InvestmentsLecture12FuturesandOptions:ObjectivesUnderstandusesoffuturesandoptions;Understandcharacteristicsoffuturesandoptionsinvestments,including:RiskPricingHedgingUnderstandandapplymathematicalandstatisticaltechniquesrequiredtoValuefuturesandoptions;Hedgeusingfuturesandoptions.DerivativeSecuritiesFinancialinstrumentsthatderivetheirvaluesfromothertradedclaimsarecalledderivatives.Typically,thevalueoftheseinstrumentsisverycloselyrelatedtothevalueoftheunderlyingasset.Asaresultderivativesareusefulfor:Speculatingontheunderlyingasset,and;Hedgingtheunderlyingasset.Furthermore,arbitrageopportunitiesmaybepossibleiftheunderlyingassetandthederivativeassetarenotpricedconsistently.SpeculatorsandHedgersSpeculatorsareindividualshopetomakeaprofitbyclosingouttheirpositionsatapricethatisbetterthantheinitialprice.Theydonotproduceorusetheassetintheirdailycourseofbusiness.Hedgersareindividualswhousederivativestooffsetanotherwiseriskypositionintheunderlyingasset.Theyeitherproduceorusetheassetintheirdailycourseofbusiness.Example:WheatForwardsInawheatforwardcontracttwocounter-partiesagreetoexchangesomequantityofwheatatsomedateinthefutureatapricenegotiatedtoday.Awheatfarmerhasexposuretothefuturespotpriceofwheat.Thespotpriceisthemarketpriceofwheatforimmediatedelivery.Thecropplantedinthespringandharvestedinthefallwillbesoldatfallspotprices.Sincethesespotpricesareuncertain,theprofitsonthefarmer’scroparerisky.Arisk-aversefarmercanhedgethisriskbysellingwheatnowusingaforwardcontract.WheatForwardsWhomighttaketheoppositesideofthistrade?Abreadproducermaywishtohedgeproductioncosts.Aweatherforecastermayspeculatethatthefuturespotpricewillbewellabovetheforwardpriceandthereforeusethiscontractaspartofatradingstrategy(buyusingtheforwardcontractandsellinthefuturespotmarket).ArbitrageursTwobasictypesofarbitragetrades:Investnothingandmakepositivefutureprofits;Receiveprofitstodaywithoutanyfutureobligations.Arbitrageursusederivativecontractstoextractarbitrageprofits.Theiractions,alongwithnormalsupplyanddemandforces,ensureconsistentrelationshipsamongtheunderlyingassetpricesandthederivativesecurityprices.WheatForwardsWhomightbeinapositiontoderivearbitrageprofitsfromwheatforwardcontracts?Ifyouhaveatechnologyforstoringwheatandtheforwardpriceishighrelativetotoday’sspotprice,youmaywantto:Borrowmoneytobuywheatnow,Sellitwiththeforwardcontract,Storeituntilthefall,Deliverthewheatandusetheproceedstopaybackyourlenders.Noticethatthecostofstorageandlendingrateswillplaceaboundonhowhightheforwardpricecanbe(asortofno-arbitragebound).FuturesFuturesContracts-DefinitionAfuturescontractisanagreementbetweentwopartiestobuyorsellanassetatacertaintimeinthefutureforacertainprice.Characteristicsoffuturescontracts:Tradedonanexchange;Contractsarestandardized;Clearinghouseseliminatedefaultrisk;Marginisrequired.Futures-CommonExamplesCommodityfutures:Wheat;Crudeoil;Gold;Livecattle.Financialfutures:S&P500indexfutures;T-billfutures;Futures–ContractSpecificationComponentsofcontractspecification:Asset;Contractsize;Deliveryarrangements;Cashorphysicaldelivery;Place;Time.Pricequotes;Pricemovementlimits;Positionlimits.FuturesContracts-MarginWhenyouenterintoafuturescontract,thebrokertypicallyrequirethatyoudepositfundsintoamarginaccount.Youmayormaynotearninterestonthisaccount.Atcontractinitiationyoudeposittheinitialmargin(alsocalledperformancemargin).Thisaccountis“marked-to-market”periodically.Periodicprofits,asrepresentedbychangesinthefuturesprice,arecreditedtoordebitedfromyouraccount.Margin(cont’d)Youcanwithdrawanyfundsinexcessoftheperformancemargin.Ifyourmarginaccountbalancefallsbelowthemaintenancemarginyouwillreceiveamargincall,inwhichcaseyoumustdepositadditionalfunds(thevariationmargin)tobringyourbalancebacktotheinitialmarginlevel.Ifyoudonothonouramargincall,yourpositionisclosedout.MarginExampleTimefromcontractinitiationMarginAccountBalanceInitialMaintenanceMarginExampleFuturesonindexTwocontracts,contractsize=$50Initialmargin-$1500/contractMaintenancemargin-$1000

ImportantPointsAsaninvestoryouhavemorethanyourinitialmarginatriskpriortomaturity.Previousexample,couldlooseupto1000inindextermsor(1000x$50x2)=$100,000Youcanwithdrawexcessmargin.FuturesPayoutAlthough,inpractice,thelifetimepayoutfromafuturesaccruesovertimeandaccumulatesinthemarginaccount,itisusefultothinkofthepayoutfromthecontractasbeingreceivedatthematuritydate.Itisimportanttounderstandtherelationshipbetweenthishypotheticalpayoutandthespotprice.FuturesPayoffsLetFbethefuturespriceandST

bethespotpriceatmaturity.Thepayoutfromalongpositioninafuturescontract: ST–FThepayoutfromashortpositioninafuturescontract: F–STGraphicallyRiskofaFuturesSupposethatyouarelonganS&P500E-Miniindexfutureswithonemonthtomaturity.Thecurrentindexvalueis1242.98andthefuturespriceis1250.75Theinitialmarginis$4,313andearnstherisk-freerateofinterest.Youmakenomarginpaymentsuntilmaturitywhenyousettletheposition.WhatistheBetaoftheinvestment?BetaofFuturesThereturnontheinvestment:TheBetaisE-MiniBetaInthiscase,thebetaofthecontract,withmargin,is:Yourinvestmentisalmost15timesmoreriskythananinvestmentinthemarket!OptionsOptionContractsInanoptioncontractthewritergrantsthebuyertheoption,butnottheobligation,tobuyfromortoselltothewriteraspecificassetataspecificprice(calledthestrikeorexerciseprice)withinaspecifiedperiodoftime.CommonOptionsCallOptionBuyerhastherighttobuytheassetatagivenprice(theexerciseprice)atagivendate.Writerhascommitmenttoselltheunderlyingassettotheholderattheexercisepriceifexercised.PutOptionBuyerhastherighttoselltheassetatagivenprice(theexerciseprice)atagivendate.Writerhascommitmenttobuytheunderlyingassetfortheholderattheexercisepriceifexercised.TerminologyAmericanOptionOptionthatcanbeexercisedatanytimepriortoexpirationdate.EuropeanOptionOptionthatcanbeexercisedonlyatexpirationdate.In-the-moneystockprice>exercisepriceOut-of-the-moneystockprice<exercisepriceAtthemoneystockpriceexercisepriceNotationS(orSt):Currentstockpriceattimet.K:exerciseorstrikeprice.T:timetoexpiration(maturity)C(c):valueofAmerican(European)call.P(p):valueofAmerican(European)put.ValuesofoptionsatexpirationAPayoutGraphshowsthecashflowsresultingfromapositioninanoptionasafunctionofoftheunderlyingasset’sprice.eg.CallOptions:IfST<Kthenpayofftocallowner=0IfST

Kthenpayofftocallowner=ST-KWhereST=valueofstockatexpiration,andK=StrikePriceOptionValueifExercisedKSharepriceBUYCALLKSharepriceSELLCALLExample:CallOptionCalloptiononMOT:K=$90Valueofoptionatdifferentstockprices: StockPrice $80 $90 $100 $110 $120 OptionPayoff $0 $0 $10 $20 $30OptionValueifExercisedValueoftheputoptionatexpiration: Payofftoputowner=K-STifST<K Payofftoputowner=0ifST

KSharepriceKKBUYPUTSELLPUTSharepriceOptionValueifExercisedOptionValueifExercisedExample:PutOptionPayoffsFortheownerofacalloption:Fortheownerofaputoption:ExoticOptionsAsianOptionsBarrierOptionsLookbackOptionsCurrencyTranslatedOptionsBinaryOptionsDerivativeStrategiesDerivativeStrategiesWewillexaminehowderivativesmaybeusedto:Eliminaterisk(hedge);Modifyrisk(partiallyhedge);Replicateotherpayoffsandcreatesyntheticpayoffs.Hedging:DeterminingExposureConsideranindividualwhowishestohedge.Inordertodetermineexactlyhowtohedge,wemustdeterminethesensitivityoftheirpayoffstotheunderlyinghedginginstrument.(i.e.calculatethehedgeratio.)Payoffdiagramsareusefulfordeterminingtheseratios.HedgingExample1Supposeyouareanindexportfoliomanagerwith$4,000,000investedintheS&P500stocks.YouhavechosentoeliminateanyriskintheportfoliobyusingtheE-miniS&P500contract(rememberthateachcontractprovidesapayoutequalto$50timestheS&P500level).Determinethenumberofcontractsyoushouldbuyorsellinordertoachievethehedge.Step1:DescribethePayoffsPortfolioreturn:FuturesContractPayoff:Step1:DescribePayoffsGraphically:Step2:DeterminetheHedgeRatioIfthesensitivityistobezero,weneedthepayofftobezero,regardlessoftherealizedreturnonthemarket.Inequationterms:HedgingExample2Supposeyouareanindexportfoliomanagerwith$4,000,000investedinastockportfolio.YouhavechosentoeliminatethemarketriskintheportfoliobyusingtheE-miniS&P500contract.Usingamarketmodelregressionyouhavedeterminedthattheportfoliobetais1.175.Howmanycontractsyoushouldbuyorsellinordertoachievethehedge.PayoffsPartialHedgingExampleYouneednotoffsetalltheriskintheportfolio.Infact,byusingindexfuturesyoucantuneyourportfoliotohaveanyexposureyoudesire.Example:Giventhedatainthepreviousexample,describehowyoucanusethefuturescontracttochangetheportfoliobetato0.5.GraphicallyCreatingSyntheticPayoffsDerivativescanbeusedtoreplicatepayoffs.Example1:Describehowyouwoulduseafuturescontractthatmimicsthepayofftoinvesting$50,000intheS&P500.PayoffGraphExampleDescribehowyoucanuseafuturescontractandinvestmentsint-billstocreateawelldiversifiedportfoliowithabetaof0.5.OptionsStrategiesBasicoptionpayoffscanbeaddedtogethertocreatecompoundpayoffsthathavealmostarbitrarycharacteristics.We’llexamine:Put-Callparity;Coveredstrategies;Spreads.Put-CallParitySupposeyousimultaneouslybuyacallandwriteaput,bothhavingastrikepriceKandmaturityT.SummaryofpayoffsfromthispositionatT:SothetotalpayoffstothesepositionswillbeST–K.ST<KST>KPayofffromcallpurchased0ST–KPayofffromputwritten-(K–ST)0TotalST–KST–K51AnAlternativeBorrowtodayandrepayKatmaturityBuy1shareofstockYourpayoffsfromthesepositionsattimeT: Payoff=Asset-Liability=ST–KThetwopositionsgiveyouidenticalpayoffs.Bynoarbitrage,thecostsofestablishingthesepositionsmustbeidentical52ThePut-CallParityRelationshipCosttoestablishoptionpositions:PurchasecalloptionforC0SellputoptionforP0Totalcostofestablishingposition:C0-P0Costtoestablishleveredequityposition:Costofstock:S0Borrowedfunds:Totalcostofestablishingleveredstockposition:Costsofestablishingidenticalpayoffpositionsmustbeidentical:53IfthePut-CallParitydoesnothold,thenarbitrageopportunitieswillarise.Example:Supposethatthesecuritypriceis$31,theexercisepriceis$30,therisk-freerateis10%perannum,thepriceofa3-monthEuropeancalloptionis$3,andthepriceofa3-monthEuropeanputoptionis$2.25.Dowehavemispricing?

Ifyes,couldwemakearbitrageprofits?ValueofPortfolioA:ValueofPortfolioB:PortfolioBisoverpricedrelativetoportfolioA.

Whatisthearbitragestrategy?BuysecuritiesinPortfolioAandshortthesecuritiesinPortfolioB.CoveredStrategiesProtectivePutBuyingaputonalongpositionintheunderlyi