Solutions for Derivatives 2nd Us Edition by Sundaram by 6alsm

Derivatives:Principles&Practice

RangarajanK.Sundaram NewYorkUniversity SanjivR.Das SantaClaraUniversity

March17,2015

1 Pleasedobringanyerrorsoromissionstoourattentionatrsundara@stern.nyu.eduorsrdas@scu.edu.

Contents

Chapter1.Futures&Options-Overview3

Chapter2.FuturesMarkets7

Chapter3.PricingForwards&FuturesI17

Chapter4.PricingForwards&FuturesII32

Chapter5.HedgingwithFutures&Forwards40

Chapter6.InterestRateForwards&Futures54

Chapter7.OptionsMarkets69

Chapter8.OptionPayoﬀs&TradingStrategies75

Chapter9.No-AbitrageRestrictionsonOptionPrices85

Chapter10.Early-Exercise&Put-CallParity97

Chapter11.OptionPricing:AnIntroduction107

Chapter12.BinomialOptionPricing123

Chapter13.ImplementingtheBinomialModel145

Chapter14.Black&ScholesModel155

Chapter15.TheMathematicsbehindBlack-Scholes166

Chapter16.OptionsModeling:BeyondBlack-Scholes179

Chapter17.SensitivityAnalysis:Greeks193

Chapter18.ExoticOptionsI:Path-IndependentOptions207

Chapter19.ExoticOptionsII:Path-DependentOptions225

Chapter20.ValueatRisk245

Chapter21.ConvertibleBonds261

Chapter22.RealOptions272

Chapter23.SwapsandFloatingRateProducts279

Chapter24.EquitySwaps291

Chapter25.CurrencyandCommoditySwaps299

Chapter26.TermStructureofInterestRates:Concepts309

Chapter27.EstimatingtheYieldCurve321

Chapter28.ModelingTermStructureMovements340

Chapter29.FactorModelsoftheTermStructure350

Chapter30.TheHeath-Jarrow-MortonandLiborMarketModels363

Chapter31.CreditDerivativeProducts378

Chapter32.StructuralModelsofDefaultRisk385

Chapter33.Reduced-formModelsofDefaultRisk394

Chapter34.ModelingCorrelatedDefault406

Chapter35.DerivativePricingwithFinite-Diﬀerencing417

Chapter36.DerivativePricingwithMonteCarloSimulation421 appendix:Using Octave 431

Chapter1.Futures&Options-Overview

1. Question: Whatisaderivativesecurity?

Answer: Aderivativesecurityisaﬁnancialsecuritywhosevaluedependson(orderivesfrom) other,morefundamental,underlyingvariablessuchasthepriceofastock,acommodityprice,an interestrate,anexchangerate,oranindexlevel.Theunderlyingmayalsobeaderivativesecurity itself–therearemanyderivativesthatarewrittenonotherderivatives.

2. Question: Giveanexampleofasecuritythatisnotaderivative.

Answer: Aninterestrateisnotaderivative.Itsisafundamentaleconomicquantityreﬂecting thevalueofmoney.

Astockisalsotypicallyviewedasa“primitive”(ratherthanaderivative)security.However,a stockmayalsobeviewedasaderivative:itrepresentsaclaimonthecashflowsgeneratedbythe assetsoftheissuingfirm.Indeed,viewingequityasaderivativewrittenontheunderlyingfirm’s assetvalueisfundamentaltothe“structuralmodel”approachtostudyingcorporatecreditrisk. WeexaminestructuralmodelsinChapter32.

3. Question: Canaderivativesecuritybetheunderlyingforanotherderivativesecurity?Givean example.

Answer: Yes,itcan.Thesimplestexampleisanoption(say,acall)thatgivesyoutheright topurchaseanotheroption(say,aputwrittenonsomeunderlyingstock).Inthiscase,theﬁrst optioniscalleda“compound”option;itisanoptiononanoption,inthiscase,acallonaput. CompoundoptionsarestudiedinChapter18.Compoundoptionsalsoplayaroleinthestructural modelsstudiedinChapter32.

4. Question: Derivativesmaybeusedforbothhedgingandinsurance.Whatisthediﬀerencein thesetwomotives?

Answer: Theobjectiveofhedging,whetherwithaderivativeorotherwise,istoeliminatethe riskassociatedwithanexistingmarketcommitmentandtocreateanetpositionthatis“riskfree.”Thatis,thehedgenulliﬁesexistingrisk;insodoing,iteliminatesbothupsideanddownside potentialfrommarketmoves.Futuresandforwardsareinstrumentsforhedgingrisk.

“Insurance”isone-sidedprotection.Itguardsagainstunfavorableevents(“downsiderisk”)even whileallowingfullupsidepotentialtoberealized.Optionsareinstrumentsthatprovideﬁnancial insurance.

5. Question: Deﬁneaforwardcontract.Explainatwhattimearecashﬂowsgeneratedforthis contract.Howissettlementdetermined?

Answer: Aforwardcontractisanagreementtobuyorsellanassetatafuturedate(denoted T ), ataspecifiedpricecalledthedeliveryprice(denoted F ).Denotetheinitialdate(theinception dateorthedateoftheagreement)by t =0.Atinceptiontherearenocashflowsonaforward contract.Atmaturity,ifthethen-prevailingspotprice ST oftheunderlyingassetisgreaterthan F , thenthebuyer(the“longposition”)hasgained ST F viatheforwardwhiletheseller(the“short position”)hascorrespondinglylost ST F .Dependingoncontractspecifications,thesettlement mayeitherbeincash(thesellerpaysthebuyer ST F )orphysical(thesellerdeliverstheasset andreceives F ).If ST <F ,thebuyerloses F ST andthesellergainsthisquantity.

6. Question: Explainwhobearsdefaultriskinaforwardcontract.

Answer: Defaultarisesif,atmaturity,oneofthepartiesfailstofulfilltheirobligationsunderthe contract.Defaultriskonlymattersforthepartythatis”inthemoney”atmaturity,thatis,that standstoprofitatthelocked-inpriceinthecontract.(Ifthespotpriceatmaturityissuchthata partywouldlosefromperformingontheobligationinthecontract,counterpartydefaultisnota problem.)Priortomaturity,sinceeitherpartymayfinishin-the-money,bothpartiesareexposed todefaultrisk.

7. Question: Whatrisksarebeingmanagedbytradingderivativesonexchanges?

Answer: Animportantoneiscounterpartydefaultrisk.Inatypicalfuturesexchange,theexchangeinterposesitselfbetweenbuyerandsellerandguaranteesperformanceonthecontract.This reducessigniﬁcantlythedefaultriskexposureofbothparties.Further,dailysettlingofmarked-tomarketgainsandlossesensuresthatthelosstotheexchangefromaninvestor’sdefaultislimited toatmostoneday’ssettlementamount(andbecauseofmaintenancemarginsisusuallylessthan eventhis;seeChapter2foradescriptionofthemarginingprocess).

8. Question: Explainthediﬀerencebetweenaforwardcontractandanoption.

Answer: Aforwardcontractisanagreementtobuyorsellanassetatafuturedate(denoted T ), ataspecifieddeliveryprice(denoted F ).Theagreementismadeattime t =0forsettlementat maturity T .Anoptionistherightbutnottheobligationtobuy(a“call”option)orsell(a“put” option)anassetataspecifiedstrikepriceonorbeforeaspecifiedmaturitydate T .Incomparing alongforwardcontracttoacalloption,themaindifferenceliesinthefactthattheforwardbuyer hastobuythestockattheforwardpriceatmaturity,whereasinacalloption,thebuyerisnot requiredtocarryoutthepurchaseifitisnotinhisinteresttodoso.Theforwardcontractconfers theobligationtobuy,whereastheoptioncontractprovidesthisrightwithnoattendantobligation.

9. Question: Whatisthediﬀerencebetweenvalueandpayoﬀinthecontextofderivativesecurities.

Answer: The value ofaderivativeisitscurrentfairpriceoritsworth.The payoff (orpayoffs) referstothe cashflows generatedbythederivativeatvarioustimesduringitslife.Forexample, thevalueofaforwardcontractatinceptioniszero:neitherpartypaysanythingtoenterintothe contract.Butthepayoffsfromthecontractatmaturitytoeitherpartycouldbepositive,negative, orzerodependingonwherethespotpriceoftheassetisatthatpointrelativetothelocked-in deliveryprice.

10. Question: Whatisashortpositioninaforwardcontract?Drawthepayoﬀdiagramforashort positionataforwardpriceof$103,ifthepossiblerangeoftheunderlyingstockpriceis$50-150.

Answer: Ashortpositioninaforwardiswhereyouaretheselleroftheforwardcontract.Inthis case,yougainwhenthepriceoftheunderlyingassetatmaturityisbelowthelocked-indelivery price.Thepayoﬀdiagramforthiscontractisasshowninthefollowingpicture.Whenthepriceof thestockatmaturityisthedeliverypriceof$103,thereareneithergainsnorlosses.

11. Question: Forwardpricesmaybederivedusingthenotionofabsenceofarbitrage,andmarket eﬃciencyisnotnecessary.Whatisthediﬀerencebetweenthesetwoconcepts?

Answer: Absenceofarbitragemeansthatatradingstrategycannotbefoundthatcreatescash inflowswithoutanycashoutflows,i.e.,createssomethingoutofnothing.Efficiency,asthattermis usedbyfinancialeconomists,impliesmore:notonlytheabsenceofarbitragebutthatassetprices reflectallrelevantinformation.

12. Question: Supposeyouareholdingastockposition,andwishtohedgeit.Whatforwardcontract wouldyouuse,alongorashort?Whatoptioncontractmightyouuse?Comparetheforwardversus theoptiononthefollowingthreecriteria:(a)uncertaintyofhedgedpositioncash-ﬂow,(b)Up-front cash-ﬂowand(c)maturity-timeregret.

Answer: Ifaforwardcontractistobeused,thenashortforwardisrequired.Alternatively,aput optionmayalsobeused.Thefollowingdescribesthethreecriteriaforthechoiceoftheforward versustheoption.

• Cash-ﬂowuncertaintyislowerforthefuturescontract.

• Thefuturescontracthasnoup-frontcash-ﬂow,whereastheoptioncontracthasaninitial premiumtobepaid.

• Thereisnomaturity-timeregretwiththeoption,becauseiftheoutcomeisundesirable,the optionneednotbeexercised.Withthefuturescontractthereisapossibledownside.

13. Question: Whatderivativesstrategymightyouimplementifyouexpectedabullishtrendin stockprices?Wouldyourstrategybediﬀerentifyoualsoforecastthatthevolatilityofstockprices willdrop?

Answer: Ifyouexpectpricestorise,thereareseveraldifferentstrategiesyoucouldfollow:you couldgolongaforwardandlockinapricetodayforthefuturepurchase;youcouldbuyacall whichgivesyoutherighttobuythestockatafixedstrikeprice;oryoucouldsellaputtoday, receiveapremium,andkeepthepremiumasyourprofitifpricestrendupwardasyouexpect.

Thevolatilityissueisabittrickier.AsweexplaininChapter7,bothcallandputoptions increase invaluewithvolatility,soifyouexpectvolatilitytodecrease,youdonotwantto buy acall:when volatilitydrops,whatyouhaveboughtautomaticallybecomeslessvaluable.

14. Question: Whataretheunderlyingsinthefollowingderivativecontracts?

(a)Alifeinsurancecontract.

(b)Ahomemortgage.

(c)Employeestockoptions.

(d)Aratelockinahomeloan.

Answer: Theunderlyingsareasfollows:

(a)Alifeinsurancecontract:theeventofone’sdemise.

(b)Ahomemortgage:mortgageinterestrate.

(c)Employeestockoptions:equitypriceoftheﬁrm.

(d)Aratelockinahomeloan:mortgageinterestrate.

15. Question: Assumeyouhaveaportfoliothatcontainsstocksthattrackthemarketindex.You nowwanttochangethisportfoliotobe20%incommoditiesandonly80%inthemarketindex. Howwouldyouusederivativestoimplementyourstrategy?

Answer: Onewouldusefuturestodoso.Wewouldshortmarketindexfuturesfor20%ofthe portfolio’svalue,andgolong20%incommodityfutures.Acollectionofcommodityfuturesadding uptothe20%wouldberequired.

16. Question: Inthepreviousquestion,howdoyouimplementthesametradingideawithoutusing futurescontracts?

Answer: Futurescontractsaretradedonexchangesandareknownas“exchange-traded”securities.Analternativeapproachtoachievingthegoalwouldbetouseanover-the-counterorOTC product,forexample,anindexswapthatexchangesthereturnonthemarketindexforthereturn onabroadlydeﬁnedcommodityindex.

17. Question: YoubuyafuturescontractontheS&P500.IsthecorrelationwiththeS&P500 indexpositiveornegative?Ifthenominalvalueofthecontractis$100,000andyouarerequiredto post$10,000asmargin,howmuchleveragedoyouhave?

Answer: Thefuturescontractispositivelycorrelatedwiththestockindex.Theleverageis10 times.Thatis,forevery$1investedinmargin,yougetaccessto$10inexposure.

Chapter2.FuturesMarkets

1. Question: Whatare“deliveryoptions”inafuturescontract?Generally,whyaredeliveryoptions providedtotheshortbutnottothelongposition?

Answer: Atsettlementofafuturescontract,thecontractcallsforthebuyertopaytheseller thedeliverypriceinexchangeforthesellerdeliveringtothebuyerthespecifiedgradeofthe underlying.Deliveryoptionsallowtheshortpositiontosubstituteanalternativegradeorquality forthestandardquality,atanadjustmentinthedeliveryprice.Thefuturescontractspecification liststhealternativedeliverablegradesanddescribeshowthepricewillbeadjustedforeachgrade. Deliveryoptionsareprovidedbecausespecifyingthedeliverablegradenarrowlyinacommodity futurescontractmaylimitoverallsupplyandfacilitatemarketcornersorsqueezes.Cornersand squeezesaremarketmanipulationattemptsinwhichthemanipulatortakesonmorelongpositions inagivenfuturescontractthantheshortpositionhasabilitytomakedelivery.Thisisachieved bythelongeithercontrollingalloftheavailablespotsupply(a“corner”)oratleastasufficient quantitysothattheshortpositionhasdifficultyfindingadequatedeliverablesupply(a“squeeze”). Inasuccessfulattempt,thepriceofthecommodityisdrivenupbythelackofsupply.Theshort positionmustbuytherequiredquantityfordeliveryatahighpriceandtosellitbacktothe longpositionatthefixedpriceagreedtointhecontract(orequivalentlymustcompensatethelong positionforthedifferenceinprices).Theprovisionofdeliveryoptionsreducestheopportunityfor suchbehaviorbythelongposition.Forexactlythesamereason,deliveryoptionsareprovidedonly totheshortpositionandnotthelong.

2. Question: Howdodeliveryoptionsaﬀecttherelationshipoffuturespricestoforwardprices?

Answer: Thedeliveryoptionisanoptionavailableonlytotheshortposition.Theproﬁtopportunitypresentedbydeliveryoptionstotheshortpositioncomesattheexpenseofthelongposition. Otherthingsbeingequal,thepresenceofdeliveryoptionsmeansthatthefuturespricewillbe lowerthantheforwardpriceforacontractwrittenonthestandardgrade.Thepresenceofdelivery optionsmakesthefuturescontractmoreattractivetotheshort(whocannotlosefromhavingthis extraoption),butlessattractivetothelong.Withfewerbuyers(longpositions)andmoresellers (shortpositions),thefuturespricewillbelowerthantheforwardprice.

3. Question: Towhatdothefollowingtermsrefer: initialmargin,maintenencemargin,and variationmargin?

Answer: Aninvestoropeningafuturesaccountisrequiredtodepositaspeciﬁedamountofcash intoanaccountcalledthemarginaccount.Theamountdepositedinitiallyiscalledthe initial margin.

Attheendofeachday,thebalanceinthemarginaccountisadjustedtoreﬂecttheinvestorgains andlossesfromfuturespricemovementsovertheday.Thisprocessiscalledmarking-to-market. Thechangestothemarginaccountarecalled variationmargin

Acriticalminimumbalanceamount,calledthe maintenancemargin isspeciﬁedforthemargin account.Ifthebalanceinthemarginaccountfallsbelowthislevel,thentheinvestorreceivesa margincallrequiringtheaccounttobetoppedupbacktotheleveloftheinitialmargin;ifthe top-updoesnotoccur,theaccountisclosedout.

4. Question: Whatarepriceticks?

Answer: Exchangestypicallyplacerestrictionsontheminimumamountbywhichpricesmay changeineitherdirection.Theseareknownaspriceticksandvaryfromcontracttocontract.In theUS,thepricetickisusuallyaround$10-$25percontract.

5. Question: Explainpricelimitsandwhytheyexist.

Answer: Exchangesoftenplacerestrictionsonthemaximumamountbywhichpricesmayﬂuctuateduringatradingsession.Theseareknownaspricelimits.Pricelimitsexhibitconsiderable variationacrosscontractsrangingfromaround$1,000percontractinsomecasestoover$10,000 percontractinothers.Inyetothercases,pricelimitsmaynotexistatall,ormaynotbehard limits,operating,instead,inaﬂexiblemanner.

PricelimitsexistforthesamereasonascircuitbreakersontheNYSE.Theyareaimedatpreventing panicsinthemarket,andareafunctionoftheusuallevelsofvolatilityfortheassetunderlyingthe contract.

6. Question: Whatarepositionlimitsinfuturesmarkets?Whydoweneedthese?Aretheyeﬀective fortheobjectiveyoustate,orcanyouthinkofbetterwaystoachievetheobjective?

Answer: Toreducethepossibilityofmarketdisruptionandtopreventanysingletraderfrom excercisingundueinﬂuenceonprices,exchangesandregulatorslimitthemaximumnumberof speculativepositionsaninvestormayhold.Thesearecalledpositionlimits.Positionlimitsvary fromcontracttocontract.Animportantdeterminantofthelimitsisthelikelyphysicalsupplyof theunderlyingcommodity.Mostcommodityfuturescontractsinvolvepositionlimits,whereasin marketswheresupplyisnotaconstraint(e.g.,Treasuryorcurrencyfutures),thereareoftenno positionlimits.Also,investorswhoqualifyasbonaﬁdehedgersdonotnormallyfacepositionlimits, thoughinpracticethismaymeanthattheyareallowedmuchhigherlimitsthanspeculators.

Positionlimitsinsinglecontractsmaynotbeeﬀectiveincontrollingoverallcounterpartyrisk,since theydonotaccountfortheotheraspectsofthetrader’sportfoliowhichmaycontainrisk-enhancing ormitigatingpositions,dependingonthecorrelationofthevariouscomponentsoftheportfolio. OthermeasuressuchasValue-at-Risk(tobestudiedinChapter20)maybemoreusefulasthey reﬂectthecorrecteconomicriskincontractsareusuallybetterwaysofmanagingtotalrisk.

7. Question: Whatarethediﬀerentwaysinwhichfuturescontractsmaybesettled?Explainwhy theseexist.

Answer: Futurescontractsmaybesettled(a)bydelivery,(b)incash,and(c)byexchangeof physicals.

Themostcommonisphysicaldelivery,wheretheshortpositionactuallydeliverstheassettothe long.Forcontractssettledbyphysicaldelivery,deliverymaybeeﬀectedonanydayduringthe deliverymonth.

Forsomeﬁnancialfuturescontracts,especiallywherephysicaldeliveryisaproblem(stockindex futuresareanexample),cashsettlementisused.Incashsettlement,thetwosidessimplysettle thechangeincontractvalueincashterms.Foranumericalexampleofhowcashsettlementworks, consideraCOMEXgoldfuturescontract.(Atthetimeofwriting,COMEXgoldfuturesare physicallysettled;weassumecashsettlementonlytoillustratethecomputations.)Supposeyou enterintoalonggoldfuturescontractatafuturespriceof$930perozandthefuturespriceat maturityis$964peroz.Then,bybuyingat$930,youhavegainedanamountof$(964-930)= $34peroz.Ifthecontractiscashsettled,youwillreceive$34perozincashfromtheshortfutures position.Cashsettlementtakesplacethroughthemarginaccount.

Finally,athirdmethodcalledexchangeofphysicalsisalsopossible.Thisiswherelongandshort positionswithequalpositionsizesnegotiatepriceanddeliverytermsoﬀ-exchange,andcommunicate thedetailstotheexchange.TherearesomerestrictionsonhowEFPsmaytakeplace;inparticular, theymustinvolvephysicaldelivery.

8. Question: Whatismeantby openinterest?

Answer: Theopeninterestmeasuresthenumberoffuturespositionsthathavenotyetbeen reversed.

9. Question: Discusstheliquidityandmaturityoffuturescontracts.

Answer: Liquidityis,ingeneral,theeaseofgettinginandoutofacontract.Ingeneral,most futurescontractsarehighlyliquidatshortmaturitiesbutliquiditydriesupasmaturityincreases. Onemeasureofliquidityforfuturescontracts(butnotaninfallibleone)isthesizeofopeninterest inthatcontract;ahighopeninterestindicatesalargenumberofparticipantsandsoarelatively liquidcontract.

10. Question: DescribethestandardbondintheTreasuryBondfuturescontractontheCBoTand thedeliveryoptionregardingcoupons.

Answer: ThestandardbondintheTreasurybondfuturescontractisonewithafacevalueof $100,000,atleast15yearstomaturityorﬁrstcall,andacouponof6%.

Ofthedeliveryoptionsprovidedinthecontract,themostimportantisthe“qualityoption”that allowstheshortpositiontosubstituteanycouponforthestandard6%.Thepricethatthelong positionhastopayisthequotedfuturespricetimesaconversionfactorwhichdependsonthebond thatisactuallydelivered.Theconversionfactoriscalculatedbydiscountingthecashﬂowsfrom thedeliveredbondatthestandard6%rate.Ifthedeliveredbond(a)hasacouponequaltothe standard6%,theconversionfactorwillbeequaltoone,sincewearethendiscounting6%cashﬂows ata6%rate;(b)hasacouponhigherthanthestandard6%,theconversionfactorwillbegreater thanone;(c)hasacouponlessthanthestandard6%,theconversionfactorwillbelessthanone. SeeSection2.4andChapter6fordetailsonhowtheconversionfactorisconstructed.

11. Question: SupposethedeliveredbondintheTreasuryBondfuturescontracthasaremaining maturityof20yearsanda7%coupon.Assumethelastcouponwasjustpaid.Whatisitsconversion factor?

Answer: Theconversionfactoris

12. Question: SupposetherearetwodeliverablebondsintheTreasuryBondfuturescontract,a15year8%couponbondanda22-year8%couponbond.Assumethelastcoupononbothbondswas justpaid.Whichbondhasthehigherconversionfactor?(Guesstheanswerﬁrstandthenverifyit bycomputation.)

Answer: Exerciseforthereader.(Someexercisesolutionsarelefttothereaderintentionally.It fostersreadingthetextinsomedetail.)

13. Question: Whatismeantbythedeliverygradeinacommodityfuturescontract?Whatisthe problemwithdeﬁningthedeliverygradetoonarrowly?

Answer: Sincethereareusuallydifferentqualitiesofthesamecommodity(e.g.,differentpurities ofgold),thedeliverygradeisspecifiedinthecontracttoensurethatthebuyerdoesnotreceive substandardqualityofthecommodityondelivery.Butbyspecifyingthedeliverablegradetoo narrowly,cornersand/orsqueezesbythebuyerarefacilitated;thesearesituationsinwhichthe longpositioninthefuturescontractalsocontrolsallormostofthedeliverablegradeofthespot supplyofthecommodity).

14. Question: Identifythemaininstitutionaldiﬀerencesbetweenfuturescontractsandforwardcontracts.

Answer: Forwardsareover-the-counter(i.e.,bilateral)contracts,whilefuturesareexchangetraded.Theinterpositionoftheexchangebetweenbuyerandsellercreatessomediﬀerencesin thecontracts,asthefollowingtablesummarizes:

FuturesForwards

Exchange-tradedOver-the-counter CanreverseunilaterallyCanunwindwithoriginalcounterparty, butnotunilaterallyreverse(e.g.,by oﬀsettingwithathirdparty) DefaultrisklimitedtoexchangeCounterpartydefaultriskexists MarginpaymentsrequiredNomarginpayments,but collateralpostingmayberequired StandardizedcontractsNotstandardized;fullycustomizable LargenumbersofparticipantsParticipationgenerallylimitedto institutionsandwealthyinvestors

15. Question: Explaintheterm“deliveryoptions.”Whatistherationaleforprovidingdelivery optionstotheshortpositioninfuturescontracts?Whatdisadvantagesforhedgingarecreatedby thepresenceofdeliveryoptions?Forvaluation?

Answer: Deliveryoptionsarechoicesprovidedtotheshortpositioninfuturesmarkets.They allowtheshortpositiontosubstituteadiﬀerentgradeforthestandardizedgrade.(Thepricetobe paidbythelongpositionupondeliveryisadjustedtoreﬂectthissubstitution.)

Themainadvantagesofdeliveryoptionsarethattheymakecornersandsqueezesbythelong positionmorediﬃcult,and,therefore,increasemarketintegrity.

Fromahedgingstandpoint,deliveryoptionsdegradethequalityofhedgeprovided:thelongposition cannotbesureofthequalityofcommodityitwillreceive.

Deliveryoptionscomplicatevaluation.Thefuturespricewilldependonnotjustthestandard grade,butonthe“cheapest-to-deliver”grade(i.e.,onthegradethattheshortpositionwillﬁnd mostproﬁtabletodeliver,giventheprice-adjustmentsforeachdeliverablegradeandthespotprices ofeachgrade).

16. Question: Whatisthe“closingout”ofapositioninfuturesmarkets?Whyisclosingoutof contractspermittedinfuturesmarkets?Whyisunilateraltransferorsaleofthecontracttypically notallowedinforwardmarkets?

Answer: Tocloseoutapositioninafuturesmarket,aninvestormusttakeanoﬀsettingopposite positioninthesamecontract.(Forexample,tocloseoutalongpositionin10S&P500index futurescontractswithexpiryinMarch,aninvestormusttakeashortpositionin10S&P500index futurescontractswithexpiryinMarch.)Onceapositionisclosedout,theinvestornolongerhas anyobligationsremaining.

Creditrisk iskeytoallowinginvestorstocloseoutcontracts.Inafuturesexchange,theexchange interposesitselfbetweenbuyerandsellerastheguarantorofalltrades;thus,thereislittlecredit riskinvolved.Inforwardmarkets,allowinginvestorstounilaterallytransfertheirobligationscould exacerbatecreditrisk,soitistypicallydisallowed.

Anobligationunderaforwardcontractmaybeeliminatedinoneoftwoways:(a)thecontract maybeunwoundwiththesamecounterpartyor(b)thecontractmaybeoﬀsetbyenteringintoan equalandoppositecontractwithathirdparty.Thelatteristheanalogoftheunilateralclose-out offuturescontracts.However,whileclose-outofthefuturescontractleavestheinvestorwithno netobligations,oﬀsetofaforwardcontractleavestheinvestorwithobligationson both contracts.

17. Question: Aninvestorentersintoalongpositionin10silverfuturescontractsatafuturesprice of$4.52/ozandclosesoutthepositionatapriceof$4.46/oz.Ifonesilverfuturescontractisfor 5,000ounces,whataretheinvestor’sgainsorlosses?

Answer: Eﬀectively,theinvestorbuysat$4.52perozandsellsat$4.46peroz,sotakesalossof $0.06peroz.Percontract,thisamountstoalossof(5000 × 0.06)=$300.Over10contracts,this resultsinatotallossof$3,000.00.

18. Question: Whatisthesettlementprice?Theopeningandclosingprice?

Answer: Theopeningpriceforafuturescontractisthepriceatwhichthecontractistraded atthebeginingofatradingsession.Theclosingpriceisthelastpriceatwhichthecontractis tradedatthecloseofatradingsession.Thesettlementpriceisapricechosenbytheexchangeas arepresentativepricefromthepricesattheendofasession.Thesettlementpriceistheoﬃcial closingpriceoftheexchange;itisthepriceusedtosettlegainsandlossesfromfuturestradingand toinvoicedeliveries.

19. Question: Aninvestorentersintoashortfuturespositionin10contractsingoldatafutures priceof$276.50peroz.Thesizeofonefuturescontractis100oz.Theinitialmarginpercontract is$1,500,andthemaintenancemarginis$1,100.

(a)Whatistheinitialsizeofthemarginaccount?

(b)Supposethefuturessettlementpriceontheﬁrstdayis$278.00peroz.Whatisthenew balanceinthemarginaccount?Doesamargincalloccur?Ifso,assumethattheaccountis toppedbacktoitsoriginallevel.

(c)Thefuturessettlementpriceontheseconddayis$281.00peroz.Whatisthenewbalancein themarginaccount?Doesamargincalloccur?Ifso,assumethattheaccountistoppedback toitsoriginallevel.

(d)Onthethirdday,theinvestorclosesouttheshortpositionatafuturespriceof$276.00.What istheﬁnalbalanceinhismarginaccount?

(e)Ignoringinterestcosts,whatarehistotalgainsorlosses?

Answer: Futuresposition:short10contracts

Sizeofonecontract:100oz

Initialmarginpercontract:$1,500

Maintenancemarginpercontract:$1,100

Initialfuturesprice:$276.50peroz

(a)Initialsizeofmarginaccount=1, 500 × 10=15, 000.

(b)Ifthesettlementpriceis$278peroz,theshortpositionhaseﬀectively lost $1.50peroz.This isalossof1.50 × 100=150percontract.Sincethepositionhas10contracts,theoverallloss is150 × 10=1, 500.Thus,thenewbalanceinthemarginaccountis15, 000 1, 500=13, 500. Amargincalldoesnotoccursincethisnewbalanceislargerthanthemaintenancemarginof $11,000.

(c)Whenthesettlementpricemovesto$281peroz,theshortpositioneﬀectivelylosesanother $3peroz.Thelosspercontractis3 × 100=300,sotheoveralllossis300 × 10=3, 000. Thus,thebalanceinthemarginaccountisreducedto13, 500 3, 000=10, 500.Sincethisis lessthanthemaintenancemargin,amargincalloccurs.Assumetheaccountistoppedback to$15,000.

(d)Whenthepositionisclosedoutat$276peroz,theshortpositionmakesagainof281 276=5 peroz.Thistranslatestoagainof500percontract,and,therefore,toanoverallgainof5,000. Thus,theclosingbalanceinthemarginaccountis15, 000+5, 000=20, 000.

(e)Theinvestorbeganwithamarginaccountof$15,000,anddepositedanother$4,500tomeet themargincall,foratotaloutlayof$19,500.Sincethemarginaccountbalanceattimeof closeoutis$20,000,hisoverallgain(ignoringinterestcosts)is$500.

20. Question: Thecurrentpriceofgoldis$642pertroyounce.Assumethatyouinitiatealong positionin10COMEXgoldfuturescontractsatthispriceon7-July-2006.Theinitialmarginis5% oftheinitialpriceofthefutures,andthemaintenancemarginis3%oftheinitialprice.Assumethe followingevolutionofgoldpricesoverthenextﬁvedays,andcomputethemarginaccountassuming thatyoumeetallmargincalls.

DatePriceperOunce

7-Jul-06642

8-Jul-06640

9-Jul-06635

10-Jul-06632

11-Jul-06620

12-Jul-06625

Answer: Theinitialmarginis$321,andthemaintenancemarginis$193.Thefollowingisthe evolutionofthemarginaccount.Notethatthereisonemargincallthattakesplaceon11-July-2006.

InitiationPrice=642

InitialMargin(5%)=321

MaintenanceMargin(3%)=192.6

Numberofcontracts=10

MarginAccount

OpeningDailyProﬁtAdjustedMarginCallClosing DateGoldPriceBalanceandLossBalanceDepositBalance

7-Jul-06642

8-Jul-06640321-203010301 9-Jul-06635301-502510251 10-Jul-06632251-302210221 11-Jul-06620221-120101220321 12-Jul-06625321503710371

21. Question: Whenisafuturesmarketin“backwardation”?Whenisitin“contango”?

Answer: Afuturesmarketissaidtobein backwardation ifthefuturespriceislessthanthespot price.Itisin contango iffuturespriceisabovespot.

22. Question: SupposetherearethreedeliverablebondsinaTreasuryBondfuturescontractwhose currentcashprices(forafacevalueof$100,000)andconversionfactorsareasfollows:

(a)Bond1:Price$98,750.Conversionfactor0.9814.

(b)Bond2:Price$102,575.Conversionfactor1.018.

(c)Bond3:Price$101,150.Conversionfactor1.004.

Thefuturespriceis$100,625.Whichbondiscurrentlythecheapest-to-deliver?

Answer: Sincethelongpositionwillpaythefuturespriceof100,625timestheconversionfactor insettlement,theshortpositionpreferstodeliverthebondonwhichtheratioofthesalepriceto thepurchasepriceishighest.Essentially,thismeansthebonddeliveredischeapestrelativetothe saleprice.Wecomputethisratioforallthreebondsasfollows:

100, 625 × 0 9814

98, 750 =1.0000

100, 625 × 1 018

102, 575 =0 99865

100, 625 × 1.004

101, 150 =0 99879

Hence,theﬁrstbondisthecheapesttodeliver.

23. Question: Youenterintoashortcrudeoilfuturescontractat$43perbarrel.Theinitialmargin is$3,375andthemaintenencemarginis$2,500.Onecontractisfor1,000barrelsofoil.Byhow muchdooilpriceshavetochangebeforeyoureceiveamargincall?

Answer: Ifthemarginaccountfallstoavalueof$2500thenacallwilloccur.Therefore,the lossonthepositionmustbeequalto$3375-$2500=$875foramargincall.Solvingthefollowing equation

1000(P 43)=875

gives P =43 875,whichisthepriceatwhichamargincallwilltakeplace.

24. Question: YoutakealongfuturescontractontheS&P500whenthefuturespriceis1,107.40, andcloseitoutthreedayslateratafuturespriceof1,131.75.Onefuturescontractisfor250× the index.Ignoringinterest,whatareyourlosses/gains?

Answer: Thegainis

250(1131 75 1107 40)=$6087 50

25. Question: Aninvestorentersinto10shortfuturescontractontheDowJonesIndexatafutures priceof10,106.Eachcontractisfor10× theindex.Theinvestorclosesoutfivecontractswhenthe futurespriceis10,201,andtheremainingfivewhenitis10,074.Ignoringinterestonthemargin account,whataretheinvestor’snetprofitsorlosses?

Answer: Exerciseforthereader.

26. Question: Abakeryentersinto50longwheatfuturescontractsontheCBoTatafuturespriceof $3.52/bushel.Itclosesoutthecontractsatmaturity.Thespotpriceatthistimeis$3.59/bushel. Ignoringinterest,whatarethebakery’sgainsorlossesfromitsfuturesposition?

Answer: EachCBoTWheatcontractisfor50,000bushelsandsothesettlementgainis

50 × 50, 000 × (3.59 3.52)=$175, 000

27. Question: Anoilreﬁningcompanyentersinto1,000longone-monthcrudeoilfuturescontracts onNYMEXatafuturespriceof$43perbarrel.Atmaturityofthecontract,thecompanyrolls halfofitspositionforwardintonewone-monthfuturesandclosestheremaininghalf.Atthispoint, thespotpriceofoilis$44perbarrel,andthenewone-monthfuturespriceis$43.50perbarrel.At maturityofthissecondcontract,thecompanyclosesoutitsremainingposition.Assumethespot priceatthispointis$46perbarrel.Ignoringinterest,whatarethecompany’sgainsorlossesfrom itsfuturespositions?

Answer: Exerciseforthereader.

28. Question: Deﬁnethefollowingtermsinthecontextoffuturesmarkets:marketorders,limit orders,spreadorders,one-cancels-the-otherorders.

Answer: Seesection2.3ofthebook.

29. Question: Distinguishbetweenmarket-if-touchedordersandstoporders.

Answer: Seesection2.3ofthebook.

30. Question: Youhaveacommitmenttosupply10,000ozofgoldtoacustomerinthreemonths’ timeatsomespeciﬁedpriceandareconsideringhedgingthepriceriskthatyouface.Ineachofthe followingscenarios,describethekindoforder(market,limit,etc.)thatyouwoulduse.

(a)Youarecertainyouwishtohedgeandwanttotakeupafuturespositionregardlessofthe price.

(b)Goldfuturespriceshavebeenonanupwardtrendinrecentdaysandyouarenotsureyou wanttoenterthemarketrightnow.However,ifthetrendcontinues,youareafraidyouwill belockedintotoohighaprice.Weighingtheprosandcons,youdecideyouwanttotakea futurespositionifthepricecontinuestotrendupandcrosses$370peroz.

(c)Considerthesamescenarioasin30(b),butnowsupposealsothatyouexpectanewsannouncementthatyouthinkwilldrivegoldpricessharplylower.Ifmattersturnoutasyou anticipate,youwanttoenterintoafuturespositionatafuturespriceof$350/ozorlower. However,yourecognizethereisaprobabilitythenewsannouncementmaybeadverseand goldpricesmaycontinuetotrendup.Inthiscase,youwanttobuyfuturesandexitifprices touch$370/oz.

(d)Youwanttoinstituteahedgeonlyifyoucanobtainagoldfuturespriceof$365/ozorless.

(e)Goldfuturespriceshavebeenonadownwardtrendinthelastfewdays.Youarehopingthis continuesbutdon’tanticipatepriceswillfalltoomuchbelow$362/oz,soyouarewillingto takethebestpriceyoucangetoncepricesareat$364/oz.

Answer: Therespectiveanswersare:

(a)Marketorder.

(b)Waittillgoldreaches$370andthenplaceamarketorder.

(c)Buyfutures,placeaselllimitorderat$370andabuylimitorderat$350.Ifthebuylimit orderishit,thencloseoutthefuturesposition.

(d)Placeabuylimitorderat$365.

(e)Placeabuylimitorderat$364.

31. Question: Thespotpriceofoilis$75abarrel.Thevolatilityofoilpricesisextremelyhighat present.Youthinkyoucantakeadvantageofthisbyplacingalimitordertobuyfuturesat$70 andalimitordertosellfuturesat$80perbarrel.Explainwhenthisstrategywillworkandwhen itwillnot.

Answer: Thisstrategyhasthedeceptiveattractionofbeingabuy-lowandsell-highplay.Of course,forittosucceed,bothlimitsmustbehitsothatineﬀect,youwillhaveboughtlowand soldhigh.Ifthisdoesnotoccurthentheremightbelosses.Forexample,ifthebuyorderishitat $70astheoilpricefallsandthenkeepsfalling,andneverreturnsto$80(atleastbeforetheorder orthecontractexpire),thenoneeventuallysustainsaloss.Ontheotherhand,ifthepricehits$80 andkeepsonrising,thenagainalosswillbesustained.Thus,unlessthereishighvolatilityanda reversalofdirection,thisapproachmaynotbeproﬁtableandmightturnouttobeloss-making.

32. Question: ThespreadbetweenMayandSeptemberwheatfuturesiscurrently$0.06perbushel. Youexpectthisspreadtowidentoatleast$0.10perbushel.Howwouldyouuseaspreadorderto betonyourview?

Answer: IfthepricediﬀerentialbetweenSeptemberandMayfuturesiscurrently$0.06andis expectedtowidento$0.10,thenweshouldenterintoalongpositionintheSeptembercontract andashortpositionintheMaycontract.Whenthespreadswidenswecloseoutbothcontracts.

33. Question: Thespreadbetweenone-monthandthree-monthcrudeoilfuturesis$3perbarrel. Youexpectthisspreadtonarrowsharply.Explainhowyouwoulduseaspreadordergiventhis outlook.

Answer: Assumingthethree-monthminusone-monthspreadwillnarrow,weshouldgolongthe one-monthcontractandshortthethree-monthcontract.Whenthespreadnarrows,webuyback theshortthree-monthcontractandsellbackthelongone-monthcontract.Wecapture(ignoring interest)thediﬀerencebetween$3andthenewspread.

34. Question: Supposeyouanticipateaneedforcorninthreemonths’timeandareusingcornfutures tohedgethepriceriskthatyouface.Howisthevalueofyourpositionaﬀectedbyastrengthening ofthebasisatmaturity?

Answer: Thebasisisthefuturespriceminusthespotprice.Astrengtheningofthebasisoccursif thebasisincreases.Ifthisoccurs,thepositioninthequestionispositivelyaffectedsinceyouarelong futures.Innotationalterms,yougolongfuturestoday(atprice F0,say)andcloseitoutat T (at price FT ,say)foranetcashflowonthefuturespositionof FT F0.Inaddition,youbuythecornyou needatthetime-T spotprice ST ,leadingtoatotalnetcashflowof(FT F0) ST =(FT ST ) F0. Astrengtheningofthebasis FT ST atmaturityimprovesthiscashflow.

35. Question: A shorthedger isonewhoisshortfuturesinordertohedgeaspotcashﬂowrisk.A longhedger issimilarlyonewhogoeslongfuturestohedgeanexistingrisk.Howdoesaweakening ofthebasisaﬀectthepositionsofshortandlonghedgers?

Answer: Theshorthedgerisshortfuturesandlongspot,sogainsifthebasisweakens.Thelong hedgerislongfuturesandshortspot,solosesinthiscase.

36. Question: Supposeyoudeliveragradeotherthanthecheapest-to-delivergradeonafutures contract.Wouldtheamountyoureceive(theconversionfactortimesthefuturesprice)exceed, equal,orfallshortofthespotpriceofthegradeyoudeliver?

Answer: Bydefinition,thecheapest-to-delivergradeistheoneforwhichthedifferencebetween thepricereceived(thefuturespricetimestheconversionfactor)andthespotpriceisgreatest. Sincethefuturespricereflectsthecheapest-to-delivergrade,deliveringthatgradeisazero-profit situation.Consequently,deliveringanyothergradeshouldleadtoaloss,thatis,thetheproceeds youreceivewillbelessthanthespotpriceofthatgrade.