay be true, but the

clearinghouse will stillrequire a margin deposit foreachshortoption.Inorder toeliminate this requirement,andperhapsputthemoneytobetter use, the trader maywanttobuybacktheoptions.Of course, hewill onlywantto do this if the price isreasonable. Certainly, theprice that the trader will bewillingtopayoughttobelessthan themargin requirement.Inthesameway,atraderwho

is long very far out-of-the-money options that hebelieves are worthless willusually be happy to sell theoptions at whatever price hecan. After all, something isbetter than nothing, which iswhat the options will beworthiftheyexpireoutofthemoney.

Very often the price atwhich traders are willing tobuyorsellveryfarout-of-the

moneyoptionsislessthantheminimum price that theexchange normally allows.For this reason, manyexchanges permit options totrade at a cabinet bid, a bidusuallymadeatapriceofonecurrencyunit.Forexample,ifthe minimum price for anoptionon aU.S. exchange is$5.00, an exchange maypermit options to trade at acabinetbidof$1.00.Thiswillallow traders who are either

longorshortoptionsthattheybelieve to be worthless toremove them from theiraccounts. The conditionsunderwhich cabinet bids arepermissible are specified byeachexchange.

Now let’s consider themorecomplexpositionshownin Figure 21-9. The positionconsists of options that allexpireatthesametime,butitincludescallsandputsatfive

different exercise prices,togetherwithapositionintheunderlying contract. Asbefore, we assume that theposition has some positivetheoretical edge. Otherwise,the immediategoalwouldbeto liquidate the position inorder to avoid a loss or toalter it in order to create apositive theoretical edge.Whataretherisksofholdingthisposition?

Figure21.9

Beginning with a quicklook at the sensitivities, wecan see that we are at riskfrom a decline in theunderlying market (negativedelta), from an increase inrealized volatility (negativegamma), and from anincrease in implied volatility(negative vega). Lookingonlyat thedeltaandgamma,the most favorable outcomeseemstobeaslowdownward

move in the underlyingmarket. The least favorableoutcome seems to be a swiftupwardmove.

What else can we sayaboutthisposition?Fromthenegative delta, it’s clear thatwe would like downwardmovement in the underlyingprice.Buthowfardown?Thecurrent price is 101.25. Dowewant themarket to fall to100?To95?To90?Perhaps

wewantanunlimiteddecline.However, the negativegamma indicates that a swiftand violent downward movecannot be good for thisposition.Taken togetherwiththedelta,wecanapproximatejust how far we want theunderlying to fall if werealizethatanegativegammaposition always wants tobecome delta neutral. Theprofit resulting from anegativegammapositionwill

tendtobemaximizedwhenitisdeltaneutral.

Where will our positionbedeltaneutral if themarketstarts to fall? For each pointdecline in the underlyingmarket,wemust subtract thegamma, –25.8, from ourdelta.Bydividingthecurrentdelta by the gamma, we canestimate that the position isapproximatelydeltaneutralatanunderlyingpriceof

101.25–(297.4/24.13)=101.25–12.32=88.93

Of course, this is only anapproximation because weareassuming that thegammais constant, which it is not.An increasing or declininggamma as the underlyingprice changes will alter ourconclusion. However, if wehavetomakeaquickestimateof what we would like tooccur, a slow downward

move to around 89.00 seemsbest.

We have also surmisedthataswiftupwardmovewillhurt this position. Now boththe delta and gamma areworking against the position.Suppose that the worsthappens—the underlyingcontract suddenly leaps to150. Will the result bedisastrous for us? Here wereturn to the net contract

position: if themarketmakesadramaticmovesuchthatallcontractsmove intooroutofthe money, what are we leftwith? In a large upwardmove, all the puts willcollapse to 0, while all thecallswill eventually begin toact like underlying contracts.Our position is net short atotal of 7 calls. But we arealso long 13 underlyingcontracts.Thisgivesusanetupside contract position of

+6. If the market makes areally big upward move, wewill have a position that islong 6 underlying contracts,giving us a potentiallyunlimited profit. We canconclude that as the marketmoves up, at some point ourgamma must turn positive,causing the delta toeventuallybecomepositive.

The downside contractposition is not so favorable.

Nowallthecallswillcollapseto 0, while all the puts willact like short underlyingcontracts.We are net long 5puts,butwearealsolongthesame13underlyingcontracts.Our net downside contractposition is +8. If the marketmakes a violent downwardmove,wewillhaveapositionthat is long 8 underlyingcontracts, with potentiallydisastrousresults.

Becausewearefocusingon the risk characteristics ofour position, no prices ortheoretical values are givenfortheoptionsinFigure21-9.We have simply made theassumption that the positionhas some positive theoreticaledge. However, the size ofthe theoretical edge—howmuch,intheory,weexpecttomakewith theposition ifourvolatility estimate of 27percent iscorrect—canbean

important consideration inanalyzing the risk of theposition. For example, let’sassumethatthepositionhasapositive theoretical edge of6.00. If 27 percent turns outto be the correct volatilityover the six-week life of theposition and we go throughthe delta-neutral dynamichedging process,3 we expecttoshowaprofitof6.00.

Thetheoreticaledgeand

vegacanhelpusestimateourvolatility risk.From thevegaposition of –0.759, we knowthatany increase involatilitywill hurt. Consequently, wemight ask this question: howmuch can volatility risebefore our potential profitturns into a potential loss?For each percentage pointincrease in volatility ourpotential profit will bereducedbytheamountofthevega. By dividing the

theoretical edge by the vega,we can estimate that thepositionwill break even at avolatilityofapproximately

27.00+(6.00/0.759)=27.00+7.90=34.90(%)

Assuming a theoreticaledge of 6.00, if volatilityturnsouttobenohigherthan34.90 percent, the positionwill do no worse than breakeven. Above 34.90 percent,

the position will begin toshow a loss. We discussedthis concept—the breakevenvolatility of a position—inChapter 7. This can bethought of as the impliedvolatility of the entireposition. It tells us that wehave a margin for error of7.90 volatility points in ourvolatility estimate. Whetherthis represents a small orlargemarginoferrordependson the volatility

characteristics of thisparticularmarket.

Howcanweincreasethemargin for error in ourvolatility estimate? We candosobyeitherincreasingthetheoretical edge (withoutincreasing the vega) or byreducing the vega (withoutreducing the theoreticaledge). Ifwe can increase thetheoretical edge to 8.00without increasing the vega,

the implied volatility of thepositionwillbe

27.00+(8.00/0.759)=27.00+10.54=37.54(%)

Alternatively, if we canreduce thevega to–0.65, theimpliedvolatilitywillbe

27.00+(6.00/0.65)=27.00+9.23=36.23(%)

Unfortunately, it may not

be possible to do either. Inthis case, we will have todecide whether the vega riskof–0.759isreasonablegiventhepotentialprofitof6.00.

We know that the risksensitivitiesof theposition—delta,gamma,theta,andvega—are likely to change asmarket conditions change. Itis almost impossible to do adetailed analysis of thesechanges without computer

support.However,wemaybeable to say something abouthowthedeltachangesastimeand volatility change if werecall that delta valuesmoveeither toward 50 or awayfrom50withchangesintimetoexpirationandvolatility.

Consider what willhappen if volatility begins torise.Allcalldeltaswillmovetoward 50 and put deltastoward –50. Because we are

netshort7callsandnet long5puts,intheextreme,thecalldeltapositionwillbe

–7×50=–350

and the put delta positionwillbe

5×–50=–250

Together with the 13 longunderlyingcontracts,thetotaldeltawillbe

–350–250+1,300=+700

Of course, we wouldhave to raise volatilitydramaticallyforall thedeltasto actually approach 50.But,aswebegintoraisevolatility,thecurrentdeltaof–297willbecome less negative andeventually will turn positive.In a high-volatility market,we will prefer upwardmovement in the underlyingcontract.

What about a decline involatility or the passage oftime, both of which willcause delta values to moveaway from 50? The deltavalues of out-of-the-moneyoptions will move toward 0,while the delta values of in-the-moneyoptionswillmovetoward 100. Because we arecurrently net short 2 in-the-money calls (the 90, 95, and100calls)andnetlong20in-the-money puts (the 105 and

110puts),intheextreme,ourtotaldeltawillbe

–200–2,000+1,300=–900

If we reduce volatility ortime passes, we will preferdownward movement in theunderlyingcontract.

Foranewtrader,usingabasic knowledge of delta,gamma, theta, and vegacharacteristics to analyze therisk of a position can be a

useful exercise. However,when computer support isavailable, it is almost alwayseasier and more efficient tolook at graphs of theposition’s risk.Thishasbeendone for the current positioninFigures21-10 through 21-13.

Figure21-10Positionvalueastheunderlyingpriceandvolatilitychange.

Figure21-11Positiondeltaastheunderlyingpriceandvolatilitychange.

Figure21-12Positiongammaastheunderlyingpriceandvolatilitychange.

Figure21-13Positionvalueasvolatilitychangesandtimepasses.

InFigure21-10,we cansee that at a volatility of 27percent, the maximum profitonthedownsidewilloccurata price of approximately95.00, at which point theposition delta is 0. Thisdiffers considerably fromourestimateof88.93becausethegamma,whichwasinitially–24.13,becomesamuchlargernegative number as themarket drops. The negative

deltaof–297 ismorerapidlyoffset by the increasinggamma. In Figure 21-12, wesee thaton thedownside, thegamma reaches itsmaximumof approximately –80 at anunderlyingpriceof93.

If themarketmoves up,wewill initially losemoney.But,atanunderlyingpriceof104, the gamma becomespositive. Our negative deltabeginstoturnaroundandata

priceof112actuallybecomespositive (Figure 21-11). Wewill continue to lose moneyabove112,butatsomepointthe position will begin toshow a profit. Figure 21-10onlygoesuptoanunderlyingprice of 120, but a moreextensive analysis wouldshow that at an underlyingpriceof124,thepositionwillbegintoshowaprofit.

InChapter9,we looked

at some of the nontraditionalhigher-order risk measures.Figure 21-12 shows thatbetweentheunderlyingpricesof 93 and 114, the positionhas a positive speed; as theprice rises, the gammaincreases. Below 93 andabove114, thepositionhasanegative speed; as the pricerises, the gamma declines.Wecanalsoseethatchangingthe volatility causes thegamma and, consequently,

the delta to change at adifferent rate. Loweringvolatility causes the speed toincrease, while raisingvolatility causes the speed todecline.

Figure 21-13 shows thesensitivity of the position tochanges in implied volatility,assuming a constantunderlying price of 101.25.The position clearly has anegativevega.Anydeclinein

implied volatility will helpthe position; any increase inimpliedvolatilitywillhurttheposition. Given a theoreticaledge, we can estimate thebreakeven(implied)volatilityfor the entire position bydividing the total theoreticaledge by the vega. If, forexample,wehaveatotaledgeof6.00,weestimatedthattheposition has an impliedvolatility of approximately34.90percent.Infact,wecan

see in Figure 21-13 that theimplied volatility issomewhat higher than 34.90percent. The six-week graphcrosses –6.00, which wouldexactly offset a theoreticaledge of +6.00, at a volatilityof approximately 36 percent.The reason the breakevenvolatility is greater than ourestimate is that the six-weekgraphhasapositivevolga—itcurves upward slightly. Asvolatility rises, the vega

becomesmorepositiveorlessnegative. As volatility falls,the vega becomes morenegative or less positive.Even though the currentvolga is positive,we can seethatas timepasses, thevolgaof the position becomesslightly negative. The four-week graph is approximatelyastraightline,whilethetwo-week graph curves slightlydownward.

What should weconcludeaboutthepositioninFigure 21-9? The reason fordoingananalysisistohelpusdetermine beforehand whatactions to take to eithermaximize our profits ifconditionsmove inour favoror minimize losses ifconditions move against us.Wecurrentlyhaveanegativedelta. Ifwewish tomaintaina downward bias, then noaction is necessary. If,

however,wearetradingfroma purely theoreticalstandpoint, then perhaps weought to buy the 297 deltasthatweareshort.Theeasiestwaytodothisistobuythreeunderlyingcontracts.

If we maintain ourcurrent position and themarket begins to decline,what action should we take?If thedeclineisslow(clearlya very good outcome given

our delta and gamma) andthereisnoincreaseinimpliedvolatility, perhaps we oughtto consider buying puts atlower exercise prices. Thiswill have the effect ofoffsetting our downside netcontractriskandreducingournegative vega while lockingin some of the theoreticaledge.If,however,thedeclineis swift, we may have toignore theoreticalconsiderations and buy puts

atthemarketprice.Thismaybe the cost of having a badposition, something that willinevitably occur at somepointineverytrader’scareer.Ifweareforcedtobuyputsatinflated prices, especially ifthereisanincreaseinimpliedvolatility, we may losemoney. But if the decline isvery swift, the primaryobjective may be survival.And in the long run, simplysurviving, inorder tobeable

to take advantage of thosesubsequent occasions whenconditionswork inour favor,can mean the differencebetween success and failureinoptiontrading.

What action should wetake if the market begins torise?Weoughttobepreparedforonecourseofactionifthemove is slow (the delta isworkingagainstus,whilethegamma is working for us),

but a different course ofaction if the move is swift(the delta and gamma areinitially working against us,but if the upward move islarge enough, these numbersmay eventually work in ourfavor).

A detailed positionanalysis will help us preparefor a variety of changes inmarket conditions. But nomatter how detailed our

analysis, we may stillencounter situations whereweare inuncharted territory.When conditions do change,we can never know forcertain how the marketplacewill react. If the underlyingprice begins to rise or fall,depending on the specificmarket, we may expectimpliedvolatilitytochangeina certain way. But we mayfind that it has changed in acompletelydifferentway.We

may have to accept the factthat our analysis wasincorrect and take whateveraction we can to reduce ourlosses or maximize ourprofits under these new andunexpectedconditions.

SomeThoughtsonMarketMaking

Inordertoensureliquidity

in a market, exchanges mayappoint one or more marketmakers in a product. Amarketmakerguarantees thathe will continuously quoteboth a price at which he iswilling to buy and a price atwhichheiswillingtosell.Asa consequence, a buyer orseller can always be certainthat therewillbesomeone inthe marketplace willing totake the opposite side of thetrade.Thisdoesnotmeanthat

a customer is required totradewithamarketmaker.Ifother market participants arewilling to buy at a higherpriceor sell ata lowerprice,thecustomerisalwaysfreetotrade at the best availableprice. But by continuouslyquotingabid-askspread, themarketmaker fulfillshis roleas the buyer or seller of lastresort

A market maker must

complywithrulesestablishedby the exchange concerningthe width of the bid-askspread as well as theminimum number ofcontracts that the marketmaker must be willing totrade. If exchange rulesdictate that a market makermay quote a bid-ask spreadnowiderthan2.00,thenabidprice of 63.00 for a contractimplies an offer price that isno higher than 65.00.

Similarly, an offer price of47.00impliesabidpricethatis no lower than 45.00. Themarket maker may quote atighter bid-ask spread, forexample, 63.50–64.50 in theformercaseor45.75–46.25inthelatter,but thespreadmaybe no wider than thatspecified under the exchangerules.

In addition to quoting abid-ask spread, a market

maker must be willing totrade a minimum number ofcontractsatthequotedprices.If the exchange minimum is100 contracts, the marketmakermustbewillingtobuyor sell a minimum of 100contractsathisquotedprices.He may offer to trade morethan the minimum, in whichcasehewillusuallyquotehissize along with the bid-askspread,forexample,

63.50–64.50200×200

The market maker iswilling to buy at least 200contracts at a price of 63.50or sell at least 200 contractsat a price of 64.50. Thequoted size need not bebalanced:

63.50–64.50500×200

Here the market maker iswilling to buy 500 contractsbut only willing to sell 200contracts.

Rules governing thewidth of a market maker’sbid-ask spread usually applyonlytotheminimumsizethatthe market maker must beprepared todo. If acustomerwants to trade a very largenumberofcontracts,amarketmaker is permitted to widen

the spread because of theincreasedriskassociatedwiththe trade. In response to acustomerwhowants to trade1,000 contracts, a marketmaker might quote a spreadof 62.00–66.00. To facilitatetrading,whenacustomerhasa large order, hewill usuallyindicatethathewantsaquoteforsize.

In return for fulfillinghis obligations, a market

maker will receive specialconsiderations from theexchange.Thesemaycomeinthe form of very lowexchange fees or preferentialtreatment when competingagainst other marketparticipants. If a customer iswilling to sell at the marketmaker’s bid price and twoother market participants arealso quoting the same bidprice, themarketmakermaybe entitled to 50 percent of

theorder,whiletheothertwobidders may onlybe entitledto25percenteach.

Unlike investors,speculators, or hedgers, whocanchoose thestrategies thatbest fit their needs and whocan also determine when toenter and exit a market, amarketmakerhaslesscontrolover the positions he takes.This does not mean that amarketmakeristotallyatthe

mercy of his customers. Hemay be forced to take on aposition, but he at least hassomechoiceastothepriceatwhichhedoes so.Moreover,by adjusting his bid-askspread,hecantosomeextentdetermine the types ofpositions he acquires. Buthaving done so, he may stillfind that he has taken on aposition that hewould prefernottohave.

Althoughmarketmakerstypically represent only asmall percentage of optionmarket participants, they canplayacrucial role in trading,oftendeterminingthesuccessor failure of an exchange-listed product.4 For thisreason, it may be useful totake a closer look at how anoption market maker goesabouthisbusiness.

A successful market

maker must ask threequestions:

1. What does themarketplace thinkanoptionisworth?2. What do I (themarket maker)think the option isworth?3. What positionsam I currentlycarrying?

The answers to thesequestionswilldeterminehowamarketmakerpricesoptionsandhowhemanagesrisk.

The answer to the firstquestion—what does themarketplace think an optionisworth?—isthebasisforthesimplestofallmarket-makingtechniques. In this approach,themarketmakerattempts toprofitsolelyfromthebid-askspread, constantly buying at

thebidpriceandsellingattheoffer price. No specialknowledge of option pricingtheory is required, but inorder to succeed, the marketmaker must be able toidentify an equilibrium pricearound which there are anequal number of buyers andsellers.5 If he can correctlydetermine this equilibriumprice,heisinapositiontoactas a middleman, showing a

small profit on each tradewhile carrying positions foronlyshortperiodsoftime.Ofcourse, the equilibrium priceisconstantlychangingasnewbuyers and sellers enter themarket. Although a marketmakerwillconstantlymonitormarket activity to determinechangesinbuyingandsellingpressure,evenanexperiencedmarketmakerwillsometimesfind,especiallyinaveryfast-moving market, that he has

the wrong equilibrium price.When this occurs, he mayfindthathehaseitherboughtor sold many more contractsthanhedesires.

In addition to profitingfrom the bid-ask spread, byanswering the secondquestion—whatdoIthinktheoption is worth?—an optionmarketmakerwillalso try toprofit from a theoreticallymispriced option. The

mispricing may be the resultof an unbalanced arbitragerelationship, in which casethemarketmakerwillattemptto “lock in” the profit bycompleting the arbitrage. Orthe mispricing may be theresult of using a theoreticalpricingmodel.Inthiscase,ifthe market maker buys at apricebeloworsellsatapriceabove his presumedtheoretical value, he candynamically hedge the

positiontoexpirationoruntilthe option is again trading attheoretical value. If histheoretical value is correct,the dynamic hedging processshould, in theory, result in aprofit.

Once the market makerbegins to acquire positions,he must consider thepossibility that marketconditions might moveagainsthim.Thisbringsusto

the final question—whatpositions am I currentlycarrying? Although there issome risk associated withevery position, if the riskbecomestoogreat,anadversechange in market conditionsmight put the market makerin a situation where he isunable to freely trade andtherefore unable to benefitfromhispositionasamarketmaker.Inanextremecase,hemaybeforcedoutofbusiness

because he is no longer ableto fulfill his obligations as amarketmaker.

A market maker mustconsider a variety of risks.Initially, he will probablydetermineamaximumriskheis willing to carry undercurrent market conditions.This may mean limiting thesize of his position withrespect to the various riskparameters—delta, gamma,

theta, vega, and rho.When alimit is reached, the marketmakerwill begin to focusonmaking markets that willhave the effect of reducinghis risk. Ifamarketmaker isapproaching the maximumnegativegammaposition thathe iswilling to accept, as hegets closer to this limit, hewill increasingly focus onreducing or at least limitingthis risk.Asamarketmaker,hestillmustquotebothabid

price and an offer price, buthewouldmuchprefer tobuyoptionsbecausethiswillhavethe effect of reducing hisnegative gamma position.Under normal conditions, ifasked to make a market, hewill likely do so around thepresumedtheoreticalvalue.Ifthe value of the option is64.00, he might quote amarketof63.00–65.00,butifthemarketmakerisintentonreducinghisnegativegamma

risk, hewill clearly prefer tobuy options rather than sell.To reflect this preference, hecanadjusthisbid-askspread,perhaps quoting a market of63.50–65.50.Thefact thathehas raised both his bid andoffer makes it more likelythathewillbuyoptionsratherthan sell. Of course, he maystill be required to sell if theoffer of 65.50 is accepted.Butatleasthehasdonesoatamoreadvantageousprice.

A market maker mustconsider not only the risksunder current marketconditionsbutalsohowthoserisksmightchangeasmarketconditions change. Supposethat in a rising market themarketmakerhasreachedthemaximumnegativegammaheiswillingtoaccept.However,in analyzing the position, hehas also noted that if theunderlyingcontractcontinuesto rise, the gamma risk will

begin to decline.6 If theunderlying does move, themarket maker may still behurt because he has anegativegammaposition.Buthe may decide that he canlivewiththisriskbecausethegamma risk will begin todecline.

Inadditiontomonitoringthevariousrisksensitivities,amarket maker must alsointelligently manage his

inventory. As conditionschange, a position thatincludes a concentrated riskmay evolve into a seriousthreat to the market maker.Consideramarketmakerwhohas the following gammaposition spread out over 10differentexerciseprices:

Even if the total gammarisk is relatively small(indeed, the total gamma inthis case is 0), the fact thatsucha largenegativegammais concentrated at oneexerciseprice,95,islikelytobe of concern to the marketmaker. If theseare long-termoptions,thesituationmaynotbe critical today.But as timepasses, if the underlyingmarket approaches 95, theposition will take on

increasingly greater risk.Rather than let this riskincrease,anintelligentmarketmaker will focus onspreading out his risk moreevenlyacrossexerciseprices.In the same way that a wiseinvestorwillseektodiversifyhis risk, amarketmakerwillstriveforasimilargoal.

In thisexample, the riskwas concentrated at oneexercise price. But any

concentration of risk at aspecific exercise price orexpirationdateor intermsofa single large risk sensitivityshould be a cause forconcern.Itmaynotalwaysbefeasible because marketconditions do not alwayscooperate, but a marketmaker’s ultimate objectiveshould be to diversify hispositionasmuchaspossible.

Consider the stock

option position shown inFigure 21-14. This positiondoes not fall into any easilyrecognizable category andrepresents the type of mixedcollection of options that amarket maker mightaccumulate over time as aresult of buying and sellingby customers.7 The currentmarket conditions (i.e.,underlying share price, timeto expiration, implied

volatility, and expecteddividends) are also shown inFigure21-14.

Figure21-14

To fully analyze theposition, we will need tomake some assumptionsabout the term structure ofimplied volatility. Here wewill assume that April is theprimary month and that themean volatility for thismarketis30percent.Wewillalso assume that the impliedvolatility for June changes at75 percent of the rate ofchange in April and the

implied volatility for Augustchanges at 50 percent of therate of change inApril.8Wecan see that the currentimplied volatilities areconsistent with this termstructure:

April (primarymonth)impliedvolatility =34.27%

Difference

from themean =34.27% –30.00% =4.27%

June impliedvolatility =33.20%

Differencefrom themean =33.20% –30.00% =

3.20% ≈0.75 ×4.27%

Augustimpliedvolatility =32.14%

Differencefrom themean =32.14% –30.00% =2.14% ≈

0.50 ×4.27%

The primary riskcharacteristics of theposition9—theoretical profitand loss (P&L), delta,gamma, and vega—areshown in Figures 21-15through 21-18.10 From thesegraphs, it is evident that therisks of the position canchange significantly asmarket conditions change,

with the delta, gamma, andvega gyrating betweenpositive and negative. Giventhis, how should we analyzetheposition?

Figure21-15Positionvalueastheunderlyingpriceandvolatilitychange.

Figure21-16Positiondeltaastheunderlyingpriceandvolatilitychange.

Figure21-17Positiongammaastheunderlyingpriceandvolatilitychange.

Figure21-18Positionvegaastheunderlyingpriceandvolatilitychange.

A market maker’sultimate goal is to establishpositions with a positiveprofit expectation whileintelligently managing risk.Indeed, were it not for thecomplexities of themarketplace and the uniquecharacteristics of options, amarketmaker’s lifemightbethought of as quite boringbecauseheistryingtodothesamethingoverandover:

Once a position with apositive theoretical edge hasbeen established, the marketmaker ideally would like toreduce all risks to 0 withoutgivingupanypotentialprofit.This would be identical toturning the graph in Figure

21-15intoasinglehorizontalline with a positivetheoretical P&L. In reality,with a large and complexposition, it is virtuallyimpossible to achieve such agoal. A more practicalapproach is to ask whatchanges inmarket conditionsrepresent the greatestimmediate threat to theposition and what steps canbe taken to mitigate thoserisks. Even this will depend

on many subjective factors:the trader’s appetite for risk,his capitalization, the extentofhistradingexperience,andhis familiarity with themarket. Unfortunately, thereare very few easy answerswhen it comes to riskanalysis.

Some risk limitationswill be set by the firm forwhich the traderworksorbythetrader’sclearingfirm.For

example,aclearingfirmmayrequire that a tradermaintainenoughcapital towithstanda20 percent move in theunderlying contract in eitherdirection. Or the firm mayrequire enough capital towithstand a doubling ofimplied volatility. If thetrader currently hasinsufficient capital to meetthese requirements, he musteither deposit additionalmoneywith theclearingfirm

or reduce the size of thepositionsothatitfallswithintheclearingfirm’sguidelines.

How should we analyzethe risk of the position inFigure 21-14? Risk analysisis important because itenablesatradertoplanahead—to decide what course ofaction is best—given achange in market conditions.Anoptiontradermayhavetoconsider many different

market scenarios, but it isoftenbesttobeginwiththreebasicquestions:

1. Whatwill I doifmarketconditionsmoveagainstme?2. Whatwill I doifmarketconditionsmove inmy favor?(Risk analysisshould focus notonly on protectingagainst the bad

things that mightoccur but also ontakingadvantageofthegoodthings.)3. What can I donow to avoid theadverse effects ofconditions movingagainstmeatalatertime?

What are the bad thingsthat can happen to theposition?Clearly,thegreatest

threat is a violent upwardmove.Aboveastockpriceof85,thepositionwilltakeonanegative delta and from thatpointonwillcontinuetolosemoney as the market rises(Figures 21-15 and 21-16).The upside contract position(the sum of all calls andunderlyingcontracts)is–76.

With a current delta of+203, there is also some riskof a declining stock price.

Thismaynotbeofimmediateconcern, but note that as thestock price declines toward62, the position takes on anincreasingly negative vega(Figure 21-18). This meansthat the position is at risk ifthe stock price fallsmoderately while impliedvolatilityrises.

InFigure21-17,we cansee that the position has amaximum positive gamma at

stockpricesofapproximately53and72.Ifthemarketwereto approach either of theseprices and remain there, theposition would most likelytakeonitsmaximumnegativetheta and consequently begintodecayveryrapidly.

Given the various risks,whatshouldbetheimmediateconcern? The answer mustnecessarily be subjective andwill depend on what this

trader knows about thecharacteristicsofthisstock.Ifthere is somepossibilityof areallylargeupwardmove,forexample, the company is atakeover target, it isincumbent on the trader tocover at least some of hisupside risk, perhaps bypurchasing higher-exercise-pricecalls.Admittedly, if theprices of the upside calls areinflatedbecausethecompanyis known to be a takeover

target, the cost of protectingtheupsidemaybehigh.But,if a takeover could result inthe trader’s demise, thismaybeapricethathewillhavetopay.

Of course, the tradermay believe that a largeupward move is so unlikelythatheiswillingtoaccepttherisk. Then he may want tofocus on some of the lesserthreatstotheposition.Ifheis

a disciplined theoreticaltrader, hemaywant to coverhis current delta position of+203, although this too mayrepresent such a small riskthat it is not of immediateconcern. Otherwise, he maywant to sell approximately200deltasinsomeform—sellstock, sell calls, or buy puts.The last choice, buying puts,especiallythosewithexerciseprices of 60 or 65,will havethe effect not only of

reducing the delta but alsoreducingthenegativevegaintherangeof60to65.Ifgiventhe choice, the purchase ofApril 60 or 65 puts willprobably show the greatestbenefit to the position. If thestock price does decline tobetween 60 and 65, theseoptionswillbeat themoney,and at-the-money short-termoptions have the greatestgamma.Assuch,theywilldothemosttooffsetthenegative

gammainthisrange.What changes inmarket

conditions might help theposition?Belowastockpriceof 55, the position will takeon a negative delta, so acollapse in the stock pricewill obviously provebeneficial. The downsidecontract position (the sum ofall puts and underlyingcontracts) is–29.If thestockpriceshouldclimbtoward85,

especially with fallingimplied volatility, this willalso be very favorable.Indeed,almostanydeclineinimplied volatility will helptheposition,asshownbythevegainFigure21-18.

Even though time decaymay not be an immediateconcern,itmaystillbeworthconsidering how the passageof time will affect theposition. The position has a

negative theta (consistentwith a positive gamma), sothepassageoftimewillworkagainstthepositionifthereisno change in the underlyingstockprice.Thetotalthetaof–1.90maybesmall,butnotethat most of the theta isconcentrated in April. AndtheApril position consists ofa large longposition inApril70 calls. As time passes, thetheta of these options,whichareclosetoatthemoney,will

accelerate, causing theposition to lose value at anincreasingly greater rate. Ifthe market remains close to70, it is also likely that therewill be a decline in impliedvolatility. Given theposition’s negative vega, thiswill work in the position’sfavor. Still, it may be worththinkingaboutwhatactiontotakeifthestockpriceremainsclose to70.Thevalueof theposition after the passage of

one and twoweeks is showninFigure21-19.

Figure21-19Positionvalueastheunderlyingpricechangesandtimepasses.

What else might hurtthis position? We haveassumed that the stock willpay a dividend of 0.58 in 10weeks. If the company hasnot officially announced thedividend, perhaps the actualdividendwillbemorethanorless than this amount. TheApriloptions,whichexpireinfour weeks, will beunaffectedbyachangeinthedividend. But how will the

overall position be affected?We can run a computersimulationathigheror lowerdividend amounts, butperhapsaneasierapproachisto note that the position islong 3,300 shares of stock.Because we own stock andtherefore receive thedividend, any increase in thedividend will cause thepositionvaluetorise,andanydecrease will cause theposition value to fall. The

change in value will beapproximately equal to thechange in the dividendmultiplied by the number ofshares of stock, in this case,3,300.

If there is a realpossibility that the dividendwill be reduced, one way toeliminatetheriskistoreplacethe long stock position withsynthetic long stock: sell thestock, and buy calls and sell

puts at the same exerciseprice. This is similar toreducing the risk of aconversion or reverseconversion by turning theposition into a box (seeChapter15).

The total rho of +12.70also indicates that there issome risk of falling interestrates. For each full-pointdecline(100basispoints11)ininterest rates, the position

valuewillfallby12.70.It is usually easiest to

analyze risk by generatinggraphs of a position’scharacteristics, as we havedone inFigures21-15 to 21-19. However, some tradersprefer to create a tableshowing the risk sensitivitiesat various underlying prices.Thishasbeendone inFigure21-20,beginningwithastockpriceof45andcontinuingat

five-point increments up to astock price of 95. The tableincludes not only thetraditional risk measures butalsothenontraditionalhigher-order measures discussed inChapter9.Thesehigher-ordermeasures can often give atrader a more completepictureofhowtherisksofhisposition will change asmarket conditions change.Forconvenience,welistthesemeasuresbelow:

Figure21-20Risksensitivitiesastheunderlyingpricechanges

StockSplits

To conclude our

discussion, let’s consideronelast change in marketconditions—a stock split.This often happens when acompany wants to reduce itsstockpricetopromotetradingin the stock or to encouragewiderownershipofthestock.If the stock price remainshigh, tradingactivitytendstobelimited,withownershipofthe stock concentrated infewerhands.

Supposethatthestockinour example splits 2 for 1,resultinginanewstockpriceof68.76/2=34.38.Whatwillhappen to the position?Where the trader previouslyowned 3,300 shares, he willnow own 2 × 3,300 = 6,600shares.Tomaintain the samerelationship between eachexercise price and the stockprice, as a result of the split,the clearinghouse will divideall the exercise prices by 2.

The 55 exercise price willbecome27½, the60exercisepricewill become30, and soon. The underlying contractwill remain 100 shares ofstock,butinordertomaintainequity,theclearinghousewilldoublethetrader’spositionateachexerciseprice.Insteadofbeing long 77April 55 calls,the trader will now be long154 April 27½ calls. Insteadof being short 162 April 60calls, the trader will now be

short324April30calls.How will the trader’s

risk position look now? Inorder to understand whathappens, let’s consider asimple example. With anunderlying stock trading at60.00,weownaMay60callwith a delta of 50 and agamma of 5. If the stocksplits 2 for 1, our positionwillnowbe

Because the option is stillatthemoney,thedeltawillbe50. But now we own twocalls, so our delta position

willdoubleto+100.What about thegamma?

Because the gamma is thechange in delta per pointchange in the underlyingstock price, if we candetermine the new deltapositionatastockpriceof31,we will know the gamma.Suppose that the stock pricerisesto31.Thisisequivalenttothestockpricerisingto62prior to the split. At a stock

priceof62(priortothesplit),ourdeltapositionwouldhavebeen+50+(2×5)=+60.Butthe stock split caused ourdelta to double, so the newdeltapositionatastockpriceof 31 must be 2 × +60 =+120. If the delta rises from100to120withastockpricechange from 30 to 31, thegamma of the position mustbe+20.

If a stock splitsY forX

(eachXnumberofshareswillbereplacedwithYnumberofshares), we can summarizethe new conditions asfollows:

These calculations holdtrue as long as the split isYfor 1, where Y is a wholenumber(e.g.,2for1,3for1,4 for 1, etc.). If Y is not awholenumber,thenumberofshares in the underlyingcontract may have to beadjusted. For example, usingour stock price of 60, whatwill happen if the stock issplit3 for2?NowY isnotawhole number because thesplitisequivalentto1½to1.

IfweownaMay60call,wecan make the followingcalculations:

Theproblemhereisthatthe clearinghouse does not

allow fractional optionpositions(+1½May40calls).In order to eliminate thefraction, the clearinghousewillreplaceeachMay60callbeforethesplitwithoneMay40 call after the split. At thesame time, the underlyingcontract will be adjusted sothat the new underlyingcontract is equal to the oldunderlying contractmultipliedbythesplitratio

100shares×3/2=150shares

Using these adjustments,the delta and gamma nowmake sense. The option is atthe money, so it should beequivalent to approximately50 percent of the underlyingcontract, or 75 shares. If thestock price rises to 41, equalto a price of 61½ before thesplit,theolddeltawouldhavebeen

50+(1.5×5)=57.5

The option would havebeen equivalent to 57.5percent of the underlyingcontract. Therefore, the newoption(the40call)shouldbeequivalentto

0.575–150shares=86.25shares

As expected, this is thesame as the delta (75) plus

thegamma(11.25).What happens to the

other risk measures—theta,vega, and rho—if a stocksplits?Thesenumbersremainunchanged. The passage oftime, changes in volatility,and changes in interest rateshave the same effect on apositionafterasplitasbeforea split. Only the delta andgamma must be adjusted.Indeed, assuming that all

other conditions remainunchanged, a stock split hasno real effect on a trader’sposition. It simply results inanaccountingchangeinsucha way that equity ismaintained. Of course, allother conditions may notremain unchanged. When astocksplits,wemightassumethat thedividendalsowillbesplit proportionally. But thisisnotnecessarily thecase.Astocksplitoftenindicatesthat

acompanyisdoingwell,andit is not unusual for the splitto be accompanied by anincreaseinthedividend.Anychange in the expecteddividend will change thevalue of an option position.Figure 21-21 shows thecharacteristicsofouroriginalposition after a 2-for-1 stocksplit with no change in theexpecteddividend.

Figure21-21Theeffectofa2-for-1stocksplit.

1Inordertofocusonthevolatilitycharacteristicsofthepositions,weassumeaninterestrateof0inthisandotherexamples.2Somereadersmayrecognizethispositionasariskreversalorsplitstrikeconversion.MoreonthisinChapter24.3Thepositionis,ofcourse,notcurrentlydeltaneutral.Ifwewanttodynamicallyhedgetheposition,wemustbeginbyoffsettingthecurrentdeltaof–297,perhapsbypurchasingthreeunderlyingcontracts.4Customerssometimesbelievethatmarketmakers“fix”thepricesofexchange-tradedcontracts.Thismaybetrueforshortperiodsoftime,usuallyat

thebeginningofthetradingdaywhenverylittleinformationisavailable.Ultimately,however,amarketmaker’squotesreflectcurrentmarketactivity.Amarketmakerdoesnotsetpricesanymorethanathermometersetsthetemperature.5Atraderwhotriestoprofitsolelyfromthebid-askspread,buyingatthebidpriceandsellingattheaskpricewithoutregardtotheoreticalvalue,issometimesreferredtoasascalper.Scalpingisacommontradingtechniqueinopen-outcrymarkets.6Inthiscase,themarketmakerhasapositivespeedposition.Hisgammapositionbecomesmorepositiveorlessnegativeasthepriceoftheunderlying

contractrises.7Anactivemarketmaker’spositionislikelytobemuchlargerthanthepositionshown,withhundredsoreventhousandsofoptionsateachexerciseprice.Forsimplicity,thepositionshownhasbeenscaleddown.Buttherisk-analysisprocesswillbethesame.8Wemightalsomakeassumptionsaboutthetermstructureofinterestrates,aswellashowimpliedvolatilityisdistributedacrossexerciseprices.Inordernottooverlycomplicatethecurrentexample,wewillassumeaconstantinterestrateacrossexpirationmonths,aswellasuniformimpliedvolatilitiesacrossexerciseprices.Weleavethediscussionofvolatilityskews

toalaterchapter.9Inthisexample,wehaveassumedthattheoptionsareEuropeanandhavemadeallcalculationsusingtheBlack-Scholesmodel.Therisk-analysisprocesswouldbethesameiftheoptionswereAmerican,althoughthecalculationsnecessarilywouldhavetobemadeusinganAmericanpricingmodel.10Becauseofthetermstructureofvolatility,thechangesinvolatilityinFigures21-15,through21-18areexpressedinpercenttermsratherthanpercentagepoints.Givenourassumptions(i.e.,meanvolatility=30percent,Juneimpliedvolatilitychanges75percentasfastasApril,August

impliedvolatilitychanges50percentasfastasApril),a20percentincreaseinvolatilityfromthecurrentlevelsresultsin

11Traderscommonlyexpresschangesininterestratesinbasispoints.Onebasispointisequalto1/100ofa

percentagepoint,or0.0001.

22

StockIndexFuturesand

Options

Because stock indexfutures and options areamong the most activelytraded of all derivatives, it

will be worthwhile to take acloserlookatthesecontracts.Eventhoughthefocusofthisbook is primarily options,stock index futures andoptionsaresocloselyrelated,and so many strategiesinvolvebothcontracts, that itis almost impossible todiscuss one withoutdiscussing theother.Wewilltherefore include bothinstrumentsinourdiscussion.

WhatIsanIndex?

An index is a number thatrepresents the compositevalueof agroupof items. Inthecaseofastockindex, thevalue of the index isdetermined by the marketpricesofthestocksthatmakeuptheindex.Asthestocksinthe index rise in price, thevalue of the index rises; asthe stocks fall in price, the

value of the index falls. Ifsomestocks in the index risewhile others fall, theoffsetting changes in stockpricesmayresultintheindexitself remaining unchanged,even though the price ofeverystock in the indexmayhave changed. Although theindex is made up ofindividualstocks,thevalueofthe index always reflects thetotal value of the stocks thatmakeuptheindex.

Stock indexes are oftenclassified as being eitherbroadbasedornarrowbased.A broad-based index isusually made up of a largenumber of stocks and isintended to represent thevalue of the market as awhole or at least a largeportionof themarket.Beloware some widely followedbroad-basedindexes.

The designation of anindex as broad based can besomewhatsubjective.Even ifan index is composed of asmaller number of stocks, itmaystillbeconsideredbroadbased if the companies thatmakeuptheindexrepresentawide cross section of theeconomy in a country orregion.

Anarrow-based index isusually composed of a smallnumberofstocksandreflectsthe value of a particularmarketsegment.

CalculatinganIndex

There are several methodsthat can be used to calculatethevalueofastockindex,butthe most common methodsfocus on either the prices ofthestocks in the indexor thecapitalization of thecompanies that make up theindex. To see how thesemethods work, consider theABC Index composed of thefollowingthreestocks:

The market capitalizationof each company is equal tothe stock pricemultiplied bythe number of outstandingshares. This represents thetotalvalueof all stock in the

company.If an index is price

weighted, the value of theindex is the sum of theindividualstockprices

ABCIndex(priceweighted)=∑pricei=80+20+50=

150

Ifanindexiscapitalizationweighted (cap weighted forshort), thevalueof the indexis the sum of the individual

capitalizations

ABCIndex(capweighted)=∑(pricei×sharesi)=8,000+40,000+20,000=68,000

SupposethatthepriceofStock A rises 10 percent to88.HowwillthevalueoftheABC Index change if theindex is priceweighted?Thenewindexvaluewillbe

88+20+50=158

Inpercent terms, this isanincreaseof

8/150=5.33%

We can make the samecalculation for the price-weighted index if Stock Brises 10 percent to 22 or ifStock C rises 10 percent to55. The percent increases intheindexare

StockB:2/150

=1.33%StockC:5/150=3.33%

In percent terms, changesin the highest-priced stock,Stock A, have the greatesteffect on the value of theindex. Stock A has thegreatest index weighting—itaccounts for the largestportion of the index.We cancalculate the role that eachstock plays in the index by

calculating the individualweightings:

We can also calculatetheweightingsforeachstock(with small rounding errors)if the ABC Index iscapitalizationweighted:

NowStockB,thestockwiththegreatestcapitalization,hasthegreatestindexweighting.

In a price-weightedindex,stockswiththehighestprice have the greatest indexweighting.Inacapitalization-weighted index, stocks with

the greatest capitalization(stocks with a large numberof outstanding shares) havethegreatestweighting.

We can also create anequal-weighted index where,in percent terms, each stockplaysexactlythesameroleinthe index.Wecando thisbymaking the initialcontribution of each stock tothe index identical. Forexample, suppose that

initially the value of ourindexis

∑(pricei/pricei)=1+1+1=3

Hereeachstockcontributesexactly 33.33 percent to theindex. Of course, if wealways divide each stock byitself, the value of the indexwillneverchange.Butthisisonlythevaluewhentheindexis first introduced.

Subsequently, as the price ofeach stock changes, the newprice is divided by the oldprice to determine the newvalueoftheindex.Ifanyonestock in the index rises 10percent, the effect on theindex will be the samebecause

88/80=22/20=55/50

If all three stocks rise 10percent, thenewvalueof the

indexwillbe

88/80+22/20+55/50=1.10+1.10+1.10=3.30

The indexwill riseexactly10percent.1

Astimepassesandsomestocks in an equal-weightedindex outperform otherstocks, the weighting of thestockswillchangesothattheindexwillnolongerbeequalweighted. In order to ensure

that each stock in the indexaccounts for approximatelythe same value, equal-weighted indexes areperiodicallyrebalanced.

Suppose that at a laterdate the prices of Stocks A,B, andC are 76, 25, and51,respectively.Thevalueoftheequal-weighted index nowwillbe

76/80+25/20+51/50=0.95+1.25+1.02=3.22

StockBnowaccountsforagreater portion of the indexthan either StockA or StockC. To ensure that all stocksagain have an equalweighting, the index is nowrebalanced

76/76+25/25+51/51=3.00

Of course, the index valueof 3.00 seems inconsistentwith the preceding indexvalue of 3.22. In order to

generate a continuous indexvalue, the index value afterthe rebalancing must bemultiplied by the percentincrease in the index duringthe previous rebalancingperiod. In our example, theindex after the rebalancing,we must multiply the indexvalueby

3.22/3.00=1.0733

because the index rose by

7.33 percent over the lastrebalancingperiod.

It is a relatively easytask to add up a list ofindividual stock prices.Consequently, the earliestindexeswere priceweighted.The Dow Jones IndustrialAverage, introduced in 1896,isprobablythebestknownofall price-weighted indexes.However, a capitalization-weighted index gives amore

accurate picture of eachcompany’s value. With theadvent of computertechnology to make thecalculations, most widelyfollowed indexes are nowcapitalizationweighted.

The total capitalizationofacompanydependsonthenumberofoutstandingsharesin the company. However,company restrictions mayprevent some of these shares

from being available fortrading. Shares held in thecompany treasury, bycompany officers, or inemployee investment plansmay not be available to thepublic. The shares that areavailable for trading arereferred to as the free float,anditisthenumberofsharesinthefreefloat that typicallyisused to calculate thevalueof a capitalization-weightedindex.

TheIndexDivisorWhen an index is first

introduced, it is common toset the value of the index tosomeroundnumber.Supposethat we initially want thevalueoftheABCIndextobe100. To accomplish this, wemust adjust the raw indexprice of either 150 (priceweighted) or 68,000 (capweighted) by using a divisortoachieveourtargetvalueof

100.Because

Rawindexvalue/divisor=targetindexvalue

thedivisormustbe

Divisor=targetindexvalue/rawindexvalue

For our ABC Indexes, therespectivedivisorsare

Once the divisor hasbeen determined, allsubsequentindexcalculationsaremadebydividingtherawindexvalueby thedivisor. Ifthe price of Stock B rises to25, the price-weighted indexvalue, which was initially100,willnowbe

(80+25+50)/1.50=

155/1.50=103.33

The capitalization ofCompanyBwillnowbe25×2,000= 50,000, and the cap-weightedindexvaluewillbe

(8,000+50,000+20,000)/680=78,000/680=

114.71

It is sometimesnecessarytoadjustthedivisorto ensure that the index

accurately reflects theperformance of thecomponent stocks. Considerwhatwill happen ifStockA,which was trading at 80,splits2for1.Thestockpriceis now 40, but with 200shares outstanding. StockpricetotalSharesoutstandingMarketCapitalization

If the ABC Index is priceweighted, the index value,which was previously 100(using our divisor of 1.50),willnowbe

(40+20+50)/1.50=110/1.50=73.33

But is this logical? Fromthe point of view of aninvestor in Company A, thevalueofhisholdingshasnotchanged, so why should theindex value change? Togenerate a continuous andlogicalindexvalue,theindexdivisor must be adjusted.With a new raw index valueof 110 and a target index

value of 100 (assuming thatno other price changesoccurred), the new indexdivisorwillbe

Newdivisor=110/100=1.10

When an index divisor isadjusted, the organizationresponsibleforcalculatingtheindex will typically issue apress release announcing thenew divisor and the reasonfortheadjustment:“Thenew

ABCIndexdivisor is1.10asa result of the 2 for 1 stocksplitofCompanyA.”

How will the 2 for 1stock split affect the divisorin our cap-weighted ABCIndex? We can see that thecapitalization of Stock A isunchanged at 8,000.Therefore, no adjustment isrequired. The divisor is still680.

The component stocks

thatmakeupanindexarenotpermanent. A company maycease to exist because it hasgone out of business orbecause it has been takenoverbyanothercompany.Ora company may no longermeetthecriteriaforinclusionin an index because its priceor capitalization has droppedbelow some threshold. Tomaintain a constant numberof index components, acompany that is removed

from an index must bereplaced with a newcompany.Thiswillrequireanadjustmenttothedivisor.

Suppose that CompanyC is replaced in the ABCIndex with Company D,currently trading at 75 with500sharesoutstanding:

Thenewdivisors forABCIndexwillbe

Total-ReturnIndexesInatraditionalstockindex,

when the price of acomponent stock falls, theprice of the index will fall.This is true even if the pricedecline is the result of adividend payout. In a total-returnindex,alldividendsareassumed to be immediatelyreinvested in the index.Consequently, stock pricedeclines resulting from a

dividendpayoutdonotcausetheindexvaluetodecline.

The value of the price-weighted ABC Indexcomposed of our originalthreestockswithadivisorof1.50is

(80+20+50)/1.50=100.00

IfStockApaysadividendof 1.00 and opens at a priceof79ontheex-dividendday,the opening index value will

be

(79+20+50)/1.50=99.33

But if theABC Index is atotal-return index, theopening index value willremain at 100 because the1.00 decline in Stock A wassolely the result of thedividendpayout.Tomaintainan index value of 100, theindex divisor must beadjustedto1.49because

(79+20+50)/1.49=100.00

Whenever a componentstock in a total-return indexpays a dividend, the divisorwillbeadjustedtoreflect thedividendpayout.

Although they are lesscommon than traditionalindexes, there are somewidely followed total-returnindexes. The best known oftheseisprobablytheGermanDAXIndex.Occasionally,an

index, such as the StandardandPoor’s (S&P)500 Index,will be published in twoversions,asbothatraditionalindexandatotal-returnindex.The former, however, ismuchmorewidelyfollowed.

ImpactofIndividualStockpriceChangesonanIndex

Ifanindividualcomponentstockpricechanges,howwillthis affect the value of anindex? Suppose that thecurrent value of the price-weighted ABC Index is I. IfthepriceofStockA(priceA)changes by an amount a,whatwillbethenewvalueofthe index? The raw value ofthe index should rise by abecause

(A+a)+B+C=I+a

In percent terms, thechange in the index is a/I.Supposethatwerewritea/Iinaslightlydifferentform

a/I=(a/A)×(A/I)

a/Aisthepercentchangeinthe stock price, while A/I isthe stock’s weighting in theindex.Thepercent change inthe index must therefore beequaltothepercentchangeinthe stock multiplied by the

stock’s weighting in theindex.This is true regardlessofwhether the index is priceweighted, cap weighted, orequalweighted.

We can confirm thisthroughanexample.Supposethat Stock A in the price-weighted ABC Index,currently trading at 100witha divisor of 1.50, rises onepoint.Thenewindexvalueis

(81+20+50)/1.50=

151/1.50=100.67

The weighting of Stock A(beforeitsone-pointrise)was53.33 percent, so the percentchangeintheindexshouldbe

0.5333×(1/80)=0.5333×0.0125=0.0067

From this we get a newindexvalueof

(1+0.0067)×100=100.67

For the cap-weightedABCIndex,currently tradingat 100with a divisor of 680,thenewindexvaluewillbe

[(81×100)+40,000+20,000]/680=68,100/680=

100.147

The weighting of Stock A(beforeitsone-pointrise)was11.76 percent, so the percentchangeintheindexshouldbe

0.1176×(1/80)=0.1176×0.0125=0.00147

From this we get a newindexvalueof

(1+0.00147)×100=100.147

As the price of each stockchanges, thenew indexpriceis

Oldindexprice×(1+∑

percentchangei×weighti)

For a price-weightedindex, if we know the indexdivisor, we can simplify thiscalculationbynoting that thechange in the index per one-pointmove in any individualcomponent is always givenby

Changeinstockprice/divisor

Each 1.00 change in a

stock in the price-weightedABC Index will cause theindextochangeby

1.00/1.50=0.67

It may seem odd thatevery point change in acomponent stock has thesameeffectonan index. If acomponent stock rises onepointand then risesa secondpoint, the second point willcause a smaller percent

increase in the stock price.One might therefore expectthe second point to have asmaller effect on the index.But this is offset by the factthateachpointincreaseinthestock also increases thestock’s weighting in theindex. Taken together, thepercent change in the stockand itsweightingcombine toyieldaconstantpointchangeintheindex.

A trader canoccasionally use theforegoing calculations tomake a more accurateestimate of an index’s truevalue. Most indexes arecalculated from the last tradeprice of each componentstock.Butthelasttradepricemay not be an accuratereflection of where the stockis currently trading. Supposethat trading in an indexcomponent stock has been

temporarily halted.2 Theindexvaluewill be based onthe last trade price of thehalted stock, but this lastpricemay differ significantlyfromtheexpectedpricewhentradinginthestockresumes.

Suppose that the currentvalueofan index is1,425.50and that the last trade pricefor a component stock is63.00. However, trading inthe stock has been halted

pendingnewsthatisexpectedtocausethestockpricetorisesignificantly. Although noone knows the exact price atwhich the stock will reopen,the indication (very oftendisseminated by theexchange) is somewherebetween 67.50 and 68.00. Iftheweighting of the stock inthe index is 2.5 percent, anindextradermightuseapriceof 67.75 to estimate the newindex price when the stock

reopens,thatis,

1,425.50×[1+.025×(67.75–63.00)/63.00]=1,428.19

Alternatively, the tradermayhavealreadydeterminedthat eachpoint change in thestock price will cause achange of 0.57 in the indexvalue, yielding a new indexestimateof

1,425.50+(4.75×.57)=

1,428.21

Either estimatewill enablethe trader to make a moreinformeddecision.

Volume-WeightedAveragePriceThe indexvalueat theend

of a trading day is usuallydetermined by the last priceof each component stock

when trading closes. But thelast trade price may notaccurately reflect tradingactivityinthestock.Supposethatatthecloseofthetradingdaythequotedbid-askspreadforastockis43.10–43.30andthat thevery last trade in thestockwasfor300sharesataprice of 43.30. Suppose,however,thatjustpriortothelasttrade,2,400sharestradedat43.15,andjustpriortothat,another1,800sharestradedat

43.10.Thelasttradeof43.30seems tobe an anomaly, andlogic suggests that perhapsoneof theotherpricesoughtto be used for the indexcalculation. To solve thisproblem,someexchangesusea volume-weighted averageprice (VWAP) over adesignated period prior toclosing.Inourexample,ifthelast three trades during theVWAP period are those justgiven, the closing price for

thestockwillbe

[(300×43.30)+(2,400×43.15)+(1,800×43.10)]/(300+2,400+1,800)=

43.14

The volume-weightedaveragepriceof43.14willbeused to calculate the indexvalue.

StockIndexFutures

In theory,onecancreateafutures contract on a stockindexinexactlythesamewaythat futures contracts arecreated on traditionalcommodities. At expiration,the holder of a long stockindexfuturespositionwillberequired to take delivery ofall the stocks that make upthe index in their correctproportions. The holder of ashortpositionwillberequiredto make delivery of the

stocks.In fact, no stock index

futures contracts are settledthrough the physical deliveryofthestocksthatmakeuptheindex. Such a process,requiring the delivery of thecorrect number of shares ofmany different stocks,wouldbe unmanageable for mostclearing organizations.Moreover, settlement mightrequire the delivery of

fractional shares of stock,which is not possible. Forthese reasons, exchangestypically settle stock indexfutures at expiration in cashrather than through physicaldelivery of the componentstocks.

Aswithallfutures,stockindex futures are subject tomargin and variation, with afinal cash payment equal tothe difference between the

expiration value of the indexandthepreviousday’sfuturessettlement price. If the indexvalue at the moment ofexpiration is 462.50 and thepreceding day’s settlementprice for the futures contractwas 461.00, the holder of alongpositionwillbecreditedwitha finalpaymentof1.50.If the value of each indexpointis$100,thelongfuturespositionwillbecreditedwith$100× 1.50= $150, and the

short futures positionwill bedebited by an equal amount.Once this final payment hasbeen made, both parties areout of the market andunaffectedbyanysubsequentindexmovement

What should be the fairprice for a stock indexforward contract? In Chapter2, we calculated the forwardprice for an individual stockbyaddingtheinterestcoststo

the stock price (the cost ofbuying now) and subtractingthe expected dividends (thebenefitofbuyingnow)

F=S×(1+r×t)–D

The forward price for theindexcanbecalculatedusingthe same procedure.We addtheinterestcosttothecurrentindex price and subtract thetotaldividends that the indexcomponents are expected to

pay prior to maturity. Butunlike an individual stock,where dividends are paid inone lump sum, the dividendpayments for an index arelikely to be spread out overtime.An exact forward pricecalculation requires us toknow the amount of thedividend for each stock, thepayment date, and theweighting of the stock in theindex. From this, we cancalculatethetotalvalueofall

the dividends, including theinterestthatcanbeearnedoneach dividend payment fromthe payment date tomaturityoftheforwardcontract.

Clearly, calculation ofthe dividend payout and,consequently, calculation ofthe forward price can berather complex. To simplifythiscalculation,many tradersuse an approximation bytreating the dividend flow as

if it were a negative interestrate

F=S×[1+(r–d)×t]

where d is the averageannualized dividend, inpercent terms, for the index.If

Current indexprice=100.00Time tomaturityofthe

forwardcontract = 4monthsInterest rate =6.00percentAverageannualizeddividendpayout = 2.25percent

the three-month forwardpriceoughttobe

100.00×[1+(0.06–0.0225)×4/12]=100.00×1.0125=

101.25

For long-term forwardcontracts, this approximationrepresents a reasonabletradeoff between ease ofcalculation and accuracy.Unfortunately, for short-termcontracts, the fact thatdividend payments come indiscrete bundles that arespread out unevenly over the

life of the forward contractcanresult in largeerrors.Wecan see this in Figure 22-1,which shows the dailydividend payout of the DowJones Industrial Index over athree-monthperiod.The totalannualized dividend isapproximately 2.75 percent,butdependingon the time tomaturity of a forwardcontract, thisvaluecaneitheroverstate or understate thetruedividendpayout.

Figure22-1dowJonesIndustrialIndexdailydividendpayout,october–december2012.

Suppose that a forwardcontractmaturesattheendofthe three-month dividendcycle. If a position in theforward contract is taken atthe beginning of this period,the 2.75 percent estimate ofthe dividend flow is areasonablyaccuratereflectionoftheactualdividendpayout.However, if the position istaken toward the end of thethree-month period, after all

thedividendshavebeenpaid,2.75 percent is a grossoverstatement; the truedividendpayoutiscloseto0.ThedottedlineinFigure22-1shows the true dividendpayout, on an annualizedbasis, from thatpoint in timeto maturity. If a position istakenwhen thedotted line isbelow 2.75 percent, thisestimate overstates the truedividendpayout.Ifapositionis takenwhen thedotted line

is above 2.75 percent, thisestimate understates the truedividendpayout.

IndexArbitrageIn February 1982, the

Kansas City Board of Tradebegan trading futures on theValueLineStockIndex.Thiswas the first exchange-tradedstock index futures contractlisted in the United States.

Two months later, in April1982,theChicagoMercantileExchange began tradingfutures on the S&P 500Index.

In theory, the price of afuturescontractshouldreflectthe fair value of holding thefutures contract rather thanholdingthestocksmakingupthe index. If the futurescontract is not trading at fairvalue,atradercanexecutean

arbitrage by purchasing oneasset, either the basket ofstocksorthefuturescontract,andselling theother. If thereare no other considerations,the trader should realize aprofitequal to themispricingof the futures contract.However,thisprofitwillonlybefullyrealizedatexpirationof the futures contract, atwhich time the futurescontract and indexvaluewillconverge. At expiration, the

value of the futures contractwill automatically be settledin cash, but the trader willhave to place an order toliquidate the stock position.He will want to do this insuch away that the prices atwhichthebasketofstocksaretradeddeterminethevalueofthe index at the moment ofexpiration. This can be doneby placing amarket-on-closeorder, guaranteeing that thelasttradepriceforeachstock,

which determines the finalindex value, will be theliquidation price for thetrader’sstockholdings.

Index arbitrage entailsrisks similar to any stockfutures arbitrage strategy. Ifthe trade has not beenexecuted at a fixed interestrate, any change in ratesrepresents a risk to theposition. If dividends havebeen incorrectly estimated,

this will also affect theprofitability of the strategy.Moreover, if the strategyinvolves selling stock short,theremaybe restrictions thatmakethestrategyimpractical.Andevenifstockcanbesoldshort, the short interest ratemay make the strategyunprofitable. This type ofstrategy,where a trader buysor sells a mispriced stockindex futures contract andtakesanopposingposition in

theunderlyingstocks,isoftenreferredtoasindexarbitrage.Because computers can beprogrammed to calculate thefairvalueofafuturescontractand to execute the arbitragewhen the futures contract ismispriced,suchstrategiesarealso known as programtrading.

With the advent ofcomputer-driven trading,index arbitrage has become

an increasingly popularstrategy. When a computerdetects an index futurescontract that is mispricedwith respect to the indexitself, the computer can sendorders to either sell futurescontracts and buy thecomponent stocks (a buyprogram) or buy futurescontracts and sell thecomponent stocks (a sellprogram). Once the strategyhas been executed, it will

usually be carried toexpiration, atwhich time theposition will be liquidatedthrough a market-on-closeordertoeitherbuyorsellthecomponent stocks. Initially,exchanges were able toprocess market-on-closeorders resulting from indexarbitrage strategies withoutsignificant problems.However,asthepopularityofprogram trading increased,exchanges found that as the

close of business approachedon the last day of trading,they were receiving ever-largermarket-on-closeorders.These large orders oftencaused disruptions in thenormal trading process, withunexpected jumps in theprices of component stocks.For this reason, manyderivative exchanges, at thebehest of the relevant stockexchanges, agreed to settleindex futures contracts at

expiration based on theopening prices of thecomponent stocks rather thanthe closing prices. Thiseliminated a last-minute rushto buy or sell stock andenabled stock exchanges tomoreeasilymatchupbuyandsellorders.

Settlement at expirationbased on opening pricesrather than closing prices isnow used for most stock

index futures and optioncontracts. This settlementprocedure is sometimesreferred to asAM expiration.PM expiration, where thesettlement value isdetermined by closing pricesat theendof the tradingday,is still used for a smallnumber of stock indexcontracts.3

ReplicatinganIndexSometimes a trader will

want to create a holding ofstocks that exactly replicatesthevalueoftheindex.Hecandothisbyholdinganamountof each stock in the exactproportion to the stock’sweightintheindex.

Returning to our ABCIndex, we had the followingvalues:

If a trader wants toreplicate the price-weighted

ABCIndex,53.33percentofhis holdings should be inStock A, 13.33 percent inStockB,and33.33percentinStockC. If a traderwants toreplicate the capitalization-weighted ABC Index, 11.76percentofhisholdingsshouldbe inStockA, 58.82 percentinStockB,and29.41percentin Stock C. If the trader has$100,000 to invest, he needstoholdthefollowingnumberofsharesineachstock:

Because the weightingof each stock in a price-weighted index isproportional to its price, wecanreplicateaprice-weighted

indexbypurchasinganequalnumber of shares of eachcomponent stock. The same,however, is not true for thecapitalization-weightedindex,wheretheweightingofeach stock is proportional toitstotalcapitalization.Inbothcases, however, we canconfirm that the propernumber of shares willreplicate a $100,000investmentintheindex

(667×80)+(667×20)+(667×50)≈$100,000(price

weighted)(147×80)+(2,941×20)+(588×50)≈$100,000(capitalizationweighted)

Why might someonewant to replicate an index?An investor may want to doso in order to earn a returnequal to that of the index.This is a commonmethodofdiversifying investments.

Indeed, the investor mayfurther diversify byreplicating several indexesrepresenting various marketsegments. A trader may alsowant to replicate an index inorder to take advantage of amispriced arbitragerelationship. If a stock indexfutures contract istheoretically overpriced, thetrader may seek to sell thefutures contract andpurchaseall the component stocks.He

will need to do so in such awaythatheexactlyreplicatestheindexfuturescontract.

Theamountofstockthatthe trader will need topurchase will depend on thesize,ornotionalvalue,of thefuturescontract.This,inturn,will depend on the indexmultiplier that the exchangehas assigned to the futurescontract. Suppose that ourcapitalization-weighted index

with a divisor of 68,000 iscurrently trading at 100.00and that the exchange hasassigned a multiplier of$1,000 to each point. Thenotional value of the futurescontractistherefore100.00×$1,000 = $100,000. Giventhis,itmayseemthatatraderwho is able to sell anoverpriced index futurescontract can offset thisposition by purchasing 147shares of Stock A, 2,941

shares of Stock B, and 588shares of Stock C. Theproblemwiththisapproachisthat the trader needs toreplicate the futures contract,not theactual index.And thefutures contract and indexmay have differentcharacteristics.

To understand whyreplicating the indexwill notexactly offset the futuresposition, consider what will

happen over the life of thefutures contract while thetrader is waiting forexpiration, when the futuresprice and index price willconverge. The prices of thestocks will surely fluctuate,resulting in either a profit orlosstohisstockposition.Butthis profit or loss will beunrealized because the tradermust hold the position toexpiration to ensure anarbitrage profit. At the same

time, the profit or lossresulting from futurescontract will be immediatelyrealized, resulting in avariation credit or debit eachday. If there is a variationcredit, the trader will earninterest;ifthereisavariationdebit, the trader must payinterest. In either case, theresulting interest will changethe arbitrage profit that thetrader originally expected.This is another example of

settlement risk, which wediscussed in Chapter 15. Aposition that exactlyreplicates the index is animperfect hedge against thefutures contract because oneside is subject to stock-typesettlement, while the othersideissubject tofutures-typesettlement. Given this, whatshouldbethecorrecthedge?

Ignoring dividends, thefair value of a stock index

forwardcontractis

F=S×(1+r×t)

For each point increase inthe index, the index futurescontractshouldriseby1+r×t. If we think of the cashindex as the underlyingcontract, we can apply theconcept of the delta to thefutures contract in much thesamewaywedotoanoptioncontract.Thedelta is therate

at which the value of acontract will change withrespect to movement in theunderlying contract. If thegoalistobedeltaneutral,foreachfuturescontractwehold,we must hold an opposingcashindexpositionequalto1+r×t.

The magnitude of thefutures delta will depend onboth the amount of timeremaining to expiration and

thelevelofinterestrates.Fora long-term futures contractin a high-interest-rateenvironment, the requiredholdings in the componentstocks may be considerablygreater than the equivalentfutures position. Asexpirationapproachesor inalow-interest-rateenvironment, the futures andstockholdingswillbealmostidentical. Consequently, anindex arbitrage strategy

requires an adjustment to thestock position as time passesorinterestrateschange.

Suppose that there arefour months to expiration ofour ABC Index futurescontract and that the annualinterestrateis6.00percent.Ifwe sell an overpriced futurescontract, we must offset thiswithalongstockholdingof1+ 0.06 × 4/12 = 1.02, or 2percent greater than the

holding required foranexactindex replication. If amonthpasses so that there are nowonly three months toexpiration, we should reduceourstockholdingto1+0.06×3/12=1.015,or1½percentgreater than the sharesrequired for an exact indexreplication. The requiredholdings for thecapitalization-weighted ABCIndexareasfollows:

A change in interestrates will not only affect thedelta of the futures contractbut can also affect theprofitability of an index

arbitrage strategy. If a traderinitiates a buy program (i.e.,buystocks,sellfutures),heiseffectivelyborrowingcash inorder to purchase the stocks.Ifthecostoffundsistiedtoafloatinginterestrate,anyrateincreasewillhurthisposition,andanydecreasewillhelp.Ifhe institutes a sell program(i.e.,sellstocks,buyfutures),heiseffectivelylendingcash.Now any rate increase willhelp his position, and any

decrease will hurt it. If thechange in interest rates issufficiently large, an initiallyprofitable strategy mightbecome unprofitable. This isespeciallytrueiftheprogramtrade consists of long-termfutures contracts. In such acase, the interestconsiderations are magnifiedbecauseofthegreatercostsofborrowing or lending overextendedperiods.Inthesameway, because of reduced

interest considerations,changes in interest rates areunlikely to affect programtrades consisting of short-termfutures.

We have also assumedthatthedividendpayoutofallthestocksinanindexremainsconstant. But this is notnecessarily true. Companiescanhavegoodyearsandbadyears, and their dividendpolicies can change

accordingly. In a buyprogram(i.e.,buy thestocks,sell the futures),any increasein dividends will help theposition, and any decreasewill hurt. In a sell program(i.e., sell the stocks, buy thefutures), the opposite is true.In a broadly based indexconsisting of hundreds ofstocks, it is unlikely that achangeinthedividendpolicyof any one company or evenseveralcompanieswillhavea

significant impact on theprofitability of a programtrade. But in a narrow-basedindexconsistingofonlyafewstocks, a change in theexpected dividend payout ofeven one firm can alter thepotential profitability of thetrade. In such a case, thetradermustcarefullyconsiderbeforehand the possibility ofa dividend change for thecompanies that make up theindex.

BiasintheFuturesMarketStock index futures are

among the most liquid andactively traded of all futurescontracts. These marketsenable all types of traders tomake decisions based ongeneral market conditionsrather than on uniqueconditions that might affectan individual stock. Most

traders believe that thegeneralmarketis lesssubjectto manipulation thanindividual stocks and thatindex markets offer a morelevelplayingfield.

One especially activeparticipant in the stock indexmarket is the portfoliomanager whose goal istypically to generate amaximum return on capitalwith a minimum amount of

risk. Historically, a portfoliomanager has achieved thisgoal intheequitymarketsbymaintaining a portfolio ofstocks that the managerbelieves will outperform thegeneral market. As themanageridentifiesnewstocksthat meet this criterion, headds them to the portfoliowhileatthesametimesellingoffstocksthathaveeithermethisperformancegoalsorhaveceased to perform as

expected.Occasionally,amanager

with an equity portfoliomaywant to protect his holdingagainst an expected short-term decline in the generalmarket. Prior to theintroductionof index futures,theonlywaytodothiswastosell off the stocks in theportfolio and then buy thembackata laterdate.Notonlywas this time consuming but

the transaction costs alsotendedtoreducetheexpectedprofits fromtheposition.Butwiththeintroductionofindexfutures a manager with abroadly based portfolio maydecide that his holdings tendto mimic an index on whichfutures are available. If themanager believes that thecharacteristicsofhisportfolioare sufficiently similar to theindex, index futures offer amethodofhedgingthestocks

in the portfolio without thetime-consuming and costlyprocess of selling eachindividual stock in theportfolio.

The effect of portfoliohedging strategies on stockindex futures tends to resultinaone-sidedmarketbecausethe vast majority of equityportfolio managers take longpositions in equities. Even ifa manager believes that a

stock will underperform themarket, it is much lesscommonforamanagertosellstockshort(sellstockthathedoes not own) as part of hisinvestmentprogram.Hence,aportfolio manager is almostalwaystryingtohedgealongposition in the market. Toachieve this, a portfoliomanagerismostoftensellingfutures contracts. Thisconstant selling pressuretends to depress the price of

futures contracts comparedwiththeoreticalvalue.

If therewereasurewaytoprofit from thisdownwardbias in the market,arbitrageurs would take theopposite position in theunderlying index. But wehave seen that replicating anindexwithabasketofstocksis not always possible.Moreover,whentheportfoliomanager protects his long

equity position by sellingfutures, a market maker orarbitrageurendsuptakingtheopposite position; he isbuyingfutures.Ifhewantstohedge his position with anunderlying basket of stocks,he must sell stocks short. Insome markets, the short saleof stock may be prohibited,but even if short sales arepermitted,sellingstocksshortis never as easy as buyingstocks. Moreover, the short

sale of a stock, as discussedin Chapter 2, may not earnfullinterest.

Given all these factors,buyingandsellingpressureinthe stock index futuresmarket is not symmetrical.Many more factors seem toresult in downward pressureonfuturespricesthanupwardpressure.Thisdoesnotmeanthat such markets can neverbecome inflated,with futures

contracts trading at pricesgreater than fair value, butthisisbyfartheexception.Instock index markets aroundthe world, there tends to beconstant downward pressureonfuturesprices.

StockIndexoptions

There are really two typesofstockindexoptions—thosewhere the underlying is an

index futures contract andthosewheretheunderlyingisa cash index. Although theyare alike in many respects,they also have uniquecharacteristics that set themapartfromeachother.4

OptionsonStockIndexFuturesExchange-traded options

on stock index futures were

first listed in the UnitedStates inJanuary1983,whenthe Chicago MercantileExchange began tradingoptions on S&P 500 futurescontracts. Options on stockindexfuturesareevaluatedinthe same way as any otherfutures option. Exercise orassignment results in afutures position, which isimmediately subject tomargin and variation. Theonly time exercise or

assignmentdoesnot result inafuturespositioniswhentheoptions and the underlyingfutures contract expire at thesame time. Because moststock index futures trade onthe March-June-September-December quarterly cycle,therearefourtimeseachyearwhen stock index futures,options on futures, andoptions on the cash index allexpireat the same time.Thistriple witching typically

occurs on the thirdFriday ofthe contract month, when allexpiring stock indexcontracts, both futures andoptions,aresettledincash.

Consider a trader whoowns a February 1,000 callon a stock index futurescontract.BecauseFebruaryisa serial month (there are noFebruary futures), theunderlying contract is theMarch future. If the March

future is trading at 1,025 atFebruary expiration, thetrader will exercise theFebruary1,000call, resultingin a long March futuresposition. Unless the traderimmediately sells the Marchfuture, the position will besubject to a marginrequirement that the tradermust deposit with theclearinghouse. At the sametime, the trader, throughexercise, will buy a March

futures contract at 1,000.Withthefuturescontractnowtrading at 1,025, the trader’saccountwillbecreditedwith25.00 points times the indexpointvalue.Ifthepointvalueis $100, the trader’s accountwill be credited with 25 ×$100 = $2,500. In the sameway,atraderwhoisassignedonaFebruary1,000callwillhave a short March futuresposition. Unless the traderbuys back the March future,

he will also be required topostmargin, and his accountwill be debited by $2,500.Boththetraderwhoexercisesandthetraderwhoisassignedstill have market positions.One traderhasa longfuturesposition and therefore wantsthemarket to rise. The othertrader has a short futuresposition and therefore wantsthemarkettodecline.

Now consider what will

happen at expiration to atrader who owns a March1,000 call in the same indexfutures market. Unlike theFebruary option, which issubject toPMexpiration(theoption essentially expires atthe close of business onexpirationFriday),theMarchoption is subject to AMexpirationbecause theMarchfuture is subject to AMexpiration. The value of theMarch future will be

determined by the openingprices of all the componentstocks on expiration Friday,and this, in turn, willdetermine the value of theMarch1,000call.Ifthecallisout of the money, it willexpireworthless.Ifthecallisin the money, the exchangewill automatically settle allexpiring in-the-moneyoptions in cash. The traderwho owns the call will becredited with an amount

equal to the differencebetween the exercise priceand the opening index valuetimes the indexmultiplier. Ifthe opening index value is1,040 and the multiplier isagain$100, thetraderwhoislong the option will becredited with $4,000. At thesame time, the trader who isshort the option will bedebited by an equal amount.Moreover, once this cashtransfer takes place, both

tradersareoutof themarket.Whether the indexsubsequently rises or falls isof no consequence becauseno market position resultsfromthecashsettlement.

Options on stock indexfutures, like most futuresoptions, are American andtherefore carry the right ofearly exercise. If the optionsare subject to stock-typesettlement, as they are in the

United States, there may besome early exercise valueover an equivalent Europeanoption, as described inChapter 16, although thisextra value will usually besmall. If the options aresubject to futures-typesettlement, as they are onmost exchanges in Europeand the Far East, there iseffectively no additionalvalue over an equivalentEuropeanoption.

OptionsonaCashIndexThefirstcashoptionsona

stock index began trading atthe Chicago Board OptionsExchange (CBOE) in March1983. The exchange hadwanted to listoptionsononeof the widely followedindexes,suchastheS&P500or Dow Jones IndustrialsAverage, but was initially

unable toobtain the rights totradeanyoftheseindexes.Asaresult,theCBOEdecidedtocreate its own OptionsExchange Index (with tickersymbolOEX)madeupof100of the largest U.S.companies.5 Because allindividual equity optionstraded at the CBOE at thattimewereAmerican,withtheright of early exercise, itseemedlogical tomakeOEX

options American as well.However,oncetradingbegan,it became obvious that theearlyexercisefeatureresultedin additional and unforeseenrisks and also greatlycomplicated theoreticalevaluation. As a result, allexchange-traded cash indexoptions are now European,with no possibility of earlyexercise.

For stock index options

on a cash index,6 nounderlying position resultsfrom exercise. At expiration,the exchange automaticallysettles all options in cash,with a cash credit to thepurchaserofan in-the-moneyoptionequaltothedifferencebetween the exercise priceand index price and cashdebit of an equal amount tothe seller of the option. Thisisthesameprocedureusedto

settleexpiringfuturesoptionswhen the underlying contractfor the option is the expiringfutures month. Cash indexoptions are typically subjectto AM expiration, with thevalue of the index, andconsequentlythevalueoftheoptions, being determined bythe opening prices of all theindexcomponents.

How should a traderhedgeapositionincashindex

options?Intheory,onemightbuy or sell all the stocks inthe index in the rightproportion to hedge such aposition.However,thiswouldrequire trades in manydifferent stocks and, intheory, might require thepurchaseor saleof fractionalshares.Moreover,asthedeltaof the option positionchanged, the trader wouldhavetoperiodicallyadjustthestock holdings. Given these

drawbacks, hedging aposition with a basket ofcomponent stocks isimpractical for most traders.What most traders want is ahedging instrument that iseasily traded and correlatesclosely with the cash index.Thecontract thatmeets theserequirements is a futurescontract on the same stockindexasthecashoptions.

Assuming that futures

contracts on an index areavailable, a trader in a cashindex option market willhedge his position with thefutures contract that expiresat the same time as theoptions. If no correspondingfuturesmonthisavailable,thenearest futures contractbeyond the option expirationis used as the hedginginstrument. For index futurestradingonthequarterlycycle,we can summarize the

underlying hedginginstrumentasfollows:

Clearly, this is not a

perfect solution to thehedging problembecause thefutures contract and the cashindex are not identical.Indeed, a futures contractmaytradeatapriceaboveorbelow its theoretical valuecompared with the cashindex. But for most traders,using the futures contractrepresentsapracticalsolutiontothehedgingproblem.

Even ifweusean index

futures contract as thehedging instrument, we stillneed an underlying price toevaluate options. For March,June, September, andDecember options, if aposition is carried toexpiration, a trader can becertain that at themomentofexpiration the cash value oftheindexandthevalueofthecorresponding futurescontract will converge.Consequently, a trader can

treat the futures contract asthe underlying contract. Notonlydoes thismakepracticalsense, but it also makestheoretical sense becauseoption values are derivedfromtheforwardpriceof theunderlying contract, and thefuturescontract is simply thetraded form of the forwardprice.Moreover, ifbothcashoptions and futures optionsareavailableonanindexandalloptionsexpireatthesame

time, there is effectively nodifference between theoptions.Theywillessentiallytradeatthesameprices.7

The question of whatunderlyingprice tousewhenevaluating a cash indexoption is somewhat morecomplex for serial monthoptions, where there is nocorresponding futuresmonth.If December futures areavailable, we can always

priceDecemberoptionsusingthe December futures price.We may also use theDecember futures contract tohedge an October orNovember option position ifno corresponding October orNovember futures contract isavailable.But theOctober orNovember forward pricewilldiffer from the Decemberforward price, so using theDecemberfuturespriceastheunderlying price cannot be

correct.If we assume that the

December futures contractrepresents the correctDecember forward price,what should be the correctNovemberforwardprice?Wemight work backwardsbecause

FDec=FNov×(1+r×t)–D

Then

However, this requires usto estimate the dividendsexpected between Novemberand December expirations.An easier method used bymost traders is to determinethe November forward priceimplied by option prices inthe marketplace. We can dothisbyobservingthepricesofaNovembercallandput that

areclosetoatthemoneyandwhose prices willconsequently be similar andthen use put-call parity tocalculate the implied forwardprice.Forexample,

November1,000 call =34.85November1,000 put =29.90Time to

Novemberexpiration = 2monthsAnnualinterest rate =6.00percent

Because

then

F=(C–P)×(1+r×t)+X

FNov=(34.80–29.85)×1.01+1,000=1,005

The implied Novemberforwardpriceis1,005.00.

Now suppose that whenwe calculate the impliedNovember forward price, theDecember futures price is1,010.00. This means thatthere should be a differencebetween the Novemberforward price and the

December forward price of5.00. As the price of theDecember futures contractfluctuates, if we want tocalculate theoretical valuesfor November cash options,wecanuseas theunderlyingprice, the December futuresprice,less5.00.

We might also use put-call parity to calculate theimplied December forwardprice. But this is not really

necessary because we havethe implied DecemberforwardpriceintheformofaDecember futures contract.Still,wemightchecktoseeifDecember option prices areconsistentwiththeDecemberfuturesprice.If

Decemberfutures price=1,010Time toDecember

expiration = 3monthsAnnualinterest rate =6.00percent

from put-call-parity weknow that the December1,000 combo (the differencebetween the prices of theDecember 1,000 call and1,000put)shouldbe

If theDecember1,000callis tradingatapriceof44.60,the December 1,000 putshouldbetradingatapriceof44.60–9.85=34.75.

The price of theNovember/December 1,000roll (i.e., the differencebetween the December andNovember 1,000 synthetics)is

(44.60–34.75)–(34.80–29.85)=9.85–4.95=4.90

1Alesscommonvariationonanequal-weightedindexinvolvesweightingthestocksgeometricallyratherthanarithmetically.Thevalueofageometric-weightedindexmadeupofnstocksisthenthrootoftheproductofthepriceratios.IfourABCIndexisgeometricweighted,theinitialindexvaluewillbe

(80/80×20/20×50/50)1/3=1.00Asthepricesofthe

componentstockschange,thevalueoftheindexwillbe[Π(today’spricei/yesterday’s

pricei)]2Tradinginastockcanbehaltedforavarietyofreasons,butitoccursmostoftenwhenthereisimportantnewspendingconcerningthecompany.Byhaltingtrading,theexchangehopestogiveinvestorstimetoabsorbthenew

informationandtherebymakeabetterassessmentofitsimpactonthemarket.3Optionsonexchange-tradedfunds,whichareoftendesignedtomimicastockindex,aresubjecttotraditionalPMexpiration.Thevalueoftheoptiondependsonclosingstockpricesattheendoftradingonexpirationday.4Wemightalsoincludeoptionsonexchange-tradedfunds.However,exchange-tradedfundsareissuedinsharesandthereforetendtotradelikeindividualequityoptions.5TheCBOEsubsequentlyreachedanagreementwithStandardandPoor’sallowingtheexchangetotradeoptionsontheS&P500Index.Aspartofthe

agreement,StandardandPoor’sassumedtheresponsibilityforcalculatinganddisseminatingOEXvalues.Atthesametime,theOEXwasrenamedtheS&P100Index,althoughitstillretainsitsoriginaltickersymbolOEX.6TickersymbolsforcashindexesveryoftenendwiththeletterX,forexample,SPX(StandardandPoors500Index),DJX(DowJonesIndustrialIndex),DAX(DeutscheAktienIndex—theGermanStockIndex),AEX(AmsterdamExchangeIndex),OMX30(StockholmOptionsMarketIndex),andASX200(AustralianStockExchangeIndex).7Fordeeplyin-the-moneyoptionson

futures,whicharetypicallyAmerican,theremaybeaveryslightadditionalearlyexercisevalue.

23

ModelsandtheRealWorld

A trader who uses atheoretical pricing model isexposed to two types of risk—the risk that the trader hasthe wrong inputs into themodel and the risk that the

modelitselfiswrongbecauseit is based on false orunrealisticassumptions.Thusfarwehavefocusedprimarilyon the first area, the riskassociated with the inputsinto themodel.A traderwilltypically deal with this riskby paying close attention tothe sensitivities of an optionposition (i.e., delta, gamma,theta,vega,andrho), therebypreparing to take protectiveaction when market

conditionsmoveagainsthim.While any of the inputs intothe model may represent arisk, we have placed specialemphasis on volatilitybecause it is the one inputthat cannot be directlyobservedinthemarketplace.

However, an activeoptiontradercannotaffordtoignore the second type ofrisk, the possibility that theassumptions on which the

modelisbasedareinaccurateor unrealistic. Some of theseassumptions pertain to theway business is transacted inthemarketplace,whileotherspertain to themathematicsofthemodel.

To begin, we might listthe most importantassumptions built intotraditionalpricingmodels1:

1. Markets arefrictionless.

A. Theunderlyingcontractcanbefreely boughtor sold,withoutrestriction.B. Unlimitedmoney can beborrowed orlent, and thesame interestrate applies to

alltransactions.C. There areno transactioncosts.D. There areno taxconsequences.

2. Interest ratesare constant overthe life of anoption.3. Volatility is

constant over thelifeofanoption.4. Trading iscontinuous,withnogaps in thepriceofan underlyingcontract.5. Volatility isindependent of theprice of theunderlyingcontract.6. Over smallperiodsof time, the

percent pricechanges in anunderlying contractare normallydistributed,resulting in alognormaldistribution ofunderlyingpricesatexpiration.

The reader may alreadyhave an opinion about thevalidityoftheseassumptions,

butlet’sconsiderthemonebyone.

MarketsareFrictionless

In Chapter 8, we came tothe obvious conclusion thatmarkets are not frictionless.The underlying contractcannot always be freelybought or sold; there are

sometimes tax consequences;a trader cannot alwaysborrow and lend moneyfreely, nor at the same rate;and there are alwaystransactioncosts.

In futures markets, theunderlying cannot always befreelyboughtorsoldbecausean exchangemay set a dailyprice limit beyond which afutures contract is notpermittedtotrade.Whenthat

limitisreached,themarketitlocked, and trading is halteduntilthemarketcomesoffitslimit. If it does not come offits limit, trading does notresumeuntilthenextbusinessday.

Even ifa futuresmarketis locked, itmay be possibleforatradertocircumventthetrading restriction. Instead ofbuying or selling futurescontracts, a trader might be

able to trade in the cashmarket. Or the trader mightbe able to trade a futuresspreadwhere one side of thespread is not locked. Forexample, a traderwhowantstobuyaJunefuturescontractthat is up its allowable limitmay be able to buy aJune/March spread (i.e., buyJune, sell March). If theMarchfuturescontractisstilltradingbecauseitisnotupitslimit, the trader can then go

backintothemarketandbuyback the March futurescontract.ThisleaveshimlongaJunefuturescontract,whichwas his original intention. Iftheunderlyingfuturesmarketis locked but the optionmarketisnotlocked,atradermight be able to buy or sellsyntheticfuturescontracts.

Trading can also behaltedonastockexchangeifa designated stock index

either rises or falls during atrading day by apredeterminedamount.Whenthis limit is reached, theexchangewillhalttradingforsome period of time. Theexchange’s circuit breakersspecifyhowlongtradingwillbe halted for a given percentchange in a stock marketindex

In Chapter 2, we notedthat an exchange or

regulatory authority mayplacerestrictionsontheshortsale of stock—the sale ofstock that a trader does notactually own. Even if shortsalesarepermitted,theremaybe restrictions on when suchsalescanbemade.Ifatradercannot freely sell stock, putprices will tend to becomeinflated compared with callprices, and arbitragerelationships, such asconversions and reversals,

will appear to be mispriced.Manystockoptiontraders,asa matter of good tradingpractice, will try to carrysome long stock so that theywillalwaysbeinapositiontosellstockiftheneedarises.

The assumption that atrader can always borrow orlend money freely is a moreserious weakness in pricingmodels. Even if a trader hassufficient funds to initiate a

trade, he may find at somelater date that he needsadditional funds to meetincreased marginrequirements.2Ifmoneywerefreely available, marginwouldneverbeaproblem.Atrader could always borrowmargin money and depositthe money with theclearinghouse. Because theborrowing and lending ratesare assumed to be the same,

and because theclearinghouse,intheory,paysinterest on the margindeposit,therewouldneverbea problem obtaining marginmoney, norwould there everbeacostassociatedwithit.

Intherealworld,tradersdo not have unlimitedborrowing capacity. If atrader cannot meet a marginrequirement, he may beforced to liquidate a position

prior to expiration. Becauseall models, even those thatallow for early exercise,assume that a trader willalways have the choice ofholding a position toexpiration, the inability tomeet margin requirementsand therefore maintain theposition canmake the valuesgenerated by the theoreticalpricing model less reliable.Anexperienced tradershouldalways consider the risk of a

position not only in terms ofhowmuchthepositionmightlosein totalbutalsoin termsof how much margin mightbe required to maintain thepositionovertime.

Even if a trader hasunlimitedborrowingcapacity,the fact that formost traders,borrowing and lending ratesare not the same can alsocause problems withstrategies based on model-

generated values. A traderwho borrows margin moneyat one rate will almostcertainly receive a lower ratewhenhedeposits thismoneywith the clearinghouse. Thedifferencebetweentheseratesis something of which themodel is unaware. And thegreater the differencebetween borrowing andlendingrates,thelessreliablewill be the values generatedbythemodel.

Although there areoccasionally taxconsiderations, for mosttraders, these are usuallysecondary. For a given astrategy, a trader is unlikelytoaskhimself,“Ifthistradeisprofitable or unprofitable,what will be the taxconsequences?” Differencesin tax consequences rarelymakeonestrategybetterthananother.3

Lastly, the assumptionthat there are no transactioncosts is a serious flaw in thefrictionless marketshypothesis. While a strategymay or may not be affectedby tax or interest-rateconsiderations, there arealways transaction costs.These costs can come in theform of brokerage fees,clearing fees,oranexchangemembership. For somemarket participants,

transaction costs may beprohibitive, and a strategythat looks sensible based onmodel-generated values maynot be worth doing whentransaction costs are alsotaken into consideration.Moreover, transaction costscanaccruenotonlywhenthestrategy is initiated orliquidated but also wheneveran adjustment is made. If astrategy will require manyadjustments because it has a

high gamma and the traderintends to remainapproximately delta neutral,thetransactioncostscanhavea significant impact onmodel-generatedvalues.

InterestRatesareConstantovertheLifeofanOption

When a trader feeds an

interest rate into a pricingmodel, the model assumesthatthisonerateappliestoalltransactions over the entirelife of the option. Whatevercash flows result from anoption trade will be eitherinvested, if a credit, orborrowed, if a debit, at oneconstant rate. In reality, veryfew traders initiate one tradeand simply hold the positionto expiration. As tradersinitiatenewpositionsorclose

out existing ones, they areconstantly borrowing andlendingmoney.Moreover, infutures options markets,traders are subject tochanging margin andvariation requirements. Forallthesereasons,mosttradersrequire a degree of cashliquidity that is incompatiblewith borrowing or lending atone fixed rate over longperiods of time. To achievethe required liquidity, traders

commonly finance theirtradingactivitybyborrowingfrom or lending to theirclearing firm at a variablerate.Theclearingfirmactsasa bank, informing the traderof the effective rate or ratesthatapplyonanygivenday.

Evenifatraderisabletonegotiate a fixed rate oversomeperiod of time, there isstill the problem ofdetermining which of the

various rates apply: is thetrader borrowing money (aborrowing rate), lendingmoney (a lending rate), orreceiving interest on a shortstock position (a short stockrebate). In the last case, therate that the trader receiveswill often depend on thedifficulty of borrowing thestock.

Although changinginterest rates will cause the

value of a trader’s optionposition to change, interestrates tend to be a lesser riskfor most traders, at least forshort-term option strategies.The impact of changinginterest rates is a function oftime to expiration. Becausemost actively traded optionstend to be short term, withexpirations of less than oneyear, interest rates wouldhave to change dramaticallytohaveanimpactonanybut

themostdeeplyin-the-moneyoptions. Changing interestrates become even less of aconcern when one considershow much more sensitiveoption values are to changesinthepriceof theunderlyinginstrument or to changes involatility.

This is not to say that atrader should completelyignore interest-rate risk. Forstock options especially,

raisinginterestratesraisestheforward price, which raisesthe value of calls and lowersthevalueofputs.Theoptionsthataremostsensitivetothischange are deeply in-the-money long-term options.Such options will have thegreatest interest-ratesensitivity, as reflected bytheir high rho values. Withmany exchanges now listinglong-term options, a tradershouldbeawareoftheimpact

of changing rates on suchoptions. Figure 23-1 showsthe effect of rising interestrates on long-term stockoptions. Figure 23-2 showsthe effect on rho values forstock options as we increasetimetoexpiration.

Figure23-1Theoreticalvaluesasinterestrateschange.

Figure23-2Rhovaluesastimetoexpirationchanges.

VolatilityIsConstantovertheLifeoftheOption

When a trader feeds avolatility into a theoreticalpricing model, he isspecifying themagnitudeandfrequency of price changesthatwilloccuroverthelifeofthe option. Because these

pricechangesareassumed tobe normally distributed, themodel recognizes that therewill be somenumber of one,two,three,andsoonstandarddeviation occurrences andthattheseoccurrenceswillbeevenly distributed over thelife of the option. Twostandard deviation pricechanges will be evenlydistributed among the onestandard deviation pricechanges; three standard

deviation price changes willbe evenly distributed amongthe one and two standarddeviation price changes; andsoon.

In the real world,however, price changes areunlikely to be evenlydistributed. Over the life ofan option, a trader willencounter periods of highvolatility, where large pricechanges will dominate,

together with periods of lowvolatility, where small pricechanges dominate. Thecombination of these high-and low-volatility periodswill result in one volatility.But a theoretical pricingmodelisindifferentastohowthe volatility unfolds. Themodelseesonevolatilityandevaluates optionsaccordingly.

Figures 23-3 and 23-4

are daily high/low/close barcharts for a hypotheticalunderlying contract over aperiod of 80 trading days.Both bar charts representexactly the same close-to-close realized volatility overthe period in question, 28percent. But the order inwhichthevolatilityunfoldsisdifferent. In Figure 23-3,volatility is clearlydeclining,with larger price changesoccurringearly in the80-day

period and smaller pricechangesoccurringlaterintheperiod. In Figure 23-4, theopposite is true. Volatility isrising, with smaller pricechanges occurring early andlarger changes occurringlater. The reader may havealready guessed that thecharts are in fact mirrorimages of each other andtherefore must represent thesame volatility. Even thoughthevolatilityunfolded in two

completely differentscenarios, in both cases, apricing model will use thesamevolatility,28percent,tomakeallcalculations.

Figure23-3Fallingvolatility.

Figure23-4Risingvolatility.

InbothFigures23-3and23-4, the beginning andending price is 100. Supposethat a trader buys a 100straddle and assumes,correctly, a volatility of 28percent. What should thisstraddle be worth? Tosimplify the example, let’sassume that there are 80calendar days to expirationandthateverydayisatradingday (hence noweekends and

holidays). To focus only onvolatility, let’s also assumethat the interest rate is 0.Under theseassumptions, theBlack-Scholes model willgenerate a value for both the100callandputof5.23,foratotalstraddlevalueof10.46.

Alternatively, supposethatwecalculatethevalueofthe 100 call and put byrunning a simulation of thedynamic hedging process.

Using the closing price eachday, the number of daysremainingtoexpiration,andaknown volatility of 28percent,we can calculate thedelta at the end of eachtradingday.Wecanthenbuyorselltherequirednumberofunderlying contracts toremain delta neutral. (This isthe same approach used toexplain the dynamic hedgingprocess in Chapter 8.) Theresults of such a simulation

show that if the volatility isfalling(Figure23-3), the100call and put are worth 2.97each,foratotalstraddlevalueof 5.94. But, if volatility isrising (Figure 23-4), the 100call and put are worth 6.41each,foratotalstraddlevalueof 12.82. Why do thesevalues differ so dramaticallyfromtheBlack-Scholesvalueof10.46?

A strategy that will be

helped by higher realizedvolatility, such as a longstraddle, will benefit most ifperiods of high in volatilityoccur when the gamma isgreatest. The high gammawill magnify the changes inthe delta as the underlyingprice changes, resulting ingreater profit from thedynamic hedging process.Because the 100 straddle isessentially at the money andthe gamma of an at-the-

money option increases asexpiration approaches, anyincrease in volatility close toexpiration will have adisproportionately greaterimpact on the option’s valuethan a similar increase involatilityearlyintheoption’slife.Consequently,therising-volatility scenario increasesthe value of the 100 straddlewellabovetheBlack-Scholesvalue. Of course, the highergamma close to expiration

goes hand in hand with ahigher theta. With nounderlying movement closeto expiration, the option willdecay at an accelerated rate.Therefore, the falling-volatility scenario has aninordinately negative impacton the value of the 100straddle, causing thevalue tofall below the Black-Scholesvalue.

For out-of-the-money

options, the effect is just theopposite. The gamma of anout-of-the-money option islargest early in its life, so aperiodofhighvolatilityearlyin the option’s life willincrease itsvalue.Anout-of-the-money option will beworthmorethanthepredictedBlack-Scholes value in afalling-volatility scenario andworth less in a rising-volatility scenario. This isconfirmedby the resultsof a

dynamic hedging simulationforthe80putand120call.Atavolatilityof28percent, theBlack-Scholesvaluesare0.21forthe80putand0.54forthe120 call. If, however,volatilityisfalling,thevaluesare0.44and0.89.Ifvolatilityis rising, the values are 0.05and0.14.

Option values under ourthree different volatilityscenarios for exercise prices

from70 to 130 are shown inFigure23-5.Withthepriceofthe underlying remaininggenerally between 95 and105, options with exercisepricesof95,100,and105areworth more than the Black-Scholes value in a rising-volatilitymarketandlessthanthe Black-Scholes value in afalling-volatility market. Theopposite is true for exercisepricesbelow90orabove110.They are worth more in a

falling-volatility market andless in a rising-volatilitymarket.

Figure23-5optionvaluesunderthreedifferentvolatilityscenarios.

If an option is held toexpiration with noaccompanying dynamic deltahedging, the value of theoption depends solely on theunderlying price atexpiration.Theoption’svalueis independentof thepathbywhichtheunderlyingcontractreaches its terminal value.But the preceding examplesmake it clear that in aworldwhere a trader dynamically

hedgesanoptionposition,thevalue of the option is in factpath dependent. Even if weassumeasinglevolatility,theroute that the underlyingtakes can have a significantimpact on the value of theoption.

Because the value of anoption seems to be pathdependent, one mightconclude that the Black-Scholes model is unreliable.

Indeed, for any one random-walk scenario, the valueresulting from the dynamichedging process will almostcertainlydifferfromaBlack-Scholesvalue.ButtheBlack-Scholes model is aprobabilistic model. A givenvolatility will, on average,resultinagivenvaluefortheoption. In our example, weconsidered only twoalternative volatilityscenarios, where volatility is

either rising or falling. Butthere are an almost infinitenumber of paths that theunderlyingpricemightfollowover the life of an option. Ifwe were to generate a largenumber of random pricepaths, all with normallydistributed price changes andwiththesamevolatilityof28percent and if we were tothen simulate the dynamichedging process, we wouldfind that, on average, each

exercise price is worthsomething very close to thevaluepredictedbytheBlack-Scholesmodel.

Although the Black-Scholes model assumes thatprices follow a randomwalkthrough time with constantvolatility, we might insteadassumethatvolatilityis itselfrandom. Several models thatassume stochastic volatilityhave been proposed and

might,insomecases,bemoresuitable than a traditionalpricing model. At the sametime, such models add anadditional dimension ofcomplexity to a trader’s lifeand for this reason are notwidelyused.

Somecontracts,by theirvery nature, are known tochange their volatilitycharacteristics over time.Interest-rate products in

particular fall into thiscategory. As a bondapproaches maturity, theprice of the bond movesinexorably toward par. Atmaturity, regardless ofinterest rates, the bond willhave a fixed and knownvalue. Clearly, one cannotassume that the price of thebond follows a randomwalkthrough time. Even if oneassumes that interest ratesmove randomly and that the

volatility of interest rates isconstant, interest-rateinstrumentswill change theirvolatility over time becauseinstruments of differentmaturities have differentsensitivities to changes ininterest rates. Ifwe take intoconsideration the fact thatinterest rates also vary fordifferent maturities, atraditional Black-Scholestype model is obviously notwell suited to the evaluation

ofsuchproducts.Thishasledtothedevelopmentofspecialmodels to evaluate interest-rateinstruments.

TradingIsContinuous

Tomodel option values, amodel must make someassumptions about how theprice of an underlying

contract changes over time.One possible assumption isthat prices follow acontinuous diffusion process.Under this assumption, pricechanges are continuous,withno gaps permitted betweenconsecutive prices. Anexample of a typicalcontinuous diffusion processmight be the temperaturereadings in a specificlocation. Although thetemperature can change very

quickly, there will never beany gaps. If the temperatureis initially 25 degrees butlaterdropsto22degrees,thenat some intermediate time,even if only very briefly, thetemperature must have alsobeen 24 degrees and then 23degrees.

The Black-Scholesmodel assumes that theunderlyingcontract followsacontinuous diffusion process.

Trading proceeds 24 hoursper day, 7 days per week,withoutinterruption,andwithno gaps in the price of theunderlying contract. If acontracttradesat46.05andatsome later time trades at46.08, then at someintermediatetimeitmustalsohave traded, even if onlybriefly,at46.06and46.07.Ifone were to graph with penand paper the prices of anunderlying contract that

followacontinuousdiffusionprocess, onewouldnever liftthe pen from the paper. Anexample of this is shown inFigure23-6a.

Figure23-6(a)diffusionprocess.(b)Jumpprocess.(c)Jump-diffusionprocess.

If we assume that theunderlyingcontract followsacontinuous diffusion process,we can also assume that thedynamichedgingprocesscanbe carried out continuously.This is fundamental tocapturing an option’stheoretical value. The Black-Scholesmodelassumesthataposition can be rehedged toremain delta neutral at everypossiblemomentintime.

A continuous diffusionprocess may be a reasonableapproximation of how priceschange in the realworld, butit is clearly not perfect. Anexchange-traded contractcannot follow a purediffusion process if theexchange is not open 24hours per day.At the end ofthe trading day, a contractmay close at one price andthen open the next day at adifferent price. This causes a

price gap, something that adiffusion process does notpermit. Even during normaltradinghours,newsmightbereleased, theimpactofwhichcan be almost instantaneous,causingthepriceofacontracttogapeitherupordown.

Instead of a diffusionprocess,pricesmightfollowajump process. In a jumpprocess, the price of acontract remains fixed for a

period of time and theninstantaneously jumps to anew price, where it againremains fixed until a newjump occurs. The way inwhich central banks setinterest rates is typical of ajump process. In the UnitedStates, when the FederalReservesetsthediscountrate,it remainsfixeduntil theFedannounces a change. Thediscount rate then jumps toanew level. A typical jump

process, shown inFigure 23-6b, is a combinationof fixedprices and instantaneousjumps.

In the realworld, pricesof most underlying contractsfollow neither a purediffusion process nor a purejumpprocess.Therealworldseemstobeacombinationofthe two—a jump-diffusionprocess. Most of the time,trading proceeds normally

with no price gaps.Occasionally, though, anunexpected change inmarketconditions occurs that causesthe underlying contract toinstantaneously gap to a newprice. Such a process isshowninFigure23-6c.

If a theoretical pricingmodel assumes that pricesfollow a diffusion processwhen in fact theydon’t, howis this likely to affect values

generated by the model? Tounderstand the effect of agap, consider a trader whosellsanat-the-moneystraddlewith the underlying contracttrading at 100. Howwill thetrader feel if the underlyingcontract suddenly gaps up to105?Clearly,thisisnotwhatthe trader was hoping for.Suchalargemovemightwellbe accompanied by anincrease in impliedvolatility,which will also hurt the

trader’s position. But even ifimplied volatility does notchange, because of thenegative gamma associatedwith the short straddle, thelargemove in the underlyingcontract will clearly workagainst the trader. How badwill the damage be? If theoptions are relatively longterm,say,oneyear,thegapinthe underlying price isunlikely to be the end of theworld. After all, with one

year remaining to expiration,the underlying market couldcertainly fall back to 100.While the gap has hurt thetrader, it is probably notdisastrous. But, if the gapoccurswithonlyaveryshorttime remaining to expiration,say, one day, the trader isnow in a much worsesituation.With only one dayto expiration, there is notenoughtimeforthemarkettoretrace its movement. The

100 calls that the trader soldas part of the short straddlewill immediately go deeplyinto the money, acting likeshort underlying contracts.Thestraddlemayhavebegunapproximately delta neutral,but after the gap, the traderwill find himself naked shortdeeply in-the-money calls,eachwithadeltaof100.Thevalueof theone-daystraddlewill increase dramaticallycompared with the value of

theone-yearstraddle.The reason the effect of

thegapismuchgreaterifthestraddle is short term ratherthan long term is a result ofhowthegammachangesovertime. We know that asexpiration approaches, thegamma of an at-the-moneyoption increases, causing thedelta to change much morerapidly when the underlyingprice moves. The dynamic

hedging process can reducesome of the damage if thetrader is able to buyunderlying contracts as theunderlying price rises. But agapisaninstantaneousmove;there is no opportunity toadjust.Theveryhighgamma,combinedwithaninabilitytomake any adjustment, makesthe consequences of the gapmuchmoredramaticclose toexpiration.

Not only does thegamma of an at-the-moneyoption increase as expirationapproaches, but it alsoincreases as we reducevolatility. Consequently, theimpactofagapwillbemuchgreater in a low-volatilitymarket than in a high-volatility market. If weconsider these two traitstogether, we can concludethat at-the-money optionsclose to expiration in a low-

volatility market are amongtheriskiestofoptions.

Figure 23-7 shows thechange in value for a 100straddle if themarket shouldgapasexpirationapproaches.The chart shows the changeunder two volatilityscenarios,15and25percent.Notethegreaterchangeinthestraddle value close toexpiration, as well as thegreater change in a low-

volatilitymarket.Figure23-7effectofagaponthe

valueofa100straddle.

Optionshave theuniquecharacteristicofautomaticallyand continuously rehedgingthemselves by changing theirdeltas as the price of theunderlying contract changes.It is this characteristic forwhich buyers of options arepaying. A trader who uses atheoretical pricing modelattemptstotakeadvantageofa mispriced option byhedging the option position,

delta neutral, with theunderlying contract and thenmanually performing therehedging process himselfoverthelifeoftheoption.Ifamodel assumes that pricesfollow a diffusion process,the model also assumes thatone can continuouslymaintain a delta-neutralhedge. But when the marketgaps, the assumptions onwhich themodel isbasedareviolated. Consequently, the

values generated by themodel are rendered invalid.This problem extends to anyapplication that attempts toreplicate optioncharacteristics through acontinuous rehedging in theunderlying market. Theproponents of portfolioinsurance (see Chapter 17)suffered their greatestsetbacks on October 19 and20, 1987, when the marketmade several large-gap

moves. Because of the gaps,the portfolio insurers wereunable to make continuousdelta adjustments to theirpositions. As a consequence,they found that the cost ofprotection offered byportfolioinsurancewasmuchgreater than they hadexpected.

To more accuratelyevaluate options, a variationon the Black-Scholes model

has been proposed thatincludes the possibility ofgaps in the price of theunderlying contract. Thisjump-diffusion model, intheory, generates values thatare more accurate thantraditional Black-Scholesvalues,4 but the model isconsiderably more complexmathematically and alsorequires twonewinputs—theaveragesizeofa jumpin the

underlying market and thefrequency with which jumpsarelikelytooccur.Unlesstheuser can accurately estimatethese new inputs, the valuesgeneratedbyajump-diffusionmodelmaybenobetter—andmight be worse—than thosegenerated by a traditionalmodel.Many traders take theview that whateverweaknesses are encounteredin a traditionalmodel can bebestoffset through intelligent

decision making based onactual trading experienceratherthanthroughtheuseofa more complex jump-diffusionmodel.

Assuming that a traderhas a delta-neutral positionthatheintendstodynamicallyhedge, any gap will have anegative impact on a traderwho has a negative gammaposition because the traderwill not have an opportunity

to adjust as the marketmoves. The same gap willhave a positive impact on atraderwithapositivegammaposition because hewill alsonot have an opportunity toadjust as the market moves.In the latter case, this workstothetrader’sadvantage.

Because a gap in themarket will have its greatesteffect on high-gammaoptions, and because at-the-

money options close toexpiration have the highestgamma, it is these optionsthat are most likely to bemispriced by a traditionaltheoretical pricing model.Consequently, as expirationapproaches, experiencedtraders will tend to rely lessand less on model-generatedvaluesandmoreontheirownexperienceandintuition.Thisis not to suggest that underthese circumstances a model

isofnovalue,butoneneedsto make adjustments whenthe model is known to beincorrect.

As a result of the gapsthat occur in the real world,bothatrader’sexperienceandempirical evidence seem toindicate that a traditionalmodel, with its built-indiffusionassumption,tendstoundervalueoptionsintherealworld. If one compares the

averagehistoricalvolatilityofanunderlyingmarketwiththeaverage implied volatilityoverlongperiodsoftime,theaverage implied volatility isalmost always greater. Thisseems to indicate that buyersof options are overpaying.Part of this may be due tohedgers willing to pay anadditional premium forprotective options. But theimplied volatility is derivedfrom a theoretical pricing

model that does not includethe possibility of gaps in theunderlying price. Thepossibilityofthesegapstendsto indicate that perhaps thevalues of options are in factgreater in therealworld thanis predicted by a traditionaltheoreticalpricingmodel.

Wehaveseenthatagapwill have the greatest impactonanoptionpositionclosetoexpiration,particularlyforat-

the-money options becausethese options have thegreatest gamma. From a riskstandpoint, thismeans that itcanbeverydangeroustosella large number of at-the-money options close toexpirationbecauseanygapinthe underlying market canhavedevastatingresults.Newtraders in particular areadvised to avoid suchpositions. No risk managerwill appreciate even

experienced traders beingshortlargenumbersofat-the-money options as expirationapproaches.

ExpirationStraddles

Ifitisdangeroustosellat-the-money options close toexpiration, perhaps there issome sense in taking theopposite position bypurchasing at-the-money

options as expirationapproaches.Thismayseemtocontradict conventionaloption wisdom, whichfocuses on the rapid timedecay associated with suchoptions.Butthereisalwaysatradeoff between risk andreward. If one sells at-the-money options, the rewardmay be an accelerated profitif the market doesn’t move(high positive theta), but theriskisanincreasedlossifthe

market does move (highnegative gamma). Becausethe model does not knowabout thepossibilityofagapin theunderlyingmarket, therisk is often greater than thereward. If one sells at-the-money options, the lossesfrom an unexpected gap canmore than offset the profitsresulting from increased timedecay.Anexperienced tradermay therefore take theopposite position by

purchasing at-the-moneyoptionsclosetoexpiration.

This is not to suggestthat every time expirationapproaches, a trader shouldbuyat-the-moneyoptions.Aswith any strategy, conditionsmust make the strategy lookattractive. But because manytraders are intent on sellingtime premium as expirationapproaches, it is oftenpossible to findcheapat-the-

money options. Suppose thatwith three days remaining toexpiration, theBlack-Scholesmodel generates a value foran at-the-money call of 0.75.What can we say about thiscall? Although we may notknowtheexactvaluebecausewedon’tknowthetruefuturevolatility, there is highlikelihood that in the realworld the call is worthmorethan 0.75 because the modeldoesn’t know about the

possibility of a gap in themarket. If,on topof this, thecallistradingatapricebelowits model-generated value,say, 0.65, it is likely to be agoodbuy.

As with any strategybasedonvolatility, the traderwhobuys these callswill tryto establish a delta-neutralposition. Because of thesynthetic relationship, if thecallsareunderpriced,theputs

atthesameexercisepricewillalsobeunder-priced.Thus, alogical strategy might be thepurchase of at-the-moneystraddles. This enables atrader to buy bothunderpriced calls andunderpricedputsandtoprofitiftheunderlyingmarketgapseitherupordown.

In theory, all volatilitystrategies, including anexpiration straddle, ought to

be adjusted periodically toremain delta neutral.However, with little timeremaining to expiration, themodel is not only unreliablewith respect to theoreticalvalues, but also unreliablewith respect to deltas.Because it is impossible tosaywhat the right delta is, itisalsoimpossibletosaywhatthecorrectadjustment is.Forthis reason, traders who buyexpiration straddles often

abandon any attempt toremain delta neutral andsimply sit on the position toexpiration. This may not bethe theoretically correct wayto manage a volatilityposition, but given all theuncertainties associated withtheoretical evaluation asexpirationapproaches,itmaybeapracticalchoice.

Evenifatradercarefullychooses his expiration

straddles, the great majorityof time no gapwill occur inthe market. In any singlecase,thetraderismorelikelyto show a loss than a profit.But the primary concern isnot the profit or loss5 fromany single trade, but whathappens in the long run.Returning to the rouletteexample in Chapter 5, aplayerwhochoosesanumberat a roulette table can expect

towinonaverageonly1timein 38. But, if the theoreticalvalue of the bet is 95 centsandtheplayercanbuythebetfor less than 95 cents, heexpects tobeawinner in thelongrun.Evenifheisabletopay a very low price for thebet, say, 50 cents, he stillexpects to lose 37 times outof38.Butnowthebetisveryattractive. Even if he onlywins 1 time in 38, this willstill more than offset the

small losses he takes eachtimeheloses.Thesamelogicistrueofexpirationstraddles.A trader may lose severaltimes before winning. Butwhen he does win, he canexpect a return that is greatenoughtomorethanoffsetallthesmalllosses.

The fact that an at-the-moneystraddlemaybecheapdoes not mean that a tradershouldbuy these straddles in

large numbers. Suchstrategies are likely to resultin a loss more often than aprofit, soan intelligent tradershouldonlyinvestanamountthat he can afford to lose.However, when conditionsareright,atraderoughttobewilling to make theinvestment. Even if he losesseveraltimesinsuccession,inthe long run, he willencounter gaps in themarketorlargeincreasesinvolatility

often enough to make suchstrategiesprofitable.

VolatilityIsIndependentofthePriceoftheUnderlyingContract

When a trader feeds avolatility into a theoreticalpricing model, the volatilitydefines a one standard

deviationpricechangeatanytime during the life of theoption regardless of whetherthe underlying contracthappenstoberisingorfallingin price. If a contract iscurrently at 100 and weassume a volatility of 20percent, a one standarddeviation price change isalways based on thisvolatilityof20percent. If, atsomelatertimeduringthelifeof the option, the contract

should move up to 125 ordown to 75, the effectivevolatility is still assumed tobe20percent.

In many markets,however, this assumptionappears to be inconsistentwith most traders’experience.Ifoneweretoaska stock index trader whetherhis market becomes morevolatile when rising orfalling, he would probably

say that it becomes morevolatilewhen falling.On theotherhand,ifoneweretoaskacommodity trader the samequestion,helikelywouldgivethe opposite answer. Hismarket will tend to becomemorevolatilewhen rising. Inother words, the volatility ofa market is not independentofthepriceoftheunderlyingcontract.Onthecontrary, thevolatility over time seems todepend on the direction of

movement in the underlyingcontract. In some cases, atrader expects the market tobecome more volatile if themovement is downward andless volatile if themovementis upward; in other cases, atrader expects the market tobecome more volatile if themovementisupwardandlessvolatile if the movement isdownward.

Because volatility in

some markets seems todepend on the direction ofprice movement in theunderlyingcontract, a furthervariation of the Black-Scholes model has beenproposed. The constant-elasticity of variance (CEV)model6 is based on theassumption that volatilitychanges as the price of theunderlying contract changes.Price changes are still

assumed tobe random in theCEV model, but thevolatility, and consequentlythe magnitude of the pricechanges,varieswiththepriceoftheunderlyingcontract.

Like the jump-diffusionmodel, the CEV model isbothmathematicallycomplexandrequiresadditionalinputsintheformofamathematicalrelationship between thevolatilityandpricemovement

in the underlying contract.Given these difficulties, theCEV model has not foundwide acceptance amongoptiontraders.

UnderlyingPricesatExpirationAreLognormallyDistributed

Intherealworld,doprices

at expiration form alognormal distribution? Wemight try to answer thisquestion by asking how thepercent price changes aredistributed.Ifthisdistributionis normal, the continuouscompounding of pricechangesislikelytoresultinalognormal distribution ofprices.

Figure 23-8a is ahistogram of daily Standard

and Poor’s (S&P) 500 Indexpricechangesfor the10-yearperiod from 2003 through2012.Eachbarrepresentsthenumber of occurrences of agiven price change roundedto the nearest ¼ percent. Asonewouldexpect,mostofthechanges are relatively smalland close to 0. As we moveaway from the 0 in eitherdirection,weencounterfewerand fewer occurrences. Thedistribution seems to have

manyofthecharacteristicsofanormaldistribution.Butisitreally a normal distribution,and ifnot,howdoes itdifferfrom a true normaldistribution?

Figure23-8(a)s&P500dailypricechanges:January2003–december2012.(b)Crudeoildailypricechanges:January2003–december2012.(c)euro(versusdollar)dailypricechanges:January2003–december2012.(d)Bunddailypricechanges:January2003–december2012.

If the frequencydistribution conforms exactlyto a normal distribution, thetops of the bars shouldcoincide exactly with a truenormal distribution. To findout if this is the case, themean (+0.0296 percent) andstandard deviation (1.31percent)havebeencalculatedfor all 2,535 daily pricechanges over the 10-yearperiod. From these numbers,

a best-fit normal distributionhas been overlaid on thefrequency chart. The actualfrequency distribution issimilar to the normaldistribution, but there aresome clear differences.Becausethebarsrepresentingthe small price changes riseabove thenormaldistributioncurve, there seem tobemoredayswithsmallpricechangesthanonewouldexpectfromatrue normal distribution.

Although they are not asobvious,therearealsoseverallarge price changes, oroutliers, that rise above theextreme tails of the normaldistribution. These outliersseemtosuggestthattherearemore large moves in ourfrequency distribution thanonewouldexpectfromatruenormal distribution. Finally,in the midsections, betweenthe peak of the distributionand the extreme tails, there

seemtobefeweroccurrencesthanonewouldexpect.

One might surmise thatthe differences in Figure 23-8a between the S&P 500frequencydistributionandthetrue normal distribution areeitheruniquetotheS&P500or an aberration of the 10-year period in question,which admittedly includedthe financial crisis of 2008.However, studies tend to

indicate that price-changedistributions for almost allexchange-traded underlyingmarkets exhibitcharacteristics that are verysimilar to the S&P 500distribution.Therearealwaysmoredayswithsmallmoves,more dayswith largemoves,and fewer days withintermediate moves than arepredicted by a true normaldistribution. The differencesbetween the actual and

theoretical distributions canalso be seen in several otherhistogramscoveringthesameperiod of time: crude oil(Figure 23-8b), the euro(Figure23-8c), and theBund(Figure23-8d).

SkewnessandKurtosis

Distributions suchas those

in Figure 23-8a through 23-8d are approximately normalbut still differ from a truenormal distribution. If one istrying to make decisionsbasedonthecharacteristicsofa distribution, it might beusefultoknowhowtheactualdistribution differs from thenormal. A perfectly normaldistribution can be fullydescribed by its mean andstandard deviation. But twoother numbers, the skewness

andkurtosis,areoftenusedtodescribe the extent to whichan actual distribution differsfrom a true normaldistribution.7

The skewness of adistribution(Figure23-9)canbe thought of as the lop-sidedness of the distribution,ortheextenttowhichonetailislongerthantheothertail.Ina positively skeweddistribution, the right tail is

longer than the left tail. (Thelognormal distribution shownin Figure 6-7 is positivelyskewed.) In a negativelyskewed distribution, the lefttail is longer than the righttail. A perfectly normaldistributionhasaskewnessof0.The frequency distributionin Figure 23-8c (euro) ispositively skewed, while thedistributionsinFigures23-8a(S&P500),23-8b(crudeoil),and 23-8d (Bund) are

negativelyskewed.Figure23-9skewness—thedegree

towhichonetailofadistributionislongerthantheothertail.

The kurtosis of adistribution (Figure 23-10) istheextenttowhichthecenterof the distribution is eitherunusually tall or unusuallyflat. A distribution with apositive kurtosis has a tallpeak (leptokurtic), whiles adistribution with a negativekurtosishasaloworflatpeak(platykurtic). A perfectlynormal distribution has akurtosisof0(mesokurtic).8

Figure23-10Kurtosis—thedegreetowhichadistributionhasatallerpeakandwidertails.

At first sight, a positivekurtosis distribution lookssimilar to a low standarddeviationdistributionbecauseboth have high peaks. But adistribution with a lowstandard deviation also hasshort tails, while adistribution with a positivekurtosis has elongated tails.Onemightthinkofapositivekurtosis distribution as anormaldistributionwherethe

midsection to the left andright of the peak has beensqueezed inward.This forcesthe peak of the distributionupwardandthetailsoutward.ThefrequencydistributionsinFigures 23-8a through 23-8dall exhibit the same positivekurtosis, which is typical ofalmost all exchange-tradedunderlying markets. Theyhavehigherpeaks(moredayswithsmallmoves),elongatedtails (more days with big

moves), and narrowmidsections (fewerdayswithintermediatemoves) than arepredicted by a true normaldistribution. Traderssometimes refer to these as“fattail”distributions.

The S&P 500distribution has an unusuallylarge kurtosis value of10.415. To see the extent towhich the tails of thisdistribution are abnormally

fat, we can express thebiggest up and down movesin standard deviations andthen consider the chances ofthese moves occurring underthe assumption of a normaldistribution. The biggest upmoveintheS&Poverthe10-year period was 11.58percent. With a standarddeviationof1.31percent,thistranslates into an 8.84standard deviationoccurrence. The probability

of such an occurrence isapproximately 1 chance in2,000,000,000,000,000,000(2quintillion,foranyonewhoiscounting). The biggest downmove,9.03percent,translatesintoa6.75standarddeviationoccurrence,withaprobabilityequal to approximately 1chance in 350,000,000,000(350billion).Simplyput, thelikelihood of either of theseoccurrences is so small thatthey will essentially never

occur.9The kurtosis values for

crude oil, the euro, and theBund are not as dramatic asthe S&P 500. But even inthese markets under theassumptions of a normaldistribution,wewouldexpectto see the biggest up anddown moves only once inmany millions ofoccurrences.Keepinginmindthatthedatacoveredaperiod

of between 2,500 and 2,600days, we can see how muchmore often big moves occurare in the real worldcompared with what ispredicted by a normaldistribution.Theprobabilitiesassociated with the largestmoves in our sampledistributions are shown inFigure23-11.

Figure23-11Probabilitiesassociatedwiththebiggestupanddownmoves.

1Bytraditionalpricingmodelwemeanthosethataremostcommonlyused:theBlack-ScholesmodelanditsvariationsortheCox-Ross-Rubinsteinmodel.2Thepossibilitythatatraderinafuturesoptionmarketmayalsohavetocomeupwithadditionalvariationmoney,asopposedtomarginmoney,afterestablishinganoptionpositionisincorporatedintomostmodels.Thisiswhyaconversionorreversalinafuturesoptionmarketmaynotbedeltaneutral.3Thisisnottosaythattaxconsequencesarealwaysinsignificant.Taxconsiderationscanplayaroleinportfoliomanagementorinoption

strategiesinvolvingdividendswhenthedividendsaresubjecttotaxrulesdifferentfromthegainsorlossesfromstockoroptions.4Adiscussionofthejump-diffusionmodelcanbefoundinmostadvancedtextsonoptionpricing.Foradditionalinformation,seeRobertMerton,“OptionPricingwhenUnderlyingStockReturnsAreDiscontinuous,”JournalofFinancialEconomics3(March):125–144,1976;StanBeckers,“ANoteonEstimatingtheParametersintheJump-DiffusionModelofStockReturns,”JournalofFinancialandQuantitativeAnalysis,March1981,pp.127–140;andEspenGaarderHaug,TheCompleteGuidetoOptionPricingFormulas,2nd

ed.(NewYork:McGraw-Hill,2006).5Thisassumes,ofcourse,thatthetraderisabletoabsorbthelossandstillstayinbusinessforthelongrun.6ForinformationontheCEVmodel,seeJohnC.CoxandStephenA.Ross,“TheValuationofOptionsforAlternativeStochasticProcesses,”JournalofFinancialEconomics3(March):145–166,1976;StanBeckers,“TheConstantElasticityofVarianceModelandItsImplicationsforOptionPricing,”JournalofFinance,June1980,pp.661–673;MarkSchroder,“ComputingtheConstantElasticityofVarianceOptionPricingModel,”JournalofFinance44(1):211–219,1989;andEspenGaarderHaug,The

CompleteGuidetoOptionPricingFormulas,2nded.(NewYork:McGraw-Hill,2006).7Theskewnessandkurtosisfunctionsareincludedinmostcommonlyusedspreadsheets.Theirformulascanbefoundinastatisticsorprobabilitytextbook.8Mathematically,atruenormaldistributionhasakurtosisof3.However,ascommonlyexpressed,3isusuallysubtractedfromthekurtosisvalue,soatruenormaldistributionhasakurtosisvalueof0.9NassimTalebhasreferredtosuchunlikelyoccurrencesas“blackswans.”SeeNassimNicholasTaleb,TheBlack

Swan:TheImpactoftheHighlyImprobable(NewYork:PenguinBooks,2008).

24

VolatilitySkews

There are clearly realproblems associated with theuseofatraditionaltheoreticalpricing model. Markets arenot frictionless,pricesdonotalways follow a diffusion

process, volatility may varyoverthelifeofanoption,therealworldmaynotlooklikealognormal distribution. Withall these weaknesses, onemight wonder whethertheoretical pricing modelshave any practical value atall. Infact,most tradershavefound that pricing models,while not perfect, are aninvaluable tool for makingdecisions in the optionmarket.Evenifamodeldoes

not work perfectly, tradershave found that using amodel, even a flawedone, isusually better than using nomodelatall.

Still,a traderwhowantsto make the best possibledecisions cannot afford toignore the problemsassociated with a theoreticalpricingmodel.Consequently,a trader who uses a pricingmodel might look for a way

to reduce the potential errorsresulting from theseweaknesses. Initially, onemightsimplylookforabettertheoretical pricing model. Ifsuch a model exists, it willcertainly be worth replacingthe old model with the newone. But better is a relativeterm. A model might bebetter in the sense that itgives slightly more accuratetheoretical values. But if themodel is extremely complex

and difficult to use, or if itrequires additional inputs ofwhicha tradercannotalwaysbe certain, then the modelmaymerelysubstituteonesetof problems for another.Given the fact that mosttradersarenottheoreticians,amore realistic solution mightbe to use a less complexmodel and somehow fine-tune it so that it is consistentwith the realities of themarketplace.

A trader trying tocompensateforweaknessesina pricing model might makethe assumption that themarketplaceisusingthesamemodel as the trader and thenask how the marketplace isdealing with the weaknessesin the model. This issomewhat analogous tocalculating implied volatilitywhere we assume thateveryone is using the samemodel, that the price of the

option is known, and thateveryone agrees on all theinputsexceptvolatility.Fromthese assumptions, we areable to determine thevolatilitythatthemarketplaceis implying to theunderlyingcontract. We can take thesame general approach butask instead what weaknessesthe marketplace is implyingtothemodel.

Figure 24-1 shows the

implied volatilities acrossexerciseprices forJune2012FTSE 100 Index1 optionstraded on the LondonInternational FinancialExchangeonMarch16,2012.Calculationsweremadeattheend of the trading day fromthe average of the bid-askspread using the Black-Scholes model. It isimmediately apparent thatimplied volatilities vary

across exercise prices. If weassume that the exerciseprice, time to expiration,underlyingprice, and interestrate are known, thetheoreticalvalueofanoptioninaBlack-Scholesworldwilldepend solely on thevolatility of the underlyingcontract over the life of theoption. Of course, we won’tknow what that volatility isuntil we reach expiration, atwhichtimewecanlookback

and calculate the historicalvolatility over the 13-weekperiodfromMarch16toJuneexpiration.ButtheFTSE100Index can have only onevolatility over this period.Becausetheunderlyingindexis the same for alloptions, itdoesn’t make sense in aperfect Black-Scholes worldfor every exercise price tohave a different impliedvolatility.Iftheactivityinthemarketplace were a result of

everyone believing in theefficiency of the Black-Scholesmodel, the sellingofoverpriced options and thebuyingofunderpricedoptionswouldeventuallycauseeveryoption to have the sameimplied volatility. Yet thisalmost never happens in anymarket.

Figure24-1June2012FTSE100impliedvolatilities:March16,2012.

The distribution ofimplied volatilities acrossexercise prices is oftenreferredtoasavolatilityskewor,possibly,avolatilitysmileor volatility smirk dependingontheshapeoftheskew.Onelikely explanation for thedistribution of impliedvolatilitieshas todowith theway in which options areusedasahedginginstrument.In the stock market, most

investors are long stock2 andarethereforeconcernedaboutadeclineinstockprices.Thetwo most common hedgingstrategies to protect a longunderlying position, asdescribed in Chapter 17, arethe purchase of protectiveputs and the sale of coveredcalls.

If a stock investordecides to purchase aprotectiveput,whichexercise

pricewillhechoose?Anout-of-the-money put costs lessthanan in-the-moneyputbutalso offers less protectionagainst a down move.However,iftheinvestorissoworried about a downwardmove that he needs theprotection afforded by an in-the-money put, he ought tosimply sell the stock. Theresult is that most protectiveputs are purchased at lowerexerciseprices.

If, instead, the investordecidestosellacoveredcall,hewillalmostalwaysdosoata higher exercise price. Thiswillofferlessprotectionthanthe sale of an in-the-moneycall, but presumably theinvestor holds the stockbecause he believes that thestockpricewillrise.Ifitdoesrise, he will want toparticipateinatleastsomeofthe upside profit potential. Ifthe stock rises and the

investor has sold an in-the-moneycall, the stockwill bequickly called away, limitingany upside profit. The resultis thatmostcoveredcallsaresoldathigherexerciseprices.

As a result of hedgingactivity, in the stock optionmarket there tends to bebuyingpressureon the lowerexercise prices (the purchaseofprotectiveputs)andsellingpressure on the higher

exercise prices (the sale ofcovered calls). This causesthe implied volatilities atlower exercise prices to riseand the impliedvolatilitiesathigher exercise prices to fall.The resulting skew, such asthat in Figure 24-1, issometimes referred to as aninvestment skew. It occurs inmarkets in which peoplefreely invest, the mostobvious example being thestock market. Traders

sometimes describe aninvestment skew by sayingthatthe“skewistotheputs,”indicating that put impliedvolatilities are inflated. Butput-call parity dictates that ifaputpriceisinflated,thecallprice at the same exercisepricemustalsobeinflated,soperhapsitismoreaccuratetosay that the “skew is to thedownside.”

While investors in the

stock market may worryabout falling stock prices, inother markets hedgers mayworry about rising prices.This is often the case incommodity markets whereend users try to protectthemselves against risingprices by either buyingprotective calls at higherexercise prices or sellingcovered puts at lowerexercise prices. In theresulting demand or

commodity skew (there is ademand for the commodity),lower exercise prices havelowerimpliedvolatilities,andhigher exercise prices havehigherimpliedvolatilities.Ofcourse, commodityproducers, such as farmers,mining companies, and oildrillingcompanies, are likelyto worry about fallingcommodityprices,soitmightseem that there ought to beequal hedging activity

betweenthelongs(producers)and the shorts (end users).But inmanymarkets theendusers tend to dominate,perhaps because highercommodity prices, and theconcomitant inflationarypressures, are perceived ashaving a negative effect onthe entire economy.Moreover, in somecountries,the government has aprogramofpricesupportsforagricultural products, so

growers have less to worryabout from fallingagriculturalcommoditypricesthan end users have fromrisingprices.

Finally, there aremarketswherebothlongsandshorts are equally worried.Consider a U.S. companybuying goods in Europe thatmust be paid for on somefuture date in euros. Thecompany clearly is worried

abouta risingeurocomparedwith the dollar. At the sametime, a European companymaybuygoods in theUnitedStatesthatmustbepaidforindollars. This company isworried about a falling eurocompared with the dollar. Ifboth companies choose tohedge their risk in thecurrency option market, thehedging activity will tend toresult in a balanced skew,where there is no obvious

domination of impliedvolatilitiesat eitherhigherorlower exercise prices. Thisdoes not mean that impliedvolatilities will necessarilyform a flat skew, but thedistribution of impliedvolatilities is likely to besymmetrical around thecurrentunderlyingprice.ThethreecommontypesofskewsareshowninFigure24-2.

Figure24-2(a)Investmentskew.(b)Demandskew.(c)Balancedskew.

Inadditiontodistortionscaused by hedging activity,we also know from Chapter23 that there are inherentweaknesses in many models.For example, most tradersbelieve that stock marketsbecome more volatile whenthey are falling and lessvolatilewhen theyare rising.Wealsoknow that anoptionismost sensitive to volatilitychanges (it has its highest

vega)whenitisatthemoney.If an underlying stock istradingat100andthemarketbeginstofall,thevegaofthe95 putwill rise because it isbecomingmoreatthemoney.If the market also becomesmore volatile because thestockpriceisfalling,thiswillincrease the volatility valueof the 95 put. But, if themarketbeginstorise,the105call, even though its vega isrising,will not benefit to the

same extent as the 95 putbecause the market isbecoming less volatile. So itshouldnotcomeasasurprisethat the 95 put carries ahigher implied volatility andthe 105 call a lower impliedvolatility than expected.Thisis consistent with aninvestmentskew.

The marketplace, likeevery individual trader, istrying to evaluate options as

efficiently as possible givenall available information.Whether one believes thatmarkets are efficient or not,one can argue that themarketplace is trying to beefficient. From the widerange of implied volatilitiesfound in almost every optionmarket, we can reasonablyinfer that the marketplacedoes not think the Black-Scholes model is perfectlyefficient. Unfortunately,

trying to identify the sourceoftheinefficiencymaynotbepossible. Itmight have to dowithhowoptionsareused inhedging strategies. Or itmight have to do withweaknesses in the theoreticalpricing model. Whatever thereason, we can make theassumption that at anymoment in time themarketplace believes thatoptionsarepricedefficiently,evenifthosepriceshappento

differ from model-generatedvalues.

Anoptiontraderusingatheoretical pricing modelmight take the view that thevolatility skew containsusefulinformationthatcanbeused in the decision-makingprocess. By treating thevolatility skew as anadditional input into thetheoreticalpricingmodel, theskew becomes an important

aid in generating theoreticalvalues and managing risk.Moreover, analysis of theskewcanformthebasisforavarietyofoptionstrategies.

ModelingtheSkew

If we want to include askew in our model, we needto do it in a way that themodel understands. This istypically done using a

mathematical function thatgenerates a best fit for theskew

f(x)=y

where y is the impliedvolatility at each exerciseprice x. A trader can chooseany function that seems toyield a good fit, but manytraders use a polynomialfunctionoftheforma+bx+cx2 + dx3 + …. A best-fit

function for the impliedvolatilities in Figure 24-1 isshowninFigure24-3.

Figure24-3June2012FTSE100impliedvolatilities:March16,2012.

If we think of the skewas an input into the model,then, as with all inputs, weneed to ask how changes inthe input will affect aposition. If we can modelpossible changes in the skewasmarket conditions change,wewillbeinabetterpositionto assess the risk associatedwith an option position. Inparticular, as marketconditions change, we will

want to model both thelocation and shape of theskew.

Ofcourse,wemighttakethe position that the locationand shape of the volatilityskew will remain fixed.Under this sticky-strikeassumption, the current skewdetermines the impliedvolatility at each strikeregardless of how marketconditionschange.

Unfortunately, a sticky-strike skew, with its fixedvolatilities at each exerciseprice, is not consistent withtheobserveddynamicsofthemarketplace. In most optionmarkets, the skew will shiftastheunderlyingpricemovesor impliedvolatility changes.An alternative approach is touseafloatingskew,wheretheentire skew is shiftedhorizontallyastheunderlyingprice rises or falls or

verticallyasimpliedvolatilityrises or falls. The shift isequaltotheamountofchangein either the price orvolatility. If the underlyingprice rises five points, theskewisshiftedtotherightbyfive points. If impliedvolatility falls by twopercentagepoints,theskewisshifted downward by twopercentage points. This typeof skew is shown in Figure24-4.

Figure24-4(a)Asimplefloatingskewastheunderlyingpricechanges.(b)Asimplefloatingskewasimpliedvolatilitychanges.

Shifting the entire skewmight be a reasonableapproach if a trader believesthat the shape of the skewwill remain unchangedregardlessofchangingmarketconditions.But is this likely?The implied volatilities atdifferent exercise prices arelikely to depend on how themarketplace views thelikelihood of either larger orsmallermovesinthepriceof

the underlying contract. Butall moves are relative withrespecttoboththeunderlyingprice and the time toexpiration.Inrelativeterms,aprice change of 10.00 isgreater with an underlyingprice of 100 (a 10 percentmove) than with anunderlying price of 200 (a 5percent move). In the sameway, a 10 percent move isgreater over a one-weekperiod than the same 10

percent move over a one-monthperiod.

A first step in adjustingfor the relativemagnitude ofprice changes is to expresseach exercise price along thex axis in terms of itsmoneyness—how far in themoney or out of the moneythe exercise price is as apercent of the underlyingprice. The 90 exercise pricewith the underlying price at

100willhaveamoneynessof0.90. This is the samemoneyness as the 180exercise price with anunderlying price of 200. Wecanmakeafurtherrefinementby expressing each exerciseprice in logarithmic terms ln(X/S), where S is theunderlying or spot price andXistheexerciseprice.Thisisconsistent with theassumption that underlyingprices are lognormally

distributed.Howwill thepassageof

time affect the shape of thevolatility skew? Consider a90 put with the underlyingcontract trading at 100. Astimepasses,inrelativeterms,the 90 put is moving furtherout of the money. In aninvestment skew, as theoption moves further out ofthe money, its impliedvolatilitywillrise.Inasense,

it is moving “up the skew.”This will cause the skew toappear more severe as timepasses, with lower exerciseprices carrying increasinglyhigher implied volatilities.Higher exercise prices mayalso be affected by thepassage of time because anout-of-the-money call willalso go further out of themoney. Depending on theshapeof the skewandwherean exercise price falls along

the skew, its impliedvolatility may rise, fall, orremain the same. If noadjustmentismade,theeffectof timepassing on theFTSE100 skew is shown inFigure24-5.

Figure24-5FTSE100optionimpliedvolatilities,March16,2012(FTSE=5965.58).

To compare volatilityskews for differentexpirations, we need todetermine theoretically howfarinthemoneyoroutofthemoney an option is. Perhapstheeasiestwaytodothisistoexpresseachexercisepriceinterms of standard deviationsaway from at the money.Recalling the square-rootrelationshipbetweentimeandvolatility, and using our

logarithmicscale,thenumberof standard deviations in themoney or out of the moneyfor each exercise price isgivenby

with an exactly at-the-money3 option having astandard deviation of 0.Skews for several FTSE 100option expirations as of

March16,2012,areshowninFigure24-6.Whenexpressedin this format, the skew issometimes referred to as asticky-delta skewbecause thedelta is an approximation ofhow far in themoney or outofthemoneyanoptionis.

Figure24-6FTSE100impliedvolatilities,March16,2012.

TheskewsinFigure24-6 appear to be similar, buttheyareclearlynot identical.Alladjustmentsthusfarhavebeen to the x axis, changingthecalibration tomoreeasilycompare exercise prices. Butwe might also adjust the yaxis, the volatility. When atrader refers to the overallimpliedvolatilityinamarket,he is almost always referringtotheimpliedvolatilityofat-

the-money options. Whetherthe implied volatility at anyexercise price is high or lowwill depend on whether it ishigh or low compared withthe at-the-money impliedvolatility. As a result, manytraders recalibrate the y axisin terms of how the impliedvolatility at an exercise pricecompares with the at-the-money impliedvolatility.Wecan do this by expressing yvalues as the difference

betweentheimpliedvolatilityof an at-the-money optionand the implied volatility ateachexerciseprice. If theat-the-money implied volatilityis 20 (percent) and theimplied volatility at anexercisepriceis25(percent),theyvalueis20–25=–5.Ifthe implied volatility at adifferent exercise price is 18(percent), they value is 20–18=2.

This method may besatisfactory if impliedvolatilities remain relativelyconstant,butsupposethattheat-the-money impliedvolatility doubles from 20 to40 percent. We might alsoexpect the volatility at eachexercise to double. Anexerciseprice thatpreviouslyhad an implied volatility of25 percent will now have animplied volatility of 50percent,andanexerciseprice

that previously had animplied volatility of 18percent will now have animplied volatility of 36percent. We can bettercalibrate the y axis byexpressing the volatility ateach exercise price as apercent of at-the-moneyimpliedvolatility.Withanat-the-money implied volatilityof 20 percent, an impliedvolatilityof25percentwouldbe expressed as 25/20 = 125

percent.Animpliedvolatilityof 18 percent would beexpressed as 18/20 = 90percent. And an impliedvolatility equal to the at-the-money implied volatilitywould be expressed as 20/20=100percent.InFigure24-7,the y axis for the sampleFTSE 100 skews has beenrecalibrated using thisapproach.

Figure24-7FTSE100impliedvolatilities,March16,2012.

Figure24-7 is typicalofmany stock indexes,exhibitingaverypronouncedinvestment skew, with lowerexercise prices significantlyinflatedcomparedwithhigherexercise prices. A differentset of skews, for wheatoptions, is shown in Figure24-8. In this example, theskewsexhibitmorecurvaturebut with higher exerciseprices somewhat more

inflated, as is often the casewithademandorcommodityskew.Theskewsalsoseemtoexhibit less consistencyacross different expirationmonths than the FTSE 100.While skews in a financialproduct tend to be similaracross expiration months,skewsinacommoditymarketcan often vary acrossdifferent expirations, perhapsowing to seasonal volatilityconsiderations or because of

short-term supply anddemandimbalances.

Figure24-8wheatimpliedvolatilities,January27,2012.

Theforegoingmethodofmodeling a skew is used bymanytradersbutisinnowaymeant to be definitive.Adjustments are oftenrequiredtopreventthemodelfrom generating illogicalvolatilities or theoreticalvalues. For example, as wereduce volatility, an out-of-the-money option goesfurther out of the moneybecauseitisagreaternumber

of standard deviations awayfrom the underlying price.Butinaninvestmentskew,asa put goes further out of themoney,it’svolatilityisrising—it is “climbing the skew.”If the skew is sufficientlysteep, the increase involatility may in fact causethe theoretical value of theput to rise.This is inherentlyillogical because we expectalloptionvaluestodeclineifwereducevolatility.

Skewnessandkurtosis

The shape of the skew isnot constant. As marketconditions change, optionpriceswillalsochange,oftencausingtheshapeoftheskewto change. Two commonchanges have to do with thetiltandcurvatureoftheskew.The tilt, which defines howmuch the implied volatilities

oflowerexercisepricesdifferfrom the implied volatilitiesof higher exercise prices, isoften referred to as theskewness. This followslogically from the definitionof skewness in Chapter 23(see Figure 23-10). If theprobability distribution has alonger left tail (negativeskewness), there is greaterlikelihood of large downmoves, resulting in greaterdemand for lower exercise

prices. If the distribution hasa longer right tail (positiveskewness), there is greaterlikelihood of large upmovesand, consequently, greaterdemand for higher exerciseprices. Examples of positiveand negative skewness areshowninFigure24-9.

Figure24-9Skewness.

Figure24-10kurtosis.

The curvature, orkurtosis, defines how muchthe implied volatilities ofboth higher and lowerexercise prices are inflatedcompared with the at-the-money implied volatility.This also follows logicallyfromthedefinitioninChapter23 (see Figure 23-11). If theprobability distribution has“fat tails,” there is a greaterlikelihood of large moves in

either direction.Consequently, therewillbeagreater demand for out-of-the-money options (positivekurtosis). Examples ofincreasing positive kurtosisareshowninFigure24-10.4

Figure24-11Theskewasamodelinput.

We might think of theskew as an input into atheoretical pricing model(Figure 24-11), but the skewis input into the model as aformula rather than as asingle number. As with anyinput, it will be useful todetermine how sensitive anoption value or optionposition is to changes in theshapeoftheskew.

The sensitivities

associated with a skew willdepend on the skew modelthat is used. For example,let’s assume a very simplesecond-degree-polynomialmodel where the volatility yatanexercisepricexisgivenby

y=a+bx+cx2

In thismodel, thevalueofa is the base volatility,usually the implied volatility

of the at-the-money options.The values of b and crepresent the skewness andkurtosis, respectively, of thevolatility skew.Wecan raiseor lower the value of a asimplied volatility rises orfalls. We can raise or lowerthe value of b to increase ordecrease the skewness. Andwe can raise or lower thevalue of c to increase ordecrease the kurtosis. b canbeeitherpositiveornegative

depending onwhether higheror lower exercise prices areinflated. For exchange-tradedmarkets, the value of c isalmost always positivebecause the probabilitydistributionsof thesemarketsalways exhibit some fat-tailcharacteristics.

The sensitivity of anoption’stheoreticalvaluetoachange in skewness orkurtosis will depend on how

theoption’svaluechangesasweraiseorlowerthevalueofbandc.Ifraisingthevalueofb by one unit will cause theoption to fall by 0.15, thenthe option has a skewnesssensitivityof–0.15.Ifraisingthevalueofcbyoneunitwillcause the option to rise by0.08,theoptionhasakurtosissensitivityof0.08.Foractivetraders who carry very largeoption positions, theskewness and kurtosis

sensitivities can representsignificant risks and, as withall risks, must be monitoredto ensure that that theyremain within acceptablebounds.

The units used toexpress skewness andkurtosis sensitivity willdepend on how the skewmodel has been constructed.Most traders choose a unitthat represents a common

change in the skewness andkurtosisvalues.For example,if the value of b commonlyranges from 0.20 to 0.40, alogical unit for b might be0.01. If the unit value is anunwieldy number, the valuecan be adjusted by includingamultiplier. If theunit valueforb is0.001butwewish toexpress the unit value as awhole number,we can use amultiplierof0.001 toyield aunit value of 1. The model

willthenbeexpressedas

y=a+0.001bx+cx2

Ifwe raise the skewvalueof b by 1, we are reallyraisingitby0.001.Thesameapproachcan alsobeused toexpresscinsimpleunits.5

In most skew models,the at-the-money exerciseprice acts as a pivot point sothat an option that is exactlyat themoneyhas a skewness

and kurtosis sensitivity of 0.Optionsthatareinthemoneyoroutofthemoneycanhaveeitherapositiveoranegativeskewness sensitivity. If weincrease the skewness input,the volatility of higherexercise prices will rise,whiles the volatility of lowerexercise prices will fall.Consequently,higherexerciseprices will have positiveskewness sensitivity values,andlowerexercisepriceswill

have negative sensitivity. Ifwe increase the kurtosisinput,thevolatilityofoptionsat both higher and lowerexercise prices will rise.Consequently,anyoptionthatis not exactly at the moneywill have a positive kurtosissensitivity.

Which options are themost sensitive to changes inskewnessandkurtosis?Thereis no definitive answer

because it depends on thevolatility characteristics ofthe market as well as theskewmodel that is used.Butin many skew models putswith deltas of –25 and callswith deltas of +25 tend tohave the greatest skewnesssensitivity. For this reason, acommon measure ofskewness is the differencebetweentheimpliedvolatilityof the –25 delta put and the+25 delta call. There is no

similar benchmark forkurtosis, but for manymodels, puts with deltas ofapproximately –5 and callswithdeltasof+5tendtohavethe greatest sensitivity to achangeinkurtosis.

SkewedRiskMeasures

How we model the

volatility skew also willaffect the risk measuresgenerated by a model—thedelta, gamma, theta, andvega. Look again at Figure24-4a, where the floatingskewisshiftedeitherrightorleft as the underlying pricerises or falls. As the skew isshifted, thevolatilityat someexercise prices will rise,while the volatility at otherexercise priceswill fall.Thischangeinvolatilitycancause

anoption’svalueand its risksensitivities to change eithermoreor less thanexpected iftherewerenoskew.

For example, consideranout-of-the-moneyputwitha delta of –20. Ignoring thegamma, if the underlyingpricerises1.00,weexpecttheoption value to decline by0.20. But in an investmentskew, such as in Figure 24-4a, as the underlying price

rises, thevolatilityofanout-of-the-money putwill rise asit moves further out of themoney. If the option has avega of 0.10 and the shift inthe skew causes the impliedvolatilityoftheoptiontorise0.5 percent, the highervolatility will cause theoption’svaluetoriseby0.5×0.10 = 0.05. Consequently,the option will only declineby0.15,adeclineof0.20duetoachangeintheunderlying

price combined with anincrease of 0.05 due to theincrease in impliedvolatility.The option has a skewed oradjusteddeltaof–15.

The inclusion of avolatility skew in a pricingmodel will affect thecalculation of all option riskmeasures and can greatlycomplicate a trader’s abilityto manage risk. For manytraders,itmaybebesttokeep

thingssimple,perhapsusingaskew model to generatetheoreticalvalueswhileusinga traditional model tocalculate the delta, gamma,theta,andvega.Foranactivetrader who carries largeoption positions, calculatingaccurate skewed sensitivitiesbecomes much moreimportant because the totalvalue of the position canchange very quickly asmarket conditions change.

Financial engineers atprofessional option tradingfirms are often responsiblefor developing methods toaccurately calculatetheoretical values and risksensitivities using a volatilityskew model. But even themost sophisticated model isunlikely to generate valuesthat exactly model optionprices under all marketconditions.Amodelcanhelp,but it will always have

limitations.By combining the

volatility term structureacross expiration dates withthe volatility skew acrossexercise prices, we can forma volatility surface. Whilesometimes difficult tovisualize, a volatility surfacemay enable a trader to moreeasily see the basic volatilitycharacteristics of an optionmarket. The more exercise

prices and expiration datesthat are available, the moreaccuratewillbe thevolatilitysurface. Sample volatilitysurfaces for the FTSE 100optionsandwheatoptionsareshown in Figures 24-12 and24-13.Atthetime, theFTSE100 Index was trading at5,966, and the front-monthwheat futures contract wastradingat647.

Figure24-12FTSE100volatilitysurface,March16,2012.

Figure24-13wheatvolatilitysurface,January27,2012.

ShiftingtheVolatility

Traders have long notedthat inmany optionmarkets,implied volatility tends tochange as the price of theunderlying contract changes.Somemarketsexhibitadirectrelationship betweenmovement in the underlyingprice and changes in impliedvolatility: when the

underlying price rises,implied volatility tends torise; when the underlyingprice falls, implied volatilitytendstofall.Thisistypicalofmarketswithademandskew,such as agricultural andenergy products. Othermarkets may exhibit aninverserelationship:whentheunderlying price rises,implied volatility tends tofall; when the underlyingprice falls, implied volatility

tendstorise.Thisistypicalofmarkets with an investmentskew,suchasstockandstockindexmarkets.

For purposes of bothoption evaluation and riskmanagement, many traderswill attempt to incorporatethis characteristic into anoption pricing model. Onepossible theoretical solutionis theCEVmodel referred toinChapter23.Butthismodel

can be mathematicallycomplex and requiresadditionalinputs,allofwhichmake it difficult to use.Alternatively, many traderssimply use a “home grown”modelthatshiftsthevolatilityup or down in a way that isconsistent with the observedvolatility characteristics of amarket. However, no modelwill generate accurate valuesunder all conditions becauseimplied volatility often

changesinwaysthatseemtodefyeventhebestmodel.

A shift in volatility canalso affect the risksassociated with a position.Consider a trader who buysan at-the-money straddle.Ignoring interestconsiderations and slightadjustments for a lognormaldistribution, the trader’sposition is approximatelydelta neutral: the call has a

deltaof50,andtheputhasadeltaof–50.Butdeltaneutralmeans that the trader has noparticular preference formarket movement in onedirection or the other. Is thisreally true? If thisposition istakeninastockindexmarket,the trader actually has apreference for downwardmovementbecauseheprefershigher volatility, somethingthatismorelikelytooccurinafallingmarket.Eventhough

the position may be deltaneutralinatheoreticalworld,in the real world, it is deltanegative. On the other hand,if this position is taken in acommoditymarket,wherethemarket is likely to becomemore volatile when pricesrise, the trader really has apositive delta position. Ofcourse, itmay be difficult todeterminethereal-worlddeltaof either position. That willdependonhowfastvolatility

risesorfallsastheunderlyingprice changes. But in neithercaseisthepositiontrulydeltaneutral.

SkewnessandkurtosisStrategies

Just as a trader may haveanopinionaboutthedirectionof movement in a market(delta strategies) or about

impliedandrealizedvolatility(vegaandgammastrategies),a trader may also have anopinionabouttheshapeofthevolatilityskew.WecanseeinFigure 24-14 that, dependingon the type of skew andwhether a trader expects theskew to become steeper orflatter,thetraderwillwanttobuylowerexercisepricesandsellhigherexerciseprices,orvice versa. This is mostcommonlydoneusingout-of-

the-moneyoptions,veryoften25 delta calls and –25 deltaputs because these optionstend to be most sensitive tochanges in the slope of theskew.

Figure24-14(a)Decliningskewness.(b)Increasingskewness.

If a skew trade is nothedged, the position willclearly have a positive delta(longcalls and short puts)ornegative delta (long puts andshort calls). A trader whowants to focus solely on“buying skew” or “sellingskew” must offset the deltaposition, most commonlywith an opposing deltaposition in the underlyingcontract. When this is done,

the entire strategy is usuallyreferred toasarisk reversal.With the underlying contracttradingatapricecloseto100,the following are typical riskreversals (delta values are inparentheses):

+10 June 95puts(–25)–10 June 105calls(+25)+5 underlyingcontracts

or

–30 December90puts(–15)+30December110calls(+15)–9 underlyingcontracts

In these examples, thecalls and puts have the samedelta, but this is not arequirement. Morecommonly,callsandputsare

chosen with the same vegavalues.6Thisensures that theposition is vega neutral atinception and thereforeprimarilysensitivetochangesintheslopeoftheskewratherthan changes in overallimplied volatility. Of course,asmarket conditions change,thedelta,gamma,andvegaofthe position will almostcertainly change. When thisoccurs, a trader will have to

decide whether to maintainthe position and, if so, howbest to manage the delta,gamma, and vega risk. Therisk characteristics of atypical risk reversal werediscussedinChapter21.

Justasatradermayhavean opinion about skewness,theslopeofavolatilityskew,a trader may also have anopinion about kurtosis, thecurvatureofavolatilityskew.

If the kurtosis is expected toincrease,thepricesofoptionsat both lower and higherexercise prices will increase.A trader will therefore wantto buy strangles bypurchasing both out-of-the-money calls and out-of-the-moneyputs.If thekurtosis isexpected to decrease, optionprices at both lower andhigher exercise prices willdecline. A trader will thenwant to sell strangles by

sellingbothout-of-the-moneycalls and out-of-the-moneyputs.This isshowninFigure24-15.

Figure24-15Risingandfallingkurtosis.

If a trader “buys”kurtosis by purchasingstrangles or “sells” kurtosisby selling strangles, thepositionwillalsobesensitiveto overall changes involatilitybecausethepositionwill have a very pronouncedpositive or negative vega.Evenifthetraderiscorrectinhisassessmentofkurtosis,theposition can be negativelyaffectedbyoverallchangesin

implied volatility. If thetrader wishes to focus solelyon kurtosis, he will need toneutralize his vega positionwithoutchangingthekurtosisof the position. Because at-the-moneyoptionsareneutralwith respect to kurtosis, atrader can achieve this bytaking an offsetting vegaposition in at-the-moneystraddles. Assuming that theselected strangles andstraddlesaredeltaneutral,the

entire position will also bedelta neutral. With theunderlyingcontracttradingata price close to 100, thefollowingare typicalkurtosispositions (vega values are inparentheses):

If a kurtosis position ismadeupofastranglethathas

exactly half the vega of thestraddle, a vega-neutralposition will consist of twostrangles for each straddle.Whendoneinthis2×1ratio,the position is sometimesreferredtoasadragonfly.

An opinion about skewand kurtosis can also beincorporated into otherstrategies.Considertheskewsin Figure 24-16 on the sameunderlying product but for

different expiration months.If a trader has noopiniononwhether either skew ismispriced individually butbelieves that the skews aremispriced with respect toeach other, a logical strategymight be to take a skewposition in one expirationmonthandanopposingskewposition in the other month.For example, a trader mightbuyout-of-the-moneyputs inJune and sell out-of-the-

moneyputs inMarch.At thesame time, the trader mightsellout-of-the-moneycalls inJune and buy out-of-the-money calls in March. Thetrader has, in effect, boughtputcalendarspreadsandsoldcall calendar spreads. If theskew is the onlyconsideration (the trader hasno opinion on whetherimplied volatility is high orlow), the trader will try totake a position that is vega

neutral by choosing calendarspreads that haveapproximatelythesamevega.Any residual deltas can behedged away with theunderlyingcontract.

Figure24-16Buyingandsellingskewindifferentexpirationmonths.

If, in addition to anopinion about the skew, atrader also has an opinionabout the relative impliedvolatility in differentexpiration months, he cantake this into considerationwhenchoosingastrategy.Ifatrader believes that impliedvolatility in June is lowcompared with impliedvolatilityinMarch,thetraderwillconsiderbuyingcalendar

spreads—buy June optionsandsellMarchoptions. If, atthesametime,thetraderalsobelieves that the skews aremispriced with respect toeach other, as in Figure 24-16, he will choose calendarspreads that take bothrelationships intoconsideration. Now he willwant to buy put calendarspreads—buy out-of-the-moneyJuneputsandsellout-of-the-moneyMarchputs.By

doing so, the trader takesadvantage of both impliedvolatilityandskew.Notethatthe trader will avoid callcalendar spreads because thevolatility and skew will tendtooffseteachother.TheJunecalls are too expensive withrespect to skew, but theMarchcallsaretooexpensivewith respect to impliedvolatility. If the traderbelieves that June impliedvolatility is high compared

with March, now he willchoose to sell call calendarspreadsbecausetheJunecallsare too expensive withrespect to both impliedvolatilityandskew.

The same approach canbeusedwhenkurtosis in twodifferent expiration monthsseems to be mispriced, asshown inFigure 24-17.Nowa trader might considerbuying June strangles and

selling March strangles. Ifthis is a simple kurtosisstrategy within a singleexpiration month, it will benecessary to offset the vegaby purchasing at-the-moneystraddles. But a trader canavoid this complication bychoosingstrangles in the twodifferent expiration monthsthat have approximately thesame vega values. Thisensures that the entirestrategy is sensitive only to

changes inkurtosis. If, at thesame time, June impliedvolatility seems lowcompared with March, thestrategy has an addedadvantage. June options arecheap compared with Marchoptions with respect to bothvolatilityandkurtosis.

Figure24-17Buyingandsellingkurtosisindifferentexpirationmonths.

ImpliedDistributions

In a perfect Black-Scholesworld, the prices of theunderlying contract areassumed to be lognormallydistributed at expiration, andevery option with the sameexpirationdateought tohavethe same implied volatility.The fact that options acrossdifferent exercisepriceshave

different implied volatilitiesmust mean that themarketplace believes that thedistribution of underlyingprices at expiration is notlognormal. Exactly whatprobability distribution is themarketplace implying to theunderlying contract atexpiration? We can estimatethis implied distribution bylooking at the prices ofbutterfliesinthemarketplace.

Atexpiration,abutterflyhas aminimumvalue of 0 ifthe underlying price is at oroutside the wings and amaximum value of theamount between exerciseprices if the underlying priceis exactly at the body, ormidpoint,of thebutterfly.Atexpiration, the 95/100/105butterfly (i.e., buy a 95 call,sell two100calls,buya105call) will have a minimumvalue of 0 if the underlying

priceisatorbelow95oratorabove105,amaximumvalueof5.00iftheunderlyingpriceis exactly 100, or someamountbetween0and5.00ifthe underlying price isbetween 95 and 100 orbetween100and105.

Suppose that exerciseprices at five-point intervalsare available extending from0toinfinity:

…,70,75,80,85,90,95,

100,105,110,115,120,125,130,…

What will be the value ofthe position at expiration ifwe buy every five-pointbutterfly?

Regardless of theunderlying price at

expiration, theentirepositionwill always have a value ofexactly 5.00. As a result, ifweaddupthepricesofallthebutterflies, the total valuemustbe5.00.7

Suppose that we makethe assumption that the onlyprices that are possible atexpiration are prices that areequaltoanexerciseprice

…,70,75,80,85,90,95,100,105,110,115,120,125,

130,…

The probability of eachunderlying price occurringmustbeequal to thepriceofthat butterfly, where theinside exercise price is equalto the underlying pricedivided by 5.00. If the priceof the 75/80/85 butterfly is0.15, the probability of anunderlying price of 80 atexpirationmustbe

0.15/5.00=0.03(3%)

If the price of the90/95/100 butterfly is 0.50,the probability of anunderlying price of 95 atexpirationmustbe

0.50/5.00=0.10(10%)

Figure 24-18 shows aseries of call values togetherwith the resulting butterflyvalues for our series of

exercise prices.8 Theprobability associated witheach underlying price isdetermined by dividing thebutterfly value by the totalvalueofallbutterflies,whichweknowmust be 5.00. (Thereader may wish to confirmthatallthebutterflyvaluesdoindeed sum to 5.00 and thattheprobabilitiessumto1.00,or 100 percent.) Theunderlying prices and their

associated probabilities areshown inFigure 24-19.Notethat these values form aprobabilitydistributionthatisskewed to the right. Thisshould come as no surprisebecause the values werederived from the Black-Scholes model, whichassumes a lognormaldistribution of underlyingprices.

Figure24-18Butterflyvaluesandprobabilities.

Figure24-19Adiscreteprobabilitydistributionimpliedfromthepricesofbutterflies.

Of course, thedistributioninFigure24-19isonly an approximationbecause it includes a limitednumber of underlying prices.A more exact distributionrequires us to consider moreandmoreexerciseprices.Wecan do this by reducing thewidth of the butterflies.Instead of using incrementsof 5.00, we might useincrements of 2.00, 1.00, or

0.50. Indeed, if we useincrements that areinfinitesimally small, thebutterflyvalueswillenableusto construct a continuousprobability distribution.Figure 24-20 shows theprobability distribution withthe increment betweenexercise prices reduced to0.10. With such a smallincrement, the distributionappearsalmostcontinuous.

Figure24-20Acontinuouslognormalprobabilitydistributionimpliedfromthepricesofbutterflies.

How does thedistributionimpliedbyoptionprices compare with atraditional lognormaldistribution? The implieddistribution will change asoptionpriceschange,sotherecannot be one implieddistribution under all marketconditions.Butwemight getsomesenseofthedistributionthat the marketplace isimplying by using the prices

of butterflies generated by avolatility skew to derive adistribution and comparing itwith a Black-Scholesdistribution with a constantvolatility.InFigure24-21,wehavetakenthevolatilityskewfor the FTSE 100 optionsshown in Figure 24-3 andcreatedtwodistributions,onefrom the prices generatedfrom the skew and one fromprices generated from aconstant volatility across

every exercise price. Whatcanwe infer fromFigure24-21?

Figure24-21Three-monthpricedistributionimpliedfromFTSE100optionprices,March16,2012(FTSE100Index=5,965.58).

Compared with atraditional lognormaldistribution, the marketplaceseems to be implying thefollowing:

1. A greaterprobability of asmall tointermediateupwardmove2. A greaterprobability of a

large downwardmove3. A smallerprobability of asmall tointermediatedownwardmove4. A smallerprobability of alargeupwardmove

Thisimplieddistributionistypicalofmoststockindex

markets,andmanyofthesepointsseemtobeconsistentwiththeS&P500histograminFigure23-8a.Theredoseemtobemoresmallmovesintherealworldthanispredictedbyatheoreticaldistribution.Therealsoseemtobemorelargedownwardmovesandfewerintermediatedownwardmoves.Butthehistogramalsoshowsmorebigupwardmoves,whichisnotconsistentwiththe

implieddistribution.Figure 24-22 shows the

three-month distributionimplied from the prices ofoptions on wheat futures onJanuary 27, 2012. Thisimplied distribution seems toconform more closely to atheoretical lognormaldistribution than does thedistribution in the FTSE 100example. However, themarketplace is still implying

more small moves, slightlyfewer intermediate upwardmoves, and slightly morelarge upward moves than atruelognormaldistribution.

Figure24-22Three-monthpricedistributionimpliedfromwheatoptionprices,January27,2012(withthree-monthwheatfuturesat661.75).

Ofcourse,Figures24-21and 24-22 are snapshots ofmarkets at one moment intime,and itwouldbeunwiseto draw any sweepingconclusions from theseexamples.Nonetheless,itcanoftenbeusefulforatradertocomparehisopinionsaboutaprobability distribution withthat implied by prices in themarketplace.Ifthereisacleardisagreement, it may point

the way to a potentiallyprofitablestrategy.

1TheFinancialTimesStockExchange100Index(theFTSE100)isthemostwidelyfollowedindexofU.K.stockprices.2Thereare,ofcourse,investorsandtraderswhotakeshortstockpositions,buttheyarerelativelysmallinnumbercomparedwiththosewhoarelongstock.3AmoretheoreticallycorrectapproachinvolvesusingtheforwardpriceFratherthanthespotprice,S

Inthiscase,theat-the-forward

optionhasastandarddeviationof0.

4Becauseallexchange-tradedmarketsseemtoexhibitpositivekurtosis,weignorenegativekurtosisskews.5Whentradersusethetermsskewnessandkurtosis(orskewandkurtforshort),itisnotalwaysclearwhethertheyarereferringtotheinputsintothemodel(thevaluesofbandcinourexample)orthesensitivityoftheoption’svaluetoachangeintheseinputs.Typically,atraderwillrefertothesensitivitiesastheoption’sskewnessorkurtosis.Orthetraderwillrefertohisskewnessandkurtosisposition:thesensitivityofhisentirepositiontoaone-unitchangeinthe

skewnessorkurtosisinputs.6Ariskreversalthatisveganeutralwilltendtobegammaneutral,althoughthiswillnotalwaysbethecase.Atradermayhavetodecidewhetheritismoreimportantfortheriskreversaltobeveganeutralorgammaneutral.7Ifweincludeinterestrates,andtheoptionsaresubjecttostock-typesettlement,thetotalwillbethepresentvalueof5.00.8ThevaluesinFigure24-18correspond,approximately,toBlack-Scholesvaluesusinganunderlyingpriceof100,threemonthstoexpiration,avolatilityof20percent,andaninterestrateof0.

25

VolatilityContracts

Volatility contracts havebeenoneofthemajorsuccessstories in the derivativesmarket. They enable marketparticipants to pursuestrategies that were

previously either impossibleor, even under the bestconditions, difficult toexecute. But volatilitycontracts have unusualcharacteristics,andanytraderhoping to make the best useof these contracts must befully familiar with thesecharacteristics.

Prior to the introductionofoptionsandoptionpricingmodels, there was no

effective way for a trader tocapture volatility value or toprofit from a perceivedmispricingofvolatilityinthemarketplace. Once listedoptions were introduced,however, it became possibletousetheimpliedvolatilityinanoptionmarkettodeterminehow the marketplace waspricing volatility. In Chapter8, we showed that a tradercouldthencaptureavolatilitymispricing by either buying

or selling options anddynamically hedging theposition over the life of theoption.

This all sounds verygoodintheory,butintherealworld, things are not sosimple. Even if we aresomehowabletolookintothefutureanddetermine the truevolatility of the underlyingcontract over the life of theoption, the actual results of

any single dynamic hedgingstrategywill almost certainlydiffer from the resultspredicted by the theoreticalpricing model. This is oftendue to the weaknesses intraditional theoretical pricingmodels, many of which wetouchedoninChapter23:

Theorderinwhichpricechangesoccurcanaffecttheresultsof

adynamichedgingstrategy.Ifgapsoccurin

theunderlyingprice,itmaynotbepossibletobuyorselltheunderlyingcontractinawaythatisconsistentwiththedynamichedgingprocess.Thereturnsfor

anunderlying

contractmaynotbenormallydistributed.

In addition to weaknessin the model, the costs ofdynamically hedging aposition may be significant.Each time the position isrehedged, a trader may havetogiveupthebid-askspread,and there will also bebrokerageandexchangefees.These costs will certainly

reduce, and may even erase,anyexpectedprofit.

Even ifone is interestedin trading volatility, thedrawbacks of using adynamic hedging approachwilloftendetera trader fromusing options to tradevolatility. To overcome thisobstacle, traders have soughta lesscomplicatedmethodofimplementing volatilitystrategies.Thishasledtothe

development of volatilitycontracts, contracts thatenable a trader to take aposition on volatilitywithoutgoingthroughacomplexandcostly dynamic hedgingprocess. At expiration, thevalue of these contractsdependssolelyonarelativelystraightforward volatilitycalculation.

There are two primarytypes of volatility—realized

volatility and impliedvolatility. Consequently,there are two types ofvolatility contracts. Realizedvolatilitycontracts settle intothe realized volatility of anunderlying contract over aspecified period of time.Implied volatility contractssettle into the impliedvolatility of options on anunderlying contract on aspecifieddate.

RealizedVolatilityContracts

At expiration, the value ofarealizedvolatilitycontractisequal to the annualizedstandard deviation oflogarithmicpricereturnsoverthe life of the contract. Thereturns are typicallycalculated from dailysettlement prices on theprimary exchange on which

the contract is traded. Thismeans that the annualizationfactor will depend on thenumber of trading days in ayear on that particularexchange. If there are 252trading days, the settlementvolatilitywillbe

whereeachdatapointxi is

equal to the daily pricereturns pi/pi–1 (today’ssettlement price divided byyesterday’s settlement price),andnisthenumberoftradingdaysinthecalculationperiod.

There are two points ofparticular note. First, theexpiration value representsthe true volatility over thecalculationperiod rather thana volatility estimate. Wetherefore use the population

standarddeviationratherthanthe sample standarddeviation, dividing by nratherthann–1.Second,thevolatility calculation isindependent of any trend inprices.Wethereforeassumea0mean, using ln(xi) for eachdatapointratherthanln(xi)–µ. These calculationconventions are common tomost realized volatilitycontracts.

The profit or loss atexpiration for a realizedvolatilitycontractwilldependon the price at which theinitial trade was made, thenotional amount of the trade,and the value of the contractatexpiration.Ifthebuyerofarealized volatility contractentersintothetradeatapriceof20percentwithanagreed-on notional amount equal to$1,000 per volatility pointand the realized volatility

over the calculation periodturnsouttobe23.75percent,the buyer will show a profitof

$1,000×(23.75–20.00)=$3,750

If the realized volatilityturnsouttobe18.60percent,thebuyerwillshowalossof

$1,000×(18.60–20.00)=–$1,400

Realized volatilitycontracts are most oftentraded in the off-exchangemarket, with banks andproprietary trading firmsacting as market makers.1Quotes for realized volatilitycontracts typically include aprice, quoted in volatilitypoints, and a volatilityexposure, quoted as notionalvega. A market maker whooffers a quote for realized

volatilityof19.50–20.50 for$10,000 notional vega iswilling tobuy thecontractata volatility of 19.50 percentand sell the contract at avolatility of 20.50 percent,with every volatility pointhavingavalueof$10,000.Inthe same way, a client mayput in an order to buy$25,000 notional vega at 30.The client is prepared to paya volatility of 30 percent,with every volatility point

havingavalueof$25,000.In these examples, the

priceofthevolatilitycontractwas quoted in volatilitypointsandsettledinvolatilitypoints. In fact, most realizedvolatilitycontractsaresettledin variance points, wherevarianceisequaltothesquareofvolatility

For this reason, realized

volatility contracts are oftenreferred to as variancecontracts or, morecommonly,varianceswaps.

Why settle a volatilitycontract in variance pointsrather than volatility points?As we shall see later, forpurposes of hedging avolatility contract, it ismucheasier to replicate a varianceposition than a volatilityposition. Additionally, the

reader may recall from thediscussion of forwardvolatility in Chapter 20 thatvariance has the verydesirable characteristic that itisproportional to time. If thevariance over some timeperiod t1 is equal to σ12 andthe variance over a secondsuccessive time period t2 isequaltoσ22,thenthevarianceover the combined timeperiodsis

This means that variancecontracts can be easilycombined to coverconsecutive time periods,even if the time periods arenotofequallength.

For example, if theannualized volatility over atwo-month time period is 25(expressing the volatility in

points) and the annualizedvolatility over the followingone-month timeperiod is 22,the annualized variance overthe entire three-month periodis

If avolatilitycontract isquoted in volatility pointswithanotionalvegaamount,but settlement is in variance

points, how much is eachvariance point worth?Without going into themathematics, by convention,each variance point is equalto the notional amountdividedbytwicethevolatilityprice

If the buyer of a volatilitycontract pays 20 for $10,000

vega notional, but thecontract issettled invariancepoints, each variance pointhasavalueof

$10,000/2×20=$250

If the realized volatilityover the life of the contractturnsouttobe19percent,thebuyerwillshowalossof

$250×(192–202)=$250×(361–400)=$9,750

If the realized volatilityover the life of the contractturnsouttobe23percent,thebuyerwillshowaprofitof

$250×(232–202)=$250×(529–400)=$32,250

Because variance is thesquare of volatility, if acontract issettled invariancepoints,thevalueatsettlementcan quickly escalate withhigher volatilities. If a single

dramatic event occurs thatcauses the underlyingcontract to make anunexpectedly large move,resulting in a volatility overthe calculation period of 50,theprofit to thebuyer in ourexamplewillbe

$250×(502–202)=$250×(2,500–400)=$525,000

Of course, the seller willhave an equal loss. Indeed,

the seller of a variance swapmaynotbewillingtotakeonthe risk of a one-timedramatic event that causesvolatility to skyrocket.Manyvarianceswapsthereforehavea cap that limits theexpiration value of thecontract.Ifthecontracttradesat a price of 20 and has avolatilitycapof40(equaltoavarianceof1,600),nomatterhow high volatility goes, theprofittothebuyerandriskto

thesellercanneverbegreaterthan

$250×(402–202)=$250×(1,600–400)=$300,000

Caps are most commonfor variance swaps onindividual stocks, where aone-timeeventcanresultinadramatic increase involatility. Caps are lesscommon for variance swapson broad-based indexes,

where a one-time eventaffecting one indexcomponentisunlikelytohaveacorrespondinglylargeeffecton the volatility of the entireindex. Of course, varianceswaps are primarily an off-exchange product.The buyerandselleroftheswaparefreeto negotiate any contractspecifications, including acap, that are mutuallyagreeabletobothparties.

ImpliedVolatilityContracts

Realized volatility is animportant consideration inoption pricing, but it issomething that cannot bedirectly observed, at least atany given moment in time.When option traders talkaboutvolatility,theyaremostoften referring to impliedvolatility,whichissomething

that can be observed. Theimplied volatility is aconsensus, derived from theprices of options in themarketplace, of what thevolatility of the underlyingcontract will be over someperiod in the future.Becauseoptionpricescanbeobservedat any moment in time,implied volatility at anymoment can likewise beobserved.

In the early days ofexchange-traded options, theconcept of implied volatilitywas not well understood, atleast not among mostnonprofessional traders.However, as option tradingincreased in popularity, allmarket participants, bothprofessional andnonprofessional,begantopaycloserattentiontotheimpliedvolatility in option markets.As a means of promoting a

better understanding ofoptionsandasanaid tobothtraders and end users,exchanges begandisseminating impliedvolatility data.With growingpublic interest in options,these numbers began toappear with increasingfrequency in financial newsreports.

There are, of course,many different implied

volatilities.Notonlyaretheremany different underlyingmarkets, but for eachunderlying, there are manydifferent exercise prices andexpiration months. Whatexchanges wanted was onenumber that reflected thegeneral implied volatilityenvironment. This led theChicago Board OptionsExchange (CBOE) to focuson the implied volatility of abroad-based index,

specifically its most activelytraded product, the OptionsExchange Index, with tickersymbol OEX. In 1993, theCBOE began disseminatingvaluesforthevolatilityindex(VIX), a theoretical 30-dayimplied volatility calculatedfromthepricesofoptionsontheOEX.TheVIXeventuallydeveloped into a widelyrecognizedfinancialindicatornot only in the optioncommunity but also in the

financial world in general.Other exchanges havefollowed suit by creatingvolatility indexes of theirown,buttheVIXremainsthebest known of all impliedvolatilityindexes.

As the VIX becamemore widely recognized, theCBOE began to consider thepossibility of creating atradablecontractbasedontheVIX. This necessitated two

major changes in the index.The first change had to dowith the underlying contract.Initially, VIX valuesdisseminated by the CBOEwere derived from the pricesof OEX options. However,the CBOE subsequentlyintroduced options on theStandard and Poor’s (S&P)500Index,withtickersymbolSPX, and these eventuallyreplaced the OEX as theexchange’s most actively

traded index product. That,combined with the fact thatthe S&P 500 was a muchmore widely followed indexthan the OEX, led theexchange in 2003 to begincalculating theVIX from theprices of S&P 500 optionsratherthanOEXoptions.

The second change hadto do with the calculationmethod. The original VIXwascalculatedfromcallsand

putsatthetwoexercisepricesthatbracketedtheindexprice—essentially the at-the-money options. For a givenexpirationmonth,thecallandput implied volatilities ateach exercise price wereaveraged,andthesewerethenweighted by the differencebetween the exercise priceand the index price to yieldan implied volatility for thatexpirationmonth. In order todetermine a theoretical 30-

dayimpliedvolatility,thetwonear-term expiration monthswere weighted to derive afinalvalue.2Anexamplemayhelp to clarify themethodology.

Assume that the indexprice is 863.40 and that thetwo exercise prices thatbracket this number are 860and870.Assumealsothatthenearest option contract,Month 1, has 14 days

remaining to expiration andthat the second optioncontract, Month 2, has 42days remaining. Impliedvolatilities for the twoexercisepricesineachmonthareasfollows:

The average impliedvolatilities for each exercisepriceandmonthare

The implied volatility ineachmonthistheinterpolatedimpliedvolatilitybetweenthetwo exercise prices—theimplied volatilities weightedby their distance from theindex price. The closer theexercise price is to the indexprice, the greater is the

weighting:

The VIX value is theinterpolatedimpliedvolatilitybetween the two expirationmonths—the impliedvolatilities weighted by how

close their expirations are to30days.Thecloserto30,thegreater the weighting. Withexpirationsof14and42days,thefinalVIXvalueis

The final VIX valuedisseminatedbytheexchangeis the calculated VIX valuerounded to two decimal

points,inthiscase21.02.When the exchange

beganplanningfor trading inVIX-related products, therewere twomajorobjections tothe original calculationmethodology. First, anyexchange-traded product hasto have a very well-definedvalueonwhicheveryonecanagree. If there are significantdisagreementsastothevalueof a contract, especially at

expiration, some traders willfeel that they are beingtreated unfairly. This willcertainlyinhibittradingintheproduct and might, in somesituations,leadtolegalactionagainst the exchange. Theoriginal VIX calculationrequired a theoretical pricingmodel to determine impliedvolatilities. This in itself canresult in disagreements.Whichmodelshouldbeused?The Black-Scholes model?

The binomial model? Someother more exotic model?(Recall fromChapter 22 thatthe OEX is an Americanoption, carrying with it therightofearlyexercise.)Evenif there is general agreementon an appropriate model,there may be disagreementsas to the inputs into themodel. What interest rateought to be used? Whatdividend assumptions shouldbe made? The exchange

concludedthatifitwantedtointroduce trading in VIX-related products, it would benecessary to improve on theexisting calculationmethodology.

The second objectionhad to do with the fact thatonly at-the-money optionswere used to calculate VIXvalues. As traders becamemore knowledgeable aboutoptions, the volatility skew,

orsmile,becameincreasinglyimportant in describing thevolatility environment and indetermining appropriatestrategies.Traderswanted animplied volatility index thatwould encompass not onlythe implied volatility of at-the-money options but alsotheimpliedvolatilityacrossabroad range of exerciseprices.

The VIX calculation

methodology that waseventually chosen to replacethe old methodology wassuggestedinaresearchpaperfrom Goldman Sachspublishedin1999.3Thepaperessentially asked thisquestion: is it possible tocreateanoptionposition thatwillcapturethetruevolatilityof the underlying contractunder all possible volatilityscenarios?

In theory, ifwewant totake a volatility position, wecaneitherbuyoptions(alongvolatility position) or selloptions (a short volatilityposition) and thendynamically hedge theposition to expiration. Forexample, we might take along volatility position bypurchasing one or more at-the-moneyoptionsandsellinga delta-neutral amount of the

underlying contract.4 Byperiodically rehedging theposition to remain deltaneutral, we will capture thevolatility value of theunderlyingcontract.

Thisallsoundsverynicein theory,but it almostneverworks out exactly asexpected.Perhapsthegreatestdrawback to the strategy isthe fact that exposure to thevolatility of the underlying

market, as measured by thevega, will change over thelife of the strategy. Anoption’s vega value isgreatestwhentheoptionisatthe money, but even if webegin by purchasing at-the-money options, the optionswill almost certainly notremain at themoney. As theunderlyingpricerisesorfalls,theoptionswilleithergointothe money or out of themoney, and the vega of the

position will decline. Thiswas discussed in Chapter 9and is shownagain inFigure25-1.

Figure25-1Thevega(volatilitysensitivity)ofanoption.

If we want to create along volatility position, wewant a constant exposure tovolatility regardless ofchanges in the price of theunderlying contract. Wemight try to accomplish thisby purchasing options acrossa broad range of exerciseprices. In this scenario,shown in Figure 25-2, oneexercise pricewill always beat the money. Unfortunately,

this will still not result in aconstant vega exposurebecauseat-the-moneyoptionswith higher exercise priceshavegreatervegavaluesthanat-the-money options withlower exercise prices. If weaddupall thevegavalues inFigure 25-2 at eachunderlying price, the totalvega will be lower at lowerunderlying prices and higherathigherunderlyingprices.

Figure25-2Volatilityexposureifwepurchaseoneoptionateachexerciseprice.

To achieve a constantvolatility exposure, we needto buy more options withlower exercise prices andfewer options with higherexercise prices. How manyoptionsateachexercisepriceshould we buy? It turns outthat the proper proportion ofeachexercisepriceneededtocreate a position withconstantvolatilityexposureisinversely proportional to the

squareoftheexerciseprice

1/X2

The result of doing this isshowninFigure25-3.

Figure25-3Purchasing1/X2optionsateachexerciseprice.

Of course, to exactlyreplicate avolatilityposition,we would have to purchaseoptions, in the correctproportion, at every possibleexerciseprice—essentiallyaninfinite number of options.Exchanges,however,onlylista finite number of exerciseprices. Still, it might bepossible to use the exercisepricesthatarelistedtocreatea position that closely

approximates a theoreticallyconstant volatility position.This is thebasis for theVIXcalculationmethodologyusedbytheCBOE.

Essentially, the value ofthe VIX is the cost ofpurchasing a strip consistingof options at every availableexercise price. Because theVIX represents a 30-dayimplied volatility, the valueof the VIX is derived from

strips of options in the twomonthly expirations thatbracket 30 days. The valuesofthestripsarethenweightedbyhowcloseeachexpirationis to 30 days.Without goinginto the complete derivationof the VIX,5 there are someimportant aspects that areworthpointingout:

1.ThevalueoftheVIX is derivedfrom the volatility

value (time value)oftheoptionsintheunderlying index,not the intrinsicvalue. Therefore,only the prices ofout-of-the-moneyoptions (comparedwith the forwardprice)areused.2. The (implied)forward price forthe index is

determined usingput-call parity forthe closest-to-the-money exerciseprice.3. The optionvalue used at eachexercisepriceistheaverage of thequoted bid priceandaskprice.4. When twoexercisepriceswith

a nonzero bid areencountered, nolower exerciseprices for puts orhigher exerciseprices for calls areincluded in thecalculation5. Becauseonlyafinite number ofexercise prices areavailable, thecontributionofeach

option to the finalVIX calculation isadjusted based onthe distancebetweenconsecutiveexerciseprices.Thegreater the distancebetween exerciseprices,thegreateristheweightingintheindex for a specificoption.

Note that the VIXcalculation depends only onthe prices of options—notheoretical pricing model isrequired. Other than optionprices,theonlyotherrequiredinputisaninterestrate,whichis necessary to determine theindex forward price underput-call parity as well as theinterestcostofpurchasingtheoptions. For this, the CBOEuses the risk-free rate—theU.S. Treasury bill rate with

maturityclosest to theoptionexpiration. Otherwise,calculation of the VIX isrelativelystraightforward.

Because the VIXrepresents a theoretical 30-day implied volatility, tradedcontracts on the VIXtypicallyexpire30dayspriorto expiration of the optionsusedtocalculateVIXvalues,usually the third Wednesdayof the previous month. VIX

January contracts expire 30days prior to expiration ofFebruary SPX options; VIXFebruary contracts expire 30days prior to expiration ofMarch SPX options; and soon.

With exactly 30 daysremaining to expiration ofSPXoptions,thevalueoftheVIX at expiration isdetermined solely by theprices of SPX options in the

expiration month. Forpurposesofsettlement,ratherthanusing theaverageof thebid and ask, the expirationvalueoftheVIXiscalculatedfromtheactualopeningtradeprices of SPX options onexpiration Wednesday. Thetrade prices are determinedthrough a special openingrotation where standing buyand sell orders areautomatically matched todetermine one opening trade

price for each option. If notrade takes place for anoption,thepriceusedforthatoption reverts to the averageof the bid and ask. Thisprocedure can sometimescause unusual jumps in theVIXvalueatexpiration.Ifalloptions trade at the askpriceon theopening (abuy print),the expiration value is likelytobehigherthanexpected.Ifall options trade at the bidprice on the opening (a sell

print), theexpirationvalue islikely to be lower thanexpected. Immediately afterthe VIX expiration value isdetermined by the specialopening rotation, calculationreverts to its normalmethodology using theaverageofthebid-askspread.

SomeVIXCharacteristics

While volatility is, intheory, independent of thedirection in which theunderlying contract ismoving, in the real world,traders have long recognizedthat some markets tend tobecome more volatile as theunderlying price rises, whileothermarketstendtobecomemore volatile as theunderlying price falls. Thereis a widely held belief thatstock index markets exhibit

the latter characteristic. Itshould therefore not come asa surprise that the VIX isgenerally negativelycorrelated with the S&P 500Index.When the index falls,the VIX tends to rise; whentheindexrises,theVIXtendsto fall. This inversecorrelation between changesin the S&P 500 and changesin the VIX for the 10-yearperiodfrom2003to2012canbe seen in Figures 25-4 and

25-5. Figure 25-4 confirmsthe tendency of S&P 500prices and VIX prices tomove in opposite directions.Figure25-5shows thestronginversecorrelationvalueof–0.7444 between percentchanges in the values of thetwoindexes.Figure25-5alsoincludesabest-fitlineforthetwosetsofvalues:thepercentchange in the VIX isapproximately 5.7 timesgreater than the percent

changeintheS&P500,butintheoppositedirection.

Figure25-4s&P500andViXprices:2003–2012.

Figure25-5Dailys&P500indexchangesversusdailyViXchanges:2003–2012.

Given the apparentinverse correlation betweenthe S&P 500 Index and theVIX, one might wonderwhether this is actuallysupported by market data. Ifthe VIX rises, will the S&P500 Index become morevolatile?IftheVIXfalls,willthe index become lessvolatile? Because the VIXrepresents a 30-day impliedvolatility, if the marketplace

is correct,whenever theVIXrises, the next 30 days oughtto be more volatile than theprevious 30 days, andwhenever the VIX falls, thenext30daysoughttobelessvolatile than the previous 30days.ThemoretheVIXrisesorfalls,thegreatershouldbethe change in realizedvolatility. The actual resultsover the sample 10-yearperiod are shown in Figure25-6.

Figure25-6DoesachangeintheViXpredictachangeinrealizedvolatility?

If there is a correlationbetween changes in the VIXand changes in realizedvolatility, it is not apparentfromthedata.SometimestheVIX rises and sometimes itfalls, but there is no obviousincrease or decline involatility over the following30-day period. (There is avery small but probablyinsignificant positivecorrelation of +0.1561.)

Therefore, one mightconcludethattheVIXhasnopredictive value as anindicator of rising or fallingrealized volatility. PerhapswhatdrivestheVIXisnottheexpectationof future realizedvolatility, but the desire tobuy protection in a fallingstock market. In a fallingmarket, more hedgers enterthemarket,andtheyareoftenwilling to pay higher pricesforprotectiveoptionswithout

regard to considerations ofrealized volatility. They aredriven by the fear of furtherdeclines in the market. Forthis reason, the VIX issometimes referred to as thefearindex.

We have also noted thewidely held belief that stockindexmarketstendtobecomemore volatile as theunderlyingpricefallsandlessvolatile as the underlying

price rises. We might askwhether this assumption isborne out by the availabledata. Figure 25-7 shows thechange in the price of thes&P500Indexovera30-dayperiod compared with therealized volatility over thesameperiod.Iftheconjectureis true, more data pointsoughttofallinboththeupperleft portion (a falling indextogether with highervolatility)andthelowerright

portion (a rising indextogether with lowervolatility).

Figure25-7Arefallingstockmarketsmorevolatilethanrisingmarkets?

Here there is somereason to believe that fallingstockmarketsdo indeed tendtobemorevolatilethanrisingstock markets. We can seefrom the sample period(2003–2012) that there aremore high-volatilityoccurrences to the left of the0lineandmorelow-volatilityoccurrencestotherightofthe0 line. There is a moderateinverse correlation of –

0.3895.

TradingtheVIX

As with all indexes, theVIX is composed ofcomponents, with eachcomponent having a weightwithintheindex

An index can often bereplicated by purchasing allor a large number of theindex components in thecorrect proportion. This iscommonly done in the stockindex market to create aportfolio that tracks an indexor as part of an arbitragestrategy. But unlike a stockindex, it is not easy toreplicatetheVIX.Asoptionsgointoandoutofthemoney,the index components and

theirweightswithintheindexare constantly changing. Formost traders, the onlypracticalmethodofbuyingorsellingtheVIXis throughitsderivative products: futuresand options or productslinked to these contracts.BecausetheVIXitselfcannotbeeasilyboughtorsold,VIXderivatives do not alwaystrack the indexorperformasexpected,andnewtradersareoften surprised by the results

ofVIX-relatedstrategies.

VIXFuturesThe CBOE began trading

VIX futures contracts (withticker symbol UX or VXdepending on the quotevendor) in 2004. The futurescontractssettle into thevalueof theVIXat theopeningoftrading on expirationWednesday, with each

volatilitypointhavingavalueof$1,000.

VIX futures haveunusual characteristics whencompared with moretraditional futures markets,and traders who enter theVIX futures market for thefirst time are often surprisedand frequently disappointedattheresultsofaVIXfuturesstrategy. There are twoprimary reasons for this.

First, VIX futures exhibit aterm structure, which canaffect how futures priceschange as market conditionschange. Second, unlike aposition in other futuresmarkets, an underlyingpositionintheVIXcannotbeeasily replicated. In a stockindexfuturesmarket,atradercan replicate an underlyingindex position by buying orsellingthecomponentstocks.In a physical commodity

futures market, a trader canreplicate a long underlyingposition by purchasing thecommodity. But for mosttraders, replicating anunderlying VIX positiondirectly using options fromwhich the index is calculatedis usually not a practicalchoice.

VIX futures tend toreflect the term structure ofimplied volatility in the s&P

500 discussed in Chapter 20and shown in Figure 20-13.Most often VIX futuresexhibit a contango (upward-sloping) relationship, wherelong-term maturities trade athigher prices than short-termmaturities. A typical VIXcontango structure, forfutures during August 2012,is shown in Figure 25-8.Although less common, VIXfutures can also exhibit abackward (downward-

sloping) relationship. Such astructure for futures pricesone year earlier, in August2011, is shown inFigure 25-9. Figure 25-10 shows theVIX moving from contangoto backward during thefinancial crisis in the latterhalfof2008.

Figure25-8ViXfuturesincontango(upwardsloping).

Figure25-9ViXfuturesinbackwardation(downwardsloping).

Figure25-10ViXfuturesmoveddramaticallyfromcontangotobackwardduringthefinancialcrisisinlate2008.

WhenVIXfuturesareina normal contangorelationship, as in Figure 25-8, if there is no change inmarket conditions, as timepasses, the futures contractwill move down the term-structure curve, graduallylosing value as time passes.How does this affect tradingdecisions in the VIX futuresmarket?

Logically, a trader will

wanttobuyafuturescontractwhen he believes that thefuturespricewillriseandsella futures contract when hebelieves that the price willfall.Mosttradersassumethatwhen an underlying indexrisesorfalls,futurescontractsonthatindexwillalsoriseorfall,andthisisgenerallytrueof the VIX—when the indexrises,VIX futures rise;whenthe index falls, VIX futuresfall.Mosttradersalsoassume

that when an index rises orfalls, futures prices will riseor fall by approximately thesame amount. But VIXfutures prices reflect wherethe marketplace thinks SPXimplied volatility will be atmaturity of the futurescontract.Impliedvolatility,asreflected in the index value,maybehighorlowtoday.If,however, the marketplacebelieves that impliedvolatilitywillchangebetween

now and expiration of thefutures contract, the futurescontract will be pricedaccordingly.A traderwill bedisappointed indeed if hebuys a VIX futures contract,seesanincreaseintheindex,but finds that there is nocorresponding increase in thefuturesprice.

Suppose that VIXfutures are in a normalcontangorelationshipandthat

a traderbelieves that there islikelytobeariseinthevalueoftheVIXinthenearfuture.If he buys a futures contractand the expected increase intheVIXoccurs,whatwillbethe result? The trader mightassume that the futures pricewill increase by the sameamount as the index,but thiswillnotnecessarilybetrue.Ifthe increase in the VIXoccurswell before expirationof the futures contract, the

futures price may rise muchless than the index price.Such a scenario is shown inFigure25-11.Overafour-dayperiodinJuly2011,theindexvalue rose fromapproximately 19.4 to 23.7,an increase of 4.4 indexpoints. But over the sameperiod, the front-monthAugust futures contract,withapproximately three weeksremaining to expiration, roseonly 2.0, from 19.3 to 21.3.

Indeed, over the last twodays, even though the indexrose from 23.0 to 23.8,futurespriceshardlychangedatall.AtraderwhoownedanAugust futures contractwould have shown a profitbecausethefuturespricerose.ButseeingtheincreaseintheVIX value without a similarincrease in the futures price,the trader would almostcertainly have beendisappointedattheresult.

Figure25-11ViXfuturespricesdonotchangeasquicklyastheindex.

A similar situation canoccur ifVIX futures are in abackward structure and theindex begins to fall. Figure25-12 shows the change inVIX prices over a four-dayperiod in December 2008.During this period, the VIXfell from approximately 52.4to44.9,adeclineof7.5indexpoints. But the front-monthJanuaryfuturespricefellonly5.0, from 52.4 to 47.4. A

trader who sold Januaryfutures would likewise bedisappointedwiththeresults.

Figure25-12ViXfuturespricesdonotchangeasquicklyastheindex.

In a traditional futuresmarket, where it is usuallypossible to take a long orshort position in theunderlying index orcommodity, futures pricesmustchangeatapproximatelythe same rate as underlyingprices. If this were not true,there would be an arbitrageopportunity available. In astock index market, if thefutures price rises faster than

the index price, traders willsell the futures contract andbuy the component stocks; iftheindexrisesfasterthanthefuturesprice,traderswillbuythe futures contract and sellthe component stocks. Atradercanholdbothpositionsto maturity, knowing that atmaturitytheindexandfuturespricesmust converge.Unlikea stock index, though, theVIX is not easily tradable.Consequently, VIX futures

pricesneednotchangeat thesamerateas the index. If theindexpricerisesorfalls,VIXfuturesmaynotriseorfallbythe same amount. Indeed,futures prices might notchangeatall.

At expiration, the priceofaVIXfuturescontractwillsettle into the index valueregardless of any term-structure considerations.Therefore, the closer the

futurescontracttoexpiration,the more closely it willrespond to anychange in theindex value.A change in theindexvalueatexpirationwillbe reflected immediately inthefuturesprice.

Given the foregoingdiscussion, when choosing asimple futures strategy, atradershouldalwayskeepthefollowinginmind:

1. When theVIX

term structure iscontango, as timepasses with nochangeintheindexvalue, VIX futuresprices willinevitablydecline.2. The price of aVIX futurescontractwillalmostnever change asquicklyastheindexprice.

3. Futures pricesand index pricesmust converge atfuturesexpiration.4. For mosttraders, replicatingthe index is not arealistic choice.Therefore, futurespricesmustoftenbeevaluatedindependent of theindexprice.

Because of its unusualcharacteristics, trading VIXfutures may sound complex.But VIX futures are notnecessarily more complexthan other futures markets.They are simply different,and a trader must recognizethese differences. Buying aVIX futures contract can beprofitable if a trader believesthat an increase in the indexvaluewilloccur,especiallyifthe increase occurs close to

expiration, or if the traderbelieves that there will be alarge increase in thevalueoftheindex,perhapsresultinginan inversion of the term-structurecurvefromcontangoto backward. In the sameway, selling a VIX futurescontractcanbeprofitableifatrader believes that a declinein the index valuewill occurclose to expiration or if thetraderbelieves that therewillbealargedeclineinthevalue

of the index, perhapsresulting in an inversion ofthe term structure frombackwardtocontango.Butinboth cases the trader mustalso temper his expectations,knowing that the change inthe futures price will almostalways be less than thechangeintheindexprice.

Insteadofsimplybuyingor selling a single futuresmonth, a trader might

consider a futures spread,buyingonefuturesmonthandselling a different month.VIX futures spreads, likeindividual futures, aresensitivetothetermstructureof the futures market. In theunlikely situation where theterm structure is a straightline with constant slope,regardless ofwhether futuresprices rise or fall, the spreadvaluewillremainunchanged.Evenifbothfuturescontracts

lose value as time passes (acontango term structure) orgain value as time passes (abackward term structure),their relationshipwill remainconstant. They will lose orgainvalueatexactlythesamerate. If, however, the termstructure is curved, a muchmore common situation, theshort-term futures contractwill change value morequickly than the long-termfutures contract. Under these

conditions,iftheshapeoftheterm structure remainsunchanged, thepurchaseofalong-term futures contractand the sale of a short-termfuture contract will beprofitable in a contangomarket,andthepurchaseofashort-term futures contractand the sale of a long-termfuture contract will beprofitable in a backwardmarket. Examples of this areshownforacontangomarket

inFigure25-13.Figure25-13afuturesspreadina

contangomarket.

Of course, it is unlikelythat the term structure willremain constant. As marketconditions change, thestructure can alternatebetween contango andbackward, with varyingdegreesofcurvature foreachstructure. Because a short-term futures contract willalmost always change morequickly than a long-termcontract, if a trader believes

thatacontangostructurewillbecome less curved or willmove toward a backwardstructure,thesaleofafuturesspread (i.e., sell long term,buyshortterm)islikelytobeprofitable. If the traderbelieves that a backwardstructure will become lesscurvedorwillmovetowardacontango structure, thepurchase of a futures spread(i.e.,buylongterm,sellshortterm) is likely to be

profitable. These twoscenarios are shown inFigures25-14and25-15.

Figure25-14afuturesspreadwhenthetermstructuremovesfromcontangotowardbackward.

Figure25-15afuturesspreadwhenthetermstructuremovesfrombackwardtowardcontango.

VIXOptionsThe CBOE began trading

VIX options in 2006. Theoptions are European (noearlyexercise)andsettleintothe value of the VIX at theopening of trading onexpiration Wednesday, witheachvolatilitypointhavingavalueof$100.

Compared with other

financial indexes, theVIX ishighly volatile. From Figure25-4,it’sevidentthattheVIXcan double or even triple inprice over short periods oftime. The volatile nature ofthe VIX is confirmed inFigure 25-16, the 50- and250-day volatilities of theVIXfrom2003to2012.Overthe 10-year sample period,the 50-day volatilityoccasionallyreachedhighsofalmost 200 percent, while it

rarely fell below 50 percent.A trader might assume thatoptions on the VIX will bepriced accordingly, withimpliedvolatilitiesthatreflectthe highly volatile nature oftheindex.Thiswouldbetrueif one could hedge a VIXoptionpositionwiththeVIX.But because the index itselfcannot be easily bought orsold, the instrument that ismost commonly used tohedge a VIX option position

is a VIX futures contract.VIX futures, however, areless volatile than the indexbecausefuturespricestendtochange at a slower rate thanthepriceof theindex.Figure25-17 shows the 50-dayvolatility of the indexcompared with the same 50-dayvolatilityofthefirstthreefuturesmonths.Rarely is thefront-month futures contractas volatile as the index.Moreover, back months

become progressively lessvolatile, reflecting theconverging term structure oftheindex.

Figure25-16ViX50-and250-dayhistoricalvolatility:2003–2012.

Figure25-17Fifty-dayhistoricalvolatilityoftheViXandthefirstthreefuturesmonths.

Just as a VIX futurestrader is likely to bedisappointed when a futurescontract fails to move asmuch as the index, a VIXoptions trader is likely to bedisappointed when the valueofanoptiondoesnotreacttothefullvolatilityoftheindex.For a theoretical trader whofollows a dynamic hedgingprocedure,expectationsaboutVIX volatility should focus

onthevolatilityofthefuturescontract used to hedge theoption position, not thevolatilityoftheindex.

NotonlydoVIXoptionstend to carry lower impliedvolatilities than one wouldexpect from the volatility oftheindex,butthedistributionof implied volatilities differssignificantly from otheroptionmarkets.Thepriceofatraditional stock or

commodity can, in theory,rise without limit.Moreover,over long periods of time,there is an expectation thatthe prices of many tradedstocks and commodities willappreciate, with longer timeperiods accompanied bygreater appreciation. This isthe philosophy behind long-term investing. But, unlikethe price of a stock orcommodity, over any givenperiod of time, there are

practicallimitsbeyondwhichimplied volatility is unlikelytogo.Anoptiontraderwouldbe surprised indeed to seeimplied volatility in a stockindex market fall below 5percent, no matter how longhe observed the market. Atrader would likewise besurprised to see impliedvolatility rise above 100percent. Moreover, the valueof the VIX is influenced bythe mean reverting

characteristics of volatility.When the VIX is at a verylow level, there is a greaterlikelihood that it will rise;whenitisataveryhighlevel,there is a greater likelihoodthatitwillfall.Consequently,expectations about VIXprices will differ fromexpectations about the priceof traditional underlyingcontracts. These expectationsare reflected in the volatilityskew, the distribution of

implied volatilities for VIXoptions across exerciseprices.

The volatility skews forVIX options on March 19,2012,areshowninFigure25-18.Theshapeoftheseskewsisconsiderablydifferentfromtheskewforatypicalstockorcommodity. With somevariation, in most stock andcommodity option markets,exercisepricesthatarefarther

away from the currentunderlyingpricetendtocarryincreasingly higher impliedvolatilities—hence the termvolatilitysmile. But, forVIXoptions,theimpliedvolatilityoflowerexercisepricesdropsoff very quickly. Whilehigher exercise prices carryhigher implied volatilities, atsome point on the upside,implied volatilities stopincreasingand tend to flattenout. Rather than being a

smile, the shape of the skewmight be described as a halffrown.

VIX options seem to beimplying a price distributionthat is different from atraditional stock orcommodity. Using optionprices and the butterflyapproach described inChapter24,wecanconstructan implied price distributionfortheVIX.Thisdistribution

is shown inFigure25-19 forJune options on March 19,2012 with approximatelythree months remaining toexpiration. At the time, JuneVIX futures were trading at23.95. Compared with atraditional lognormaldistribution, the left tail ismuch more restricted,reflectingabeliefthatthereisalmost no chance that theVIX will be below 10.00 atJuneexpiration.Therighttail

is also more restricted,perhaps reflecting a beliefthat large upward moves arelesslikelythaninalognormaldistribution.Although itmaybe difficult to discern fromthe graph, the marketplacealso seems to be implying aslightly better chance of avery large upward move atthefarendoftherighttail.

Figure25-18ViXoptionimpliedvolatilityskews,March19,2012.

Figure25-19three-monthpricedistributionimpliedfromViXoptionprices,March19,2012[withthethree-month(June)futureat23.95].

ReplicatingaVolatilityContract

Even though replication ofa realized variance or VIXposition is not a practicalchoice for most traders, intheory,itispossibletocreatesuchaposition.Howcanthisbedone?

Suppose that a trader

sells a realized variancecontract at a volatility of 20(percent), equal toavarianceof 202 = 400. If the actualrealizedvolatilityoverthelifeofthecontractisgreaterthan20 percent, the trader willlose money; if the actualrealizedvolatility is less than20 percent, the trader willmake money. How can atrader hedge this position?Avariance position can be

replicated by purchasing astrip of options across allexercise prices. To create aposition with a constant-variance exposure, it isnecessary to purchase 1/X2(where X is the exerciseprice) of each option. Then,by dynamically hedging theentire position in order toremain delta neutralthroughout the life of thevarianceswap,thetotalvalue

of the strategy will exactlymatch the actual realizedvariance of the variancecontract.

It may seem that avolatility position can bereplicated using the sameapproach. But the fact thatvolatilityisthesquarerootofvariance means that if thevarianceexposureisconstant,thevolatilityexposurecannotbeconstant.Let’sreturntoan

earlier example where arealized volatility contractwith a vega exposure of$10,000 was purchased at apriceof20.Wecancomparetheoutcomesintwodifferentcases. In the first case, thecontract issettled invariancepoints,witheachpointhavinga value equal to the notionalvega divided by twice thevolatility price: $10,000/(2 ×20) = $250. In the secondcase,thecontractissettledin

volatility points with eachpoint have a value of$10,000.

At a realized variance of400 (a realized volatility of20),thevarianceP&Landthevolatility P&L are the same.However, as the differencebetweenthecontractvariancepriceof400(avolatilitypriceof 20) and the realizedvariance increases, thedifference between thevariance P&L and volatilityP&Lincreases.

A strip of options done

in the correct proportion of1/X2 yields a constantexposure tovariance.But thesamestripofoptionsdoesnotyield a constant exposure tovolatility. As realizedvolatility rises or falls, atrader who uses a strip ofoptions to hedge a volatilitypositioncannotbecertainthatthe strip will exactly offsethis position.This uncertaintymakes it difficult to hedge

volatility exposure, which iswhy such contracts areusually settled in variancepoints.

In our example, if thetrader can create the hedgethat replicates a longvolatilitypositionatapriceof19 (percent), the trader willhave a certain profit in theformofanarbitrage.Hehasashort volatility position at aprice of 20 (a variance of

400) and a long varianceposition at a price 19 (avariance of 361). If thecontract issettled invariancepoints,hemustshowaprofitof39×$250=$9,750.

Ifatraderbuystheentirestrip of options in order toachieve a constant-varianceexposure, how can hedetermine thevolatilityvalueof the strip? Did he buy thestrip at a volatility of 19

percent, or 20 percent, or 21percent, or some othervolatility? The methodologyused by the CBOE tocalculate the VIX isessentially a way of turningthe cost of the strip into avolatility value. This isanalogous to taking the priceof an option and turning itintoanimpliedvolatility.TheVIX methodology takes thepricesofalltheoptionsinthestrip and turns them into an

implied volatility position,but one with constant-varianceexposure.

At expiration, the valueof a VIX contract isdetermined by a single stripofSPXoptionsthatexpire30days in the future. But theVIXrepresentsaconstant30-day implied volatility, andprior to expiration, there areno options that expire inexactly 30 days.

Consequently, two strips ofoptions that bracket 30 daysare required to calculate theVIX, with appropriateweighting of each strip toyield a 30-day impliedvolatility. In theory, eachstrip must be dynamicallyhedged to remain deltaneutral. But VIX replicationrequires the purchase of onestrip and the sale of the onestrip, and it turns out thatgamma values of each strip

will approximately offseteach other. With a totalgamma close to 0, no delta-neutral rehedging isnecessary. The position canbecarriedtoexpirationoftheVIX contract, at which timethe long-term strip can beclosed out at the optionmarket prices that willdetermine the expirationvalueoftheVIX.

Unfortunately, there are

several problems with thisstrategy.When the long-termstrip is closed, a trader willalso want to close the short-term strip. However, whilethe long-term strip is closedby the special openingrotation on VIX expirationWednesday, the short-termstrip actually expires oneithertheFridayimmediatelypreceding or immediatelyfollowing VIX expiration. Ifthe options thatmake up the

short-termstripexpireon theFriday afterWednesdayVIXexpiration, the trader can tryto close the short-term striphimself on expirationWednesday. But in order todoso,hewillhavetogiveupthe bid-ask spread on everyoption,andthiscanbecostly.If the options that make uptheshort-termstripexpireonthe Friday prior toWednesday VIX expiration,thetraderwillhavetocarrya

naked position in the long-term strip for an additionalfive days. This can also becostly. How can the traderdeal with the risk that theshort-andlong-termstripsdonot expire at the same time?Unfortunately, there is nogoodsolutiontothisproblem,which is one reason whyreplicating the VIX is sodifficult.

An additional problem

arises because the value ofthe VIX is calculated fromthe prices of out-of-the-money options. If a traderreplicates theVIXbybuyingone strip and selling another,someoftheoptionsthatwerepreviously out of the moneywill almost certainly go intothemoneyoverthelifeofthestrip.To have a position thatis equal to the VIX value atexpiration, the in-the-moneyoptionsmust be converted to

out-of-the-money options.Weknowfromsyntheticsthatan in-the-money optionhedged with an underlyingcontract is equivalent to anout-of-the-money option ofthe opposite type. Therefore,for each option that is in themoney, a trader can buy orsell, as necessary, oneunderlying contract. Whenthe entire position, includingthe underlying contracts, isclosed at expiration, it will

exactly equal the expirationvalue of the VIX. The onlyproblemwiththisapproachisthat there is no easily tradedunderlying for SPX optionsbecause the underlyingconsistsofabasketofthe500stocks thatmake up the s&P500. If there are s&P 500futuresavailablethatexpireatthesametimeastheVIX,thefuturescanbeusedasaproxyfor the underlying contract.Otherwise,atradermayhave

to create a proxy underlying,perhaps in the form ofcombos (i.e., long call/shortput or short call/long put,with thesameexerciseprice)expiring at the same time astheshort-termstrip.

While a professionalderivativestradingfirmmightin some cases seek toreplicate a realized varianceorVIXcontract intheoptionmarket, for most traders,

given the complexities,replicating these contracts isnotarealisticpossibility.

VolatilityContractApplications

Certainly the mostcommon use of VIX andvariance contracts is tospeculate on volatility. Atraderwhohasanopinionon

whether realized volatilitywill riseor fall canspeculateby buying or selling avariance swap.A traderwhohas an opinion on whetherimplied volatilitywill rise orfall can speculate by buyingor selling a VIX contract. Inthe latter case, a trader canspeculate directly on impliedvolatility by trading VIXfutures or speculate on VIXvolatility by trading VIXoptions.

Volatility contracts canalso be used as a hedginginstrument. Market makersand hedge fund managerssometimes acquire volatilitypositions, perhapsunintentionally, asa resultoftheirmarketactivities.Iftheywant to hedge away someofthis volatility risk, varianceand VIX contracts offer asimple way of doing this. Atrader who has a realizedvolatility position, either a

positive or negative gamma,can trade variance contractsto hedge his realizedvolatility risk. A trader whohas an implied volatilityposition, either a positive ornegativevega,can tradeVIXcontractstohedgehisimpliedvolatilityrisk.

In addition to hedging avolatility position, VIXcontracts can sometimes beused as a hedge against a

market position, especially amarket position thatapproximates a broad-basedportfolio.Because there isaninverse correlation betweenmovement in the stockmarket and changes inimpliedvolatility(seeFigures25-4 and 25-5), a portfoliomanagerwhois longequitiesmight take a longposition intheVIXbyeitherbuyingVIXfutures, buyingVIX calls, orselling VIX puts. If stock

prices decline, there is anexpectation that impliedvolatility will rise, and theresulting increase in value oftheVIXpositionwilloffsetatleastsomeofthelossesinthestockmarket.

Although volatilitycontractsareusedmostoftento address direct volatilityconcerns, market participantssometimes take on indirectvolatility positions, positions

that have volatilityimplications that are notimmediately apparent. Forexample, an option marketmaker typically profits fromhigheroptiontradingvolume.But higher volume is oftentheresultofhighervolatility.When there is greatervolatility, there is greaterdemandforoptions.Assuch,the market maker has anindirect long volatilityposition. He would like

volatility to increase notbecause he has intentionallytaken a long volatilitypositionbutbecauseheisinabusiness where highervolatility tends to result inhigher profits. To hedge thisindirect long volatilityposition, market makerssometimes take a shortvolatilitypositioninvolatilitycontracts,mostcommonlytheVIX. Of course, the marketmaker is really hedging

trading volume, and heshould not take such a largeVIX position that hisattention is diverted fromhisprimary market-makingactivities.

Another type of indirectvolatility position is one inwhich a portfoliomanager isrequired to periodicallyrebalanceaportfolio.Thereisa cost to the rebalancingprocess, and the cost is

typically higher in times ofhigh volatility when bid-askspreads tend to widen. Theportfolio manager thereforetakes on a short volatilityposition as the rebalancingperiod approaches. He canhedge this short volatilityposition by taking a longvolatilitypositionintheVIX.

Finally, there are somepositionsthataretakenintheoption market that are not

usually thought of asvolatility positions but thathave volatility implications.Perhaps the most commonoptionhedgingstrategyisthecovered call, the sale of calloptions against a longunderlyingposition.Consideraportfoliomanagerwhosellsindex calls against a broad-based portfolio of stocks.What are his goals? First, hewants the value of hisportfolio to increase.Second,

hewants tooutperformsomebenchmark againstwhich hisperformance is measured,perhaps a broad-based indexsuchasthes&P500.

Ifthemanagersellscallsagainst his portfolio holdingsand themarket rises, he willachievehisfirstgoalbecausethe portfolio will increase invalue.But,ifthemarketrisestoofar,eventuallythecallshesold will be exercised,

limiting the upside profitpotential. If the marketcontinues to rise, hewill failinhissecondgoalbecausethebenchmark index willeventually outperform theportfolio.

Ifthemanagersellscallsagainst his portfolio and themarket declines, he willachieve his second goal ofoutperforming the indexbecausehewillhavetakenin

premium through the sale ofcalls. But, if the decline isgreat enough, he will fail inhis first goal because thecovered calls offer only apartial hedge against adecliningmarket.

From the portfoliomanager’s point of view, thecovered call strategy willperform best, and he willachieve both his goals whenthe market either doesn’t

move or moves very little.Theportfoliowill increase invalue as a result of thepremium received for thecovered calls. And theportfolio will outperform abenchmarkindexthatconsistsonlyofstocks.Iftheportfoliomanagerwants themarket tosit still, he has a shortvolatility position. He canhedge away someof the riskof a short volatility positionby taking a long VIX

position, usually by buyingVIXfutures.

1Whenacontractistradedbetweenprivatepartieswithoutanexchangeasanintermediary,thepossibilityofonepartydefaultingonitsobligationsaddsanadditionalriskdimensiontothetrade.Counterpartyriskcanbeanimportantconsiderationintheoff-exchangemarket.2ForadescriptionoftheoriginalVIXmethodology,seeRobertWhaley,“DerivativesonMarketVolatility:HedgingToolsLongOverdue,”JournalofDerivatives,Fall1993,pp.71–84.3KresimirDemeterfi,EmmanuelDerman,MichaelKamal,andJosephZou,“MorethanYouEverWantedtoKnowaboutVolatilitySwaps,”

GoldmanSachsQuantitativeStrategiesResearchNotes,NewYork,March1999.4Thisisessentiallyequivalenttobuyingat-the-moneystraddles.5ForadetaileddescriptionoftheVIXcalculationmethodology,see“TheCBOEVolatilityIndex,”availableat:https://www.cboe.com/micro/vix/vixwhite.pdf

AFinalThought

Because the use of atheoretical pricing modelrequires a trader to make somanydifferentdecisionswithrespecttoboththeinputsintothe model and the reliabilityof the assumptions on whichthe model is based, a newoption trader may feel that

making the right decisions iseither an impossible task orsimply amatter of luck. It istrue that a trader using amodel will almost certainlybewrongaboutat leastsomeof the inputs into themodel,and luck undoubtedly doesplay a role in the short run.But in the long run traderswho arewilling to put in theeffort required to understandhow a model works,including its strengths and

weaknesses, always seem tocomeoutahead.Experiencedtradersknowthatundermostconditions, using a model,with all its problems, is stillthe best way to evaluateoptionsandmanagerisk.

Regardless ofwhether amodelissimpleorcomplex,atrader who uses a modelneeds to have faith in themodel.Otherwise,whyuseamodel at all? Indeed, for

traders who are notmathematically proficient,usingamodel isoftena leapoffaith.Buthavingfaithinamodel does notmean havingblind, unquestioning faith. Ifa model returns values thatare clearly inconsistent withcommon sense, or if marketconditions are changing soquickly that it is impracticaltousethemodelinitscurrentform, a trader may have todecide whether to adjust the

model, if that is possible, orsimply to stop using themodel. Although we haveemphasizedtheimportanceofmodels,tradingisbothanartand a science. Experiencedtraders know that there aretimeswhen it isperhapsbestto put the model aside andmake decisions based onother intangible assets,whether intuition, “marketfeel,”orexperience.A traderwhoslavishlyusesamodelto

make every trading decisionisheadingfordisaster.Onlyatrader who fully understandswhatamodelcanandcannotdo will be able to make themodelhis servant rather thanhismaster.

GlossaryofOption

Terminology

This glossary includesoption-related terms as they

are most commonly used.However, the reader shouldbe aware that optionterminology is not uniform.Tradersmay sometimes referto different strategies oroptioncharacteristicswiththesame term. They maysometimes refer to the samestrategyorcharacteristicwithdifferentterms.

All or None (AON) Anorderthatmustbefilledinits

entiretyornotatall.

American Option Anoption that can be exercisedat any time prior toexpiration.

Arbitrage The purchaseand sale of the same orclosely related products indifferent markets to takeadvantageofapricedisparitybetweenthetwomarkets.

AsianOptionSeeAveragePriceOption.

Assignment The processby which the seller of anoption is notified of thebuyer’s intention to exercise.The seller is required to takea short position in theunderlying position in thecase of a call or a longpositioninthecaseofaput.

AttheForwardAnoption

whoseexerciseprice isequalto the forward price of theunderlying contract.Sometimes referred to asAt-the-MoneyForward.

At the Money An optionwhoseexerciseprice isequalto the current price of theunderlyingcontract.Onlistedoptionexchanges, the termismorecommonlyusedtoreferto the option whose exerciseprice is closest to the current

price of the underlyingcontract.

Automatic Exercise Theexerciseby theclearinghouseof an in-the-money option atexpiration unless the holderoftheoptionsubmitsspecificinstructionstothecontrary.

AveragePriceOption Anoption whose value atexpiration is determined bythe average price of the

underlying instrument oversome period of time. AlsoknownasanAsianOption.

Backspread A spread,usually delta neutral, wheremore options are purchasedthan sold, where all optionsare the same type and expireatthesametime.

Backward A futuresmarket where the long-termdelivery months trade at a

discount to the short-termdeliverymonths.

BarrierOption A type ofexotic option that will eitherbecome effective or cease toexist if the underlyinginstrument trades at orbeyond some predeterminedpricepriortoexpiration.

Bear Spread Any spreadthatwilltheoreticallyincreaseinvaluewithadeclinein the

price of the underlyingcontract.

Bermuda Option Anoption that can be exercisedprior to expiration, but onlyduring a predeterminedperiod or window. Alsoknown as a Mid-AtlanticOption.

Binary Option An optionthat, if in the money atexpiration, makes one

predetermined payout. AlsoknownasaDigitalOption.

Box A long call and shortput at one exercise price,togetherwithashortcallandlong put at a differentexercise price. All optionsmust have the sameunderlying contract andexpireatthesametime.

Bull Spread Any spreadthatwilltheoreticallyincrease

in value with a rise in theprice of the underlyingcontract.

Butterfly The sale(purchase) of two optionswith the same exercise price,together with the purchase(sale) of one option with alower exercise price and oneoptionwithahigherexerciseprice.All optionsmust be ofthesametype,havethesameunderlying contract, and

expire at the same time, andthere must be an equalincrement between exerciseprices.

Buy/Write The purchaseof an underlying contracttogetherwiththesaleofacalloptiononthatcontract.

Cabinet Bid An optionprice that is smaller than thenormallyallowableminimumprice. On some exchanges, a

cabinet bid is permissiblebetween traders desiring tocloseoutpositionsinveryfarout-of-the-moneyoptions.

Calendar Spread Thepurchase (sale)ofoneoptionexpiring on one date and thesale (purchase) of anotheroptionexpiringonadifferentdate. Typically, both optionsareofthesametype,havethesameexerciseprice,andhavethe same underlying stock or

commodity.AlsoknownasaTime Spread or HorizontalSpread.

Call Option A contractbetween a buyer and a sellerwhereby the buyer acquiresthe right, but not theobligation, to purchase aspecified underlying contractatafixedpriceonorbeforeaspecified date. The seller ofthe call option assumes theobligation of delivering the

underlying contract shouldthebuyerwishtoexercisehisoption.

Cap A contract between aborrower and a lender offloating-rate funds wherebythe borrower is assured ofpaying no more than somemaximum interest rate forborrowed funds. This isanalogous to a call optionwhere the underlyinginstrument is an interest rate

onborrowedfunds.

Charm The sensitivity ofan option’s delta to thepassageoftime.

Chooser Option Astraddle where the ownermust decide by somepredetermined date whetherto keep either the call or theput.

ChristmasTree A spread

involving three exerciseprices. One or more calls(puts) are purchased at thelowest (highest) exerciseprice, and one or more calls(puts) are sold at eachof thehigher (lower) exerciseprices. All options mustexpireatthesametime,beofthe same type, and have thesame underlying contract.AlsoknownasaLadder.

Class All options of the

same type with the sameexpiration date and sameunderlyinginstrument.

Clearinghouse Theorganization that guaranteesthe integrity of all tradesmadeonanexchange.

Clearing Member Amember firm of an exchangethat is authorized by theclearinghouse to processtrades for its customers and

that guarantees, through thecollection of margin andvariation, the integrity of itscustomers’trades.

Collar A long (short)underlying position that ishedged with both a long(short) out-of-the-money putandashort (long)out-of-the-money call.All optionsmustexpireatthesametime.Alsoknown as aCylinder, Fence,orRangeForward.

ColorThesensitivityofanoption’s gamma to thepassageoftime.

Combination (Combo) Atwo-sided option spread thatdoes not fall into any well-defined category of spreads.Mostcommonly,itreferstoalong call and short put orshortcallandlongput,whichtogethermake up a syntheticposition in the underlyingcontract.

Compound Option Anoptiontopurchaseanoption.

Condor The sale(purchase) of two optionswithdifferentexerciseprices,together with the purchase(sale) of one option with alower exercise price and oneoptionwithahigherexerciseprice.All optionsmust be ofthesametype,havethesameunderlying contract, andexpire at the same time, and

there must be an equalincrement between exerciseprices.

Contango A futuresmarket where the long-termdelivery months trade at apremium to the short-termdeliverymonths.

Contingency Order Anorder that becomes effectiveonly on the fulfillment ofsome predetermined

condition(s) in themarketplace.

Conversion A longunderlying position togetherwith a synthetic shortunderlying position. Thesynthetic position consists ofa short call and long put,where both options have thesame exercise price andexpire at the same time.Sometimes referred to as aForwardConversion.

CoveredWriteThesaleofa call (put) option against anexisting long (short) positionintheunderlyingcontract.

CylinderSeeCollar.

Delta(Δ)Thesensitivityofan option’s theoretical valuetoachangeinthepriceoftheunderlying contract. AlsoknownastheHedgeRatio.

Delta Neutral A position

wherethesumtotalofallthedeltas add up toapproximately 0. Undercurrentmarketconditions,thepositionhasnopreferenceasto thedirectionofmovementintheunderlyingmarket.

Diagonal Roll See TimeBox.

Diagonal Spread A longoption at one exercise priceand expiration date, together

with a short option at adifferent exercise price andexpiration date. All optionsmustbethesametype.Thisisthesameasacalendarspreadusing different exerciseprices.

DigitalOptionSeeBinaryOption.

Dragonfly A long (short)straddle, together with twoshort (long) strangles at the

sameexerciseprice,wherealloptions expire at the sametime and have the sameunderlying contract. Theexercise price of the straddlewill usually fall as close aspossible to the midpointbetweentheexercisepricesofthestrangles.

Dynamic Hedging Aprocess in which theunderlying contract isperiodicallyboughtorsoldin

order to maintain a desiredposition in a market.Dynamic hedging is mostoften used to maintain adelta-neutraloptionposition.

Efficiency A number thatrepresents the relative riskand reward of a potentialoption strategy. The risk andreward are typicallyrepresented by the totalgamma,theta,andvegaofthestrategy. The efficiency is

generated by dividing onesensitivitybyanother.

Elasticity The percentchange in an option’s valueforagivenpercentchangeinthe value of the underlyinginstrument. Sometimesreferred to as an option’sLeverage Value. Theelasticity is sometimesdenoted by the Greek letterLambda(Λ).

Eurocurrency Currencydeposited in a bank outsidethecurrency’shomecountry.

Eurocurrency rate Theinterest ratepaidoncurrencydeposited in a bank outsidethecurrency’shomecountry.

European Option Anoption that may only beexercisedatexpiration.

Exchange Option An

option to exchange one assetforanotherasset.

Ex-DividendThe firstdayon which a dividend-payingstock is trading without therighttoreceivethedividend.

Exercise The process bywhichtheholderofanoptionnotifies the seller of hisintention to take a longposition in the underlyingcontract in the case of a call

or a short position in theunderlying contract in thecaseofaput.

ExercisePriceThepriceatwhichtheunderlyingcontractwillbedeliveredintheeventan option is exercised. AlsoknownastheStrikePrice.

Expiration (Expiry) Thedate and time afterwhich anoption may no longer beexercised.

Exotic Option An optionwith nonstandard contractspecifications. Sometimesreferred to as a Second-Generation Option. Exoticoptions are usually traded inthe over-the-counter (off -exchange)market.

ExtrinsicValue See TimeValue.

Fair Value SeeTheoreticalValue.

FenceSeeCollar.

Fill or Kill (FOK) Anorder that will automaticallybe canceled unless it can beexecuted immediately and initsentirety.

FlexOptionAnexchange-tradedoptionwherethebuyerand seller are permitted tonegotiate the exact terms oftheoptioncontract.Typically,this includes the exercise

price,theexpirationdate,andthe exercise style (eitherEuropeanorAmerican).

FloorAcontractbetweenaborrower and a lender offloating-rate funds wherebythe lender is assured ofreceiving no less than someminimum interest rate forloaned funds. This isanalogous to a put optionwhere the underlyinginstrument is an interest rate

onloanedfunds.

Forward Contract Anagreement between a buyerand a seller to exchangemoney for goods at somelater date. At maturity, thebuyer is obligated to takedelivery and the seller isobligatedtomakedelivery.

Forward Conversion SeeConversion.

Forward Price The pricethat the buyer of a forwardcontract agrees to pay atmaturityofthecontract.

ForwardStartOptionAnoption thatbecomeseffectiveonly on some futurepredetermineddate.

Front Spread A spread,usually delta neutral, wheremore options are sold thanpurchased, where all options

are the same type, and allexpireatthesametime.

Fugit Assuming that allmarket conditions remainunchanged, the expectedamount of time remaining tooptimal early exercise of anAmericanoption.

Futures Contract Anexchange-traded forwardcontract.

Futures-Type SettlementA settlement procedure usedby commodity exchangeswhereby an initial margindeposit is made but underwhich no immediate cashpaymentismadebythebuyerto the seller.Cash settlementtakesplaceattheendofeachtrading day based on thedifference between theoriginal trade price or theprevious day’s settlementprice and the current day’s

settlementprice.

Gamma (Γ) Thesensitivity of an option’sdelta toachange in thepriceoftheunderlyingcontract.

Good‘tilCanceled(GTC)An order that remains activeuntilitcaneitherbeexecutedor is canceled by thecustomer.

Guts A strangle where

both the call and the put areinthemoney.

Haircut On a securitiesexchange, money that aprofessionaltraderisrequiredto keep in his account inorder to cover the riskof hisposition. Haircutrequirements are normallydetermined by the regulatoryauthority under which theexchangeoperates.

HedgeRatioSeeDelta.

Hedger A trader whoenters the market with thespecific intent of protectingan existing position in anunderlyingcontract.

Horizontal Spread SeeCalendarSpread.

Immediate or Cancel(IOC) An order that willautomaticallybecanceledifit

cannot be executedimmediately. An IOC orderneed not be filled in itsentirety.

Implied VolatilityAssumingthatallotherinputsare known, the volatility thatwouldhavetobeinputintoatheoretical pricing model toyield a theoretical value thatisidenticaltothepriceoftheoptioninthemarketplace.

In-OptionAbarrieroptionthatbecomeseffectiveonlyifthe underlying instrumenttrades at or through somepredetermined price prior toexpiration. Also known as aKnock-InOption.

In-Price The price atwhich the underlyinginstrument must trade beforean in-option becomeseffective.

In the Money An optionthathasintrinsicvaluegreaterthan0.Acalloptionisinthemoney if its exerciseprice islower than the current priceof theunderlying contract.Aput option is in themoney ifits exercise price is higherthan the current price of theunderlying contract. Anoption may also be In theMoney Forward if it hasintrinsic value greater than 0when compared with the

forward price of theunderlyingcontract.

Index Arbitrage Astrategy which attempts toprofit from the relative mis-pricing of options, futurescontracts, or the componentstockswhichmakeupastockindex.

Intermarket Spread Aspreadconsistingofopposingmarket positions in two or

more different underlyingsecuritiesorcommodities.

IntrinsicValueFor an in-the-money option, thedifference between theexercise price and theunderlying price. Out-of-the-money options have nointrinsic value. An optionwhose price is equal to itsintrinsic value is said to betradingatParity.

Iron Butterfly A long(short)straddle,togetherwithashort(long)strangle,wherealloptionsexpireatthesametime and have the sameunderlying contract. Theexercise price of the straddleis located at the midpointbetweentheexercisepricesofthestrangle.

Iron Condor A long(short)stranglewithnarrowerexerciseprices, togetherwith

a short (long) strangle withwider exercise prices, wherealloptionsexpireatthesametime and have the sameunderlying contract. Thenarrower strangle is centeredbetweentheexercisepricesofthewiderstrangle.

JellyRollSeeRoll.

Kappa(K)SeeVega. TheGreek letter kappa issometimes used to denote an

option’sexerciseprice.

Knock-In Option See In-Option.

Knock-Out Option SeeOut-Option.

Ladder See ChristmasTree.Alternatively, a typeofexotic option whoseminimum value increases asthe underlying contract goesthrough a series of

predetermined prices, orrungs, over the life of theoption.

Lambda (Λ) SeeElasticity.

LEAP (Long-TermEquity AnticipationSecurity) A long-term(usuallymore than one year)exchange-traded equityoption.

Leg One side of a spreadposition.

Leverage Value SeeElasticity.

Limit The maximumallowable price movementoversome timeperiodforanexchange-tradedcontract.

Limit Order An order tobe executed at a specifiedpriceorbetter.

Local An independenttrader on a commodityexchange. Locals performfunctions similar to marketmakers on stock and stockoptionexchanges.

Locked Market Anexchange-traded marketwheretradinghasbeenhaltedbecause prices have reachedthe limit permitted by theexchange.

Long A position resultingfrom the purchase of acontract. The term is alsoused to describe a positionthatwilltheoreticallyincrease(decrease)invalueshouldtheprice of the underlyingcontractrise(fall).Notethatalong (short) put position is ashort(long)marketposition.

LongPremiumApositionthatwilltheoreticallyincreasein value should the

underlying contract make alarge or swiftmove in eitherdirection. The position willtheoretically decrease invalue should the underlyingmarket fail tomove ormovevery slowly. The term mayalso refer to a position thatwill increase in value shouldimpliedvolatilityrise.

Long Ratio Spread Aspread where more optionsarepurchasedthansold.

Lookback Option Anexotic option whose exerciseprice will be equal to eitherthe lowest price of theunderlying instrument in thecase of a call or the highestprice of the underlyinginstrumentinthecaseofaputover the lifeof theoption.Alookback option can alsohave a fixed strike, inwhichcase its value at expirationwill be determined by themaximumunderlyingpricein

the case of a call or theminimumunderlyingprice inthecaseofaputoverthelifeoftheoption.

Margin Money depositedby a trader with the clearinghouse to ensure the integrityofhistrades.

Market-if-Touched(MIT) A contingency orderthat becomes a market orderif the contract trades at or

beyondaspecifiedprice.

Market maker Anindependent trader or tradingfirm,usuallyappointedbyanexchange, that is prepared tobothbuyandsellcontractsina designated market. Amarket maker is required toquote both a bid and offerprice in his designatedcontract.

Market-on-Close (MOC)

Anordertobeexecutedatthemarket price at the close ofthatday’strading.

MarketOrderAnordertobe executed immediately atthecurrentmarketprice.

Mark-to-market Amethod of valuing a positionbased on the current marketprice of all contracts whichmakeuptheposition.

Married Put A long(short) put together with along (short) underlyingposition.

Mid-Atlantic Option SeeBermudaOption.

Midcurve Option Infutures option markets, ashort-term option on a long-term futures contract.Midcurve options are mostcommon in euro-currency

futures markets, such asEurodollarsandEuribor.

Naked A long (short)market position with nooffsettingshort (long)marketposition.

Neutral Spread A spreadthat isneutralwithrespect tosome risk measure, mostcommonlythedelta.Aspreadmayalsobelotneutral,wherethe total number of long and

short contracts of the sametypeareequal.

Not Held An ordersubmittedtoabrokerbutoverwhich the broker hasdiscretion as to when andhowtheorderisexecuted.

Omega (Ω) The Greekletter sometimes used todenote an option’s elasticity.Analternativetolambda(Λ).

One-Cancels-the-Other(OCO)Twoorderssubmittedsimultaneously, either ofwhich may be executed. Ifone order is executed, theother is automaticallycanceled.

Order Book Official(OBO) An exchange officialresponsible for executingmarket or limit orders forpubliccustomers.

Out of the Money Anoption that currently has nointrinsic value. A call is outof the money if its exercisepriceismorethanthecurrentprice of the underlyingcontract. A put is out of themoney if its exerciseprice isless than the current price ofthe underlying contract. AnoptionmayalsobeOutoftheMoney Forward if it has nointrinsic value whencompared with the forward

price of the underlyingcontract.

Out-Option A type ofbarrier option that is deemedto have expired if theunderlying instrument tradesat some predetermined priceprior to expiration. Alsoknown as a Knock-OutOption.

Out-Price The price atwhich the underlying

instrument must trade beforean out-option is deemed tohaveexpired.

Out-Trade A trade thatcannot be processed by theclearinghouse due toconflicting informationreportedbythetwopartiestothetrade.

Overwrite The sale of anoption against an existingposition in the underlying

contract.

ParitySeeIntrinsicValue.

Phi (Ф) For foreign-currency options, thesensitivity of the option’svalue to a change in theforeign interest rate.Sometimes referred to asRho2.

Pin Risk The risk to theseller of an option that at

expiration will be exactly atthemoney.Thesellerwillnotknowwhethertheoptionwillbeexercised.

Portfolio Insurance Aprocessinwhichthequantityof holdings in an underlyinginstrument is periodicallyadjusted to replicate thecharacteristicsofanoptiononthe underlying instrument.This is similar to the delta-neutral dynamic hedging

process used to capture thevalueofamispricedoption.

PositionThesumtotalofatrader’s open contracts in aparticularunderlyingmarket.

Position Limit For anindividual trader or firm, themaximum number of opencontracts in the sameunderlying market permittedby an exchange orclearinghouse.

Premium The price of anoption.

Program Trading Anarbitrage strategy involvingthepurchaseorsaleofastockindexfuturescontractagainstan opposing position in thecomponent stocks that makeuptheindex.

Put Option A contractbetween a buyer and a sellerwhereby the buyer acquires

the right but not theobligation to sell a specifiedunderlyingcontractatafixedpriceonorbeforeaspecifieddate. The seller of the putoptionassumestheobligationof taking delivery of theunderlying contract shouldthebuyerwishtoexercisehisoption.

Range Forward SeeCollar.

RatchetOptionA typeofexotic option whoseminimumvalueisdeterminedby the underlying price at aseries of predetermined timeintervals over the life of theoption.

Ratio Spread Any spreadwhere the number of longmarket contracts (longunderlying,longcall,orshortput) and short marketcontracts (short underlying,

short call, or long put) areunequal.

Ratio Write The sale ofmultiple options against anexisting position in anunderlyingcontract.

Reversal See ReverseConversion.

Reverse Conversion Ashort underlying positiontogetherwithasyntheticlong

underlying position. Thesynthetic position consists ofa long call and short put,where both options have thesame exercise price andexpireatthesametime.AlsoknownasaReversal.

Rho(P) The sensitivity ofan option’s theoretical valuetoachangeininterestrates.

Risk Reversal A long(short) underlying position

together with a long (short)out-of-the-money put and ashort(long)out-of-the-moneycall.Bothoptionsmustexpireatthesametime.Alsoknownas a Split-Strike Conversion.ThepositionisequivalenttoaCollar.

RollA long call and shortputwith one expiration date,togetherwithashortcallandlong put with a differentexpiration date. All four

options must have the sameexercise price and the sameunderlying stock orcommodity. In slang terms,sometimes referred to as aJellyRoll.

Scalper A floor trader onan exchange who hopes toprofit by continually buyingat thebidpriceandsellingatthe offer price in a specificmarket. Scalpers usually tryto close out all positions at

theendofeachtradingday.

Second-GenerationOptionSeeExoticOption.

Serial Option On futuresexchanges, an optionexpiration with nocorresponding futuresexpiration. The underlyingcontract for a serial option isthe nearest futures contractbeyondtheoptionexpiration.

SeriesAlloptionswiththesame underlying contract,sameexerciseprice,andsameexpirationdate.

Short A position resultingfrom the sale of a contract.The term is also used todescribe a position that willtheoretically increase(decrease)invalueshouldtheprice of the underlyingcontractfall(rise).Notethatashort (long) put position is a

long(short)marketposition.

ShortPremiumApositionthatwilltheoreticallyincreasein value should theunderlying contract fail tomove or move very slowly.Thepositionwilltheoreticallydecrease in value should theunderlying market make alarge or swiftmove in eitherdirection. The termmay alsorefer to a position that willincrease in value should

impliedvolatilityfall.

Short Ratio Spread Aspread where more optionsaresoldthanpurchased.

ShortSqueezeA situationin the stock option market,usually resulting from apartial tenderoffer,wherenostock can be borrowed tomaintain a short stockposition. If assigned on ashort call position, a trader

may be forced to exercise acall early to fulfill hisdelivery obligations, eventhoughthecallstillhassometimevalueremaining.

Sigma (σ) The commonlyused notation for standarddeviation. Because volatilityis usually expressed as astandard deviation, the samenotation is often used todenotevolatility.

SpecialistAmarketmakergiven exclusive rights by anexchangetomakeamarketina specified contract or groupofcontracts.Aspecialistmaybuy or sell for his ownaccountoractasabrokerforothers. In return, a specialistis required tomaintain a fairandorderlymarket.

Speculator A trader whohopestoprofitfromaspecificdirectional move in an

underlyingcontract.

SpeedThesensitivityofanoption’s gamma to a changeintheunderlyingprice.

Spread A long marketposition and an offsettingshortmarketpositionusually,but not always, in contractswith the same underlyingmarket.

Split-Strike Conversion

SeeRiskReversal.

Stock-Type Settlement Asettlementprocedureinwhichthe purchase of a contractrequires full and immediatepayment by the buyer to theseller. All profits or lossesfrom the trade are unrealizeduntil the position isliquidated.

Stop-Limit Order Acontingency order that

becomes a limit order if thecontract trades at a specifiedprice.

Stop (Loss) Order Acontingency order thatbecomesamarketorderifthecontract trades at a specifiedprice.

Straddle A long (short)call and a long (short) putwhere both options have thesameunderlyingcontract, the

sameexpirationdate,and thesameexerciseprice.

Strangle A long (short)call and a long (short) putwhere both options have thesameunderlyingcontract, thesame expiration date, butdifferentexerciseprices.

Strap An archaic term fora position consisting of twolong (short) calls and onelong (short) put where all

options have the sameunderlyingcontract, thesameexpirationdate,and thesameexerciseprice.

Strike Price (Strike) SeeExercisePrice.

StripAnarchaictermforaposition consisting of onelong(short)callandtwolong(short)putswherealloptionshave the same underlyingcontract, the same expiration

date, and the same exerciseprice. Alternatively, inEurocurrency markets, aseries of futures or futuresoptions designed to replicatethe characteristics of a long-terminterest-rateposition.

Swap An agreement toexchange cash flows. Mostcommonly, a swap involvesexchanging variable-interest-rate payments for fixed-interest-ratepayments.

Swaption An option toenterintoaswapagreement.

Synthetic A combinationof contracts that togetherhave approximately the samecharacteristics as some othercontract.

Synthetic Call A long(short) underlying positiontogether with a long (short)put.

Synthetic Put A short(long) underlying positiontogether with a long (short)call.

Synthetic Underlying Along (short) call and short(long)putwherebothoptionshave the same underlyingcontract, the same expirationdate, and the same exerciseprice.

Tau (τ) The commonly

used notation for the amountof time remaining toexpiration. Some traders alsouse the term to refer to thesensitivity of an option’stheoretical value to a changeinvolatility(equivalenttothevega)

Term Structure Thedistribution of impliedvolatilities across differentexpirationmonthsinthesameunderlyingmarket.

Theoretical Value Anoption value generated by amathematical model givencertain prior assumptionsaboutthetermsoftheoption,the characteristics of theunderlying contract, andprevailing interest rates.AlsoknownasFairValue.

Theta (Θ) The sensitivityof an option’s theoreticalvalue to a change in theamount of time remaining to

expiration.

Three-Way A positionsimilar to a conversion orreversalbutwherethelongorshort position in theunderlying instrument hasbeen replaced with a verydeeply in-the-money call orput.

TimeBoxA longcall andshort put with the sameexercise price and expiration

datetogetherwithashortcalland long put at a differentexercise price and expirationdate. This is simply a rollusing different exerciseprices. Also known as aDiagonalRoll.

Time Premium See TimeValue.

Time Value The price ofan option less its intrinsicvalue.Thepriceofanout-of-

the-money option consistssolely of time value. AlsoknownasExtrinisicValue orTimePremium.

Time Spread SeeCalendarSpread.

Type The designation ofanoptionaseitheracalloraput.

Underlying Theinstrument to be delivered in

the event an option isexercised.

VanillaOptionAnoption,usuallyexchangetraded,withstandardized and traditionalcontract specifications asopposedtoanexoticoption.

Vanna The sensitivity ofanoption’sdelta to a changeinvolatility.

Variation The daily cash

flow resulting from changesin the settlement price of afuturescontract.

Vega.Thesensitivityofanoption’stheoreticalvaluetoachange in volatility. AlsoknownasKappa.

Vega Decay Thesensitivityofanoption’svegatothepassageoftime.

Vertical Spread The

purchase of an option at oneexercisepriceand thesaleofan option at a differentexercise price where bothoptions areof the same type,have the same underlyingcontract, and expire at thesametime.

Volatility The degree towhich the price of a contracttendstofluctuateovertime.

Volatility Skew The

tendency of options atdifferent exercise prices totrade at different impliedvolatilities. Also known as aVolatilitySmile.

Volatility Smile SeeVolatilitySkew.

VolgaThesensitivityofanoption’s vega to a change involatility. Also known asVomma.

VommaSeeVolga.

Warrant A long-term calloption.Theexpirationdateofa warrant may under somecircumstancesbeextendedbytheissuer.

WriteTosellanoption.

Zero-CostCollarAcollarwhere the prices of thepurchased and sold optionsareequal.

Zomma The sensitivity ofan option’s gamma to achangeinvolatility.

SomeUsefulMath

The mathematicalfunctions and calculationsreferred to in this text areincluded in almost all

commonlyusedspreadsheets,andformost traders, it isnotnecessary to know exactlyhow the calculations aremade. Of far greaterimportance is the ability tointerpret the numbers thatresultfromthecalculations.

For the reader who isinterested, a detaileddiscussion of thesemathematicalconceptscanbefoundinanygoodstatisticsor

finance textbook. Forconvenience, we include anoverview of these conceptsandapplications.

Rate-of-ReturnCalculations

Aninterestrateisthemostcommon rate of return. Thetotalinterestcanbecomputedin three ways: simple,

compound,andcontinuous.If

then,forsimpleinterest,

forcompoundinterest,

andforcontinuousinterest,

Because volatility is acontinuously compoundedrateofreturn,wecanusetheexponential and logarithmicfunctions to do similarcalculationsforvolatility.If

t = time to

expiration, inyearsF = a forwardprice after theperiod of timetσ = annualvolatility orstandarddeviationX = anoption’sexerciseprice

then a price range of nstandarddeviationsis

The number of standarddeviations required to reachanexercisepriceis

NormalDistributionsandStandardDeviation

If

xi = each datapointn = number ofdatapointsσ = standarddeviation or

volatilityµ=averageormean

thenthemeanoraverageµis

When calculating thestandard deviation from theentire population, σ is givenby

When estimating thestandard deviation from asample of the entirepopulation,σisgivenby1

The normal distributioncurven(x)isgivenby

In a standard normaldistribution,µ=0andσ=1.

Many of the measuresassociatedwith a distributionare derived from a group ofnumbers called moments. Ingeneral, the jth moment mjabout the mean µ of adistributionisgivenby2

From the second, third,and fourth moments, we cancalculate the skewness andkurtosisofadistribution

A perfectly normaldistributionhasaskewnessof0 and a kurtosis of 3. Tonormalize the kurtosis suchthatanormaldistributionhasakurtosisof0, it iscommontosubtract3

Figure B-1 showscalculation of the mean andstandard deviation for the

pinball distribution in Figure6-2. The steps required tocalculate the skewness andkurtosis would require aninordinate amount of space.However,therelevantvalues,including the first threemoments,are

(The right tail of thedistribution is very slightly

longerthanthelefttail.)

(The peak of thedistribution is slightly lowerand the tails slightly shorterthan a true normaldistribution.)

figureB-1CalculationofthemeanandstandarddeviationforthedistributioninFigure6-2.

Volatility

Volatility is usuallycalculated as a samplestandard deviation. It is alsocommontoassumeameanof0. The estimated annualizedvolatilityisthengivenby

where xi = ln(pn/pn–1) =natural logarithm of thecurrent price pn divided bythepreviouspricepn–1andt=the time interval, in years,betweenpricechanges.

If the underlyingcontract isa stock, in theory,

theprice returnsxi shouldbeadjustedtoreflecttheforwardprice of pn–1 over each timeperiod. However, unlessinterestratesareveryhighorthestockwillpayadividend,using the actual price ratherthan the forward price isunlikely to significantly altertheresults.

Thevolatilitycalculationfor the stock option examplein Figure 8-1 is shown in

Figure B-2. Because pricechanges were observed atseven-day intervals (t =7/365), to annualize thevolatility, itwasnecessary todivide by √7/365. Thecalculation represents thepopulationstandarddeviation(dividingbynratherthann–1) and is basedon the actualmean of the price changes.We might also calculate thevolatility assuming a 0meanor use an estimated standard

deviation.Thevariousresultsareasfollows:

There is very littledifference between thecalculations made from theactual mean and a 0 mean.The estimated standard

deviation is always greaterthan the population standarddeviation.

FigureB-2Volatilitycalculationforthe

stockoptionexampleinFigure8-1.

1Thesamplestandarddeviationissometimesdenotedwiths(insteadofσ).2Inthesamewaywecalculateasamplestandarddeviationbydividingbyn–1,wecanalsocalculatesamplemomentsbydividingbyn–1ratherthandividingbyn.

Index

Pleasenotethatindexlinkspointtopagebeginningsfromthe print edition. Locationsareapproximateine-readers,and you may need to pagedownoneormoretimesafterclicking a link to get to the

indexedmaterial.

Adjusteddelta,499Adjustments

inBlack-Scholesmodel,348dynamic hedging with,121–122tooriginalhedge,123risksandneutral,246tospread,246–248trader choices for, 247–248

in volatility spreads,206–208

Allornone(AON),539American option, 32, 292,539arbitrageboundaries for,293–295,300binomial tree and, 370–371Black-Scholesmodelnotfor,309,377deltavaluesin,313dividend-paying stocks

and,373–376European optioncomparedto,315–317evaluating,314pricingof,309–317

AON.SeeAllornoneApproximations,,

for put-call-parity, 271-273forBlack-Scholesvalue,350-352

Arbitrage, 19–21, 265–266,539

boundaries, 293–299,300free,59,60index,452–454,545relationships,291risk,273–290

ARCH. See Autoregressiveconditionalheteroskedasticity

Asian option. See Averagepriceoption

Askprice,66Assignment,29–30,539

Attheforward,60,539Atthemoney,33–35,93

bull and bear strategiesand,222–226delta,136gamma,149–150straddles,477theta,141vega,145–146

Automatic exercise, 35–36,539

Autoregressive conditionalheteroskedasticity(ARCH),

394Averagepriceoption,540

Bachelier,Louis,61–62,76Backspread,179,540Backward,14,524–526,530,540

Balancedskew,487Barone-Adesi,Giovanni,309Barone-Adesi-Whaleymodel,309–310

Barrieroption,540Bearspread,222–226,540

Bermudaoption,540Bias, in futuresmarket,457–458

Bidprice,66Bid-ask spread, 251–252,427–428,485

Binaryoption,540Binomialexpansion,361Binomial model. See Cox-Ross-Rubinsteinmodel

Binomialnotation,363-365Binomialoptionpricing,358Binomialtree,359–360

American options and,370–371Black-Scholes modelvaluesin,379callvalue,364dividend-paying stockand,374–375withdividends,372notationfor,363–365one-period,360–361periodsusedin,378recombining,368

Black,Fischer,57,61,62

Blackoutperiod,304Black-Scholesmodel, 61–68,120approximationsof, 350–352binomialvaluesin,379continuous diffusionprocessassumedin,472as continuous-timemodel,84deltain,352–354dividendsin,67–68equation,339–340

as European pricingmodel,309,377formulae,349–350gamma and vega in,354–357implied volatility in,130–131,307interestratesin,66–67jump-diffusion modelvariationof,475lognormal distributionassumptionsin,85,341–344,506

not for Americanoptions,309,377outofthemoneyoptionsand,470–471as probabilistic model,471put-call parity in, 340–341theoretical valuescalculatedwith,63thetaof,354time to expiration in,65–66

underlying price in, 66,345–346variations,62volatilityin,68,339

Black-Scholes-Mertonmodel,338

BMX. See Chicago BoardOptions ExchangeBuyWriteIndex

Bodyofbutterfly,192Bondsandnotes,17Borrowingcostsrbc,24

Box,280–282,540Breakevenprices,51Breakeven volatility, 116–117,130

Broad based stock indexes,441–442

Bull andbear butterfly, 211–212

Bull and bear calendarspread,212–214

Bull and bear ratio spread,210–211,213

Bullandbearverticalspread,

215–217Bullspread,222–226,540Bullstraddle,171Bundfutures,388,390Butterfly,289,506–507,540

bullandbear,211–212callorput,290iron,260–262,290,545and implieddistributions,508long,174–175,201short,174–175,201

time,191–192Buying,284,287Buy/write,326–327,540

Cabinetbid,420,540Calendar spread, 195, 237,239–240,540bullandbear,212–214implied volatility and,405–408longandshort,189–190,204,214rollsand,284–286

Callbutterfly,290Christmastree,183,203covered,486delta,137,139lambdaand,154–156option, 3, 26, 299–302,540–541protective,322–324ratio spread, 179–181,203synthetics,288,551

theoreticalvalueof,101,108–109,111,310–311

Cap,322,514,540,541Capitalization,443Capitalization weightedindex,443,454–456

Carrytrade,20Cash, 1, 8, 31–32, 459–461,461–462

Cash-and-carryarbitrage,20Cash-and-carry strategy, 159,161

Cash-securedput,327

Castelli,Charles,61CBOE. See Chicago BoardOptionsExchange

CEV. See Constant elasticityofvariancemodel

Charm,139–141,541Chicago Board OptionsExchange (CBOE), 62,250,309,459,515,530

Chicago Board OptionsExchange BuyWrite Index(BXM),327

Chicago Mercantile

Exchange,36,67,81,452Chooseroption,541Christmastree,182–184,203,541

Circuitbreakers,464Clearingfirm,11Clearingmember,541Clearinghouse,10,11,541Closingtrade,5CMEClearingHouse,10Collar,328–330,541Color, of option, 152–153,

541Combination (Combo), 266,541

Combovalue,266,267Commoditymarkets,486Companionoption,256,259Complex positions, 45–46,410–411,420–423

Compoundoption,541Condor,176–178,541Constant elasticity ofvariance(CEV)model,478

Contango, 14, 524–526, 530,

541Contingencyorder,208,542Continuousdiffusionprocess,471–472

Continuous-timemodel,84Contract specifications, 26–30

Convenienceyield,15Conversionmarket,267Conversion, 265–266, 273,275,277–282,289,542

Counterparty,10Coveredcall,486

Coveredoption,329Covered write, 61, 324–328,542

Cox,John,309,358Cox-Ross-Rubinstein model,309–310

Crackspread,163–164Creditspread,214Curvature,105–108Cylinder.SeeCollars

DAXIndex,448Debitspread,214

Declareddate,22–23Delta(Δ),64,110,492,542

adjusted,499of American options,313inBlack-Scholesmodel,352–354of bull spread, 218-219,223call,137,139decay,139hedge ratio and, 102–103

implied,136atthemoney,136negative and positive,416position,169probabilityand,104–106put,138rateofchangeand,100–102theoretical underlyingpositionand,103–104as volatility changes,135–136

Delta(directional)risk,227Delta-neutral,103,121,129–130, 133, 353–354, 421,542

Derivativecontracts,4Dermna,Emanuel,57Diagonal ratio spread, 236–237

Diagonalroll.SeeTimeboxDiagonal spread, 196–200,542

Diffusionprocess,472–473Digital option. See Binary

optionDividendplay,318Dividendvalue,300,302Dividend-payingstocks,373–376

Dividends,279,373–379binomialtreewith,372inBlack-Scholesmodel,67–68declareddateof,22–23Dow Jones IndustrialAveragepaying,452ex-date of, 22–23, 68,

543implied,21,273payabledateof,23recorddateof,22risk and, 239–240, 278–280stock options influencedby,100calendar spreads and,192–195

Divisor,446–447Dow Jones IndustrialAverage,445,452

Downside contract position,418

Dragonfly,504,542Driftlesstheta,354Dynamic hedging, 336, 474,512,542with adjustments, 121–122delta-neutral,133processof,127–129

Earlyexercise,292ofcalloption,299–302

of future option, 305–307protective value and,308ofputoption,302–305risk,319–320short stock impact on,305strategies,317–319

Efficiency,244–246,542Efficient-market hypothesis,394

Elasticity,154,542. See also

LambdaEqual-weightedindex,444Equilibriumprice,429Equityportfolio,457Euro-bundoptions,269Eurocurrency,81–82,542Eurocurrencyrate,543Europeanbox,314European option, 32, 60, 62,253,292,543American optioncomparedto,315–317

arbitrageboundaries for,293–299arbitrage relationshipsfor,291Black-Scholes modelpricingof,309,377dividend-paying stocksand,373

Europeanpricingmodel,309,377

EuroStoxx 50 Index, 385–386,402,405,407

EWMA. See Exponentially

weightedmovingaverageExchangeoption,543Exchanges,6–9,449,464Exchange-traded contracts,10,160

Exchange-tradedfunds,454Exchange-tradedoptions,29Ex-dividend date (Ex-date),22–23,68,543

Executionrisk,273–274Exercise,29–30,543Exercisenotice,35

Exerciseprice,4,29,65,215,543

Exoticoption,182,543Expected value, 53–54, 59–60

Expiration date, 2, 4, 28–29,65–66,73,543

Expirationstraddle,476–477Expiry.SeeExpirationdateExponentially weightedmoving average (EWMA),393–394

Extreme-valuemethod,384

Extrinsic value. See Timevalue

Fair value. See Theoreticalvalue

Fearindex,523Fence.SeeCollarsFiglewski,Stephen,57Fillorkill(FOK),208,543Fixed-income markets, 161–162

Flatpeak(platykurtic),481Flatskew,487

Flexoption,27,543Floatingskew,489,490Floor,322–323,543FOK.SeeFillorkillForecasting, volatility, 391–394

Foreigncurrencies,17–18Foreigncurrencyoptions,99Forwardcontract,2,543

basis,13fairpricefor,12–14forforeigncurrency,17–

18interestraterisk,21long-term,452notionalvalue,6stockindex,455synthetics,254

Forward conversion. SeeConversion

Forward price, 2–3, 12–17,76–77,270–271,543

Forwardrate,16,408Forwardstartoption,544

Forwardvolatility,404–409Forward-rate agreement(FRA),16

Freefloat,445Frequency distribution, 480–481

Frictionless markets, 125–126,463–466

Frontspread,182,544FTSE100index,485Fugit,307,544Futurevolatility,394–397Futurescontract,2,544

early exercise of, 305–307lockedmarkets,269–270notionalvalue,455option’sdifferencewith,26–27optionson,267–269onstockindex,450–451synthetics,270,277

Futuresexchange,2Futuresmarket,457–458,464Futuresoption,19,30–31

Futures-type settlement, 7–9,36,112,544

Gamma (Γ), 110, 113, 152,544inBlack-Scholesmodel,354–357ofbullspreads,223influence of expirationandvolatility,150–151magnituderiskmeasuredby,115–116atthemoney,149–150

negative and positive,106–107, 230, 414–418,421options,154,367as option’s curvature,105–108rent,369–370spread,165thetatradeoffwith,237underlyingpricechangesand,149–150

Gamma (curvature) risk,227–228

Gammaspread,165Gaps,realworld,476GARCH. See GeneralizedAutoregressive conditionalheteroskedasticity

Garman,Mark,62,385Garman-Klass estimators,385

Garman-Kohlhagen model,62,67

Generalized Autoregressiveconditionalheteroskedasticity

(GARCH),394Geometric-weighted index,445

Golongorshort,5Gold,93–96,384,387,389Good ‘til canceled (GTC),544

Greeks,99GTC.SeeGood‘tilcanceledGuts,172,544

Haircut,8,544Half-steps,378

Hedgeratio.SeeDelta(∆)Hedgers,321,544Hedging

collarsusedin,328–330complex strategies in,331–333covered-writes used in,324–328high and low impliedvolatilityin,331natural long or short in,321protective calls and puts

in,322–324stock index futures and,457strategy summary of,330using vertical spreads,332–333using volatility spreads,332volatility contracts usedfor,536volatility reduced by,333–334

Higher-order risk measures,355

Historicalvolatility,381–386,392of Bund futures, 388,390of EuroStoxx 50, 386,402ofgold,93–96,384,389of S&P 500 Index, 87,383,389

Hockey-stickdiagrams,40Horizontal spread. See

Calendarspread

Illiquidmarket,66Immediate or cancel (IOC),544

Implieddelta,136Implieddistribution,506–510Implieddividend,21,273Impliedinterestrates,21,273Impliedspotprice,21Impliedvolatility,88–89,92–95,116,199,544assessingriskof,401

in Black-Scholesmodel,130–131,307calendar spread and,405–408contracts,512,514–523defining,514EuroStoxx50and,407floatingskewsas,490future volatilitypredictedby,394–397hedging with high andlow,331influenceof,188–189

optionpricing comparedwith,90–91reevaluationof,131relativechangesin,400sensitivitytochangesin,425–426term structure of, 397–404,430–432timedecayand,380–381

Inprice,544Inthemoney,33–35,92,545Index

arbitrage,452–454,545

broad based stock, 441–442calculation,443–445capitalization-weighted,443,454–456divisor,446–447equal-weighted,444fear,523geometric-weighted,445impact of individualstock price changes,448–450narrow based stock,

441–442price-weighted, 443–445,449,454–455rebalancing,445replicating,454–456total-return,447–448volume-weightedaverageprice,450

In-option,544In-price,544Insurancepolicies,4,321Interest, 5, 123–124, 239–240, 278–280, 303, 318–

319Interestrates,228

inBlack-Scholesmodel,66–67foreign currencyoptionsand,99forward contract andrisksof,21implied,21,273negative,83optionvalues influencedby,97–100,466-467products volatility of,

81–82stockoptionsimportanceof,305–306,466–467in theoretical pricingmodels,126theoretical value and,467volatility spreads withdividendsand,192–195

Interest-ratevalue,300Intermarket spreads, 161–162,545

Intrinsicvalue,32–34,38–40,

299–300,545Investmentskew,486–487IOC.SeeImmediateorcancelIron butterfly, 260–262, 290,545

Ironcondor,262–263,545

Jellyroll.SeeRollJumpprocess,472,473Jump-diffusion model, 475,478

Jump-diffusion process, 472,473

KansasCityBoard of Trade,452

Kappa(K),110,545Klass,Michael,385Knock-in option. See In-option

Knock-out option. See Out-option

Kohlhagen,Steven,62Kurtosis, 481–483, 495–498,501–506,555

Ladder, 545. See also

ChristmastreeLambda (Λ), 154–156. SeealsoElasticity

Lasttradingday,28LEAP.SeeLong-termEquityAnticipationSecurity

Leg,163–164,545Leveragevalue.SeeElasticityLIBOR.SeeLondon Interbank OfferedRate

Limit,546Limitorder,208,546

Liquidmarket,66Liquidity,249–250,427Local,546Lockedlimit,23Lockedmarkets,23,546Lognormal distributions, 82–83Black-Scholes modeland assumptions of, 85,341–344,506butterflieswith,508pricechangesin,84standard deviations and,

346–348of underlying price,478–481

London Interbank OfferedRate(LIBOR),67,81

Long,54,321,477butterfly,174–175,201calendar spread, 189–190,204,214call,38,42–43callChristmas tree,183,204condor,177–178,202

natural,321position,5,98,328,546premium,197,546put,39,42–43put Christmas tree, 184,203raterl,24ratiospread,546straddle,170–171,205strangle, 172–173, 201,262synthetic,253,260–261

timebutterfly,193Long-term EquityAnticipation Security(LEAP),545

Long-termoptions,27–28Lookbackoption,546

Magnituderisk,115–116Margin and variationsettlement,7

Margindeposit,7Margin of error, 239, 240–244

Margins,8,546Marketconditions

market making and,429–430option traders andchangesin,139–140optionvalues influencedby,97–98positions influenced by,435–436risk analysis andchanges in, 434–435,437

riskand,413,415–416stock splits and, 438–440theoreticaledgeand,231

Marketintegrity,10–11Marketliquidity,235Market maker, 427–430,432–433,546

Marketorder,208,546Market-if-touched(MIT),546Market-on-close(MOC),453,546

Markets,126,165–166

commodity,486declining,426–427fixed-income,161–162frictionless,463–466futures contracts andlocked, 23, 269–270,546optionpositionin,64option traders and speedof,53volatility measuringspeedof,69

Mark-to-market,546

Marriedput,547Maturity date. See expirationdate

Mean,73–77,345Mean reverting, 387–388,398

Median,345Merton,Robert,62,338Mid-Atlantic option. SeeBermudaoption

Midcurveoptions,28,547MIT.SeeMarket-if-touchedMOC.SeeMarket-on-close

Models.SeespecificmodelModes,84,345Moneyness,491

Nakedposition,164,209,547Narrow based stock index,441–442

Naturalgasfutures,403–404Naturallongsandshorts,321Negativedividendrisk,279Negative gamma position,106–107, 230, 414–418,421

Negativeinterestrates,83Negative theoretical edge,118

Negativetimevalue,109–110Negativevega,230,236Negativevolga,236Netcontractposition,417Neutral hedge. See Risklesshedge

Neutralspread,210,547New York MercantileExchange,29

Nominal value. See notional

valueNonsymmetrical strategies,185

Normal distribution, 69–73,75–77,480–481,554–556

Notheld,209,547Notionalvalue,6,455Notionalvega,513

OBO.SeeOrderbookofficialOCO. See One-cancels-the-other

OEX. See Options Exchange

IndexOmega(?),154,547One-cancels-the-other(OCO),209,547

Openinterest,5Openposition,5Openingtrade,5Option pricing, 32–36, 62,90–91,338–339,358

Option replication. SeePortfolioinsurance

Optionriskanalysis,116Option-pricingtheory,119Options Clearing

Corporation,10,36Options Exchange Index(OEX),459,515

Order book official (OBO),547

Orders, submitting spread,208–209

OTC. See Over-the-countermarket

Outofthemoney,33–35,93,135–136, 419–420, 423,470–471,547

Outliers,481

Out-option,547Out-price,547Out-trade,547Over-the-counter (OTC)market,19

Overwrite,324,547

Paralysis through analysis,118

Parity.SeeIntrinsicvalueParitygraphs,38–51Parkinson,Michael,384Partialderivatives,100

Pathdependent,471Phi(Φ),112Physicalcommodities,2,14–15

Physicalsettlement,7–8Physicalunderlying,30–31Pinrisk,274–275P&L.SeeProfitandlossPlain-vanilla-interest-rateswap,5

Population standarddeviation,382

Portfolioinsurance,335–337

Portfolio manager, 333–334,336,537

Positivedividendrisk,279Positive gamma position,106–107,415

Premium,4Pricechanges,479

individual stock, 448–450in lognormaldistributions,84underlying, 141–142,145–146,149–150

volatility and observed,80–81

Pricedistribution,73Pricemovement,65,72–75Pricevolatility,82Price weighted index, 443–445,449,454–455

Pricing,ofAmericanoptions,309–317

Pricingmodels,62,338–339American,309enteringskewinto,489–494

European,309,377volatilityskewin,499weaknesses,484–485

Probabilisticmodel,471Probabilities,4,376,483

deltaand,104–106expected value, 53–54,59–60modelsand,56–57symmetrical distributionof,60theoreticalvaluesin,54–56

Probabilityfunctions,344Profitandloss(P&L),46–51,124,125,133

Programtrading,453Protective options, 323–324,327,329

Protectiveput,486Protectivevalue,308Pseudoprobabilities,376Put

butterfly,290cash-secured,327

Christmastree,184,203delta,138lambdaand,154–156long,39,42–43married,547option,4,26,302–305protective,322–324,486ratio spread, 179–181,204short,40shortChristmastree,184synthetics,261–262,551

theoretical value of,101–102, 108–109, 111,312

Put-callparity,267–268,462inBlack-Scholesmodel,340–341for stock options, 270–273,288synthetic futurescontractand,270

Quadraticmodel.SeeBarone-AdesiWhaleymodel

Randomwalk,69–73,471Rateofreturn,82,553–554Ratio spread, 179–182, 202,210–211, 213, 236–237,546,549

Ratiostraddle,171Ratiostrategy,162Ratiowrite,332Realizedprofit,7Realized volatility, 86–87,116,512–514

Rebalancingprocess,537Recombining binomial tree,

368Recorddate,22Rehedging,129,367Reversal,273,275, 277–282,502,549

Reverse conversion, 265,268,278–279,289

Rho (P), 111–112, 113, 467,549

Rho(interest-rate)risk,228Risk,21,273–274

analysis,434–435,437arbitrage,273–290

boxes,280–282complexpositions, 422–423delta(directional),227dividends and interest,239–240,278–280earlyexercise,319–320gamma (curvature),227–228gamma spreads and,236–237interest and, 239–240,278–280

magnitude,115–116management, 97–100,135marginforerrorin,239negative and positivedividend,279neutraladjustments,246pin,274–275rho(interest-rate),228rollsand,282–285settlement,276–278spreadingstrategiesand,161,166–168

instockoptions,240strategies andconsiderations of, 234,238theoretical pricingmodelsandtypesof,463theta(timedecay),228timeboxesand,285–287vega(volatility),228volatility,228–234

Riskmeasures,99higher-order,355interpreting,112–118

skewed,498–499stock splits and, 439–440traditional andnontraditional,156–157

Riskreversal,502,549Risk-analysisprocess,432Risk-freerate,67Risklesshedge,63–64,102Risk-neutralworld,358–360Risk-reward tradeoff, 40, 97,176,244

Roll,282–286,549

Ross,Stephen,358Rubinstein,Mark,358

Sample standard deviation,382

Scalper,158,549Scholes,Myron,61,62,338Second-generation option.SeeExoticoption

Sellstockshort,23Serialcorrelated,386Serialoptions,27,549Settlementintocash,31–32

dates,22in to a futures position,30–31futures-type, 7–9, 36,112,544marginandvariation,7optionsandtypesof,36physical,7–8in to the physicalunderlying,30–31procedures,6–9risk,276–278stock-type,7–9,36,268,

316,318–319,550Sharpe,William,334Sharperatio,334Short,5

butterfly,174–175,201calendar spread, 189–190,204,214call,39callChristmas tree,183,203collars,330condor,177–178,202

naturallongor,321position,5,98,328,549premium,197,549put,40put Christmas tree, 184,203sales,23–25sellstock,23squeeze,23,195,549stock,98,305straddle,170–171,201strangle, 172–173, 201,

262Shortraters,24Shortratiospread,549Short-stockrebate,24Short-stocksqueeze,23,195,549

Short-termoptions,27–28Sigma(σ),77,549Skew

balanced,487flat,487floating,489,490

investment,486–487modelinputof,498modeling,489–494riskmeasures and, 498–499sticky-delta,492sticky-strike,489volatility,485–486,532,552

Skewness, 481–482, 495–498,501–506,555

Slope,inparitygraphs,40–46S&P 500 Index, 250, 327,

448,478–479,482,515historical volatility, 87,383,389

SPAN.SeeStandardPortfolioAnalysisofRisk

Specialopeningrotation,520Specialist,549–550Speculator,53,131,321,410,428,550

Speed,150–151,550Split-strike conversion, 328,550.SeealsoRiskreversal

Spread,158,550

adjustmentsto,246–248bear,222–226,540bid-ask,251–252bull,222–226,540commonfeaturesof,169creditanddebit,214diagonal,196–200,542diagonalratio,236–237dynamic hedging and,165efficiencyof,244–246front,182,544

for futures contracts,160–161gamma,165gammariskof,236–237intermarket andintramarket,161–162liquidityin,249–250withnegativevega,230static,165submitting order for,208–209theoretical value anddeltasin,218–219

traderbuying,164trading style and, 248–249volatility characteristicsand,233–234

Spreadingstrategiesdefining,159–164futures contracts using,160–161inoptionsmarkets,165–166risk controlled by, 166–168

risks reduced through,161synthetics used in, 259–260

Standard cumulative normaldistributionfunction,344

StandarddeviationBlack-Scholes model,344–346calculating,382lognormal distributionsand,346–348meanand,73–76

normal distribution and,554–556price movement and,74–75sample and population,382volatilityas,77–78weekly,79

Standard Portfolio AnalysisofRisk(SPAN),36

Statichedge,121Staticspread,165Sticky-deltaskew,492

Sticky-strikeskew,489Stockindex

broadandnarrowbased,441–442forwardcontract,455futures,458–459futurescontracton,450–451hedging and futures in,457implied distribution of,509options,458–462

Stockoptions,19dividends influencing,100interest rates importantin,305–306,466–467longandshortpositions,98physicalunderlyingand,30–31put-call parity for, 270–273,288

Stocksplits,438–440Stocks

dividend-paying, 373–376forwardpricefor,15–17Index impacted by pricechangesin,448–450

Stock-type settlement, 7–9,36,268,316,318–319,550

Stop(loss)order,209,550Stop-limitorder,209,550Straddles, 61, 170–171, 205,238,474,550buying,287delta neutral and, 353–

354expiration,476–477atthemoney,477syntheticsand,260–261

Strangles,171–173,238,261,550

Strap,550Strategies

bullandbear,222–226buy/write,326–327,540cash-and-carry,159,161choosing appropriate,

199–206complex hedging, 331–333earlyexercise,317–319efficiencyof,244–246hedging summary of,330nonsymmetrical,185ratio,162risk considerations in,234,238skewness and kurtosis,501–506

symmetrical,178Strike price. See Exerciseprice

Strip,519,534–536,550Swap,5,550Swaption,550Symmetricalstrategies,178Synthetics,287–290,550

call,288,551contracttypes,258forwardcontract,254futures contract, 270,

277longstraddles,260–261longunderlying,253options,255–260put,261–262,551relationships,267shortposition,264–265spreading strategiesusing,259–260underlying, 253–255,551volatility spread using,287–290

Tailing,277Tallpeak(leptokurtic),481Tau(τ),551Taxes,127,465Term structure, 390, 397–405,430–432,551

Theoreticaledge,118,227margin of error and,240–244market conditions and,231market liquidity and,235

Theoretical pricing modelsbasicinputsin,335diffusion processassumedin,472–473interestratesin,126performance of, 119–121problemswith,56–57risktypesin,463underlyingpricein,66understanding andhavingfaithin,538volatility in, 80, 89–90,

204,206,380weaknessesin,511–512

Theoretical value, 142–143,551Black-Scholes modelcalculating,63of call, 101, 108–109,111,310–311in dynamic hedging,133–134offuturescontract,316interestratesand,467ofoptions,55,58

andprobability,54–56of put, 101–102, 108–109,111,312of underlying contract,115volatility changes and,146–147

“The Theory of Options inStocksandShares,”61

The Theory of Speculation(Bachelier),61–62

Theta(Θ),113,142–143,551ofBlack-Scholesmodel,

354ofbullspreads,224–225driftless,354gamma tradeoff with,237atthemoney,141negative timevalueand,109–110ofoptions,367–368positive,417astimedecay,108–109underlyingpricechangesand,141–142

volatility changes and,144–145

Theta(timedecay)risk,228Three-way,280,551Tickersymbols,460Time, volatility scaled for,78–79

Timebox,285–287,551Timebutterfly,191–192Time decay, 108–109, 380–381

Timepremium,33,551Time spreads. See Calendar

spreadTime value, 32, 33–34, 543,551

Time-seriesanalysis,393Total-returnindex,447–448Trading, exchanges halting,449,464

Transaction costs, 126–127,129

Type,6,551

Underlying,4,27-28,551Underlyingcontract,439

synthetic options and,255–260synthetic short positionand,264–265theoreticalvalueof,115volatility independent ofpriceof,478

Underlying position, 41–43,103–104

UnderlyingpriceinBlack-Scholesmodel,66,345–346cash index option and,

461–462exercise price and, 49–51gamma and changes in,149–150lognormal distributionof,478–481optionvalueand,37–38in theoretical pricingmodels,66theta and changes in,141–142vega and changes in,

145–146Unrealizedprofits,7Upsidecontractposition,419,

ValueLineIndex,452Vanillaoption,551Vanna,139,140,146,551Variancecontract,513Varianceswap,513Variation,8,551Vega,110,113,116,552

inBlack-Scholesmodel,354–357

ofbullspreads,224decay, 147, 149, 356–357,552atthemoney,145–146negative,230,236neutral,399notional,513astimepasses,147–148underlyingpricechangesand,145–146volatility changesinfluencing,146–147

Vega(volatility)risk,228

Vertical spread, 213–226,329,332–333,552

VIX.SeeVolatilityindexVolatility,552

inBlack-Scholesmodel,68,339breakeven, 116–117,130calculating,556–557delta and changes in,135–136forecast,87forecasting,391–394

forward,404–409future,394–397gamma and changes in,150–151hedging to reduce, 333–334implied,88–95,512interpretingdataon,85–95in-the-money optionsand,92meanreverting,387–388observed price changes

and,80–81realized, 86–87, 116,512rising and falling, 468–469risk,228–234scaledfortime,78–79shifting,500–501smirk,485asspeedofmarket,69as standard deviation,77–78surface,499

termstructureof,390in theoretical pricingmodels, 80, 89–90, 204,206,380theoretical value andchangesin,146–147theta and changes in,144–145underlyingcontractpriceand,478value,299,304vega and changes in,146–147

Volatilitycontracts,512–514applications,536–537replicating,534–536usedforhedging,536

Volatility index (VIX), 515–519characteristics of, 520–523futures,524–529options,530–533trading,523–524

Volatility skew, 485–486,532,552

Volatility smile. SeeVolatilityskew

Volatilityspread,169adjustments,206–208butterfly,173–176calendar spread, 182–191Christmastree,182condor,176–178diagonal spread, 196–199hedgersusing,332interest rates and

dividendsin,192–195ratiospread,179–182straddles,170–171strangle,171–173submitting an order,208–209synthetics used in, 287–290timebutterfly,191–192

Volga,147–148,552Volume-weighted averageprice(VWAP),450

Vomma.SeeVolga

VWAP. See Volume-weightedaverageprice

Warrant,552Wilmot,Paul,57Wingsofbutterfly,192Write,325,552

Yieldvolatility,82

Zero-costcollar,330,552Zero-meanassumptions,382Zomma,152–154,552

AbouttheAuthor

SheldonNatenberg beganhis trading career in 1982 asanindependentmarketmakerin equity options at theChicago Board Options

Exchange. From 1985 to2000, he traded commodityoptions, also as anindependent floor trader, attheChicagoBoardofTrade.

While continuing totrade,Mr.Natenberghasalsobecomeactiveasaneducator.In this capacity, he hasconducted seminars foroption traders at majorexchanges and professionaltrading firms in the United

States, Europe, and the FarEast. In2000,Mr.Natenbergjoined the education team atChicagoTradingCompany,aproprietary derivativestradingfirm.