Put-call parity

Put–callparitycanbestatedinanumberofequivalentways,mostterselyas:whereCisthe(current)valueofacall,Pisthe(current)valueofaput,Disthediscountfactor,Fistheforwardpriceoftheasset,andKisthestrikeprice.Notethatthespotpriceisgivenby(spotpriceispresentvalue,forwardpriceisfuturevalue,discountfactorrelatesthese).Theleftsidecorrespondstoaportfoliooflongacallandshortaput,whiletherightsidecorrespondstoaforwardcontract.TheassetsCandPontheleftsidearegivenincurrentvalues,whiletheassetsFandKaregiveninfuturevalues(forwardpriceofasset,andstrikepricepaidatexpiry),whichthediscountfactorDconvertstopresentvalues.UsingspotpriceSinsteadofforwardpriceFyields:Rearrangingthetermsyieldsadifferentinterpretation:Inthiscasetheleft-handsideisafiduciarycall,whichislongacallandenoughcash(orbonds)topaythestrikepriceifthecallisexercised,whiletheright-handsideisaprotectiveput,whichislongaputandtheasset,sotheassetcanbesoldforthestrikepriceifthespotisbelowstrikeatexpiry.Bothsideshavepayoffmax(S(T),K)atexpiry(i.e.,atleastthestrikeprice,orthevalueoftheassetifmore),whichgivesanotherwayofprovingorinterpretingput–callparity.Inmoredetail,thisoriginalequationcanbestatedas:whereisthevalueofthecallattime,isthevalueoftheputofthesameexpirationdate,isthespotpriceoftheunderlyingasset,isthestrikeprice,andisthepresentvalueofazero-couponbondthatmaturesto$1attimeThisisthepresentvaluefactorforK.Notethattheright-handsideoftheequationisalsothepriceofbuyingaforwardcontractonthestockwithdeliverypriceK.Thusonewaytoreadtheequationisthataportfoliothatislongacallandshortaputisthesameasbeinglongaforward.Inparticular,iftheunderlyingisnottradeablebutthereexistsforwardsonit,wecanreplacetheright-hand-sideexpressionbythepriceofaforward.Ifthebondinterestrate,,isassumedtobeconstantthenNote:referstotheforceofinterest,whichisapproximatelyequaltotheeffectiveannualrateforsmallinterestrates.However,oneshouldtakecarewiththeapproximation,especiallywithlargerratesandlargertimeperiods.Tofindexactly,use,whereistheeffectiveannualinterestrate.WhenvaluingEuropeanoptionswrittenonstockswithknowndividendsthatwillbepaidoutduringthelifeoftheoption,theformulabecomes:whereD(t)representsthetotalvalueofthedividendsfromonestocksharetobepaidoutovertheremaininglifeoftheoptions,discountedtopresentvalue.Wecanrewritetheequationas:andnotethattheright-handsideisthepriceofaforwardcontractonthestockwithdeliverypriceK,asbefore.Derivation[edit]Wewillsupposethattheputandcalloptionsareontradedstocks,buttheunderlyingcanbeanyothertradeableasset.Theabilitytobuyandselltheunderlyingiscrucialtothe"noarbitrage"argumentbelow.First,notethatundertheassumptionthattherearenoarbitrageopportunities(thepricesarearbitrage-free),twoportfoliosthatalwayshavethesamepayoffattimeTmusthavethesamevalueatanypriortime.Toprovethissupposethat,atsometimetbeforeT,oneportfoliowerecheaperthantheother.Thenonecouldpurchase(golong)thecheaperportfolioandsell(goshort)themoreexpensive.AttimeT,ouroverallportfoliowould,foranyvalueoftheshareprice,havezerovalue(alltheassetsandliabilitieshavecanceledout).Theprofitwemadeattimetisthusarisklessprofit,butthisviolatesourassumptionofnoarbitrage.Wewillderivetheput-callparityrelationbycreatingtwoportfolioswiththesamepayoffs(staticreplication)andinvokingtheaboveprinciple(rationalpricing).ConsideracalloptionandaputoptionwiththesamestrikeKforexpiryatthesamedateTonsomestockS,whichpaysnodividend.Weassumetheexistenceofabondthatpays1dollaratmaturitytimeT.Thebondpricemayberandom(likethestock)butmustequal1atmaturity.LetthepriceofSbeS(t)attimet.NowassembleaportfoliobybuyingacalloptionCandsellingaputoptionPofthesamematurityTandstrikeK.ThepayoffforthisportfolioisS(T)-K.NowassembleasecondportfoliobybuyingoneshareandborrowingKbonds.NotethepayoffofthelatterportfolioisalsoS(T)-KattimeT,sinceourshareboughtforS(t)willbeworthS(T)andtheborrowedbondswillbeworthK.Byourpreliminaryobservationthatidenticalpayoffsimplythatbothportfoliosmusthavethesamepriceatageneraltime,thefollowingrelationshipexistsbetweenthevalueofthevariousinstruments:Thusgivennoarbitrageopportunities,theaboverelationship,whichisknownasput-callparity,holds,andforanythreepricesofthecall,put,bondandstockonecancomputetheimpliedpriceofthefourth.Inthecaseofdividends,themodifiedformulacanbederivedinsimilarmannertoabove,butwiththemodificationthatoneportfolioconsistsofgoinglongacall,goingshortaput,andD(T)bondsthateachpay1dollaratmaturityT(thebondswillbeworthD(t)attimet);theotherportfolioisthesameasbefore-longoneshareofstock,shortKbondsthateachpay1dollaratT.ThedifferenceisthatattimeT,thestockisnotonlyworthS(T)buthaspaidoutD(T)individends.History[edit]Formsofput-callparityappearedinpracticeasearlyasmedievalages,andwasformallydescribedbyanumberofauthorsintheearly20thcentury.MichaelKnoll,inTheAncientRootsofModernFinancialInnovation:TheEarlyHistoryofRegulatoryArbitrage,describestheimportantrolethatput-callparityplayedindevelopingtheequityofredemption,thedefiningcharacteristicofamodernmortgage,inMedievalEngland.Inthe19thcentury,financierRussellSageusedput-callparitytocreatesyntheticloans,whichhadhigherinterestratesthantheusurylawsofthetimewouldhavenormallyallowed.[citationneeded]Nelson,anoptionarbitragetraderinNewYork,publishedabook:"TheA.B.C.ofOptionsandArbitrage"in1904thatdescribestheput-callparityindetail.Hisbookwasre-discoveredbyEspenGaarderHaugintheearly2000sandmanyreferencesfromNelson'sbookaregiveninHaug'sbook"DerivativesModelsonModels".HenryDeutschdescribestheput-callparityin1910inhisbook"ArbitrageinBullion,Coins,Bills,Stocks,SharesandOptions,2ndEdition".London:EnghamWilsonbutinlessdetailthanNelson(1904).MathematicsprofessorVinzenzBronzinalsoderivestheput-callparityin1908andusesitaspartofhisarbitrageargumenttodevelopaseriesofmathematicaloptionmodelsunderaseriesofdifferentdistributions.TheworkofprofessorBronzinwasjustrecentlyrediscoveredbyprofessorWolfgangHafnerandprofessorHeinzZimmermann.TheoriginalworkofBronzinisabookwritteninGermanandisnowtranslatedandpublishedinEnglishinaneditedworkbyHafnerandZimmermann("VinzenzBronzin'soptionpricingmodels",SpringerVerlag).Itsfirstdescriptioninthemodernacademicliteratureappearstobe(Stoll1969).[1][notincitationgiven]Implications[edit]Put–callparityimplies:Equivalenceofcallsandputs:Parityimpliesthatacallandaputcanbeusedinterchangeablyinanydelta-neutralportfolio.Ifisthecall'sdelta,thenbuyingacall,andsellingsharesofstock,isthesameassellingaputandbuyingsharesofstock.Equivalenceofcallsandputsisveryimportantwhentradingoptions.Parityofimpliedvolatility:Intheabsenceofdividendsorothercostsofcarry(suchaswhenastockisdifficulttoborroworsellshort),theimpliedvolatilityofcallsandputsmustbeidentical.[2]