Incommonwithdynamichedging,statichedgingprovidesvaluationfor-

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1 StaticHedgingofBarrierSecurities BreakingBarriers: PeterCarr MorganStanley 1585Broadway,6thoor (212) NewYork,NY10036 AndrewChou MITComputerScience Cambridge,MA TechnologySquare CurrentVersion:November7,1996 (617) tothanknickfiroozye,galingeorgiev,ciamacmoallemi,andrewsmith, Thisversionisextremelypreliminary.Anyerrorsareourown.Wewouldlike andravisundaramforfruitfuldiscussions.

2 1Introduction Exoticoptionsareverysophisticatedinstruments,andthetechniquesused tohedgeandvaluethemarefairlycomplex.themostcommontypeof exoticoptionsarebarrieroptions,whichwereintroducedinamericanoverthe-countermarketsyearsbeforevanillaoptionswerelisted(seesnyder[19]). Barrieroptionsarespecialcasesofbarriersecurities,whichmayinvolve nottouched(out-barriersecurities).theyareusuallyfurtherclassiedinto singleormultiplebarriers.examplesofthelatterincludedoublebarrier options,rolldowncalls,andevenlookbackoptions.inthispaper,wefocus \downsecurities"(barrierbelowspot)and\upsecurities"(barrierabove maturityprovidedthatthebarrierhasbeentouched(in-barriersecurities)or spot). onsinglebarriersecuritiesforsimplicity,leavingmultiplebarrieroptions forfutureresearch.singlebarriersecuritiesallowforanarbitrarypayoat BreakingBarriers:StaticHedgingofBarrierSecurities outofvanillaeuropeanoptions.thisallowsustohedgepath-dependent traditionalmanner.inparticular,weshowhowbarriersecuritiescanbe brokenupintomorefundamentalsecurities,whichinturncanbecreated pliesdynamicreplicationstrategiesintheunderlyingassets.inthispaper, wehopetoaddinsightintothesestructuresbylookingattheminanon- barriersecuritieswithpath-independentvanillaoptions,withtradinginthe Thestandardmethodologyforhedgingandvaluingbarriersecuritiesapmulasforbarriersecurities.Whenbothtypesofhedgingstrategiesarecast paper,wewillalsoshowhowthevaluationformulasbasedondynamicreplicationcanbeusedtouncoverthestatichedge.sinceformulasforbarrier optionshavebeenavailablesincetheseminalpaperofmerton[13],thesefor- Incommonwithdynamichedging,statichedgingprovidesvaluationfor- inthesameeconomicmodel,theformulasresultinidenticalvalues.inthis latteroptionsoccurringonlyattheinitiationandtheexpiration1ofthe termedstatic. hedge.duetotherelativeinfrequencyoftrading,suchhedgesarecommonly mulasarenowwidelyavailable(seenelken[15]andzhang[21]forsurveyson timeofthebarrier. 1Wedenetheexpirationofthehedgeastheearlierofmaturityandthersthitting 1

3 ofexoticoptionsbeyondthoseexplicitlypresentedinthispaper.(eg.double exoticoptions).thisresultcanbeusedtodeterminestatichedgesforahost abletypesofbarriersecurities.wegiveexplicitresultsfordownbarriers, andpartialbarrieroptions.) andexhortthereadertousepublishedformulasforupbarrieroptionsto uncoverthestatichedge.specialcasesofdownsecuritiescoveredinclude downcalls,downputs,andseveraltypesofbinaryoptions.bydenition, barrierhasbeenhit,whileanamericanone-touchbinarypaysadollarat aeuropeanone-touchbinaryputpaysonedollaratmaturityifthelower Wewillillustrateourdecompositionresultswithseveralcommonlyavail- atmaturityifthebarrierisneverreached.weindicatethestatichedgefor thersthittingtime,ifany.bycontrast,ano-touchbinarypaysonedollar allthreetypesofbinaries. strategiesbothworkperfectlywellintheory,thereareacoupleofreasonsto believethatstatichedgingmaybethemethodofchoiceinpractice.first, whichwouldgenerateruinoustransactionscostsifimplementedinpractice. aliteralinterpretationofdynamicreplicationrequirescontinuoustrading, Thestandardkludgeforthisproblemistotradeperiodically,whichleadsto acceptablylowapproximationerrorwhenthegammaofthesecurityislow. Whilewewillemployamodelinwhichdynamicandstaticreplication However,barrieroptionsoftenhaveregionsofhighgamma,whichwhile catastrophictotheseperiodicallyrebalancedstrategies,isofnoconsequence replicationisdirectlyproportionaltotheoption'svega,whichisagainoften asset.theerrorarisingfromusingthewrongvolatilityrateindynamic rstpassagetimetothebarrier.jumpsacrossthebarriercaninducesub estimationofthefuturecarryingcostsandvolatilityrateoftheunderlying orsuper-replicationforbothtypesofstrategies,withthesuperiorstrategy usuallybeingidentiableinadvance.second,dynamicreplicationrequires whatsoevertothestatichedger,solongastheinvestorcantradeatthe claims,thesameresultsapplytothem.2 independentclaim.sincethispapershowsthatbarriersecuritiesmaybeviewedassuch hedgeandwhentheunderlyingisatthebarrier.therealizedvolatility duringthelifeofthehedgeisofnoconsequencetothestatichedger,except knowingtheimpliedvolatilityofthevanillaoptionsattheentryofthestatic high2forbarriersecurities.bycontrast,staticreplicationreliesonlyon 2Carr[5]showsthathighgammasimplyhighvegasforanyEuropean-stylepath-

4 partiallyreturnedifthehedgeisliquidatedatthebarrier.thus,themain disadvantageofstatichedgingoverdynamichedginginpracticeappearsto betherelativeilliquidityofthestandardoptionsmarketwhencomparedto premiumispaidattheinitiationofthestatichedge,itshouldatleastbe volatility,whichistypicallygreaterthanhistoricalvolatility.however,ifthis totheextentthatitimpactsimpliedvolatility. themarketfortheunderlyingasset.perhaps,thispaperwillhelpmitigate Theabovebenetsofstatichedgingmaywellbeembeddedintheimplied thisdisadvantage.inanycase,empiricalworkisneededtocomparethe relativeviabilityoftheseapproaches. tracedbacktoworkbyross(see[18]),whoshowedhowtheavailabil- Madan[7]extendthislineofresearchtomultipleunderlyingassets.Ho[12] ityofvanillaoptionsinallstrikesmaybeusedtostaticlyreplicatepath- independentpayos.breedenandlitzenberger[4]showedthatbuttery ArrowDebreusecurities.GreenandJarrow[11],Nachman[14],andCarrand spreadscanbeusedtoobservethepricesoffundamentalderivativescalled Likemostworkonderivatives,theliteratureonstatichedgingmaybe options.derman,ergener,andkani[9],[10]relaxthedriftrestrictionin theforegoingpapersbyintroducinganalgorithmforhedgingsinglebarrier zero.carr,ellis,andgupta[6]extendtheseresultstoasymmetricvolatility structureandmorecomplexinstruments,suchasdoubleandpartialbarrier usingtheblack[1]modelwhichrestrictsthedriftoftheunderlyingtobe andthispaperextendthislineofresearchtopath-dependentsecurities. optionsinabinomialmodel,usingoptionswithasinglestrikebutmultiple BowieandCarr[3]introducestatichedgingforsinglebarrieroptions, expiries.bycontrast,thispaperprovidesexplicitformulasforstatichedges inthestandardblackscholes[2]modelusingoptionswiththesameexpiry butmultiplestrikes. 2StaticReplicationwithBarrierOptions initiallyassumeonlythatmarketsarefrictionlessandarbitrage-free.to ever,werstdevelopasetofresultsinamoregeneralsetting.thus,we 2.1Assumptions Asstated,wewilleventuallyusethestandardBlackScholes[2]model.How- 3

5 stantdividendyield,d.theresultsinthissectioneasilyextendtostochastic constantrisklessraterandthattheunderlyingassetisastock,withacon- simplifynotation,wealsoassumethatinvestorscanborroworlendata treattheavailabilityofacontinuumofstrikesasanapproximationofthe interestratesanddividendyields. over-the-countermarketinbarrieroptions.whilewefullyrecognizethat security,butwithanypositivestrike.justascontinuoustradingisaccepted asareasonableapproximationtorealityeventhoughmarketsclosedaily,we requirealiquidmarketinknockinoptionswiththesametriggerasthebarrier tradeinallbarriersecurities.forthepurposesofthissection,hedgeswillonly Animplicationoffrictionlessmarketsisthatinvestorscancontinuously path-independentpayosariseasspecialcases3ofthepayosfrombarrier marketsforbarriersecuritieshavelimitedliquidityinpractice,wenotethat securities.giventheadventofexoptionsinthelistedmarketandthe liquidityassumptionistenableforthisimportantspecialcase.hopefully, emergenceofasubstantialover-the-countermarketinvanillaoptions,our tionscanbeusedtocreateafundamentalsecuritycalleda\down-and-in Inthissubsection,werstshowhowabutteryspreadofdown-and-inop- 2.2DecompositionintoFundamentalSecurities theskepticalreaderwillndtheresultsofthissectiononbarriersecurities Arrow".Thesesecuritiesarefundamentalinthesensethatanarbitrary tobeofinterest,evenwhenrestrictedtothepath-independentcase. lioofsuchsecurities.itfollowsthatthearbitrarydown-and-insecuritycan payofromadown-and-insecuritymaybeeasilydecomposedintoaportfo- bestaticlyhedgedandvaluedbyaportfolioofdown-and-inoptions.the nextsectionshowshowthedown-and-inoptionscaninturnbereplicated withvanillaoptions.thenetresultisthatabarriersecuritycanbestaticly hedgedwithaportfolioofvanillaoptions. out-barriersecurities,thebarrierismovedaninnitedistanceawayfromthespot. in-barriersecurities,thebarrierismovedtothespot.conversely,toobtainresultsfrom payointoastaticportfolioconsistingoflistedinstrumentssuchasbonds, Inparticular,wecandecomposeclaimswithanarbitrarypath-independent appliedtothepath-independentcasebymovingthebarrierappropriately. 3Toachieveresultsforpath-independentsecuritiesfromthecorrespondingresultsfor Ourdecompositionofbarriersecuritiesintodown-and-inoptionscanbe 4

6 combinedwithourresultsforin-securitiestoobtainthethecorresponding beingofintrinsicinterest,ourresultsonthepath-independentcasecanbe statichedgeandvaluationformulafor\out-securities". 2.3StaticReplicationwith\Arrows" anewdecompositionoftheclaimvalueintointrinsicandtimevalue.besides forwardcontracts,andvanillaoptions.aspecialcaseofthisformulaleadsto down-and-incallswiththesamebarrier: betthatalowerbarrierwashit,thebutteryspreadshouldbeformedfrom wherec(k)isthecurrentpriceofacallstruckatk.ifwewishtoalso butteryspreadwithvanillacalls,whichcosts: AsimplewaytobetthattheunderlyingwillnisharoundKistoforma DIBS(K;H)=DIC(K?4K;H)?2DIC(K;H)+DIC(K+4K;H); BS(K)=C(K?4K)?2C(K)+C(K+4K); barrierh.solongasthebarrierhasbeenhit,thenalpayoofthisposition isatriangleasshowninfigure1. wheredic(k;h)isthecurrentpriceofadown-and-incallstruckatkwith Thus,thisbetcanbenormalizedsothattheareaunderthetriangleisone: NDIBS(K;H)=DIC(K?4K;H)?2DIC(K;H)+DIC(K+4K;H) Notethattheareaunderthetriangleis: 1224K4K=(4K)2: heightgetstaller,sothattheareaismaintainedatone.thelimitingpayo approacheswhatisknownasadiracdeltafunction.thesecurityproviding As4Kapproacheszero,thebaseofthetriangularpayogetssmallerandthe winningfounders,kennetharrowandgerarddebreu.giventheirheritage thispayoiscalledanarrow-debreusecurity,namedaftertheirnobelprize : andtheirpayo,wewillrefertothesesecuritiesas\down-and-inarrows" (seefigure2). 5

7 $ 4K erlycalledasecondgenerationderivativesecurity.thenameisappropriate Sincethepayoofadown-and-inArrowisnon-standard,itmaybeprop- Figure1:PayoofaDown-and-InButterySpread. FinalStockPrice K K+4K secondderivativeofthedown-and-incallvaluewithrespecttoitsstrike: inanothersensesincethevalueofadown-and-inarrowisgivenbythe T,solongasthestockpricehashitthebarrierHpreviously.Similarly, LetDIB(H)bethevalueofadown-and-inbondwhichpaysonedollarat wheredia(k;h)denotesthecurrentvalueofadown-and-inarrowstruck atkwithbarrierh. WenextshowhowArrowscanalternativelybeobservedfromputvalues. (1) letdis(h)denotethevalueofadown-and-instock,whichpaysthestock priceatt,solongasthebarrierhasbeenhitbeforehand.thereisasimple 6

8 $ 6 generalizationofput-call-parityinvolvingthesecontracts: DIC(K;H)=DIS(H)?KDIB(H)+DIP(K;H): 0Figure2:PayoofaDown-and-InArrow. K Finalstockpricein$ DierentiatingtwicewithrespecttothestrikeKimpliesthatthedown-andinArrow'svaluecanalternativelybederivedfromdown-and-inputvalues: (2) aninvestorcansynthesizethepayof(s).absenceofarbitragethereby barrierhasbeenhit.bybuyingandholdingaportfolioofdown-and-in requiresthatthevalueofthedown-and-inclaimdiv(h)payingf(s)at ArrowsofallstrikeswiththenumberofArrowsatstrikeKgivenbyf(K)dK, Now,letf(S)denoteanarbitrarynalpayoreceivedsolongasthe (3) maturityissimply:div(h)=z1 0f(K)DIA(K;H)dK: 7 (4)

9 atk=1.thisobservationmotivatesrewriting(4)as: valueandslopeatk=0,whiledown-and-incallshavezerovalueandslope DIV(H)=Z Whenviewedasfunctionsoftheirstrike,down-and-inputshavezero using(2)yieldsthefollowingdecompositionofanarbitrarydown-and-in claimintodown-and-inversionsofzeros,forwardcontracts4,andoptions5: whereisanarbitrarypositiveconstant.integratingbypartstwiceand +Z Thus,tosynthesizethepayof(S)receivedatTifthebarrierhasbeenhit, buyandholdaportfolioconsistingoff()down-and-inbonds,f0()downand-inforwardcontractswithdeliveryprice,f00(k)dkdown-and-inputs SettingthebarrierHtoinnityin(6)yieldsthecorrespondingresultfor f00(k)dic(k;h)dk:(6) ofallstrikesk,andf00(k)dkdown-and-incallsofallstrikesk>. path-independentpayosasaspecialcase: V=f()e?rT+f0()[Se?dT?e?rT]+Z sitionofaclaimwithanarbitrarypath-independentpayointoitsintrinsic FurthersettingtotheforwardpriceFSe(r?d)Tyieldsanewdecompo- value,f(f)e?rt,anditstimevalue: V=f(F)e?rT+ZF 0f00(K)P(K)dK+Z1 0f00(K)P(K)dK+Z1 Ff00(K)C(K)dK:(8) f00(k)c(k)dk: (7) writingabarrierput. 4Notethatbarrierforwardcontractsareeasilysynthesizedbybuyingabarriercalland Itisdiculttoimaginepayosarisinginpracticenotsatisfyingtheserestrictions. 5Technically,(6)holdsonlyforpayosf(K)whichsatisfy: 8K#0f0(K)DIP(K;H)=0 K"1f0(K)DIC(K;H)=0:

10 Thetimevalueisexpressedintermsofthepricesofout-of-the-money-forward toblack'smodel,thenastheunderlyinggetsmorevolatile,theoptionvalues K),thenthetimevalueispositive.Finally,notethatifwerestrictattention ofthepayo.ifthepayoislinear,thenf00(k)=0forallkandthereis value,thetimevalueofanarbitraryclaimissimplyalinearcombinationof thetimevaluesofanoption,withcoecientsgivenbythesecondderivative putsandcalls.sinceputsandcallswiththesamestrikehavethesametime beeliminatedfromthestatichedge,providedfbehavesreasonablyatthese grow,andthereforesodoesthetimevalue. notimevalue.conversely,ifthepayoisgloballyconvex(f00(k)0forall putsfromthestatichedge,providedthatf(0)andf0(0)arebounded: extremes.forexample,settingtozeroandhtoinnityin(6)eliminates Ifwesettoinnityorzeroin(6),thencertaintypesofcontractscan selle?dtshares,andbuyacallstruckatkp.thus,(7)generatesput-callparityasaspecialcase.similarly,(7)canbeusedtogeneratethereplication payooff(s)=max(0;kp?s),(9)indicatesthatoneshouldbuykzeros, anditsderivativesmaybeneeded.forexample,toreplicateavanillaput's Iffisnotsmooth,generalizedfunctionssuchasHeavisidestepfunctions 0f00(K)C(K)dK: (9) V=f(0)B+f0(0)Se?dT+Z1 ofdigitaloptionsusingverticalspreads. tract(6)from(7)andusein-outparity: Togenerateresultsforup-securities,replaceDbyUinalltheaboveresults. DOV(H)=f()DOB(H)+f0()[DOS(H)?DOB(H)] Togeneratethecorrespondingresultsfordown-and-outsecurities,sub- 3StaticReplicationwithVanillaOptions +Z 0f00(K)DOP(K;H)dK+Z1 f00(k)doc(k;h)dk: (10) Thelastsectionshowedhowbarriersecuritiescanbestaticlyreplicatedwith barrieroptions.thissectionshowshowthesamesecuritiescanbestaticlyreplicatedwithvanillaoptions.replicabilityofthiskindisimportant sincevanillaoptionsaremoreliquidinpracticeandtheirpricesaremore 9

11 requiringtheimpositionoftherestoftheblackscholesassumptions.we transparent.however,thereplicationisachievedatthetheoreticalcostof stantvolatilityrate.importantly,thepriceprocessiscontinuous,sothat thereforeassumethatthestockpriceobeysalognormalprocesswithacon- theunderlyingcannotjumpacrossthebarrier6. identicaltoavanillacall.toreplicatethisexotic,wewantaportfolioof willexpireworthlessatmaturity.uponreachingthebarrier,itbecomes Inthissubsection,weprovidethemainintuitionforourresults.Theactual 3.1MainIntuition Europeanoptionstoimitatethisbehavior.Ifthebarrierisneverreached, Intheappendix,wegiveaformalderivationofourstatichedgingtechnique. alwaysbeequivalenttoacall. techniqueisfairlydirect,althoughitssimplicitymaybelostinthedetails. ourportfolioshouldbeworthlessatmaturity,andatthebarrier,itshould Consideradown-and-incalloption.Ifthebarrierisneverreached,it in-barrierbetouchedissuperuous,andsowecanreplicatewitheuropean optionsusingtheresultsofthelastsection.forpayosabovethebarrier, thestockpriceisatthebarrier.thus,wecanalsoreplicatethereected wewillreectthesepayosbelowthebarrier.thereectedpayoswillbe constructedtohaveavaluematchingthatoftheoriginalpayoswhenever andbelowthebarrier.forpayosbelowthebarrier,therequirementthatthe Dependingonitsstrike,adown-and-incallcanhavepayosbothabove hasthesamevalueasaportfolioofeuropeanoptionswithapayoof: appendixshowsthatthedown-and-inversionofthissecuritywithbarrierh Moregenerally,supposeaEuropeansecurityhasnalpayof(ST).The 3.2AdjustedPayo payoswitheuropeanoptionstocompleteourstatichedge. undervaluetheexotic. 6Ifjumpswerepossible,onecanforecastwhetherourstaticportfoliowillovervalueor ^f(st)(0f(st)+sthpfh2 10STifST<H, ifst>h,

12 BarrierSecurity No-touchbinaryput One-touchbinaryput(European)0 Down-and-outcall AdjustedPayo Down-and-output 1max(ST?Kc;0) 1+(ST=H)p Table1:AdjustedPayosforDownSecurities.(p=1?2(r?d) max(kp?st;0)?(st=h)pmax((h2=st)?kc;0)forst<h?(st=h)pmax(kp?(h2=st);0)forst<h forst>h wherethepowerp1?2(r?d) down-and-insecurity.foradown-and-outsecurity,in-outparityimpliesthat theadjustedpayois: ^f(st)(f(st) 2:Wecall^f(ST)theadjustedpayoforthe ifst>h, 2) InTable1andFigure3,weshowtheadjustedpayoforsomecommon putsandcallsisusuallynotpossible.however,asfigure3makesclear, wiselinear.thus,anexactreplicationusinganitenumberofeuropean securities. Uponinspectionofthetable,theadjustedpayosareusuallynotpiece-?STHpfH2 STifST<H. one-touchbinarycanbeexactlyreplicatedbytwodigitals. determinedbyrisk-neutralvaluation: thepayosareclosetolinear.furthermore,afewspecialcasesareworth mentioning.whenr=d,thenp=1andallpayosarelinear.theresultingpayosareidenticaltotheresultsgiveninbowieandcarr[3].also,for r?d=2 Giventheadjustedpayo,thevalueofthereplicatingportfoliocanbe 2,thenp=0andthebinarypayosarelinear.Inparticular,a wheretheappendixshowsthatthevalueofanarrowintheblackscholes V(S;)=Z1 0^f(K)A(S;;K)dK;S>0; 11

13 stockpricesabovethebarrier: modelis: Thevalueofthedown-and-inclaimisobtainedbyrestrictingthisvalueto A(S;;K)=e?r1 Kp22exp8<:?(ln(K=S)?(r?d?2 22 2))29=;: Inthissection,wederivethestatichedgeinanothermanner.Wesuppose 3.3DerivationfromPricingFormula thatapricingformulaforabarriersecurityisknown,eitherbecauseitexists DIV(S;;H)=V(S;);S>H: intheliterature(seeeg.[17]),orbecauseithasbeenderivedusingdynamic replicationarguments.wethenshowhowthisformulacanbeusedtogenerateastatichedgeusingvanillaoptions.theadvantageofapproachingstatic togeneratestatichedgesforawidesetofsecurities. hedginginthismanneristhatitisverysimpleandtheapproachcanbeused restrictionthatstockpricesareabovethebarrier: thereplicatingoptionportfolioforanystockpricebysimplyremovingthe pricesandthetimetomaturity.therststepistondthevalueof theformulad(s;)foradownsecurityasafunctionofthecurrentstock workbackwardsfromtheresultsoflastsection.thus,weassumeweknow Forsimplicity,weagainworkwithdownsecuritiesonly.Weessentially Thesecondstepistoobtaintheadjustedpayowhichgaverisetothisvalue. Sincevaluesconvergetotheirpayoatmaturity,simplytakethelimitofthe valueasthetimetomaturityapproacheszero: V(S;)=D(S;);S>0: ^f(s)=lim #0V(S;);S>0: (11) Thethirdstepistouse(7)with=Htouncovertherequisitestaticposition inbonds,forwardcontracts,andvanillaoptions. 12 (12)

14 Kc>H.FromMerton[13],thevaluationformulais: SKc+r?d+2 SKc+r?d?2 p 21A indicatorfunctionby1()gives: RemovingtherequirementthatS>H,letting#0,anddenotingthe 1A;S>H: lim #0DIC(S;;H)=SSHp?21 =SHpmax H2 S?Kc!1 S>Kc!?KcSHp1 S?Kc!: H2 H2 S>Kc! down-and-incall'sadjustedpayoof^f(s)=shpmax0;h2 withtable1(recallkc>h).thethirdstepistostaticlyreplicatethe listedinstruments.setting=hin(7)andreplacingfby^fgives: V0=^f(H)e?rT+^f0(H)[Se?dT?He?rT]+ZH Thus,usingin-outparity,theadjustedpayoforadown-and-outcallagrees Thereadercanverifythat: 0^f00(K)P(K)dK+Z1 S?Kcusing H^f00(K)C(K)dK: ^f00(k)=hkcp?2 ^f0(h)=0 ^f(h)=0 K?H2 Kc!+KHp?2(p?1)p?2 K?pKc H21 (13) Applying(13),ourreplicatingportfoliobecomes: 1.HKcp?2putsatstrikeH2 Kc.13 K<H2 Kc!

15 othersecurities,considerthevaluationofanamericanbinaryputpayinga dollarattherstpassagetimetoh.from[17],thevaluationformulais: 2.KHp?2(p?1)p?2 Toshowhowthisapproachcanbeusedtogenerateadjustedpayosfor K?pKc H2dKputsatstrikeKforK<H2 Kc. fors>h,where12?r?d thats>handletting#0givestheadjustedpayoas(seefigure4): lim #0ABP(S;;H)="SH++SH?#1(S<H): 2;q2+2r 2:Removingtherequirement p1a; Financialmathematicsisfortunateinthattherearemanyequivalentinterpretationsofthesamephenomena,andoneofthemethodstoevaluatederivativesecuritiesisthroughdierentialequations(asexempliedbyWilmott,et equation.dierentoptionsarecreatedbyimposingdierentinitialvalue al[20]).allpricingformulasmustsatisfythesameblackscholesdierential Anotherwaytointerpretourresultsistouseanalternativeframework. andboundaryconditions.inthissection,weareessentiallytransforming adirichletproblem(incompleteinitialvalueproblemwithboundaryconditionsatthebarrier)intoacauchyproblem(completeinitialvalueproblem). Forsinglebarrieroptions,bothtypesofproblemsgiverisetouniquesolutions.Giventhenalsolution,itisstraightforward(asillustratedabove)tand-inArrows.IntheBlackScholesmodel,thevalueofadown-and-inArrow transformbetweenthetwotypesofproblems. Alldown-and-insecuritiescanbedecomposedintoastaticportfolioofdown- 4SummaryandFutureResearch struckatsomelevelkabovethebarrierhmatchesthevalueofasuitably weightedpath-independentarrowstruckatthegeometricreectionofkin H.Itfollowsthatthevalueofadown-and-insecuritycanberepresented byastaticportfolioofarrowsstruckbelowthebarrier.sinceanysuch 14

16 portfoliocanbecreatedoutofastaticportfolioofeuropeanoptions,downand-insecuritiescanbestaticallyhedgedwithvanillaoptions.in-outparity Itfollowsthatonecanstartfromaformulaobtainedbydynamicreplication anduncovertheimplicitstatichedge. derivedviastaticreplicationmatchthoseobtainedbydynamicreplication. impliesthatthesameresultholdsforoutoptions.thevaluationformulas ofthereplicatingportfolio'spayofromthetarget.thelatterapproachis likelytopermitloweroeringpricesandatleastsomeoftheriskcanbe tradeinonlyanitenumberofstrikes.onepossibilityistoattemptsuperreplicationattheleastcost.anotheristominimizemeansquarederror replicationerrorarisingfromhedgingwithvanillaoptionswhenonecan Infutureanalyticalworkinthisarea,weplantoexploretheeectsof imposingmultiplebarriers.itwillalsobeinterestingtoexaminethestatic diversiedaway.finally,itshouldbeinterestingtoconductempiricaltests comparingstaticanddynamichedgingunderrealisticmarketconditions. 15

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18 [14]Nachman,D.,1988,\SpanningandCompletenesswithOptions",ReviewofFinancialStudies,1,3, [17]RubensteinM.andE.Reiner,1991,\BreakingdowntheBarriers",Risk, [16]Reiner,E.andM.Rubenstein,1991,\UnscramblingtheBinaryCode", [15]NelkenI.,1995,HandbookofExoticOptions,Probus,ChicagoIL. Risk,4,28{35. [18]Ross,S.,1976,\OptionsandEciency",QuarterlyJournalofEconomics,90,75{89. [19]Snyder,G.,1969,\AlternativeFormsofOptions,"FinancialAnalysts 4,8. [21]Zhang,P.,1995, [20]P.Wilmott,J.DeWynne,andS.Howison,1993, nancialpress. Journal,26Sept.-Oct., ingmanuscript. ExoticOptions:AGuidetotheSecondGenerationOptions,forthcom- OptionPricing:MathematicalModelsandComputation,OxfordFi- 17

19 securities.recallthatarrowsarefundamentalsecuritiesinthesensethat Appendix Thisappendixshowshowweobtaintheadjustedpayofordown-and-in thevalueofanarbitrarypath-independentclaimiseasilyrepresentedin termsofthem: wherethecurrentspotpricesandthetimetomaturityt?thave beenintroducedasargumentsofthearrow'svalue.dynamicreplication argumentssuchasthosegivenbycoxandross[8]implythatthevaluecan alsobecalculatedusingrisk-neutralvaluation: V(S;)=Z1 V(S;)=Z1 0f(K)e?r`(S;;K)dK; 0f(K)A(S;;K)dK; IntheBlackScholesmodel,therisk-neutraldensiyislognormalanditfollows atkatt,conditionalonstartingfromsatt.sincetheaboveequations discountedrisk-neutralprobability: holdforanypayofunctionf,itfollowsthatanarrow'spriceissimplythe where`(s;;k)istherisk-neutralprobabilitythatthestockpricenishes thatthearrow'spriceisgivenby: A(S;;K)=e?r1 Kp22exp8<:?(ln(K=S)?(r?d?2 A(S;;K)=e?r`(S;;K): riersecuritiessince: DIV(S;;H)=Z1 Recallfrom(4)thatadown-and-inArrowisfundamentalinvaluingbar- 22 2))29=;:(14) frompayosabovethebarrier.forpayosbelowthebarrier(i.e.sh), wherezr1hf(k)dia(s;;k;h)dkistheportionofthevaluearising itfollowsthatdia(s;;k;h)=a(s;;k). =ZH 0f(K)A(S;;K)dK+Z; 0f(K)DIA(S;;K;H)dK+Z1 18 (15)

20 securitieswheneverthebarrierisreached.thus, abilitydensityfunctionfortherstpassagetimetothebarrierhwhen startingfroms.then,zisthediscountedexpectedvalueofthearrow ToevaluateZmoreexplicitly,let(S;t;H)denotetherisk-neutralprob- wherei(t)r1hf(k)a(h;t;k)dk:considerthechangeofvariablesst= H2 KinI(t):I(t)=Z0 Z=Z 0e?rt(S;t;H)I(t)dt; ST!A(H;t;H2=ST)(?H2 S2TdST) (16) Using(14),wehave: H2 S2TA(H;t;H2=ST)=H2 =ZH S2Te?rt 0f ST!H2 (H2=ST)p22texp8><>:?hlnH2 S2TA(H;t;H2=ST)dST: 1 ST=H?r?d?2 (17) =e?rt1 STp22texp8><>:?hln(H=ST)?r?d?2 22t 2ti2 22t9>=>; 2ti29>=>; wherep=1?2(r?d) 2.Substitutinginto(17)gives: =e?rtst =ST HpA(H;t;ST); Hp1 STp22texp8><>:?hln(ST=H)?r?d?2 22t 2ti29>=>; I(t)=ZH 0f H2 ST!ST HpA(H;t;ST)dST; (18) andsubstitutinginto(16)gives: Z=Z 0e?rt(S;t;H)"ZH 0f 19H2 ST!ST HpA(H;t;ST)dST#dt

21 ThelastlinefollowssinceST<Hintherangeoftheintegral.Notethat =ZH 0f H2 ST!ST HpZ functionalform.substitutinginto(15)gives: althoughwehaveused()inourderivation,weneverneededtoknowits HpA(S;;ST)dST: 0e?rt(S;t;H)A(H;t;ST)dtdST payoforadown-and-insecurity. where^f(st)hf(st)+sthpfh2 DIV(S;;H)=ZH =Z1 0^f(ST)A(S;;ST)dST; 0f(ST)A(S;;ST)dST+ZH STi1(ST<H)isdenedastheadjusted 0A(S;;ST)ST Hpf H2 ST!dST 20

22 2.5 Adjusted Payoff for One touch Binary Put (European) 1.5 Adjusted Payoff for No touch Binary Put Adjusted Payoff 1 Adjusted Payoff Final Stock Price Final Stock Price Adjusted Payoff for Down and out Call Adjusted Payoff for Down and out Put Figure3:Adjustedpayosfordownsecurities(r=0:05,d=0:03,=:15, Kc=Kp=110,H=100) Final Stock Price Final Stock Price Adjusted Payoff Adjusted Payoff

23 Adjusted Payoff for One touch Binary Put (American) 2.5 =:15,H=100). Figure4:AdjustedpayoforAmericanbinaryput(r=0:05,d=0:03, Final Stock Price 22 Adjusted Payoff