option volatility and pricing

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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)