paul wilmott on quantitative finance

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contentsofvolumeoneVisualBasicCodexxvPrologtotheSecondEditionxxviiPARTONEMATHEMATICALANDFINANCIALFOUNDATIONS;BASICTHEORYOFDERIVATIVES;RISKANDRETURN11ProductsandMarkets52Derivatives253TheRandomBehaviorofAssets554ElementaryStochasticCalculus715TheBlack–ScholesModel916PartialDifferentialEquations1017TheBlack–ScholesFormulaeandthe‘Greeks’1098SimpleGeneralizationsoftheBlack–ScholesWorld1399EarlyExerciseandAmericanOptions15110ProbabilityDensityFunctionsandFirst-exitTimes16911Multi-assetOptions18312HowtoDeltaHedge19713Fixed-incomeProductsandAnalysis:Yield,DurationandConvexity22514Swaps251 viiicontents15TheBinomialModel26116HowAccurateistheNormalApproximation?29517InvestmentLessonsfromBlackjackandGambling30118PortfolioManagement31719ValueatRisk33120ForecastingtheMarkets?34321ATradingGame359 PARTONEmathematicalandfinancialfoundations;basictheoryofderivatives;riskandreturnThefirstpartofthebookcontainsthefundamentalsofderivativestheoryandpractice.Welookatbothequityandfixedincomeinstruments.Iintroducetheimportantconceptsofhedgingandnoarbitrage,onwhichmostsophisticatedfinancetheoryisbased.Wedrawsomeinsightfromideasfirstseeningambling,andwedevelopthoseintoananalysisofriskandreturn.Theassumptions,keyconceptsandresultsinPartOnemakeupwhatislooselyknownasthe‘Black–Scholesworld,’namedforFischerBlackandMyronScholeswho,togetherwithRobertMerton,firstconceivedthem.Theiroriginalworkwaspublishedin1973,aftersomeresistance(thefamousequationwasfirstwrittendownin1969).InOctober1997MyronScholesandRobertMertonwereawardedtheNobelPrizeforEconomicsfortheirwork,FischerBlackhavingdiedinAugust1995.TheNewYorkTimesofWednesday,15thOctober1997wrote:‘TwoNorthAmericanscholarswontheNobelMemorialPrizeinEconomicScienceyesterdayforworkthatenablesinvestorstopriceaccuratelytheirbetsonthefuture,abreakthroughthathashelpedpowertheexplosivegrowthinfinancialmarketssincethe1970’sandplaysaprofoundroleintheeconomicsofeverydaylife.’1PartOneisselfcontained,requiringlittleknowledgeoffinanceoranymorethanelementarycalculus.Chapter1:ProductsandMarketsAnoverviewoftheworkingsofthefinancialmarketsandtheirproducts.Achaptersuchasthisisobligatory.However,myreaderswillfallintooneoftwogroups.Eithertheywillknoweverythinginthischapterandmuch,muchmorebesides.Ortheywillknowlittle,inwhichcasewhatIwritewillnotbeenough.1We’llbehearingmoreaboutthesetwoinChapter44on‘Derivatives****Ups.’ 2PartOnemathematicalandfinancialfoundationsChapter2:DerivativesAnintroductiontooptions,optionsmarkets,marketconventions.Definitionsofthecommonterms,simplenoarbitrage,put-callparityandelementarytradingstrategies.Chapter3:TheRandomBehaviorofAssetsAnexaminationofdataforvariousfinancialquantities,leadingtoamodelfortherandombehaviorofprices.Almostallofsophisticatedfinancetheoryassumesthatpricesarerandom,thequestionishowtomodelthatrandomness.Chapter4:ElementaryStochasticCalculusWe’llneedalittlebitoftheoryformanipulatingourrandomvariables.Ikeeptherequirementsdowntothebareminimum.ThekeyconceptisIto’slemmawhichIwilltrytointroduceinasaccessibleamanneraspossible.ˆChapter5:TheBlack–ScholesModelIpresenttheclassicalmodelforthefairvalueofoptionsonstocks,currenciesandcommodities.ThisisthechapterinwhichIdescribedeltahedgingandnoarbitrageandshowhowtheyleadtoauniquepriceforanoption.ThisisthefoundationformostquantitativefinancetheoryandIwillbebuildingonthisfoundationformuch,butbynomeansall,ofthebook.Chapter6:PartialDifferentialEquationsPartialdifferentialequationsplayanimportantroleinmostphysicalappliedmathematics.Theyalsoplayaroleinfinance.Mostofmyreaderstrainedinthephysicalsciences,engineeringandappliedmathematicswillbecomfortablewiththeideathatapartialdifferentialequationisalmostthesameas‘theanswer,’thetwobeingseparatedbyatmostsomecomputercode.IfyouarenotsureofthisconnectionIhopethatyouwillperseverewiththebook.Thisrequiressomefaithonyourpart;youmayhavetoreadthebookthroughtwice:Ihavenecessarilyhadtorelegatethenumerics,thereal‘answer,’tothelastfewchapters.Chapter7:TheBlack–ScholesFormulaeandthe‘Greeks’FromtheBlack–Scholespartialdifferentialequationwecanfindformulaeforthepricesofsomeoptions.Derivativesofoptionpriceswithrespecttovariablesorparametersareimportantforhedging.Iwillexplainsomeofthemostimportantsuchderivativesandhowtheyareused.Chapter8:SimpleGeneralizationsoftheBlack–ScholesWorldSomeoftheassumptionsoftheBlack–Scholesworldcanbedroppedorstretchedwithease.Iwilldescribeseveralofthese.Laterchaptersaredevotedtomoreextensivegeneralizations.Chapter9:EarlyExerciseandAmericanOptionsEarlyexerciseisofparticularimportancefinancially.Itisalsoofgreatmathematicalinterest.Iwillexplainbothoftheseaspects.Chapter10:ProbabilityDensityFunctionsandFirst-exitTimesTherandomnatureoffinan-cialquantitiesmeansthatwecannotsaywithcertaintywhatthefutureholdsinstore.Forthatreasonweneedtobeabletodescribethatfutureinaprobabilisticsense.Chapter11:Multi-assetOptionsAnotherconceptuallysimplegeneralizationofthebasicBlack–Scholesworldistooptionsonmorethanoneunderlyingasset.Theoreticallysimple,thisextensionhasitsownparticularproblemsinpractice.Chapter12:HowtoDeltaHedgeNoteveryonebelievesinnoarbitrage,theabsenceoffreelunches.Inthischapterweseehowtoprofitifyouhaveabetterforecastforfuturevolatilitythanthemarket. mathematicalandfinancialfoundationsPartOne3Chapter13:Fixed-incomeProductsandAnalysis:Yield,DurationandConvexityThischapterisanintroductiontothesimplertechniquesandanalysescommonlyusedinthemarket.InparticularIexplaintheconceptsofyield,durationandconvexity.InthisandthenextchapterIassumethatinterestratesareknown,deterministicquantities.Chapter14:SwapsInterest-rateswapsareverycommonandveryliquid.Iexplainthebasicsandrelatethepricingofswapstothepricingoffixed-ratebonds.Chapter15:TheBinomialModelOneofthereasonsthatoptiontheoryhasbeensosuccessfulisthattheideascanbeexplainedandimplementedveryeasilywithnocomplicatedmathematics.Thebinomialmodelisthesimplestwaytoexplainthebasicideasbehindoptiontheoryusingonlybasicarithmetic.That’sagoodthing,right?Yes,butonlyifyoubearinmindthatthemodelisfordemonstrationpurposesonly,itisnottherealthing.Asamodelofthefinancialworlditistoosimplistic,asaconceptforpricingitlackstheelegancethatmakesothermethodspreferable,andasanumericalschemeitisprehistoric.Useonceandthenthrowaway,that’smyrecommendation.Chapter16:HowAccurateistheNormalApproximation?OneofthemajorassumptionsoffinancetheoryisthatreturnsareNormallydistributed.Inthischapterwetakealookatwhywemakethisassumption,andhowgooditreallyis.Chapter17:InvestmentLessonsfromBlackjackandGamblingWedrawinsightsandinspi-rationfromthenot-unrelatedworldofgamblingtohelpusinthetreatmentofrisk,return,andmoney/riskmanagement.Chapter18:PortfolioManagementIfyouarewillingtoacceptsomeriskhowshouldyouinvest?IexplaintheclassicalideasofModernPortfolioTheoryandtheCapitalAssetPricingModel.Chapter19:ValueatRiskHowriskyisyourportfolio?Howmuchmightyouconceivablyloseifthereisanadversemarketmove?Thesearethetopicsofthischapter.Chapter20:ForecastingtheMarkets?Althoughalmostallsophisticatedfinancetheoryassumesthatassetsmoverandomly,manytradersrelyontechnicalindicatorstopredictthefuturedirectionofassets.Theseindicatorsmaybesimplegeometricalconstructsoftheassetpricepathorquitecomplexalgorithms.Thehypothesisisthatinformationaboutshort-termfutureassetpricemovementsarecontainedwithinthepasthistoryofprices.Alltradersusetechnicalindicatorsatsometime.InthischapterIdescribesomeofthemorecommontech-niques.Chapter21:ATradingGameManyreadersofthisbookwillneverhavetradedanythingmoresophisticatedthanbaseballcards.TogetthemintotheswingofthesubjectfromapracticalpointofviewIincludesomesuggestionsonhowtoorganizeyourowntradinggamebasedonthebuyingandsellingofderivatives.IhadalotofhelpwiththischapterfromDavidEpsteinwhohasbeenrunningsuchgamesforseveralyears. CHAPTER1productsandmarketsInthisChapter...•thetimevalueofmoney•anintroductiontoequities,commodities,currenciesandindices•fixedandfloatinginterestrates•futuresandforwards•no-arbitrage,oneofthemainbuildingblocksoffinancetheory1.1INTRODUCTIONThisfirstchapterisaverygentleintroductiontothesubjectoffinance,andismainlyjustacollectionofdefinitionsandspecificationsconcerningthefinancialmarketsingeneral.Thereislittletechnicalmaterialhere,andtheonetechnicalissue,the‘timevalueofmoney,’isextremelysimple.Iwillgivethefirstexampleof‘noarbitrage.’Thisisimportant,beingonepartofthefoundationofderivativestheory.Whetheryoureadthischapterthoroughlyorjustskimitwilldependonyourbackground;mathematiciansnewtofinancemaywanttospendmoretimeonitthanpractitioners,say.1.2THETIMEVALUEOFMONEYThesimplestconceptinfinanceisthatofthetimevalueofmoney;$1todayisworthmorethan$1inayear’stime.Thisisbecauseofallthethingswecandowith$1overthenextyear.Attheveryleast,wecanputitunderthemattressandtakeitoutinoneyear.Butinsteadofputtingitunderthemattresswecouldinvestitinagoldmine,oranewcompany.Ifthosearetoorisky,thenlendthemoneytosomeonewhoiswillingtotaketherisksandwillgiveyoubackthedollarwithalittlebitextra,theinterest.Thatiswhatbanksdo,theyborrowyourmoneyandinvestitinvariousriskyways,butbyspreadingtheirriskovermanyinvestmentstheyreducetheiroverallrisk.Andbyborrowingmoneyfrommanypeopletheycaninvestinwaysthattheaverageindividualcannot.Thebankscompeteforyourmoneybyofferinghighinterestrates.Freemarketsandtheabilitytochangebanksquicklyandcheaplyensurethatinterestratesarefairlyconsistentfromonebanktoanother. 6PartOnemathematicalandfinancialfoundationsIamgoingtodenoteinterestratesbyr.AlthoughratesvarywithtimeIamgoingtoassumeforthemomentthattheyareconstant.Wecantalkaboutseveraltypesofinterest.Firstofallthereissimpleandcompoundinterest.Simpleinterestiswhentheinterestyoureceiveisbasedonlyontheamountyouinvestinitially,whereascompoundinterestiswhenyoualsogetinterestonyourinterest.Compoundinterestisthemaincaseofrelevance.Andcompoundinterestcomesintwoforms,discretelycompoundedandcontinuouslycompounded.Letmeillustratehowtheyeachwork.SupposeIinvest$1inabankatadiscreteinterestrateofrpaidonceperannum.Attheendofoneyearmybankaccountwillcontain$1×(1+r).Iftheinterestrateis10%Iwillhaveonedollarandtencents.AftertwoyearsIwillhave$1×(1+r)×(1+r)=(1+r)2,oronedollarandtwenty-onecents.AfternyearsIwillhave(1+r)ndollars.Thatisanexampleofdiscretecompounding.NowsupposeIreceiveminterestpaymentsatarateofr/mperannum.AfteroneyearIwillhave
rm1+.(1.1)m(Ihavedroppedthe$sign,takingitasreadfromnowon.)Iamgoingtoimaginethattheseinterestpaymentscomeatincreasinglyfrequentintervals,butatanincreasinglysmallerinterestrate:Iamgoingtotakethelimitm→∞.Thiswillleadtoarateofinterestthatispaidcontinuously.Expression(1.1)becomes
mrmlog(1+r)r1+=em∼e.mThisisasimpleapplicationofTaylorserieswhenr/missmall.AndthatishowmuchmoneyIwillhaveinthebankafteroneyeariftheinterestiscontinuouslycompounded.Similarly,afteratimetIwillhaveanamountert(1.2)inthebank.Almosteverythinginthisbookassumesthatinterestiscompoundedcontinuously.Anotherwayofderivingtheresult(1.2)isviaadifferentialequation.SupposeIhaveanamountM(t)inthebankattimet,howmuchdoesthisincreaseinvaluefromonedaytothenext?IfIlookatmybankaccountattimetandthenagainashortwhilelater,timet+dt,theamountwillhaveincreasedbydMM(t+dt)−M(t)≈dt+···,dtwheretheright-handsidecomesfromaTaylorseriesexpansionofM(t+dt).ButIalsoknowthattheinterestIreceivemustbeproportionaltotheamountIhave,M,theinterestrate,r,andthetimestep,dt.ThusdMdt=rM(t)dt.dt productsandmarketsChapter17DividingbydtgivestheordinarydifferentialequationdM=rM(t)dtthesolutionofwhichisM(t)=M(0)ert.Iftheinitialamountatt=0was$1thenIget(1.2)again.ThisequationrelatesthevalueofthemoneyIhavenowtothevalueinthefuture.Conversely,ifIknowIwillgetonedollarattimeTinthefuture,itsvalueatanearliertimetissimplye−r(T−t).Icanrelatecashflowsinthefuturetotheirpresentvaluebymultiplyingbythisfactor.Asanexample,supposethatris5%i.e.r=0.05,thenthepresentvalueof$1,000,000tobereceivedintwoyearsis$1,000,000×e−0.05×2=$904,837.Thepresentvalueisclearlylessthanthefuturevalue.Interestratesareaveryimportantfactordeterminingthepresentvalueoffuturecashflows.ForthemomentIwillonlytalkaboutoneinterestrate,andthatwillbeconstant.InlaterchaptersIwillgeneralize.ImportantAsideWhatmathematicshaveweseensofar?Togetto(1.2)allweneededtoknowaboutarethetwofunctionse(orexp)andlog,andTaylorseries.Believeitornot,youcanappre-ciatealmostallfinancetheorybyknowingthesethreethingstogetherwith‘expectations.’I’mgoingtobuilduptothebasicBlack–Scholesandderivativestheoryassumingthatyouknowallfourofthese.Don’tworryifyoudon’tknowaboutthesethingsyet,takealookatAppendixAwhereIreviewtheserequisitesandshowhowtointerpretfinancetheoryandpracticeintermsofthemostelementarymathematics.Justbecauseyoucanunderstandderivativestheoryintermsofbasicmathdoesn’tmeanthatyoushould.Ihopethatthere’senoughinthebooktopleasethePh.D.s1aswell.1.3EQUITIESThemostbasicoffinancialinstrumentsistheequity,stockorshare.Thisistheowner-shipofasmallpieceofacompany.Ifyouhaveabrightideaforanewproductorservice1AndNobellaureates. 8PartOnemathematicalandfinancialfoundationsthenyoucouldraisecapitaltorealizethisideabysellingofffutureprofitsintheformofastakeinyournewcompany.Theinvestorsmaybefriends,yourAuntJoan,abank,oraventurecapitalist.Theinvestorinthecompanygivesyousomecash,andinreturnyougivehimacontractstatinghowmuchofthecompanyheowns.Theshareholderswhoownthecompanybetweenthemthenhavesomesayintherunningofthebusiness,andtechnicallythedirectorsofthecompanyaremeanttoactinthebestinterestsoftheshareholders.Onceyourbusinessisupandrunning,youcouldraisefurthercapitalforexpansionbyissuingnewshares.Thisishowsmallbusinessesbegin.Oncethesmallbusinesshasbecomealargebusiness,yourAuntJoanmaynothaveenoughmoneyhiddenunderthemattresstoinvestinthenextexpansion.Atthispointsharesinthecompanymaybesoldtoawideraudienceoreventhegeneralpublic.Theinvestorsinthebusinessmayhavenolinkwiththefounders.Thefinalpointinthegrowthofthecompanyiswiththequotationofsharesonaregulatedstockexchangesothatsharescanbeboughtandsoldfreely,andcapitalcanberaisedefficientlyandatthelowestcost.Figures1.1and1.2showscreensfromBloomberggivingdetailsofMicrosoftstock,includingprice,highandlow,namesofkeypersonnel,weightinginvariousindices(seebelow)etc.Thereismuch,muchmoreinfoavailableonBloombergforthisandallotherstocks.We’llbeseeingmanyBloombergscreensthroughoutthisbook.Figure1.1DetailsofMicrosoftstock.Source:BloombergL.P. productsandmarketsChapter19Figure1.2DetailsofMicrosoftstockcontinued.Source:BloombergL.P.InFigure1.3IshowanexcerptfromTheWallStreetJournalEuropeof14thApril2005.ThisshowsasmallselectionofthemanystockstradedontheNewYorkStockExchange.Thelistedinformationincludeshighsandlowsforthedayaswellasthechangesincethepreviousday’sclose.Thebehaviorofthequotedpricesofstocksisfarfrombeingpredictable.InFigure1.4IshowtheDowJonesIndustrialAverageovertheperiodJanuary1950toMarch2004.InFigure1.5isatimeseriesoftheGlaxo–Wellcomeshareprice,asproducedbyBloomberg.Ifwecouldpredictthebehaviorofstockpricesinthefuturethenwecouldbecomeveryrich.Althoughmanypeoplehaveclaimedtobeabletopredictpriceswithvaryingdegreesofaccuracy,noonehasyetmadeacompletelyconvincingcase.InthisbookIamgoingtotakethepointofviewthatpriceshavealargeelementofrandomness.Thisdoesnotmeanthatwecannotmodelstockprices,butitdoesmeanthatthemodelingmustbedoneinaprobabilisticsense.Nodoubttherealityofthesituationliessomewherebetweencompletepredictabilityandperfectrandomness,notleastbecausetherehavebeenmanycasesofmarketmanipulationwherelargetradeshavemovedstockpricesinadirectionthatwasfavorabletothepersondoingthemoving.Towhetyourappetiteforthemathematicalmodelinglater,Iwanttoshowyouasimplewaytosimulatearandomwalkthatlookssomethinglikeastockprice.Oneofthesimplestrandomprocessesisthetossingofacoin.Iamgoingtouseideasrelatedtocointossingasamodelforthebehaviorofastockprice.Asasimpleexperimentstartwiththenumber100whichyou 10PartOnemathematicalandfinancialfoundationsJones&Company,Inc.tedversioninPublisher'sNote:Permissiontoreproducethisimageonlinewasnotgrantedbythecopyrightholder.Readersarekindlyrequestedtorefertotheprofthischapter.of14thApril2005.ReproducedbypermissionofDowTheWallStreetJournalEuropeFigure1.3 productsandmarketsChapter11112000100008000600040002000018-Dec-496-Mar-5823-May-669-Aug-7426-Oct-8212-Jan-9131-Mar-9917-Jun-07Figure1.4AtimeseriesoftheDowJonesIndustrialAveragefromJanuary1950toMarch2004.Figure1.5Glaxo–Wellcomeshareprice(volumebelow).Source:BloombergL.P. 12PartOnemathematicalandfinancialfoundations110108106104102100989694Numberofcointosses92020406080100Figure1.6Asimulationofanassetpricepath?shouldthinkofasthepriceofyourstock,andtossacoin.Ifyouthrowaheadmultiplythenumberby1.01,ifyouthrowatailmultiplyby0.99.Afteronetossyournumberwillbeeither99or101.Tossagain.Ifyougetaheadmultiplyyournewnumberby1.01orby0.99ifyouthrowatail.Youwillnowhaveeither1.012×100,1.01×0.99×100=0.99×1.01×100or0.992×100.Continuethisprocessandplotyourvalueonagrapheachtimeyouthrowthecoin.ResultsofoneparticularexperimentareshowninFigure1.6.Insteadofphysicallytossingacoin,theseriesusedinthisplotwasgeneratedonaspreadsheetlikethatinFigure1.7.ThisusestheExcelspreadsheetfunctionRAND()togenerateauniformlydistributedrandomnumberbetween0and1.Ifthisnumberisgreaterthanonehalfitcountsasa‘head’otherwisea‘tail.’1.3.1DividendsTheownerofthestocktheoreticallyownsapieceofthecompany.Thisownershipcanonlybeturnedintocashifheownssomanyofthestockthathecantakeoverthecompanyandkeepalltheprofitsforhimself.Thisisunrealisticformostofus.Totheaverageinvestorthevalueinholdingthestockcomesfromthedividendsandanygrowthinthestock’svalue.Dividendsarelumpsumpayments,paidouteveryquarteroreverysixmonths,totheholderofthestock.Theamountofthedividendvariesfromyeartoyeardependingontheprofitabilityofthecompany.Asageneralrulecompaniesliketotrytokeepthelevelofdividendsaboutthesameeachtime.Theamountofthedividendisdecidedbytheboardofdirectorsofthecompanyandisusuallysetamonthorsobeforethedividendisactuallypaid.Whenthestockisboughtiteithercomeswithitsentitlementtothenextdividend(cum)ornot(ex).Thereisadateataroundthetimeofthedividendpaymentwhenthestockgoes productsandmarketsChapter113ABCDE1Initialstockprice100Stock2Upmove1.011003Downmove0.991014Probabilityofup0.599.99598.9901699.98=B1798.9802899.97998.9703=D6*IF(RAND()>1-$B$4,$B$2,$B$3)1099.960011198.960411299.9500113100.94951499.940011598.940611697.951211798.930721897.941411998.920832099.910042198.910942297.921832398.901042497.912032598.891152699.8800727100.878928101.887729100.868830101.877531100.8587Figure1.7Simplespreadsheettosimulatethecoin-tossingexperiment.fromcumtoex.Theoriginalholderofthestockgetsthedividendbutthepersonwhobuysitobviouslydoesnot.Allthingsbeingequalastockthatiscumdividendisbetterthanonethatisexdividend.Thusatthetimethatthedividendispaidandthestockgoesexdividendtherewillbeadropinthevalueofthestock.Thesizeofthisdropinstockvalueoffsetsthedisadvantageofnotgettingthedividend.ThisjumpinstockpriceisinpracticemorecomplexthanIhavejustmadeout.Oftencapitalgainsduetotheriseinastockpricearetaxeddifferentlyfromadividend,whichisoftentreatedasincome.Somepeoplecanmakealotofrisk-freemoneybyexploitingtax‘inconsistencies.’IdiscussdividendsindepthinChapter8andagaininChapter64.1.3.2StockSplitsStockpricesintheUSareusuallyoftheorderofmagnitudeof$100.IntheUKtheyaretypicallyaround£1.Thereisnorealreasonforthepopularityofthenumberofdigits,afterall,ifIbuyastockIwanttoknowwhatpercentagegrowthIwillget,theabsolutelevelofthestockisirrelevanttome,itjustdetermineswhetherIhavetobuytensorthousands 14PartOnemathematicalandfinancialfoundationsFigure1.8StocksplitinfoforMicrosoft.Source:BloombergL.P.ofthestocktoinvestagivenamount.Neverthelessthereissomepsychologicalelementtothestocksize.Everynowandthenacompanywillannounceastocksplit(seeFigure1.8).Forexample,thecompanywithastockpriceof$900announcesathree-for-onestocksplit.Thissimplymeansthatinsteadofholdingonestockvaluedat$900,Iholdthreevaluedat$300each.21.4COMMODITIESCommoditiesareusuallyrawproductssuchaspreciousmetals,oil,foodproductsetc.Thepricesoftheseproductsareunpredictablebutoftenshowseasonaleffects.Scarcityoftheproductresultsinhigherprices.Commoditiesareusuallytradedbypeoplewhohavenoneedoftherawmaterial.Forexampletheymayjustbespeculatingonthedirectionofgoldwithoutwantingtostockpileitormakejewelry.Mosttradingisdoneonthefuturesmarket,makingdealstobuyorsellthecommodityatsometimeinthefuture.Thedealisthenclosedoutbeforethecommodityisduetobedelivered.Futurescontractsarediscussedbelow.Figure1.9showsatimeseriesofthepriceofpulp,usedinpapermanufacture.2IntheUKthiswouldbecalledatwo-for-onesplit. productsandmarketsChapter115Figure1.9Pulpprice.Source:BloombergL.P.1.5CURRENCIESAnotherfinancialquantityweshalldiscussistheexchangerate,therateatwhichonecurrencycanbeexchangedforanother.Thisistheworldofforeignexchange,orForexorFXforshort.Somecurrenciesarepeggedtooneanother,andothersareallowedtofloatfreely.Whatevertheexchangeratesfromonecurrencytoanother,theremustbeconsistencythroughout.Ifitispossibletoexchangedollarsforpoundsandthenthepoundsforyen,thisimpliesarelationshipbetweenthedollar/pound,pound/yenanddollar/yenexchangerates.Ifthisrelationshipmovesoutoflineitispossibletomakearbitrageprofitsbyexploitingthemispricing.Figure1.10isanexcerptfromTheWallStreetJournalEuropeof14thApril2005.Atthetopofthisexcerptisamatrixofexchangerates.AsimilarmatrixisshowninFigure1.11fromBloomberg.Althoughthefluctuationinexchangeratesisunpredictable,thereisalinkbetweenexchangeratesandtheinterestratesinthetwocountries.Iftheinterestrateondollarsisraisedwhiletheinterestrateonpoundssterlingstaysfixedwewouldexpecttoseesterlingdepreciatingagainstthedollarforawhile.Centralbankscanuseinterestratesasatoolformanipulatingexchangerates,butonlytoadegree.Atthestartof1999EurolandcurrencieswerefixedattheratesshowninFigure1.12. 16PartOnemathematicalandfinancialfoundationsPublisher'sNote:Permissiontoreproducethisimageonlinewasnotgrantedbythecopyrightholder.Readersarekindlyrequestedtorefertotheprintedversionofthischapter.Figure1.10TheWallStreetJournalEuropeof14thApril2005,currencyexchangerates.Repro-ducedbypermissionofDowJones&Company,Inc. productsandmarketsChapter117Figure1.11Keycrosscurrencyrates.Source:BloombergL.P.1.6INDICESFormeasuringhowthestockmarket/economyisdoingasawhole,therehavebeendevelopedthestockmarketindices.Atypicalindexismadeupfromtheweightedsumofaselectionorbasketofrepresentativestocks.Theselectionmaybedesignedtorepresentthewholemarket,suchastheStandard&Poor’s500(S&P500)intheUSortheFinancialTimesStockExchangeindex(FTSE100)intheUK,oraveryspecialpartofamarket.InFigure1.4wesawtheDJIA,representingmajorUSstocks.InFigure1.13isshownJPMorgan’sEmergingMarketBondIndex.TheEMBI+isanindexofemergingmarketdebtinstruments,includingexternal-currency-denominatedBradybonds,EurobondsandUSdollarlocalmarketsinstruments.ThemaincomponentsoftheindexarethethreemajorLatinAmericancountries,Argentina,BrazilandMexico.Bulgaria,Morocco,Nigeria,thePhilippines,Poland,RussiaandSouthAfricaarealsorepresented.Figure1.14showsatimeseriesoftheMAEAllBondIndexwhichincludesPesoandUSdollardenominatedbondssoldbytheArgentineGovernment.1.7FIXED-INCOMESECURITIESInlendingmoneytoabankyoumaygettochooseforhowlongyoutieyourmoneyupandwhatkindofinterestrateyoureceive.Ifyoudecideonafixed-termdepositthebankwilloffertolockinafixedrateofinterestfortheperiodofthedeposit,amonth,sixmonths,ayear,say. 18PartOnemathematicalandfinancialfoundationsFigure1.12Eurofixingrates.Source:BloombergL.P.200EMBIPlus18016014012010080604020012/31/962/25/974/18/9706/11/199708/04/19979/25/9711/19/97Figure1.13JPMorgan’sEMBIPlus. productsandmarketsChapter119Figure1.14AtimeseriesoftheMAEAllBondIndex.Source:BloombergL.P.Therateofinterestwillnotnecessarilybethesameforeachperiod,andgenerallythelongerthetimethatthemoneyistiedupthehighertherateofinterest,althoughthisisnotalwaysthecase.Often,ifyouwanttohaveimmediateaccesstoyourmoneythenyouwillbeexposedtointerestratesthatwillchangefromtimetotime,asinterestratesarenotconstant.Thesetwotypesofinterestpayments,fixedandfloating,areseeninmanyfinancialinstru-ments.Coupon-bearingbondspayoutaknownamounteverysixmonthsoryearetc.Thisisthecouponandwouldoftenbeafixedrateofinterest.Attheendofyourfixedtermyougetafinalcouponandthereturnoftheprincipal,theamountonwhichtheinterestwascalculated.Interestrateswapsareanexchangeofafixedrateofinterestforafloatingrateofinterest.Governmentsandcompaniesissuebondsasaformofborrowing.Thelesscreditworthytheissuer,thehighertheinterestthattheywillhavetopayout.Bondsareactivelytraded,withpricesthatcontinuallyfluctuate.Fixed-incomemodelingandproductsarethesubjectofChapters13and14andthewholeofPartThree.1.8INFLATION-PROOFBONDSArecentadditiontothelistofbondsissuedbytheUSgovernmentistheindex-linkedbond.ThesehavebeenaroundintheUKsince1981,andhaveprovidedaverysuccessfulwayofensuringthatincomeisnoterodedbyinflation. 20PartOnemathematicalandfinancialfoundationsPublisher'sNote:Permissiontoreproducethisimageonlinewasnotgrantedbythecopyrightholder.Readersarekindlyrequestedtorefertotheprintedversionofthischapter.Figure1.15UKgiltspricesfromTheFinancialTimesof14thApril2005.ReproducedbypermissionofTheFinancialTimes.IntheUKinflationismeasuredbytheRetailPriceIndexorRPI.Thisindexisameasureofyear-on-yearinflation,usinga‘basket’ofgoodsandservicesincludingmortgageinterestpayments.Theindexispublishedmonthly.Thecouponsandprincipaloftheindex-linkedbondsarerelatedtotheleveloftheRPI.Roughlyspeaking,theamountsofthecouponandprincipalarescaledwiththeincreaseintheRPIovertheperiodfromtheissueofthebondtothetimeofthepayment.ThereisoneslightcomplicationinthattheactualRPIlevelusedinthesecalculationsissetbackeightmonths.ThusthebasemeasurementiseightmonthsbeforeissueandthescalingofanycouponiswithrespecttotheincreaseintheRPIfromthisbasemeasurementtotheleveloftheRPIeightmonthsbeforethecouponispaid.OneofthereasonsforthiscomplexityisthattheinitialestimateoftheRPIisusuallycorrectedatalaterdate.Figure1.15showstheUKgiltspricespublishedinTheFinancialTimesof14thApril2005.Theindex-linkedbondsareontheright.IntheUStheinflationindexistheConsumerPriceIndex(CPI).AtimeseriesofthisindexisshowninFigure1.16.Thedynamicsoftherelationshipbetweeninflationandshort-terminterestratesisparticularlyinteresting.Clearlythelevelofinterestrateswillaffecttherateofinflationdirectlythroughmortgagerepayments,butalsointerestratesareoftenusedbycentralbanksasatoolforkeepinginflationdown.WelookatthemodelingofinflationinChapter71. productsandmarketsChapter121Figure1.16TheCPIindex.Source:BloombergL.P.1.9FORWARDSANDFUTURESAforwardcontractisanagreementwhereonepartypromisestobuyanassetfromanotherpartyatsomespecifiedtimeinthefutureandatsomespecifiedprice.Nomoneychangeshandsuntilthedeliverydateormaturityofthecontract.Thetermsofthecontractmakeitanobligationtobuytheassetatthedeliverydate,thereisnochoiceinthematter.Theassetcouldbeastock,acommodityoracurrency.Afuturescontractisverysimilartoaforwardcontract.Futurescontractsareusuallytradedthroughanexchange,whichstandardizesthetermsofthecontracts.Theprofitorlossfromthefuturespositioniscalculatedeverydayandthechangeinthisvalueispaidfromonepartytotheother.Thuswithfuturescontractsthereisagradualpaymentoffundsfrominitiationuntilmaturity.Forwardsandfutureshavetwomainuses,inspeculationandinhedging.Ifyoubelievethatthemarketwillriseyoucanbenefitfromthisbyenteringintoaforwardorfuturescontract. 22PartOnemathematicalandfinancialfoundationsIfyourmarketviewisrightthenalotofmoneywillchangehands(atmaturityoreveryday)inyourfavor.Thatisspeculationandisveryrisky.Hedgingistheopposite,itisavoidanceofrisk.Forexample,ifyouareexpectingtogetpaidinyeninsixmonths’time,butyouliveinAmericaandyourexpensesareallindollars,thenyoucouldenterintoafuturescontracttolockinaguaranteedexchangeratefortheamountofyouryenincome.Oncethisexchangerateislockedinyouarenolongerexposedtofluctuationsinthedollar/yenexchangerate.Butthenyouwon’tbenefitiftheyenappreciates.1.9.1AFirstExampleofNoArbitrageFuturesandforwardsprovideuswithourfirstexampleoftheno-arbitrageprinciple.Consideraforwardcontractthatobligesustohandoveranamount$FattimeTtoreceivetheunderlyingasset.Today’sdateistandthepriceoftheassetiscurrently$S(t),thisisthespotprice,theamountforwhichwecouldgetimmediatedeliveryoftheasset.Whenwegettomaturitywewillhandovertheamount$Fandreceivetheasset,thenworth$S(T).Howmuchprofitwemakecannotbeknownuntilweknowthevalue$S(T),andwecan’tknowthisuntiltimeT.FromnowonIamgoingtodropthe‘$’signfrominfrontofmonetaryamounts.WeknowallofF,S(t),tandT.Butisthereanyrelationshipbetweenthem?Youmightthinknot,sincetheforwardcontractentitlesustoreceiveanamountS(T)−Fatexpiryandthisisunknown.However,byenteringintoaspecialportfoliooftradesnowwecaneliminateallrandomnessinthefuture.Thisisdoneasfollows.Enterintotheforwardcontract.Thiscostsusnothingupfrontbutexposesustotheuncer-taintyinthevalueoftheassetatmaturity.Simultaneouslyselltheasset.Itiscalledgoingshortwhenyousellsomethingyoudon’town.Thisispossibleinmanymarkets,butwithsometimingrestrictions.WenowhaveanamountS(t)incashduetothesaleoftheasset,aforwardcontract,andashortassetposition.Butournetpositioniszero.Putthecashinthebank,toreceiveinterest.WhenwegettomaturitywehandovertheamountFandreceivetheasset,thiscan-celsourshortassetpositionregardlessofthevalueofS(T).Atmaturityweareleftwithaguaranteed−Fincashaswellasthebankaccount.Theword‘guaranteed’isimportantbecauseitemphasizesthatitisindependentofthevalueoftheasset.ThebankaccountcontainstheinitialinvestmentofanamountS(t)withaddedinterest,whichhasavalueatmaturityofS(t)er(T−t).OurnetpositionatmaturityisthereforeS(t)er(T−t)−F.Sincewebeganwithaportfolioworthzeroandweendupwithapredictableamount,thatpredictableamountshouldalsobezero.WecanconcludethatF=S(t)er(T−t).(1.3) productsandmarketsChapter123Table1.1Cashflowsinahedgedportfolioofassetandforward.HoldingWorthWorthattoday(t)maturity(T)Forward0S(T)−F−Stock−S(t)−S(T)CashS(t)S(t)er(T−t)Total0S(t)er(T−t)−F200180ForwardSpotassetprice1601401201008060Maturity40200t00.20.40.60.81Figure1.17Atimeseriesofaspotassetpriceanditsforwardprice.Thisistherelationshipbetweenthespotpriceandtheforwardprice.Itisalinearrelationship,theforwardpriceisproportionaltothespotprice.ThecashflowsinthisspecialhedgedportfolioareshowninTable1.1.Figure1.17showsapathtakenbythespotassetpriceanditsforwardprice.Aslongasinterestratesareconstant,thesetwoarerelatedby(1.3).Ifthisrelationshipisviolatedthentherewillbeanarbitrageopportunity.Toseewhatismeantbythis,imaginethatFislessthanS(t)er(T−t).Toexploitthisandmakearisklessarbitrageprofit,enterintothedealsasexplainedabove.AtmaturityyouwillhaveS(t)er(T−t)inthebank,ashortassetandalongforward.TheassetpositioncancelswhenyouhandovertheamountF,leavingyouwithaprofitofS(t)er(T−t)−F.IfFisgreaterthanthatgivenby(1.3)thenyouenterintotheoppositepositions,goingshorttheforward.Againyoumakearisklessprofit.Thestandardeconomicargumentthensaysthatinvestorswillactquicklytoexploittheopportunity,andintheprocesspriceswilladjusttoeliminateit. 24PartOnemathematicalandfinancialfoundations1.10SUMMARYTheabovedescriptionsoffinancialmarketsareenoughforthisintroductorychapter.Perhapsthemostimportantpointtotakeawaywithyouistheideaofnoarbitrage.Intheexamplehere,relatingspotpricestofuturesprices,wesawhowwecouldsetupaverysimpleportfoliowhichcompletelyeliminatedanydependenceonthefuturevalueofthestock.Whenwecometovaluederivatives,inthewaywejustvaluedafuture,wewillseethatthesameprinciplecanbeappliedalbeitinafarmoresophisticatedway.FURTHERREADING•Forgeneralfinancialnewsvisitwww.bloomberg.comandwww.reuters.com.CNNhasonlinefinancialnewsatwww.cnnfn.com.TherearealsoonlineeditionsofTheWallStreetJournal,www.wsj.com,TheFinancialTimes,www.ft.comandFuturesandOptionsWorld,www.fow.com.•FormoreinformationaboutfuturesseetheChicagoBoardofTradewebsitewww.cbot.com.•Many,manyfinanciallinkscanbefoundatWahoo!,www.io.com/˜gibbonsb/wahoo.html.•Inthemain,we’llbeassumingthatmarketsarerandom.ForinsightaboutalternativehypothesesseeSchwager(1990,1992).•SeeBrooks(1967)forhowtheraisingofcapitalforabusinessmightworkinpractice.•Cox,Ingersoll&Ross(1981)discusstherelationshipbetweenforwardandfutureprices. CHAPTER2derivativesInthisChapter...•thedefinitionsofbasicderivativeinstruments•optionjargon•howtodrawpayoffdiagrams•noarbitrageandput-callparity•simpleoptionstrategies2.1INTRODUCTIONThepreviouschapterdealtwithsomeofthebasicsoffinancialmarkets.Ididn’tgointoanydetail,justgivingthebarestoutlineandsettingthesceneforthischapter.HereIintroducethethemethatiscentraltothebook,thesubjectofoptions,a.k.a.derivativesorcontingentclaims.Thischapterisnontechnical,beingadescriptionofsomeofthemostcommonoptioncontracts,andexplanationofthemarket-standardjargon.ItisinlaterchaptersthatIstarttogettechnical.Optionshavebeenaroundformanyyears,butitwasonlyon26thApril1973thattheywerefirsttradedonanexchange.ItwasthenthatTheChicagoBoardOptionsExchange(CBOE)firstcreatedstandardized,listedoptions.Initiallytherewerejustcallson16stocks.Putsweren’tevenintroduceduntil1977.IntheUSoptionsaretradedonCBOE,theAmericanStockExchange,thePacificExchangeandthePhiladelphiaStockExchange.Worldwide,thereareover50exchangesonwhichoptionsaretraded.2.2OPTIONSIfyouarereadingthebookinalinearfashion,fromstarttofinish,thenthelasttopicsyoureadaboutwillhavebeenfuturesandforwards.Theholderoffutureorforwardcontractsisobligedtotradeatthematurityofthecontract.Unlessthepositionisclosedbeforematuritytheholdermusttakepossessionofthecommodity,currencyorwhateveristhesubjectofthecontract,regardlessofwhethertheassethasrisenorfallen.Wouldn’titbeniceifweonlyhadtotakepossessionoftheassetifithadriseninvalue? 26PartOnemathematicalandfinancialfoundationsThesimplestoptiongivestheholdertherighttotradeinthefutureatapreviouslyagreedpricebuttakesawaytheobligation.Soifthestockfalls,wedon’thavetobuyitafterall.AcalloptionistherighttobuyaparticularassetforanagreedamountataspecifiedtimeinthefutureAsanexample,considerthefollowingcalloptiononMicrosoftstock.ItgivestheholdertherighttobuyoneofMicrosoftstockforanamount$25inonemonth’stime.Today’sstockpriceis$24.5.Theamount‘25’whichwecanpayforthestockiscalledtheexercisepriceorstrikeprice.Thedateonwhichwemustexerciseouroption,ifwedecideto,iscalledtheexpiryorexpirationdate.Thestockonwhichtheoptionisbasedisknownastheunderlyingasset.Let’sconsiderwhatmayhappenoverthenextmonth,upuntilexpiry.Supposethatnothinghappens,thatthestockpriceremainsat$24.5.Whatdowedoatexpiry?Wecouldexercisetheoption,handingover$25toreceivethestock.Wouldthatbesensible?No,becausethestockisonlyworth$24.5,eitherwewouldn’texercisetheoptionorifwereallywantedthestockwewouldbuyitinthestockmarketforthe$24.5.Butwhatifthestockpricerisesto$29?Thenwe’dbelaughing,wewouldexercisetheoption,paying$25forastockthat’sworth$29,aprofitof$4.Wewouldexercisetheoptionatexpiryifthestockisabovethestrikeandnotifitisbelow.IfweuseStomeanthestockpriceandEthestrikethenatexpirytheoptionisworthmax(S−E,0).Thisfunctionoftheunderlyingassetiscalledthepayofffunction.The‘max’functionrepre-sentstheoptionality.Whywouldwebuysuchanoption?Clearly,ifyouownacalloptionyouwantthestocktoriseasmuchaspossible.Thehigherthestockpricethegreaterwillbeyourprofit.Iwilldiscussthisbelow,butourdecisionwhethertobuyitwilldependonhowmuchitcosts;theoptionisvaluable,thereisnodownsidetoitunlikeafuture.Inourexampletheoptionwasvaluedat$1.875.Wheredidthisnumbercomefrom?Thevaluationofoptionsisoneofthesubjectsofthisbook,andI’llbeshowingyouhowtofindthisvaluelateron.Whatifyoubelievethatthestockisgoingtofall,isthereacontractthatyoucanbuytobenefitfromthefallinastockprice?AputoptionistherighttosellaparticularassetforanagreedamountataspecifiedtimeinthefutureTheholderofaputoptionwantsthestockpricetofallsothathecanselltheassetformorethanitisworth.Thepayofffunctionforaputoptionismax(E−S,0).Nowtheoptionisonlyexercisedifthestockfallsbelowthestrikeprice. derivativesChapter227Publisher'sNote:Permissiontoreproducethisimageonlinewasnotgrantedbythecopyrightholder.Readersarekindlyrequestedtorefertotheprintedversionofthischapter.Figure2.1TheWallStreetJournalEuropeof14thApril2005,StockOptions.ReproducedbypermissionofDowJones&Company,Inc. 28PartOnemathematicalandfinancialfoundationsPublisher'sNote:Permissiontoreproducethisimageonlinewasnotgrantedbythecopyrightholder.Readersarekindlyrequestedtorefertotheprintedversionofthischapter.Figure2.2TheWallStreetJournalEuropeof5thJanuary2000,IndexOptions.ReproducedbypermissionofDowJones&Company,Inc. derivativesChapter229Figure2.1isanexcerptfromTheWallStreetJournalEuropeof14thApril2005showingoptionsonvariousstocks.Thetablelistsclosingpricesoftheunderlyingstocksandthelasttradedpricesoftheoptionsonthestocks.TounderstandhowtoreadthisletusexaminethepricesofoptionsonApple.Goto‘AppleC’inthelist,thereareseveralinstances.Theclosingpriceon13thApril2005was$41.35,(theLASTcolumn,secondfromtheright).Callsandputsarequotedherewithstrikesof$37.50,$40,...,$47.50,$50,othersmayexistbutarenotincludedinthenewspaper.TheexpiriesmentionedareApril,MayandJuly.Partoftheinformationincludedhereisthevolumeofthetransactionsineachseries,wewon’tworryaboutthatbutsomepeopleuseoptionvolumeasatradingindicator.Fromthedata,wecanseethattheAprilcallswithastrikeof$40wereworth$2.40.Theputswithsamestrikeandexpirywereworth$1.20.TheAprilcallswithastrikeof$42.50wereworth$1.20andtheputswithsamestrikeandexpirywereworth$2.45.Notethatthehigherthestrike,thelowerthevalueofthecallsbutthehigherthevalueoftheputs.Thismakessensewhenyourememberthatthecallallowsyoutobuytheunderlyingforthestrike,sothatthelowerthestrikepricethemorethisrightisworthtoyou.Theoppositeistrueforaputsinceitallowsyoutoselltheunderlyingforthestrikeprice.Therearemorestrikesandexpiriesavailableforoptionsonindices,solet’snowlookattheIndexOptionssectionofTheWallStreetJournalEurope5thJanuary2000,thisisshowninFigure2.2.InFigure2.3arethequotedpricesoftheMarchandJuneDJIAcallsagainstthestrikeprice.Alsoplottedisthepayofffunctioniftheunderlyingweretofinishatitscurrentvalueatexpiry,theclosingpriceoftheDJIAwas10997.93onthedaytheoptionpriceswerequoted.Thisplotreinforcesthefactthatthehigherthestrikethelowerthevalueofacalloption.Italsoappearsthatthelongerthetimetomaturitythehigherthevalueofthecall.Isitobvious98DJIAMarchcalls7DJIAJancallsPayoff65Value4321070727476788082848688StrikeFigure2.3Optionpricesversusstrike,MarchandJuneseriesofDJIA. 30PartOnemathematicalandfinancialfoundationsthatthisshouldbeso?Asthetimetoexpirydecreaseswhatwouldweseehappen?Asthereislessandlesstimefortheunderlyingtomove,sotheoptionvaluemustconvergetothepayofffunction.Oneofthemostinterestingfeaturesofcallsandputsisthattheyhaveanon-lineardependenceontheunderlyingasset.Thiscontrastswithfutureswhichhavealineardependenceontheunderlying.Thisnon-linearityisveryimportantinthepricingofoptionssincetherandomnessintheunderlyingassetandthecurvatureoftheoptionvaluewithrespecttotheassetareintimatelyrelated.Callsandputsarethetwosimplestformsofoption.Forthisreasontheyareoftenreferredtoasvanillabecauseoftheubiquityofthatflavor.Therearemany,manymorekindsofoptions,someofwhichwillbedescribedandexaminedlateron.Othertermsusedtodescribecontractswithsomedependenceonamorefundamentalassetarederivativesorcontin-gentclaims.Figure2.4showsthepricesofcalloptionsonGlaxo–WellcomeforavarietyofstrikesasofJanuary.AlltheseoptionsareexpiringinOctober.Thetableshowsmanyotherquantitiesthatwewillbeseeinglateron.Figure2.4PricesforGlaxo–WellcomecallsexpiringinOctober.Source:BloombergL.P. derivativesChapter2312.3DEFINITIONOFCOMMONTERMSThesubjectsofmathematicalfinanceandderivativestheoryarefilledwithjargon.Thejargoncomesfromboththemathe-maticalworldandthefinancialworld.Generallyspeakingthejargonfromfinanceisaimedatsimplifyingcommunication,andtoputeveryoneonthesamefooting.1Hereareafewloosedefinitionstobegoingonwith,someyouhavealreadyseenandtherewillbemanymorethroughoutthebook.•Premium:Theamountpaidforthecontractinitially.Howtofindthisvalueisthesubjectofmuchofthisbook.•Underlying(asset):Thefinancialinstrumentonwhichtheoptionvaluedepends.Stocks,commodities,currenciesandindicesaregoingtobedenotedbyS.Theoptionpayoffisdefinedassomefunctionoftheunderlyingassetatexpiry.•Strike(price)orexerciseprice:Theamountforwhichtheunderlyingcanbebought(call)orsold(put).ThiswillbedenotedbyE.Thisdefinitiononlyreallyappliestothesimplecallsandputs.Wewillseemorecomplicatedcontractsinlaterchaptersandthedefinitionofstrikeorexercisepricewillbeextended.•Expiration(date)orexpiry(date):Dateonwhichtheoptioncanbeexercisedordateonwhichtheoptionceasestoexistorgivetheholderanyrights.ThiswillbedenotedbyT.•Intrinsicvalue:Thepayoffthatwouldbereceivediftheunderlyingisatitscurrentlevelwhentheoptionexpires.•Timevalue:Anyvaluethattheoptionhasaboveitsintrinsicvalue.Theuncertaintysur-roundingthefuturevalueoftheunderlyingassetmeansthattheoptionvalueisgenerallydifferentfromtheintrinsicvalue.•Inthemoney:Anoptionwithpositiveintrinsicvalue.Acalloptionwhentheassetpriceisabovethestrike,aputoptionwhentheassetpriceisbelowthestrike.•Outofthemoney:Anoptionwithnointrinsicvalue,onlytimevalue.Acalloptionwhentheassetpriceisbelowthestrike,aputoptionwhentheassetpriceisabovethestrike.•Atthemoney:Acallorputwithastrikethatisclosetothecurrentassetlevel.•Longposition:Apositiveamountofaquantity,orapositiveexposuretoaquantity.•Shortposition:Anegativeamountofaquantity,oranegativeexposuretoaquantity.Manyassetscanbesoldshort,withsomeconstraintsonthelengthoftimebeforetheymustbeboughtback.1Ihaveseriousdoubtsaboutthepurposeofmostofthemathjargon. 32PartOnemathematicalandfinancialfoundations1009080706050Value403020Strike,E100050100150200Stockprice,SFigure2.5Payoffdiagramforacalloption.Figure2.6Bloombergoptionvaluationscreen,call.Source:BloombergL.P. derivativesChapter2332.4PAYOFFDIAGRAMSTheunderstandingofoptionsishelpedbythevisualinterpretationofanoption’svalueatexpiry.Wecanplotthevalueofanoptionatexpiryasafunctionoftheunderlyinginwhatisknownasapayoffdiagram.Atexpirytheoptionisworthaknownamount.Inthecaseofacalloptionthecontractisworthmax(S−E,0).ThisfunctionistheboldlineinFigure2.5.Figure2.6showsBloomberg’sstandardoptionvaluationscreenandFigure2.7showsthevalueagainsttheunderlyingandthepayoff.Thepayoffforaputoptionismax(E−S,0),thisistheboldlineplottedinFigure2.8.Figure2.9showsBloomberg’soptionvaluationscreenandFigure2.10showsthevalueagainsttheunderlyingandthepayoff.Thesepayoffdiagramsareusefulsincetheysimplifytheanalysisofcomplexstrategiesinvolvingmorethanoneoption.Makeanoteofthethinlinesinallofthesefigures.Themeaningofthesewillbeexplainedveryshortly.Figure2.7Bloombergscenarioanalysis,call.Source:BloombergL.P. 34PartOnemathematicalandfinancialfoundations1201008060Value40Strike,E200050100150Stockprice,SFigure2.8Payoffdiagramforaputoption.Figure2.9Bloombergoptionvaluationscreen,put.Source:BloombergL.P. derivativesChapter235Figure2.10Bloombergscenarioanalysis,put.Source:BloombergL.P.705030Value10S*050100150200S−10−30Figure2.11Profitdiagramforacalloption. 36PartOnemathematicalandfinancialfoundations2.4.1OtherRepresentationsofValueThepayoffdiagramsshownaboveonlytellyouaboutwhathappensatexpiry,howmuchmoneyyouroptioncontractisworthatthattime.Itmakesnoallowanceforhowmuchpremiumyouhadtopayfortheoption.Toadjustfortheoriginalcostoftheoption,sometimesoneplotsadiagramsuchasthatshowninFigure2.11.InthisprofitdiagramforacalloptionIhavesubtractedfromthepayoffthepremiumoriginallypaidforthecalloption.Thisfigureishelpfulbecauseitshowshowfarintothemoneytheassetmustbeatexpirybeforetheoptionbecomesprofitable.TheassetvaluemarkedS∗isthepointwhichdividesprofitfromloss;iftheassetatexpiryisabovethisvaluethenthecontracthasmadeaprofit,ifbelowthecontracthasmadealoss.Asitstands,thisprofitdiagramtakesnoaccountofthetimevalueofmoney.Thepremiumispaidupfrontbutthepayoff,ifany,isonlyreceivedatexpiry.Tobeconsistentoneshouldeitherdiscountthepayoffbymultiplyingbye−r(T−t)tovalueeverythingatthepresent,ormultiplythepremiumbyer(T−t)tovalueallcashflowsatexpiry.Figure2.12showsBloomberg’scalloptionprofitdiagram.Notethattheprofittodayiszero;ifwebuytheoptionandimmediatelysellitwemakeneitheraprofitnoraloss(thisissubjecttoissuesoftransactioncosts).Figure2.12Profitdiagramforacall.Source:BloombergL.P. derivativesChapter2372.5WRITINGOPTIONSIhavetalkedaboveabouttherightsofthepurchaseroftheoption.Butforeveryoptionthatissold,someonesomewheremustbeliableiftheoptionisexercised.IfIholdacalloptionentitlingmetobuyastocksometimeinthefuture,whodoIbuythisstockfrom?Ultimately,thestockmustbedeliveredbythepersonwhowrotetheoption.Thewriterofanoptionisthepersonwhopromisestodelivertheunderlyingasset,iftheoptionisacall,orbuyit,iftheoptionisaput.Thewriteristhepersonwhoreceivesthepremium.Inpractice,mostsimpleoptioncontractsarehandledthroughanexchangesothatthepur-chaserofanoptiondoesnotknowwhothewriteris.Theholderoftheoptioncanevenselltheoptionontosomeoneelseviatheexchangetoclosehisposition.However,regardlessofwhoholdstheoption,orwhohashandledit,thewriteristhepersonwhohastheobligationtodeliverorbuytheunderlying.Theasymmetrybetweenowningandwritingoptionsisnowclear.Thepurchaseroftheoptionhandsoverapremiuminreturnforspecialrights,andanuncertainoutcome.Thewriterreceivesaguaranteedpaymentupfront,butthenhasobligationsinthefuture.2.6MARGINWritingoptionsisveryrisky.Thedownsideofbuyinganoptionisjusttheinitialpremium,theupsidemaybeunlimited.Theupsideofwritinganoptionislimited,butthedownsidecouldbehuge.Forthisreason,tocovertheriskofdefaultintheeventofanunfavorableoutcome,theclearinghousesthatregisterandsettleoptionsinsistonthedepositofamarginbythewritersofoptions.Clearinghousesactascounterpartytoeachtransaction.Margincomesintwoforms,theinitialmarginandthemaintenancemargin.Theinitialmarginistheamountdepositedattheinitiationofthecontract.Thetotalamountheldasmarginmuststayaboveaprescribedmaintenancemargin.Ifiteverfallsbelowthislevelthenmoremoney(orequivalentinbonds,stocksetc.)mustbedeposited.Thelevelsofthesemarginsvaryfrommarkettomarket.Marginhasbeenmuchneglectedintheacademicliterature.Butapoorunderstandingofthesubjecthasledtoanumberoffamousfinancialdisasters,mostnotablyMetallgesellschaftandLongTermCapitalManagement.We’lldiscussthedetailsofthesecasesinChapter44,andwe’llalsobeseeinghowtomodelmarginandhowtomarginhedge.2.7MARKETCONVENTIONSMostofthesimpleroptionscontractsareboughtandsoldthroughexchanges.Theseexchangesmakeiteasierandmoreefficienttomatchbuyerswithsellers.Partofthissimplificationinvolvestheconventionsaboutsuchfeaturesofthecontractsastheavailablestrikesandexpiries.Forexample,vanillacallsandputscomeinseries.Thisreferstothestrikeandexpirydates. 38PartOnemathematicalandfinancialfoundationsTypicallyastockhasthreechoicesofexpiriestradingatanytime.Havingstandardizedcontractstradedthroughanexchangepromotesliquidityoftheinstruments.Someoptionsareanagreementbetweentwoparties,notthroughanexchange,butoftenbroughttogetherbyanintermediary.Theseagreementscanbeveryflexibleandthecontractdetailsdonotneedtosatisfyanyconventions.SuchcontractsareknownasoverthecounterorOTCcontracts.Igiveanexampleattheendofthischapter.2.8THEVALUEOFTHEOPTIONBEFOREEXPIRYWehaveseenhowmuchcallsandputsareworthatexpiry,anddrawnthesevaluesinpayoffdiagrams.Thequestionthatwecanask,andthequestionthatiscentraltothisbook,is‘Howmuchisthecontractworthnow,beforeexpiry?’Howmuchwouldyoupayforacontract,apieceofpaper,givingyourightsinthefuture?Youmayhavenoideawhatthestockpricewilldobetweennowandexpiryinsixmonths,say,butclearlythecontracthasvalue.Attheveryleastyouknowthatthereisnodownsidetoowningtheoption,thecontractgivesyouspecificrightsbutnoobligations.Twothingsareclearaboutthecontractvaluebeforeexpiry:thevaluewilldependonhowhightheassetpriceistodayandhowlongthereisbeforeexpiry.Thehighertheunderlyingassettoday,thehigherwemightexpecttheassettobeatexpiryoftheoptionandthereforethemorevaluablewemightexpectacalloptiontobe.Ontheotherhandaputoptionmightbecheaperbythesamereasoning.Thedependenceontimetoexpiryismoresubtle.Thelongerthetimetoexpiry,themoretimethereisfortheassettoriseorfall.Isthatgoodorbadifweownacalloption?Furthermore,thelongerwehavetowaituntilwegetanypayoff,thelessvaluablewillbethatpayoffsimplybecauseofthetimevalueofmoney.Iwillaskyoutosuspenddisbeliefforthemoment(itwon’tbethefirsttimeinthebook)andtrustmethatwewillbefindinga‘fairvalue’fortheseoptionscontracts.Theaspectoffindingthe‘fairvalue’thatIwanttofocusonnowisthedependenceontheassetpriceandtime.IamgoingtouseVtomeanthevalueoftheoption,anditwillbeafunctionofthevalueoftheunderlyingassetSattimet.ThuswecanwriteV(S,t)forthevalueofthecontract.Weknowthevalueofthecontractatexpiry.IfIuseTtodenotetheexpirydatethenatt=TthefunctionVisknown,itisjustthepayofffunction.ForexampleifwehaveacalloptionthenV(S,T)=max(S−E,0).ThisisthefunctionofSthatIplottedintheearlierpayoffdiagrams.NowIcantellyouwhatthefinelinesareinFigures2.5and2.8:theyarethevaluesofthecontractsV(S,t)atsometimebeforeexpiry,plottedagainstS.Ihavenotspecifiedhowlongbeforeexpiry,sincetheplotisforexplanatorypurposesonly.Iwillspendalotoftimeshowingyouhowtofindthesevaluesforawidevarietyofcontracts.2.9FACTORSAFFECTINGDERIVATIVESPRICESThetwomostimportantfactorsaffectingthepricesofoptionsarethevalueoftheunderlyingassetSandthetimetoexpiryt.Thesequantitiesarevariablesmeaningthattheyinevitablychangeduringthelifeofthecontract;iftheunderlyingdidnotchangethenthepricingwouldbetrivial.Thiscontrastswiththeparametersthataffectthepriceofoptions. derivativesChapter239Examplesofparametersaretheinterestrateandstrikeprice.Theinterestratewillhaveaneffectontheoptionvalueviathetimevalueofmoneysincethepayoffisreceivedinthefuture.Theinterestratealsoplaysanotherrolewhichwewillseelater.Clearlythestrikepriceisimportant;thehigherthestrikeinacall,thelowerthevalueofthecall.Ifwehaveanequityoptionthenitsvaluewilldependonanydividendsthatarepaidontheassetduringtheoption’slife.IfwehaveanFXoptionthenitsvaluewilldependontheinterestratereceivedbytheforeigncurrency.ThereisoneimportantparameterthatIhavenotmentioned,andwhichhasamajorimpactontheoptionvalue.Thatparameteristhevolatility.Volatilityisameasureoftheamountoffluctuationintheassetprice,ameasureoftherandomness.Figure2.13showstwoassetpricepaths,themorejaggedofthetwohasthehighervolatility.Thetechnicaldefinitionofvolatilityisthe‘annualizedstandarddeviationoftheassetreturns.’IwillshowhowtomeasurethisparameterinChapter3.Volatilityisaparticularlyinterestingparameterbecauseitissohardtoestimate.Andhavingestimatedit,onefindsthatitneverstaysconstantandisunpredictable.Onceyoustarttothinkofthevolatilityasvaryinginarandomfashionthenitbecomesnaturaltotreatitasavariablealso.Wewillseeexamplesofthislaterinthebook.216014012010080Stockprice604020000.511.522.5t3TimeFigure2.13Two(simulated)assetpricepaths,oneismuchmorevolatilethantheother.2Infinanceweareoverloadedwithdata,butitisneverclearhowusefulthesedatawillbeforhelpingustomodelthefuture.Contrastthiswithproblemsinthehardsciences.Forexample,theaveragenumberofsunspotsonthesunhasprobablybeenquitestableformillionsofyears,butdatagoingthatfarbackareimpossibletoget,obviously. 40PartOnemathematicalandfinancialfoundationsThedistinctionbetweenparametersandvariablesisveryimportant.Ishallbederivingequationsforthevalueofoptions,partialdifferentialequations.Theseequationswillinvolvedifferentiationwithrespecttothevariables,buttheparameters,astheirnamesuggests,remainasparametersintheequations.2.10SPECULATIONANDGEARINGIfyoubuyafarout-of-the-moneyoptionitmaynotcostverymuch,especiallyifthereisnotverylonguntilexpiry.Iftheoptionexpiresworthless,thenyoualsohaven’tlostverymuch.However,ifthereisadramaticmoveintheunderlying,sothattheoptionexpiresinthemoney,youmaymakealargeprofitrelativetotheamountoftheinvestment.Letmegiveanexample.ExampleToday’sdateis14thAprilandthepriceofWilmottInc.stockis$666.Thecostofa680calloptionwithexpiry22ndAugustis$39.IexpectthestocktorisesignificantlybetweennowandAugust,howcanIprofitifIamright?BuythestockSupposeIbuythestockfor$666.AndsupposethatbythemiddleofAugustthestockhasrisento$730.Iwillhavemadeaprofitof$64perstock.Moreimportantlymyinvestmentwillhaverisenby730−666×100=9.6%.666BuythecallIfIbuythecalloptionfor$39,thenatexpiryIcanexercisethecall,paying$680toreceivesomethingworth$730.Ihavepaid$39andIgetback$50.Thisisaprofitof$11peroption,butinpercentagetermsIhavemadevalueofassetatexpiry−strike−costofcall730−680−39×100=×100=28%.costofcall39Thisisanexampleofgearingorleverage.Theout-of-the-moneyoptionhasahighgearing,apossiblehighpayoffforasmallinvestment.Thedownsideofthisleverageisthatthecalloptionismorelikelythannottoexpirecompletelyworthlessandyouwillloseallofyourinvestment.IfWilmottInc.remainsat$666thenthestockinvestmenthasthesamevaluebutthecalloptionexperiencesa100%loss.Highly-leveragedcontractsareveryriskyforthewriteroftheoption.Thebuyerisonlyriskingasmallamount;althoughheisverylikelytolose,hisdownsideislimitedtohisinitialpremium.Butthewriterisriskingalargelossinordertomakeaprobablesmallprofit.Thewriterislikelytothinktwiceaboutsuchadealunlesshecanoffsethisriskbybuyingothercontracts.Thisoffsettingofriskbybuyingotherrelatedcontractsiscalledhedging.Gearingexplainsoneofthereasonsforbuyingoptions.Ifyouhaveastrongviewaboutthedirectionofthemarketthenyoucanexploitderivativestomakeabetterreturn,ifyouareright,thanbuyingorsellingtheunderlying. derivativesChapter2412.11EARLYEXERCISEThesimpleoptionsdescribedaboveareexamplesofEuropeanoptionsbecauseexerciseisonlypermittedatexpiry.Somecontractsallowtheholdertoexerciseatanytimebeforeexpiry,andthesearecalledAmericanoptions.AmericanoptionsgivetheholdermorerightsthantheirEuropeanequivalentandcanthereforebemorevaluable,andtheycanneverbelessvaluable.ThemainpointofinterestwithAmerican-stylecontractsisdecidingwhentoexercise.InChapter9IwilldiscussAmericanoptionsindepth,andshowhowtodeterminewhenitisoptimaltoexercise,soastogivethecontractthehighestvalue.Notethattheterms‘European’and‘American’donotinanywayrefertothecontinentsonwhichthecontractsaretraded.Finally,thereareBermudanoptions.Theseallowexerciseonspecifieddates,orinspecifiedperiods.InasensetheyarehalfwaybetweenEuropeanandAmericansinceexerciseisallowedonsomedaysandnotonothers.2.12PUT-CALLPARITYImaginethatyoubuyoneEuropeancalloptionwithastrikeofEandanexpiryofTandthatyouwriteaEuropeanputoptionwiththesamestrikeandexpiry.Today’sdateist.ThepayoffyoureceiveatTforthecallwilllooklikethelineinthefirstplotofFigure2.14.Thepayofffortheputisthelineinthesecondplotinthefigure.Notethatthesignofthepayoffisnegative;youwrotetheoptionandareliableforthepayoff.Thepayofffortheportfolioofthetwooptionsisthesumoftheindividualpayoffs,showninthethirdplot.Thepayoffforthisportfolioofoptionsismax(S(T)−E,0)−max(E−S(T),0)=S(T)−E,whereS(T)isthevalueoftheunderlyingassetattimeT.Theright-handsideofthisexpressionconsistsoftwoparts,theassetandafixedsumE.Isthereanotherwaytogetexactlythispayoff?IfIbuytheassettodayitwillcostmeS(t)andbeworthS(T)atexpiry.Idon’tknowwhatthevalueS(T)willbebutIdoknowhowtoguaranteetogetthatamount,andthatistobuytheasset.WhatabouttheEterm?TolockinapaymentofEattimeTinvolvesacashflowofEe−r(T−t)attimet.Theconclusionisthattheportfolioofalongcallandashortputgivesmeexactlythesamepayoffasalongasset,shortcashposition.Theequalityofthesecashflowsisindependentofthefuturebehaviorofthestockandismodelindependent:C−P=S−Ee−r(T−t),whereCandParetoday’svaluesofthecallandtheputrespectively.Thisrelationshipholdsatanytimeuptoexpiryandisknownasput-callparity.Ifthisrelationshipdidnotholdthentherewouldberisklessarbitrageopportunities.InTable2.1Ishowthecashflowsintheperfectlyhedgedportfolio.InthistableIhavesetupthecashflowstohaveaguaranteedvalueofzeroatexpiry. 42PartOnemathematicalandfinancialfoundationsTable2.1Cashflowsinahedgedportfolioofoptionsandasset.HoldingWorthWorthattoday(t)expiry(T)CallCmax(S(T)−E,0)−Put−P−max(E−S(T),0)−Stock−S(t)−S(T)CashEe−r(T−t)ETotalC−P−S(t)+Ee−r(T−t)010080100Longcall-1604020S00501001502000−2050100150S200−40Shortput−60−80−100−120150Call-put10050S050100150200−50−100−150Figure2.14Schematicdiagramshowingput-callparity. derivativesChapter2432.13BINARIESORDIGITALSTheoriginalandstillmostcommoncontractsarethevanillacallsandputs.Increasinglyimportantarethebinaryordigitaloptions.Thesecontractshaveapayoffatexpirythatisdiscontinuousintheunderlyingassetprice.Anexampleofthepayoffdiagramforoneoftheseoptions,abinarycall,isshowninFigure2.15.Thiscontractpays$1atexpiry,timeT,iftheassetpriceisthengreaterthantheexercisepriceE.Again,andaswiththerestofthefiguresinthischapter,theboldlineisthepayoffandthefinelineisthecontractvaluesometimebeforeexpiry.Whywouldyouinvestinabinarycall?Ifyouthinkthattheassetpricewillrisebyexpiry,tofinishabovethestrikepricethenyoumightchoosetobuyeitheravanillacallorabinarycall.Thevanillacallhasthebestupsidepotential,growinglinearlywithSbeyondthestrike.Thebinarycall,however,canneverpayoffmorethanthe$1.Ifyouexpecttheunderlyingwillrisedramaticallythenitmaybebesttobuythevanillacall.Ifyoubelievethattheassetrisewillbelessdramaticthenbuythebinarycall.Thegearingofthevanillacallisgreaterthanthatforabinarycallifthemoveintheunderlyingislarge.Figure2.16showsthepayoffdiagramforabinaryput,theholderofwhichreceives$1iftheassetisbelowEatexpiry.Thebinaryputwouldbeboughtbysomeoneexpectingamodestfallintheassetprice.Thereisaparticularlysimplebinaryput-callparityrelationship.Whatdoyougetatexpiryifyouholdbothabinarycallandabinaryputwiththesamestrikesandexpiries?Theansweristhatyouwillalwaysget$1regardlessoftheleveloftheunderlyingatexpiry.ThusBinarycall+Binaryput=e−r(T−t).1.210.80.6Value0.40.20050100150S200Figure2.15Payoffdiagramforabinarycalloption. 44PartOnemathematicalandfinancialfoundations1.210.80.6Value0.40.20050100150S200Figure2.16Payoffdiagramforabinaryputoption.Whatwouldthetableofcashflowslooklikefortheperfectlyhedgeddigitalportfolio?2.14BULLANDBEARSPREADSApayoffthatissimilartoabinaryoptioncanbemadeupwithvanillacalls.Thisisourfirstexampleofaportfolioofoptionsoranoptionstrategy.SupposeIbuyonecalloptionwithastrikeof100andwriteanotherwithastrikeof120andwiththesameexpirationasthefirstthenmyresultingportfoliohasapayoffthatisshowninFigure2.17.Thispayoffiszerobelow100,20above120andlinearinbetween.Thepayoffiscontinuous,unlikethebinarycall,buthasapayoffthatissuperficiallysimilar.Thisstrategyiscalledabullspreadoracallspread;itbenefitsfromabull,i.e.rising,market.Thepayoffforageneralbullspread,madeupofcallswithstrikesE1andE2,isgivenby1(max(S−E1,0)−max(S−E2,0)),E2−E1whereE>E.HereIhavebought/sold(E−E)−1ofeachoftheoptionssothatthe2121maximumpayoffisscaledto1.IfIwriteaputoptionwithstrike100andbuyaputwithstrike120IgetthepayoffshowninFigure2.18.Thisiscalledabearspreadoraputspread,benefittingfromabear,i.e.falling,market.Again,itisverysimilartoabinaryputexceptthatthepayoffiscontinuous.Becauseofput-callparityitispossibletobuildupthesepayoffsusingothercontracts.Astrategyinvolvingoptionsofthesametype(i.e.callsorputs)iscalledaspread. derivativesChapter245252015Value1050050100150200SFigure2.17Payoffdiagramforabullspread.252015Value1050050100150200SFigure2.18Payoffdiagramforabearspread. 46PartOnemathematicalandfinancialfoundations2.15STRADDLESANDSTRANGLESIfyouhaveapreciseviewonthebehavioroftheunderlyingassetyoumaywanttobemorepreciseinyourchoiceofoption;simplecalls,puts,andbinariesmaybetoocrude.Thestraddleconsistsofacallandaputwiththesamestrike.ThepayoffdiagramisshowninFigure2.19.Suchapositionisusuallyboughtatthemoneybysomeonewhoexpectstheunderlyingtoeitherriseorfall,butnottoremainatthesamelevel.Forexample,justbeforeananticipatedmajornewsitemstocksoftenshowa‘calmbeforethestorm.’Ontheannouncementthestocksuddenlymoveseitherupordowndependingonwhetherornotthenewswasfavorabletothecompany.Theymayalsobeboughtbytechnicaltraderswhoseethestockatakeysupportorresistancelevelandexpectthestocktoeitherbreakthroughdramaticallyorbounceback.Thestraddlewouldbesoldbysomeonewiththeoppositeview,someonewhoexpectstheunderlyingpricetoremainstable.Figure2.20showstheBloombergscreenforsettingupastraddle.Figure2.21showstheprofitandlossforthispositionatvarioustimesbeforeexpiry.Theprofit/lossistheoptionvaluelesstheupfrontpremium.Thestrangleissimilartothestraddleexceptthatthestrikesoftheputandthecallaredif-ferent.Thecontractcanbeeitheranout-of-the-moneystrangleoranin-the-moneystrangle.Thepayoffforanout-of-themoneystrangleisshowninFigure2.22.Themotivationbehindthepurchaseofthispositionissimilartothatforthepurchaseofastraddle.Thedifferenceisthatthebuyerexpectsanevenlargermoveintheunderlyingonewayortheother.Thecontractisusuallyboughtwhentheassetisaroundthemiddleofthetwostrikesandischeaperthanastraddle.Thischeapnessmeansthatthegearingfortheout-of-the-moneystrangleishigherthanthatforthestraddle.Thedownsideisthatthereisamuchgreaterrangeoverwhichthe1201008060Value40200050100150S200Figure2.19Payoffdiagramforastraddle. derivativesChapter247Figure2.20Aportfoliooftwooptionsmakingupastraddle.Source:BloombergL.P.stranglehasnopayoffatexpiry;forthestraddlethereisonlytheonepointatwhichthereisnopayoff.Thereisanotherreasonforastraddleorstrangletradethatdoesnotinvolveaviewonthedirectionoftheunderlying.Thesecontractsareboughtorsoldbythosewithaviewonthedirectionofvolatility,theyareoneofthesimplestvolatilitytrades.Becauseoftherelationshipbetweenthepriceofanoptionandthevolatilityoftheassetonecanspeculateonthedirectionofvolatility.Doyouexpectthevolatilitytorise?Ifso,howcanyoubenefitfromthis?Untilweknowmoreaboutthisrelationship,wecannotgointothisinmoredetail.Straddlesandstranglesarerarelyhelduntilexpiry.Astrategyinvolvingoptionsofdifferenttypes(i.e.bothcallsandputs)iscalledacombi-nation.2.16RISKREVERSALTheriskreversalisacombinationofalongcall,withstrikeabovethecurrentspot,andashortputwithastrikebelowthecurrentspot.Bothhavethesameexpiry.ThepayoffisshowninFigure2.23. 48PartOnemathematicalandfinancialfoundationsFigure2.21Profit/lossforthestraddleatseveraltimesbeforeexpiry.Source:BloombergL.P.1201008060Value40200050100150S200Figure2.22Payoffdiagramforastrangle. derivativesChapter249200050100150S200−20−40Value−60−80−100Figure2.23Payoffdiagramforariskreversal.Theriskreversalisaveryspecialcontract,popularwithpractitioners.Itsvalueisusuallyquitesmallandrelatedtothemarket’sexpectationsofthebehaviorofvolatility.Thisistoocomplextogointonow,butwillbeexplainedinChapter50.2.17BUTTERFLIESANDCONDORSAmorecomplicatedstrategyinvolvingthepurchaseandsaleofoptionswiththreedifferentstrikesisabutterflyspread.Buyingacallwithastrikeof90,writingtwocallsstruckat100andbuyinga110callgivesthepayoffinFigure2.24.Thisisthekindofpositionyoumightenterifyoubelievethattheassetisnotgoinganywhere,eitherupordown.Becauseithasnolargeupsidepotential(inthiscasethemaximumpayoffis10)thepositionwillberelativelycheap.Withoptions,cheapisgood.Thecondorislikeabutterflyexceptthatfourstrikes,andfourcalloptions,areused.ThepayoffisshowninFigure2.25.2.18CALENDARSPREADSAllofthestrategiesIhavedescribedabovehaveinvolvedbuyingorwritingcallsandputswithdifferentstrikesbutallwiththesameexpiration.Astrategyinvolvingoptionswithdifferentexpirydatesiscalledacalendarspread.Youmayenterintosuchapositionifyouhaveapreciseviewonthetimingofamarketmoveaswellasthedirectionofthemove.Asalwaysthemotivebehindsuchastrategyistoreducethepayoffatassetvaluesandtimeswhichyoubelieveareirrelevant,whileincreasingthepayoffwhereyouthinkitwillmatter.Anyreductioninpayoffwillreducetheoverallvalueoftheoptionposition. 50PartOnemathematicalandfinancialfoundations121086Value420050100150SFigure2.24Payoffdiagramforabutterflyspread.121086Value420050100150SFigure2.25Payoffdiagramforacondor. derivativesChapter2512.19LEAPSANDFLEXLEAPSorLong-termequityanticipationsecuritiesarelonger-datedexchange-tradedcallsandputs.TheybegantradingontheCBOEinthelate1980s.Theyarestandardizedsothattheyareavailablewithexpiriesuptothreeyears.Theycomewiththreestrikes,correspondingtoatthemoneyandapproximately20%inandoutofthemoneywithrespecttotheunderlyingassetpricewhenissued.In1993theCBOEcreatedFLEXorFLexibleEXchange-tradedoptionsonseveralindices.Theseallowadegreeofcustomization,intheexpirydate(uptofiveyears),thestrikepriceandtheexercisestyle.2.20WARRANTSAcontractthatisverysimilartoanoptionisawarrant.Warrantsarecalloptionsissuedbyacompanyonitsownequity.Themaindifferencesbetweentradedoptionsandwarrantsarethetimescalesinvolved,warrantsusuallyhavealongerlifespan,andonexercisethecompanyissuesnewstocktothewarrantholder.Onexercise,theholderofatradedoptionreceivesstockthathasalreadybeenissued.Exerciseisusuallyallowedanytimebeforeexpiry,butafteraninitialwaitingperiod.Thetypicallifespanofawarrantisfiveormoreyears.Occasionallyperpetualwarrantsareissued;thesehavenomaturity.2.21CONVERTIBLEBONDSConvertiblebondsorCBshavefeaturesofbothbondsandwarrants.Theypayastreamofcouponswithafinalrepaymentofprincipalatmaturity,buttheycanbeconvertedintotheunderlyingstockbeforeexpiry.Onconversionrightstofuturecouponsarelost.Ifthestockpriceislowthenthereislittleincentivetoconverttothestock;thecouponstreamismorevaluable.InthiscasetheCBbehaveslikeabond.IfthestockpriceishighthenconversionislikelyandtheCBrespondstothemovementintheasset.BecausetheCBcanbeconvertedintotheasset,itsvaluehastobeatleastthevalueoftheasset.ThismakesCBssimilartoAmericanoptions;earlyexerciseandconversionaremathematicallythesame.Thereareotherinterestingfeaturesofconvertiblebonds,callback,resetting,etc.andthewholeofChapter33isdevotedtotheirdescriptionandanalysis.2.22OVERTHECOUNTEROPTIONSNotalloptionsaretradedonanexchange.Some,knownasoverthecounterorOTCoptionsaresoldprivatelyfromonecounterpartytoanother.InFigure2.26isthetermsheetforanOTCputoption,havingsomespecialfeatures.AtermsheetspecifiestheprecisedetailsofanOTCcontract.InthisOTCputtheholdergetsaputoptiononS&P500,butmorecheaplythanavanillaputoption.Thiscontractischeapbecausepartofthepremiumdoesnothavetobepaiduntilandunlesstheunderlyingindextradesaboveaspecifiedlevel.Eachtimethatanewlevelisreachedanextrapaymentistriggered.Thisfeaturemeansthatthecontractisnot 52PartOnemathematicalandfinancialfoundationsFigure2.26TermsheetforanOTC‘Put.’ derivativesChapter253vanilla,andmakesthepricingmorecomplicated.Wewillbediscussingspecialfeaturesliketheonesinthiscontractinlaterchapters.Allofthetermsheetsinthisbookarereal,inthesensethatsomeoneatabankwrotethemforpossiblecommercialpurposes.However,manyofthemweregiventomebeforetheywerefinalized.Forthatreasonyouwillseethatoftentherearebits‘missing,’andthesewouldhavebeensetatthetimethatthedealwasfinallystruck.2.23SUMMARYWenowknowthebasicsofoptionsandmarkets,andafewofthesimplesttradingstrategies.Weknowsomeofthejargonandthereasonswhypeoplemightwanttobuyanoption.We’vealsoseenanotherexampleofnoarbitrageinput-callparity.Thisisjustthebeginning.Wedon’tknowhowmuchtheseinstrumentsareworth,howtheyareaffectedbythepriceoftheunderlying,howmuchriskisinvolvedinthebuyingorwritingofoptions.Andwehaveonlyseentheverysimplestofcontracts,therearemany,manymorecomplexproductstoexamine.Alloftheseissuesaregoingtobeaddressedinlaterchapters.FURTHERREADING•McMillan(1996)andOptionsInstitute(1995)describemanyoptionstrategiesusedinpractice.•Mostexchangeshavewebsites.TheLondonInternationalFinancialFuturesExchangewebsitecontainsinformationaboutthemoneymarkets,bonds,equities,indicesandcom-modities.Seewww.liffe.com.Forinformationaboutoptionsandderivativesgenerally,seewww.cboe.com,theChicagoBoardOptionsExchangewebsite.TheAmericanStockExchangeisonwww.amex.comandtheNewYorkStockExchangeonwww.nyse.com.•Derivativeshaveoftenhadbadpress(andthere’sprobablymoretocome).SeeMiller(1997)foradiscussionoftheprosandconsofderivatives.•ThebestbooksonoptionsareHull(2005)andCox&Rubinstein(1985),modestyforbidsmefrommentioningothers. CHAPTER3therandombehaviorofassetsInthisChapter...•Jensen’sinequality•morenotationcommonlyusedinmathematicalfinance•howtoexaminetime-seriesdatatomodelreturns•theWienerprocess,amathematicalmodelofrandomness•asimplemodelforequities,currencies,commoditiesandindices3.1INTRODUCTIONInthischapterIdescribeasimplecontinuous-timemodelforequitiesandotherfinancialinstruments,inspiredbyourearliercoin-tossingexperiment.ThistakesusintotheworldofstochasticcalculusandWienerprocesses.AlthoughthereisagreatdealoftheorybehindtheideasIdescribe,Iamgoingtoexplaineverythinginassimpleandaccessiblemanneraspossible.Wewillbemodelingthebehaviorofequities,currenciesandcommodities,buttheideasareapplicabletothefixed-incomeworldasweshallseeinPartThree.3.2THEPOPULARFORMSOF‘ANALYSIS’Therearethreeformsof‘analysis’commonlyusedinthefinancialworld:•Fundamental•Technical•QuantitativeFundamentalanalysisisallabouttryingtodeterminethe‘correct’worthofacompanybyanin-depthstudyofbalancesheets,managementteams,patentapplications,competitors,lawsuits,etc.Inotherwords,gettingtotheheartofthefirm,doinglotsofaccountingandprojectionsandwhat-not.Thissoundslikeareallysensiblewaytomodelacompanyandhenceitsstockprice.Well,itisandwewilltalkaboutthislaterinthesubjectofRealOptions. 56PartOnemathematicalandfinancialfoundationsHowever,thereareunfortunatelytwodifficultieswiththisapproach.Firstitisvery,veryhard.Youneedadegreeinaccountingandplentyofpatience.Andeventhenallthemostimportantstuffcanbehidden‘offbalancesheet.’Second,andmoreimportantly,‘Themarketcanstayirrationallongerthanyoucanstaysolvent’(possiblysaidbyKeynes).Inotherwords,evenifyouhavetheperfectmodelforthevalueofafirmitdoesn’tmeanyoucanmakemoney.Youhavetofindsomemispricingandthenhopethattherestoftheworldstartstoseeyourpointofview.Andthismayneverhappen.Iffundamentalanalysisishard,thenthenextformofanalysisistheexactopposite,becauseitissoeasy.Technicalanalysisiswhenyoudon’tcareanythingaboutthecompanyotherthantheinformationcontainedwithinitsstockpricehistory.Youdrawtrendlines,lookforspecificpatternsinthesharepriceandmakepredictionsaccordingly.ThisisthesubjectofChapter20.Mostacademicevidencesuggeststhatmosttechnicalanalysisisbunk.Thefinalformofanalysisistheonewearereallyconcernedwithinthisbook,andistheformthathasbeenmostsuccessfuloverthelast50years,formingasolidfoundationforport-foliotheory,derivativespricingandriskmanagement.Itisquantitativeanalysis.Quantitativeanalysisisallabouttreatingfinancialquantitiessuchasstockpricesorinterestratesasrandom,andthenchoosingthebestmodelsforthatrandomness.Let’sseewhyrandomnessisimportantandthenbuildupasimple,random,stockpricemodel.3.3WHYWENEEDAMODELFORRANDOMNESS:JENSEN’SINEQUALITYWhyis‘randomness’socrucialtomodelingtheworldofderivatives?Whycan’twejusttrytoforecastthefuturestockpriceasbestwecanandfigureouttheoption’spayoff?Tobestseetheimportanceofrandomnessinoptiontheorylet’stakealookatsomeverysimplemathematics,calledJensen’sInequality.Thestockpricetodayis$100.Let’ssupposethatinoneyear’stimeitcouldbe$50or$150,withbothequallylikely(seeFigure3.1).Howcanwevalueanoptiononthisstock,acalloptionwithastrikeof100expiringinoneyear,say?Twowaysspringtomind.Withthosetwopossiblescenarioswecouldsaythatweexpectthestockpricetobeat$100inoneyear,thisbeingtheaverageofthepossiblefuturevalues.Thepayoffforthecalloptionwouldthenbe0,sinceitisexactlyatthemoney.Andthepresentvalueofthisiszero.Couldthisbethewaytovalueanoption?Probablynot.Youexpectthevaluetobegreaterthanzero,sincehalfthetimethereissomepayoff.Alternativelywecouldlookatthetwopossiblepayoffsandthencalculatethatexpectation.Ifthestockfallsto50thenthepayoffiszero,ifitrisesto150thenthepayoffis50.Theaveragepayoffistherefore25,whichwecouldpresentvaluetogiveussomeideaoftheoption’svalue.Itturnsoutthatthesecondcalculationisclosertowhatwedoinpracticetovalueoptions(althoughitturnsoutthattheprobabilitiesdon’tcomeintothecalculation)andwe’llseelotsofthisthroughoutthebook.Butthatcalculationalsoillustratesanotherpointofgreatimportance,thattheorderinwhichwedothepayoffcalculationandtheexpectationmatters.InthisexamplewehadPayoff(Expectedstockprice)=0 therandombehaviorofassetsChapter357VS50150100Figure3.1Futurescenarios.whereas
ExpectedPayoff(Stockprice)=25.ThisisanexampleofJensen’sinequality.Let’susesomesymbols.Ifwehaveaconvexfunctionf(S)(inourexamplethepayofffunctionforacall)ofarandomvariableS(inourexamplethestockprice)then
Ef(S)≥f(E[S]).(3.1)Wecanevengetanideaofhowmuchgreatertheleft-handsideisthantheright-handsidebyusingaTaylorseriesapproximationaroundthemeanofS.WriteS=S+
,whereS=E[S],sotheE[
]=0.Then


Ef(S)=Ef(S+
)=Ef(S)+
f(S)+1
2f(S)+···2
≈f(S)+1f(S)E
22=f(E[S])+1f(E[S])E[
2].2Sotheleft-handsideof(3.1)isgreaterthantherightbyapproximately1f(E[S])E[
2].2Thisshowstheimportanceoftwoconcepts: 58PartOnemathematicalandfinancialfoundations•f(E[S]):Theconvexityofanoption.Asarulethisaddsvaluetoanoption.Italsomeansthatanyintuitionwemaygetfromlinearcontracts(forwardsandfutures)mightnotbehelpfulwithnon-linearinstrumentssuchasoptions.
•E
2:Randomnessintheunderlying,anditsvariance.Asstatedabove,modelingran-domnessisthekeytomodelingoptions.Nowthatwehaveseenahintastowhyrandomnessissoimportant,let’sstartmodelingsomeassets!3.4SIMILARITIESBETWEENEQUITIES,CURRENCIES,COMMODITIESANDINDICESWhenyouinvestinsomething,whetheritisastock,commodity,workofartoraracehorse,yourmainconcernisthatyouwillmakeacomfortablereturnonyourinvestment.Byreturnwetendtomeanthepercentagegrowthinthevalueofanasset,togetherwithaccumulateddividends,oversomeperiod:Changeinvalueoftheasset+accumulatedcashflowsReturn=.OriginalvalueoftheassetIwanttodistinguishherebetweenthepercentageorrelativegrowthandtheabsolutegrowth.Supposewecouldinvestineitheroftwostocks,bothofwhichgrowonaverageby$10perannum.StockAhasavalueof$100andstockBiscurrentlyworth$1000.Clearlytheformerisabetterinvestment,attheendoftheyearstockAwillprobablybewortharound$110(ifthepastisanythingtogoby)andstockB$1010.Bothhavegoneupby$10,butAhasrisenby10%andBbyonly1%.Ifwehave$1000toinvestwewouldbebetteroffinvestingintenofassetAthanoneofassetB.Thisillustratesthatwhenwecometomodelassets,itisthereturnthatweshouldconcentrateon.Inthisrespect,allequities,currencies,commoditiesandstockmarketindicescanbetreatedsimilarly.Whatreturndoweexpecttogetfromthem?Partofthebusinessofestimatingreturnsforeachassetistoestimatehowmuchunpredictabil-itythereisintheassetvalue.InthenextsectionIamgoingtoshowthatrandomnessplaysalargepartinfinancialmarkets,andstarttobuildupamodelforassetreturnsincorporatingthisrandomness.3.5EXAMININGRETURNSInFigure3.2IshowthequotedpriceofPerezCompanc,anArgentinianconglomer-ate,overtheperiodFebruary1995toNovember1996.Thisisaverytypicalplotofafinancialasset.Theassetshowsageneralupwardtrendovertheperiodbutthisisfarfromguaranteed.Ifyouboughtandsoldatthewrongtimesyouwouldlosealotofmoney.Theunpredictabilitythatisseeninthisfigureisthemainfeatureoffinan-cialmodeling.Becausethereissomuchrandomness,anymathematicalmodelofafinancialassetmustacknowledgetherandomnessandhaveaprobabilisticfoundation. therandombehaviorofassetsChapter3598PerezCompanc7654321020-Feb-9531-May-9508-Sep-9517-Dec-9526-Mar-9604-Jul-9612-Oct-96Figure3.2PerezCompancfromFebruary1995toNovember1996.Rememberingthatthereturnsaremoreimportanttousthantheabsoluteleveloftheassetprice,IshowinFigure3.3howtocalculatereturnsonaspreadsheet.DenotingtheassetvalueontheithdaybySi,thenthereturnfromdayitodayi+1isgivenbySi+1−Si=Ri.Si(I’veignoreddividendshere,theyareeasilyallowedfor,especiallysincetheyonlygetpaidtwoorfourtimesayeartypically.)Ofcourse,Ididn’tneedtousedataspacedatintervalsofaday,Iwillcommentonthislater.InFigure3.4IshowthedailyreturnsforPerezCompanc.Thislooksverymuchlike‘noise,’andthatisexactlyhowwearegoingtomodelit.ThemeanofthereturnsdistributionisM1R=Ri(3.2)Mi=1andthesamplestandarddeviationis1M(Ri−R)2,(3.3)M−1i=1 60PartOnemathematicalandfinancialfoundationsDatePerezReturn01-Mar-952.11Averagereturn0.00291602-Mar-951.90−0.1Standarddeviation0.02452103-Mar-952.180.14990606-Mar-952.16−0.0108107-Mar-951.91−0.11258=AVERAGE(C3:C463)08-Mar-951.86−0.0298509-Mar-951.970.06153810-Mar-952.270.15=STDEVP(C3:C463)13-Mar-952.490.09987414-Mar-952.760.10856515-Mar-952.61−0.0542616-Mar-952.670.02185817-Mar-952.64−0.010720-Mar-952.60−0.01622=(B13-B12)/B1221-Mar-952.59−0.0027522-Mar-952.59−0.0027523-Mar-952.55−0.0123224-Mar-952.730.06930727-Mar-952.910.06481528-Mar-952.920.00289929-Mar-952.92030-Mar-953.120.06936431-Mar-953.140.00540503-Apr-953.13−0.0026904-Apr-953.240.03773605-Apr-953.250.00259706-Apr-953.280.00777207-Apr-953.21−0.0205710-Apr-953.02−0.0603711-Apr-953.080.01955312-Apr-953.190.03561617-Apr-953.210.00793618-Apr-953.17−0.0131219-Apr-953.240.021277Figure3.3Spreadsheetforcalculatingassetreturns.whereMisthenumberofreturnsinthesample(onefewerthanthenumberofassetprices).Fromthedatainthisexamplewefindthatthemeanis0.002916andthestandarddeviationis0.024521.Noticehowthemeandailyreturnismuchsmallerthanthestandarddeviation.Thisisverytypicaloffinancialquantitiesovershorttimescales.Onaday-by-daybasisyouwilltendtoseethenoiseinthestockprice,andwillhavetowaitmonthsperhapsbeforeyoucanspotthetrend.Thefrequencydistributionofthistimeseriesofdailyreturnsiseasilycalculated,andveryinstructivetoplot.InExceluseTools|DataAnalysis|Histogram.InFigure3.5isshownthefrequencydistributionofdailyreturnsforPerezCompanc.Thisdistributionhasbeenscaledandtranslatedtogiveitameanofzero,astandarddeviationofoneandanareaunderthecurveofone.OnthesameplotisdrawntheprobabilitydensityfunctionforthestandardizedNormaldistributionfunction1−1φ2√e2,2πwhereφisastandardizedNormalvariable.Thetwocurvesarenotidenticalbutarefairlyclose. therandombehaviorofassetsChapter3610.2PerezCompancreturns0.150.10.050−0.0520-Feb-9531-May-9508-Sep-9517-Dec-9526-Mar-9604-Jul-9612-Oct-9620-Jan-97−0.1−0.15Figure3.4DailyreturnsofPerezCompanc.0.7Perezreturns0.6NormalProbability0.50.40.30.20.10−4.5−3.5−2.5−1.5−0.50.51.52.53.54.5Return(scaled)Figure3.5NormalizedfrequencydistributionofPerezCompancandthestandardizedNormaldistribution. 62PartOnemathematicalandfinancialfoundationsFigure3.6Glaxo–Wellcomereturnshistogram.Source:BloombergL.P.SupposingthatwebelievethattheempiricalreturnsarecloseenoughtoNormalforthistobeagoodapproximation,thenwehavecomealongwaytowardsamodel.Iamgoingtowritethereturnsasarandomvariable,drawnfromaNormaldistributionwithaknown,constant,non-zeromeanandaknown,constant,non-zerostandarddeviation:Si+1−SiRi==mean+standarddeviation×φ.SiFigure3.6showsthereturnsdistributionofGlaxo–WellcomeascalculatedbyBloomberg.Thishasnotbeennormalized.3.6TIMESCALESHowdothemeanandstandarddeviationofthereturns’timeseries,asestimatedby(3.2)and(3.3),scalewiththetimestepbetweenassetpricemeasurements?Intheexamplethetimestepisoneday,butsupposeIsampledathourlyintervalsorweekly,howwouldthisaffectthedistribution? therandombehaviorofassetsChapter363Callthetimestepδt.Themeanofthereturnscaleswiththesizeofthetimestep.Thatis,thelargerthetimebetweensamplingthemoretheassetwillhavemovedinthemeantime,onaverage.Icanwritemean=µδt,forsomeµwhichwewillassumetobeconstant.InthePerezCompancexamplewehadameanof0.002916overatimescaleofoneday,δt=1/252sothatµ=252×002916=0.735=73.5%.Ignoringrandomnessforthemoment,ourmodelissimplySi+1−Si=µδt.SiRearranging,wegetSi+1=Si(1+µδt).IftheassetbeginsatS0attimet=0thenafteronetimestept=δtandS1=S0(1+µδt).Aftertwotimestepst=2δtandS=S(1+µδt)=S(1+µδt)2,210andafterMtimestepst=Mδt=TandS=S(1+µδt)M.M0ThisisjustS=S(1+µδt)M=SeMlog(1+µδt)≈SeµMδt=SeµT.M0000InthelimitasthetimesteptendstozerowiththetotaltimeTfixed,thisapproximationbecomesexact.Thisresultisimportantfortworeasons.First,intheabsenceofanyrandomnesstheassetexhibitsexponentialgrowth,justlikecashinthebank.Second,themodelismeaningfulinthelimitasthetimesteptendstozero.IfIhadchosentoscalethemeanofthereturnsdistributionwithanyotherpowerofδtitwouldhaveresultedineitheratrivialmodel(ST=S0)orinfinitevaluesfortheasset(ST=±∞).Thesecondpointcanguideusinthechoiceofscalingfortherandomcomponentofthereturn.Howdoesthestandarddeviationofthereturnscalewiththetimestepδt?(Recallthatyouaddvariancesnotstandarddeviations.)Again,considerwhathappensafterT/δttimestepseachofsizeδt(i.e.afteratotaltimeofT).Insidethesquarerootinexpression(3.3)therearealargenumberofterms,T/δtofthem.Inorderforthestandarddeviationtoremainfiniteasweletδttendtozero,theindividualtermsintheexpressionmusteachbeofO(δt).Since 64PartOnemathematicalandfinancialfoundationseachtermisasquareofareturn,thestandarddeviationoftheassetreturnoveratimestepδtmustbeO(δt1/2):standarddeviation=σδt1/2,whereσissomeparametermeasuringtheamountofrandomness;thelargerthisparameterthemoreuncertainisthereturn.Forthemomentlet’sassumethatitisconstant.ForPerezCompancwehaveastandarddeviationof0.024521overonedaysothat√σ=252×0.024521=0.389=38.9%.PuttingthesescalingsexplicitlyintoourassetreturnmodelSi+1−Si1/2Ri==µδt+σφδt.(3.4)SiIcanrewriteEquation(3.4)asS−S=µSδt+σSφδt1/2.(3.5)i+1iiiTheleft-handsideofthisequationisthechangeintheassetpricefromtimestepitotimestepi+1.Theright-handsideisthe‘model.’Wecanthinkofthisequationasamodelforarandomwalkoftheassetprice.ThisisshownschematicallyinFigure3.7.Weknowexactlywheretheassetpriceistodaybuttomorrow’svalueisunknown.Itisdistributedabouttoday’svalueaccordingto(3.5).SAssettomorrowAssettodayDistributionofassetpricechangetFigure3.7Arepresentationoftherandomwalk. therandombehaviorofassetsChapter3653.6.1TheDriftTheparameterµiscalledthedriftrate,theexpectedreturnorthegrowthrateoftheasset.Statisticallyitisveryhardtomeasuresincethemeanscaleswiththeusuallysmallparameterδt.ItcanbeestimatedbyM1µ=Ri.Mδti=1Theunitoftimethatisusuallyusedistheyear,inwhichcaseµisquotedasanannualizedgrowthrate.Intheclassicaloptionpricingtheorythedriftplaysalmostnorole.Soeventhoughitishardtomeasure,thisdoesn’tmattertoomuch.13.6.2TheVolatilityTheparameterσiscalledthevolatilityoftheasset.Itcanbeestimatedby1M(Ri−R)2.(M−1)δti=1Again,thisisalmostalwaysquotedinannualizedterms.Thevolatilityisthemostimportantandelusivequantityinthetheoryofderivatives.Iwillcomebackagainandagaintoitsestimationandmodeling.Becauseoftheirscalingwithtime,thedriftandvolatilityhavedifferenteffectsontheassetpath.Thedriftisnotapparentovershorttimescalesforwhichthevolatilitydominates.Overlongtimescales,forinstancedecades,thedriftbecomesimportant.Figure3.8isarealizedpathofthelogarithmofanasset,togetherwithitsexpectedpathanda‘confidenceinterval.’Inthisexampletheconfidenceintervalrepresentsonestandarddeviation.WiththeassumptionofNormalitythismeansthat68%ofthetimetheassetshouldbewithinthisrange.Themeanpathisgrowinglinearlyintimeandtheconfidenceintervalgrowslikethesquarerootoftime.Thusovershorttimescalesthevolatilitydominates.23.7ESTIMATINGVOLATILITYThemostcommonestimateofvolatilityissimply1M(Ri−R)2.(M−1)δti=1IfδtissufficientlysmallthemeanreturnRtermcanbeignored.Forsmallδt1M(logS(t)−logS(t))2ii−1(M−1)δti=1canalsobeused,whereS(ti)istheclosingpriceondayti.1Innon-classicaltheoriesandinportfoliomanagement,itdoesoftenmatter,verymuch.2WhydidItakethelogarithm?Becausechangesinthelogarithmarerelatedtothereturnontheasset. 66PartOnemathematicalandfinancialfoundationsOnestandarddeviationaboveandbelowthemeanMeanStockpriceTimeFigure3.8Pathofthelogarithmofanasset,itsexpectedpathandonestandarddeviationaboveandbelow.100%90%80%70%60%50%40%Volatilityestimate30%20%10%0%TimeFigure3.9Theplateauingeffectwhenusingamovingwindowvolatilityestimate. therandombehaviorofassetsChapter367Itishighlyunlikelythatvolatilityisconstantintime.Changingeconomiccircumstances,seasonalityetc.willinevitablyresultinvolatilitychangingwithtime.Ifyouwanttoknowthevolatilitytodayyoumustusesomepastdatainthecalculation.Unfortunately,thismeansthatthereisnoguaranteethatyouareactuallycalculatingtoday’svolatility.Typicallyyouwouldusedailyclosingpricestoworkoutdailyreturnsandthenusethepast10,30,100,...dailyreturnsintheformulaabove.Oryoucouldusereturnsoverlongerorshorterperiods.Sinceallreturnsareequallyweighted,whiletheyareintheestimateofvolatility,anylargereturnwillstayintheestimateofvolatilityuntilthe10(or30or100)dayshavepassed.Thisgivesrisetoaplateauingofvolatility,andistotallyspurious.InFigure3.9isshownthespuriousplateauingeffectassociatedwithasuddenlargedropinastockprice.Sincevolatilityisnotdirectlyobservable,andbecauseoftheplateauingeffectinthesimplemeasureofvolatility,youmightwanttouseothervolatilityestimates.We’llseesomemoreinChapter49.3.8THERANDOMWALKONASPREADSHEETTherandomwalk(3.5)canbewrittenasa‘recipe’forgeneratingSi+1fromSi:S=S1+µδt+σφδt1/2.(3.6)i+1iWecaneasilysimulatethemodelusingaspreadsheet.Inthissimulationwemustinputseveralparameters,astartingvaluefortheasset,atimestepδt,thedriftrateµ,thevolatilityσandthetotalnumberoftimesteps.Then,ateachtimestep,wemustchoosearandomnumberφfromaNormaldistribution.IwilltalkaboutsimulationsindepthinChapter80dir,forthemomentletmejustsaythatanapproximationtoaNormalvariablethatisfastinaspreadsheet,andquiteaccurate,issimplytoadduptwelverandomvariablesdrawnfromauniformdistributionoverzerotoone,andsubtractsix:12RAND()−6.i=1TheExcelspreadsheetfunctionRAND()givesauniformly-distributedrandomvariable.InFigure3.10Ishowthedetailsofaspreadsheetusedforsimulatingtheassetpricerandomwalk.3.9THEWIENERPROCESSSofarwehaveamodelthatallowstheassettotakeanyvalueafteratimestep.Thisissomeprogressbutwehavestillnotreachedourgoalofcontinuoustime,westillhaveadiscretetimestep.Thissectionisabriefintroductiontothecontinuous-timelimitofequationslike(3.4).IwillstarttointroduceideasfromtheworldofstochasticmodelingandWienerprocesses,delvingmoredeeplyinChapter4.Iamnowgoingtousethenotationd·tomean‘thechangein’somequantity.ThusdSisthe‘changeintheassetprice.’Butthischangewillbeincontinuoustime.Thuswewillgoto 68PartOnemathematicalandfinancialfoundationsABCDEFGH1Asset100TimeAsset2Drift0.1501003Volatility0.250.0198.388444Timestep0.010.0294.2800550.0395.4044160.0492.79735=D4+$B$470.0593.4516880.0693.9966490.0797.66597100.0896.5231911=E7*(1+$B$2*$B$4+$B$3*SQRT($B$4)*(RAND()+RAND()+RAND()+RAND()0.0999.5841712+RAND()+RAND()+RAND()+RAND()+RAND()+RAND()+RAND()+RAND()-6))0.195.77222130.1199.60075140.1299.01974150.13100.8729160.14101.2378170.15102.4736180.16102.7694190.17100.7347200.18102.7021210.19107.3493220.2109.887230.21108.688240.22110.7826250.23112.8932260.24111.0625270.25111.6157280.26112.5443290.27111.9805300.28115.6002310.29117.9831320.3115.2694330.31117.4374Figure3.10Simulatingtherandomwalkonaspreadsheet.thelimitδt=0.Thefirstδtontheright-handsideof(3.5)becomesdtbutthesecondtermismorecomplicated.Icannotstraightforwardlywritedt1/2insteadofδt1/2.IfIdogotothezero-timesteplimitthenanyrandomdt1/2termwilldominateanydeterministicdtterm.Yetinourproblemthefactorinfrontofdt1/2hasameanofzero,somaybeitdoesnotoutweighthedriftafterall.Clearlysomethingsubtleishappeninginthelimit.Itturnsout,andwewillseethisinChapter4,thatbecausethevarianceoftherandomtermisO(δt)wecanmakeasensiblecontinuous-timelimitofourdiscrete-timemodel.ThisbringsusintotheworldofWienerprocesses.Iamgoingtowritethetermφδt1/2asdX. therandombehaviorofassetsChapter369YoucanthinkofdXasbeingarandomvariable,drawnfromaNormaldistributionwithmeanzeroandvariancedt:E[dX]=0andE[dX2]=dt.Thisisnotexactlywhatitis,butitiscloseenoughtogivetherightidea.ThisiscalledaWienerprocess.Theimportantpointisthatwecanbuildupacontinuous-timetheoryusingWienerprocessesinsteadofNormaldistributionsanddiscretetime.3.10THEWIDELYACCEPTEDMODELFOREQUITIES,CURRENCIES,COMMODITIESANDINDICESOurassetpricemodelinthecontinuous-timelimit,usingtheWienerprocessnotation,canbewrittenasdS=µSdt+σSdX.(3.7)Thisisourfirststochasticdifferentialequation.Itisacontinuous-timemodelofanassetprice.Itisthemostwidelyacceptedmodelforequities,currencies,commoditiesandindices,andthefoundationofsomuchfinancetheory.We’venowbuiltupasimplemodelforequitiesthatwearegoingtobeusingquitealot.Youcouldask,ifthestockmarketissorandomhowcanfundmanagersjustifytheirfee?Dotheymanagetooutsmartthemarket?Aretheyclairvoyantoraren’tthemarketsrandom?Well,Iwon’tswearthatmarketsarerandombutIcansaywithconfidencethatfundmanagersdon’toutperformthemarket.InFigure3.11isshownthepercentageoffundsthatoutperformanindexofallUKstocks.Whetherwelookataone-,three-,five-or10-yearhorizonwecanseethatthevastmajorityoffundscan’tevenkeepupwiththemarket.Andstatisticallyspeaking,thereareboundtobeafewthatbeatthemarket,butonlybychance.Maybeoneshouldinvestinafundthatdoestheoppositeofallotherfunds.Greatideaexceptthatthemanagementfeeandtransactioncostsprobablymeanthatthatwouldbeapoorinvestmenttoo.Thisdoesn’tprovethatmarketsarerandom,butit’ssufficientlysuggestivethatmostofmypersonalshareexposureisviaanindex-trackerfund.3.11SUMMARYInthischapterIintroducedasimplemodelfortherandomwalkofasset.InitiallyIbuiltthemodelupindiscretetime,showingwhatthevarioustermsmean,howtheyscalewiththetimestepandshowinghowtoimplementthemodelonaspreadsheet.Mostofthisbookisaboutcontinuous-timemodelsforassets.Thecontinuous-timeversionoftherandomwalkinvolvesconceptssuchasstochasticcalculusandWienerprocesses.Iintroducedthesebrieflyinthischapterandwillnowgoontoexplaintheunderlyingtheoryofstochasticcalculustogivethenecessarybackgroundfortherestofthebook. 70PartOnemathematicalandfinancialfoundations100%6%5%9%1%90%80%70%60%50%99%94%95%91%40%OutperformingAllShareIndex30%UnderperformingAllShareIndex20%10%0%1year3years5years10yearsFigure3.11FundperformancescomparedwithUKAllShareIndex.ToendDecember1998.DatasuppliedbyVirginDirect.FURTHERREADING•Mandelbrot(1963)andFama(1965)didsomeoftheearlyworkontheanalysisoffinancialdata.•ForanintroductiontorandomwalksandWienerprocesses,seeØksendal(1992)andSchuss(1980).•SomehighfrequencydatacanbeorderedthroughOlsenAssociates,www.olsen.ch.It’snotfree,butnorisitexpensive.•ThefamousbookbyMalkiel(1990)iswellworthreadingforitsinsightsintothebehaviorofthestockmarket.Readwhathehastosayaboutchimpanzees,blindfoldsanddarts.Infact,ifyouhaven’talreadyreadMalkiel’sbookmakesurethatitisthenextbookyoureadafterfinishingmine. CHAPTER4elementarystochasticcalculusInthisChapter...•allthestochasticcalculusyouneedtoknow,andnomore•themeaningofMarkovandmartingale•Brownianmotion•stochasticintegration•stochasticdifferentialequations•Ito’slemmainoneandmoredimensionsˆ4.1INTRODUCTIONStochasticcalculusisveryimportantinthemathematicalmodelingoffinancialprocesses.Thisisbecauseoftheassumedunderlyingrandomnatureoffinancialmarkets.BecausestochasticcalculusissuchanimportanttoolIwanttoensurethatitcanbeusedbyeveryone.Tothatend,Iamgoingtotrytomakethischapterasaccessibleandintuitiveaspossible.Bytheend,Ihopethatthereaderwillknowwhatvarioustechnicaltermsmean(andrarelyaretheyverycomplicated),but,moreimportantly,willalsoknowhowtousethetechniqueswiththeminimumoffuss.Mostacademicarticlesinfinancehavea‘pure’mathematicaltheme.Themathematicalrigorintheseworksisoccasionallyjustified,butmoreoftenthannotitonlysucceedsinobscuringthecontent.Whenasubjectisyoung,asismathematicalfinance(youngish),thereisatendencyfortechnicalrigortofeatureveryprominentlyinresearch.Thisisduetolackofconfidenceinthemethodsandresults.Asthesubjectages,researcherswillbecomemorecavalierintheirattitudesandwewillseemuchmorerapidprogress.4.2AMOTIVATINGEXAMPLETossacoin.EverytimeyouthrowaheadIgiveyou$1,everytimeyouthrowatailyougiveme$1.Figure4.1showshowmuchmoneyyouhaveaftersixtosses.InthisexperimentthesequencewasTHHTHT,andwefinishedeven. 72PartOnemathematicalandfinancialfoundations2100123456WinningsNumberofcointosses−1−2Figure4.1Theoutcomeofacointossingexperiment.IfIuseRitomeantherandomamount,either$1or−$1,youmakeontheithtossthenwehaveE[R]=0,E[R2]=1andE[RR]=0.iiijInthisexampleitdoesn’tmatterwhetherornottheseexpectationsareconditionalonthepast.Inotherwords,ifIthrewfiveheadsinarowitdoesnotaffecttheoutcomeofthesixthtoss.Tothegamblersoutthere,thispropertyisalsosharedbyafairdie,abalancedroulettewheel,butnotbythedeckofcardsinBlackjack.InBlackjackthesamedeckisusedforgameaftergame,theoddsduringonegamedependonwhatcardsweredealtoutfromthesamedeckinpreviousgames.ThatiswhyyoucaninthelongrunbeatthehouseatBlackjackbutnotroulette.IntroduceSitomeanthetotalamountofmoneyyouhavewonuptoandincludingtheithtosssothat
iSi=Rj.j=1LateronitwillbeusefulifwehaveS0=0,i.e.,youstartwithnomoney.IfwenowcalculateexpectationsofSiitdoesmatterwhatinformationwehave.IfwecalculateexpectationsoffutureeventsbeforetheexperimenthasevenbegunthenE[S]=0andE[S2]=E[R2+2RR+···]=i.ii112Ontheotherhand,supposetherehavebeenfivetossesalready,canIusethisinformationandwhatcanwesayaboutexpectationsforthesixthtoss?Thisistheconditionalexpectation.TheexpectationofS6conditionaluponthepreviousfivetossesgivesE[S6|R1,...,R5]=S5. elementarystochasticcalculusChapter4734.3THEMARKOVPROPERTYThisresultisspecial,thedistributionofthevalueoftherandomvariableSiconditionaluponallofthepasteventsonlydependsonthepreviousvalueSi−1.ThisistheMarkovproperty.Wesaythattherandomwalkhasnomemorybeyondwhereitisnow.Notethatitdoesn’thavetobethecasethattheexpectedvalueoftherandomvariableSiisthesameasthepreviousvalue.Thiscanbegeneralizedtosaythat,giveninformationaboutSjforsomevaluesof1≤j0is∞xW(x,τ)=Wf(x,τ;x)Payoff(e)dx.(7.6)−∞Toshowthis,Ijusthavetodemonstratethattheexpressionsatisfiestheequation(7.2)andthefinalcondition(7.5).Bothofthesearestraightforward.Theintegrationwithrespecttoxissimilartoasummation,andsinceeachindividualcomponentsatisfiestheequationsodoesthesum/integral.Alternatively,differentiate(7.6)undertheintegralsigntoseethatitsatisfiesthepartialdifferentialequation.Thatitsatisfiesthecondition(7.5)followsfromthespecialpropertiesofthefundamentalsolutionWf.Retracingourstepstowriteoursolutionintermsoftheoriginalvariables,wegete−r(T−t)∞1222dS−log(S/S)+r−2σ(T−t)2σ(T−t)V(S,t)=√ePayoff(S),(7.7)σ2π(T−t)0SwhereIhavewrittenx=logS.Thisistheexactsolutionfortheoptionvalueintermsofthearbitrarypayofffunction.InthenextsectionsIwillmanipulatethisexpressionforspecialpayofffunctions.7.2.1FormulaforaCallThecalloptionhasthepayofffunctionPayoff(S)=max(S−E,0).Expression(7.7)canthenbewrittenas−r(T−t)∞1222e−log(S/S)+r−2σ(T−t)/2σ(T−t)dS√e(S−E).σ2π(T−t)ESReturntothevariablex=logS,towritethisase−r(T−t)∞1222−−x+logS+r−2σ(T−t)/2σ(T−t)x√e(e−E)dxσ2π(T−t)logE−r(T−t)∞1222e−−x+logS+r−2σ(T−t)/2σ(T−t)x=√eedxσ2π(T−t)logE−r(T−t)∞1222e−−x+logS+r−2σ(T−t)/2σ(T−t)−E√edx.σ2π(T−t)logE theBlack–Scholesformulaeandthe‘greeks’Chapter7115Bothintegralsinthisexpressioncanbewrittenintheform∞12−xe2dxdforsomed(thesecondisjustaboutinthisformalready,andthefirstjustneedsacompletionofthesquare).Apartfromacoupleofminordifferences,thisintegralisjustlikethecumulativedistributionfunctionforthestandardizedNormaldistribution2definedbyx1−1φ2N(x)=√e2dφ.2π−∞Thisfunction,plottedinFigure7.2,istheprobabilitythataNormallydistributedvariableislessthanx.Thustheoptionpricecanbewrittenastwoseparatetermsinvolvingthecumulativedistri-butionfunctionforaNormaldistribution:Calloptionvalue=SN(d)−Ee−r(T−t)N(d)12wherelog(S/E)+(r+1σ2)(T−t)d=√21σT−t1CDF0.80.60.40.2f0−2.8−2.4−2−1.6−1.2−0.8−0.400.40.81.21.622.42.8Figure7.2ThecumulativedistributionfunctionforastandardizedNormalrandomvariable,N(x).2i.e.havingzeromeanandunitstandarddeviation. 116PartOnemathematicalandfinancialfoundationsandlog(S/E)+(r−1σ2)(T−t)d=√2.2σT−tWhenthereiscontinuousdividendyieldontheunderlying,oritisacurrency,thenCalloptionvalueSe−D(T−t)N(d)−Ee−r(T−t)N(d)12log(S/E)+(r−D+1σ2)(T−t)d=√21σT−tlog(S/E)+(r−D−1σ2)(T−t)d=√22σT−t√=d1−σT−tTheoptionvalueisshowninFigure7.3asafunctionoftheunderlyingassetatafixedtimetoexpiry.InFigure7.4thevalueoftheat-the-moneyoptionisshownasafunctionoftime,andexpiryist=1.InFigure7.5isthecallvalueasafunctionofboththeunderlyingandtime.Whentheassetis‘at-the-moneyforward,’i.e.S=Ee−(r−D)(T−t),thenthereisasimpleapproximationforthecallvalue(Brenner&Subrahmanyam,1994):√Call≈0.4Se−D(T−t)σT−t.200180160140120100Value80604020S0050100150200250300Figure7.3Thevalueofacalloptionasafunctionoftheunderlyingassetpriceatafixedtimetoexpiry. theBlack–Scholesformulaeandthe‘greeks’Chapter71171412108Value642t000.20.40.60.811.2Figure7.4Thevalueofanat-the-moneycalloptionasafunctionoftime.70605040Value30201001501350120901050.1575Asset0.360Time0.45450.6300.75150.90Figure7.5Thevalueofacalloptionasafunctionofassetandtime. 118PartOnemathematicalandfinancialfoundations7.2.2FormulaforaPutTheputoptionhaspayoffPayoff(S)=max(E−S,0).Thevalueofaputoptioncanbefoundinthesamewayasabove,orusingput-callparityPutoptionvalue=−SN(−d)+Ee−r(T−t)N(−d),12withthesamed1andd2.Whenthereiscontinuousdividendyieldontheunderlying,oritisacurrency,thenPutoptionvalue−Se−D(T−t)N(−d)+Ee−r(T−t)N(−d)12TheoptionvalueisshowninFigure7.6againsttheunder-lyingassetandinFigure7.7againsttime.InFigure7.8istheoptionvalueasafunctionofboththeunderlyingassetandtime.Whentheassetisat-the-moneyforwardthesimpleapproximationfortheputvalue(Brenner&Subrahmanyam,1994)is√Put≈0.4Se−D(T−t)σT−t.1009080706050Value40302010S0050100150200250300Figure7.6Thevalueofaputoptionasafunctionoftheunderlyingassetatafixedtimetoexpiry. theBlack–Scholesformulaeandthe‘greeks’Chapter711943.532.52Value1.510.5t000.20.40.60.811.2Figure7.7Thevalueofanat-themoneyputoptionasafunctionoftime.10090807060Value5040302010001503045600.275Asset900.4Time1050.612013510.8150Figure7.8Thevalueofaputoptionasafunctionofassetandtime.7.2.3FormulaforaBinaryCallThebinarycallhaspayoffPayoff(S)=H(S−E),whereHistheHeavisidefunctiontakingthevalueonewhenitsargumentispositiveandzerootherwise. 120PartOnemathematicalandfinancialfoundationsIncorporatingadividendyield,wecanwritetheoptionvaluease−r(T−t)∞1222−x−logS−r−D−2σ(T−t)/2σ(T−t)√edx.σ2π(T−t)logEThistermisjustlikethesecondterminthecalloptionequationandsoBinarycalloptionvaluee−r(T−t)N(d)2TheoptionvalueisshowninFigure7.9.7.2.4FormulaforaBinaryPutThebinaryputhasapayoffofoneifS0) 124PartOnemathematicalandfinancialfoundationsyoumakemoneyonthelargemovesintheunderlyingandloseitonthesmallmoves.Tobeprecise,youmakemoney32%ofthetimeandloseit68%.Butwhenyoumakeit,youmakemore.Theneteffectistogettherisk-freerateofreturnontheportfolio.Youwon’thaveacluewherethisfactcamefrom,butallwillbemadeclearinChapter47.Gammaalsoplaysanimportantrolewhenthereisamismatchbetweenthemarket’sviewofvolatilityandtheactualvolatilityoftheunderlying,againthisisdiscussedinChapter47.Sincecostscanbelargeandbecauseonewantstoreduceexposuretomodelerroritisnaturaltotrytominimizetheneedtorebalancetheportfoliotoofrequently.Sincegammaisameasureofsensitivityofthehedgeratiotothemovementintheunderlying,thehedgingrequirementcanbedecreasedbyagamma-neutralstrategy.Thismeansbuyingorsellingmoreoptions,notjusttheunderlying.Becausethegammaoftheunderlying(itssecondderivative)iszero,wecannotaddgammatoourpositionjustwiththeunderlying.Wecanhaveasmanyoptionsinourpositionaswewant;wechoosethequantitiesofeachsuchthatbothdeltaandgammaarezero.Theminimalrequirementistoholdtwodifferenttypesofoptionandtheunderlying.Inpractice,theoptionpositionisnotreadjustedtoooftenbecause,ifthecostoftransactingintheunderlyingislarge,thenthecostoftransactinginitsderivativesisevenlarger.Herearesomeformulaeforthegammasofcommoncontracts:Gammasofcommoncontractse−D(T−t)N(d)1Call√σST−te−D(T−t)N(d)1Put√σST−te−r(T−t)dN(d)12Binarycall−σ2S2(T−t)e−r(T−t)d1N(d2)Binaryputσ2S2(T−t)ExamplesofthesefunctionsareplottedinFigure7.12,withsomescalingforthebinaries.7.5THETATheta,,istherateofchangeoftheoptionpricewithtime.∂V=∂t theBlack–Scholesformulaeandthe‘greeks’Chapter7125CallandputgammaBinarycallgammaBinaryputgammaSGammaFigure7.12Thegammasofacall,aput,abinarycallandabinaryputoption.Thethetaisrelatedtotheoptionvalue,thedeltaandthegammabytheBlack–Scholesequation.Inadelta-hedgedportfoliothethetacontributestoensuringthattheportfolioearnstherisk-freerate.Butitcontributesinacompletelycertainway,unlikethegammawhichcontributestherightamountonaverage.Herearesomeformulaeforthethetasofcommoncontracts:ThetasofcommoncontractsσSe−D(T−t)N(d1)Call−√+DSN(d)e−D(T−t)−rEe−r(T−t)N(d)122T−tσSe−D(T−t)N(−d1)Put−√−DSN(−d)e−D(T−t)+rEe−r(T−t)N(−d)122T−tBinarycallre−r(T−t)N(d)+e−r(T−t)N(d)d1−r√−D222(T−t)σT−tBinaryputre−r(T−t)(1−N(d))−e−r(T−t)N(d)d1−r√−D222(T−t)σT−tThesefunctionsareplottedinFigure7.13. 126PartOnemathematicalandfinancialfoundationsCallthetaPutthetaBinarycallthetaBinaryputthetaThetaSFigure7.13Thethetasofacall,aput,abinarycallandabinaryputoption.7.6SPEEDThespeedofanoptionistherateofchangeofthegammawithrespecttothestockprice.∂3VSpeed=∂S3Tradersusethegammatoestimatehowmuchtheywillhavetorehedgebyifthestockmoves.Thestockmovesby$1sothedeltachangesbywhateverthegammais.Butthat’sonlyanapproximation.Thedeltamaychangebymoreorlessthanthis,especiallyifthestockmovesbyalargeramount,ortheoptionisclosetothestrikeandexpiration.Hencetheuseofspeedinahigher-orderTaylorseriesexpansion.Herearesomeformulaeforthespeedofcommoncontracts:Speedofcommoncontractse−D(T−t)N(d1)√Call−d1+σT−tσ2S2(T−t)e−D(T−t)N(d1)√Put−d1+σT−tσ2S2(T−t)e−r(T−t)N(d)1−dd212Binarycall−−2d1+√σ2S3(T−t)σT−te−r(T−t)N(d)1−dd212Binaryput−2d1+√σ2S3(T−t)σT−t theBlack–Scholesformulaeandthe‘greeks’Chapter71277.7VEGAVega,a.k.a.zetaandkappa,isaveryimportantbutconfusingquantity.Itisthesensitivityoftheoptionpricetovolatility.∂VVega=∂σThisiscompletelydifferentfromtheothergreeks3sinceitisaderivativewithrespecttoaparameterandnotavariable.Thismakessomethingofadifferencewhenwecometofindingnumericalsolutionsforsuchquantities.Inpractice,thevolatilityoftheunderlyingisnotknownwithcertainty.Notonlyisitverydifficulttomeasureatanytime,itisevenhardertopredictwhatitwilldointhefuture.Supposethatweputavolatilityof20%intoanoptionpric-ingformula,howsensitiveisthepricetothatnumber?That’sthevega.Aswithgammahedging,onecanvegahedgetoreducesensi-tivitytothevolatility.Thisisamajorsteptowardseliminatingsomemodelrisk,sinceitreducesdependenceonaquantitythat,tobehonest,isnotknownveryaccurately.1816141210Value8642Volatility,s00.10.150.20.250.3Figure7.14Thevalueofanat-the-moneycalloptionasafunctionofvolatility.3It’snotevenGreek.AmongotherthingsitisanAmericancar,astar(AlphaLyrae),therealnameofZorro,thereareacoupleof16thcenturySpanishauthorscalledVega,anOpartpaintingbyVasarelyandacharacterinthecomputergame‘StreetFighter.’AndwhocouldforgetVincent,andhisbrother?Thesecondderivativewithrespecttoσhasbeencalled‘vomma’andthesecond-orderderivativewithrespecttotheassetandthevolatilityhasbeencalled‘kabanga.’Idoubtthattheyrepresentwhattheirfansthinktheyrepresent,andI’mgoingtomakenofurthermentionofthem. 128PartOnemathematicalandfinancialfoundationsThereisadownsidetothemeasurementofvega.Itisonlyreallymeaningfulforoptionshavingsingle-signedgammaeverywhere.Forexampleitmakessensetomeasurevegaforcallsandputsbutnotbinarycallsandbinaryputs.Ihaveincludedtheformulaeforthevegaofsuchcontractsbelow,buttheyshouldbeusedwithcare,ifatall.Thereasonforthisisthatcallandputvalues(andoptionswithsingle-signedgamma)havevaluesthataremonotonicinthevolatility:increasethevolatilityinacallanditsvalueincreaseseverywhere.Contractswithagammathatchangessignmayhaveavegameasuredatzerobecauseasweincreasethevolatilitythepricemayrisesomewhereandfallsomewhereelse.Suchacontractisveryexposedtovolatilityriskbutthatriskisnotmeasuredbythevega.SeeChapter52formoredetails.Hereareformulaeforthevegasofcommoncontracts:Vegasofcommoncontracts√CallST−te−D(T−t)N(d)1√PutST−te−D(T−t)N(d)1−r(T−t)√d2Binarycall−eN(d2)T−t+σ−r(T−t)√d2BinaryputeN(d2)T−t+σInFigure7.14isshownthevalueofanat-the-moneycalloptionasafunctionofthevolatility.Thereisoneyeartoexpiry,thestrikeis100,theinterestrateis10%andtherearenodividends.Nomatterhowfarinoroutofthemoneythiscurveisalwaysmonotonicallyincreasingforcalloptionsandputoptions;uncertaintyaddsvaluetothecontract.Theslopeofthiscurveisthevega.0.370.3650.360.3550.35Value0.3450.340.335Volatility,s0.330.10.150.20.250.30.35Figure7.15Thevalueofanout-of-the-moneybinarycalloptionasafunctionofvolatility. theBlack–Scholesformulaeandthe‘greeks’Chapter7129InFigure7.15isshownthevalueofanout-of-the-moneybinarycalloptionasafunctionofthevolatility.Thereisoneyeartoexpiry,theassetvalueis88,strikeis100,theinterestrateis10%andtherearenodividends.Observethatthereismaximumatavolatilityofabout24%.Thevalueoftheoptionisnotmonotonicinthevolatility.Wewillseelaterwhythismakesthemeaningofvegasomewhatsuspect.7.8RHORho,ρ,isthesensitivityoftheoptionvaluetotheinterestrateusedintheBlack–Scholesformulae:∂Vρ=∂rInpracticeoneoftenusesawholetermstructureofinterestrates,meaningatime-dependentrater(t).Rhowouldthenbethesensitivitytotheleveloftheratesassumingaparallelshiftinratesatalltimes.Again,youmustbecarefulforwhichcontractsyoumeasurerho;seeChapter52formoredetails.Herearesomeformulaefortherhosofcommoncontracts:RhosofcommoncontractsCallE(T−t)e−r(T−t)N(d)2Put−E(T−t)e−r(T−t)N(−d)2√Binarycall−(T−t)e−r(T−t)N(d)+T−te−r(T−t)N(d)22σ√Binaryput−(T−t)e−r(T−t)(1−N(d))−T−te−r(T−t)N(d)22σThesensitivitiesofcommoncontracttothedividendyieldorforeigninterestratearegivenbythefollowingformulae:SensitivitytodividendforcommoncontractsCall−(T−t)Se−D(T−t)N(d)1Put(T−t)Se−D(T−t)N(−d)1√T−t−r(T−t)Binarycall−eN(d2)σ√T−t−r(T−t)BinaryputeN(d2)σ 130PartOnemathematicalandfinancialfoundations7.9IMPLIEDVOLATILITYTheBlack–Scholesformulaforacalloptiontakesasinputtheexpiry,thestrike,theunderlyingandtheinterestratetogetherwiththevolatilitytooutputtheprice.Allbutthevolatilityareeasilymeasured.Howdoweknowwhatvolatilitytoputintotheformulae?Atradercanseeonhisscreenthatacertaincalloptionwithfourmonthsuntilexpiryandastrikeof100istradingat6.51withtheunderlyingat101.5andashort-terminterestrateof8%.Canweusethisinformationinsomeway?Turntherelationshipbetweenvolatilityandanoptionpriceonitshead.Ifwecanseethepriceatwhichtheoptionistrading,wecanask‘WhatvolatilitymustIusetogetthecorrectmarketprice?’Thisiscalledtheimpliedvolatility.TheimpliedvolatilityisthevolatilityoftheunderlyingwhichwhensubstitutedintotheBlack–Scholesformulagivesatheoreticalpriceequaltothemarketprice.Inasenseitisthemarket’sviewofvolatilityoverthelifeoftheoption.Assumingthatweareusingcallpricestoestimatetheimpliedvolatilitythenprovidedtheoptionpriceislessthantheassetandgreaterthanzerothenwecanfindauniquevaluefortheimpliedvolatility.(Iftheoptionpriceisoutsidetheseboundsthenthere’saveryextremearbitrageopportunity.)BecausethereisnosimpleformulafortheimpliedvolatilityasafunctionoftheoptionvaluewemustsolvetheequationVBS(S0,t0;σ,r;E,T)=knownvalueforσ,whereVBSistheBlack–Scholesformula.Today’sassetpriceisS0,thedateist0andeverythingisknowninthisequationexceptforσ.Belowisanalgorithmforfindingtheimpliedvolatilityfromthemarketpriceofacalloptiontoanyrequireddegreeofaccuracy.ThemethodusedisNewton–Raphsonwhichusesthederivativeoftheoptionpricewithrespecttothevolatility(thevega)inthecalculation.Thismethodisparticularlygoodforsuchawell-behavedfunctionasacallvalue.FunctionImpVolCall(MktPriceAsDouble,StrikeAsDouble,ExpiryAsDouble,_AssetAsDouble,IntRateAsDouble,errorAsDouble)Volatility=0.2dv=error+1WhileAbs(dv)>errord1=Log(Asset/Strike)+(IntRate+0.5*Volatility*Volatility)*Expiryd1=d1/(Volatility*Sqr(Expiry))d2=d1-Volatility*Sqr(Expiry)PriceError=Asset*cdf(d1)-Strike*Exp(-IntRate*Expiry)_*cdf(d2)-MktPriceVega=Asset*Sqr(Expiry/3.1415926/2)*Exp(-0.5*d1*d1)dv=PriceError/VegaVolatility=Volatility-dvWendImpVolCall=VolatilityEndFunctionInthisweneedthecumulativedistributionfunctionfortheNormaldistribution.Thefollowingisasimplealgorithmwhichgivesanaccurate,andfast,approximationtothecumulative theBlack–Scholesformulaeandthe‘greeks’Chapter7131distributionfunctionofthestandardizedNormal:1−1x2
2345Forx≥0N(x)≈1−√e2a1d+a2d+a3d+a4d+a5d2πwhere1d=1+0.0.2316419xanda1=0.31938153,a2=−0.356563782,a3=1.781477937,a4=−1.821255978anda5=1.330274429.Forx<0usethefactthatN(x)+N(−x)=1.Functioncdf(xAsDouble)AsDoubleDimdAsDoubleDimtempasDoubleDima1AsDoubleDima2AsDoubleDima3AsDoubleDima4AsDoubleDima5AsDoubled=1/(1+0.2316419*Abs(x))a1=0.31938153a2=-0.356563782a3=1.781477937a4=-1.821255978a5=1.330274429temp=a5temp=a4+d*temptemp=a3+d*temptemp=a2+d*temptemp=a1+d*temptemp=d*tempcdf=1-1/Sqr(2*3.1415926)*Exp(-0.5*x*x)*tempIfx<0Thencdf=1-cdfEndFunctionInpracticeifwecalculatetheimpliedvolatilityformanydifferentstrikesandexpiriesonthesameunderlyingthenwefindthatthevolatilityisnotconstant.AtypicalresultisthatofFigure7.16whichshowstheimpliedvolatilitiesfortheS&P500on9thSeptember1999foroptionsexpiringlaterinthemonth.Theimpliedvolatilitiesforthecallsandputsshouldbeidentical,becauseofput-callparity.Thedifferencesseenherecouldbeduetobid-offerspreadorcalculationsperformedatslightlydifferenttimes.Thisshapeiscommonlyreferredtoastheskew,orthesmile,ifitisturnedupatbothends,butitcouldalsobeintheshapeofafrown.Inthisexampleit’saratherlopsidedwrygrin,anegativeskewsinceitslopesdownwards.Whatevertheshape,ittendstopersistwithtime,withcertainshapesbeingcharacteristicofcertainmarkets. 132PartOnemathematicalandfinancialfoundationsFigure7.16ImpliedvolatilitiesfortheS&P500.Source:BloombergL.P.Thedependenceoftheimpliedvolatilityonstrikeandexpirycanbeinterpretedinmanyways.Theeasiestinterpretationisthatitrepresentsthemarket’sviewoffuturevolatilityinsomecomplexway.ThisissueiscoveredindepthinChapter50.Anotherpossibilityisthatitreflectstheuncertaintyinvolatility;perhapsvolatilityisalsoastochasticvariable,seeChapter51.Intheforeignexchangemarketstheytendtoplotimpliedvolatilityversusthedeltaofanoption.Figures7.17and7.18showplotsofimpliedvolatilityversusstrike,thepictureusuallylookedatbyequityderivativestraders,andimpliedvolatilityversusdelta,thepictureusuallylookedatbyFXderivativestraders,respectively.Althoughthesetwowaysoflookingatthedataareeffectivelythesame(andthesameassimplylookingatoptionpriceversusstrikeaswell)theydoleadtodifferentdynamicsinequityandFXmarkets.Inpractice,whenthevalueoftheunderlyingchangeswewillseetheoptionpriceschangeandalsooftenseetheimpliedvolatilitieschange.Nowthislatteristotallyinconsistentwiththetheorywe’vedevelopedsofar,butit’swhathappensandwe’llbelookingatthismorecloselylateron.ButtheimportantpointisthatinequitymarketsitisusuallysaidthatimpliedvolatilitystaysunchangedasafunctionofstrikewhereasinFXmarketstheimpliedvolatilitiesstayunchangedasafunctionofdelta.Thesearecalledthestickystrikeandstickydeltamodels.Thisisprobablydueinpartsimplytothepicturesthatthedifferenttypesoftraderslookatratherthanduetoanyfundamentaldifferencebetweenequitiesandexchangerates. theBlack–Scholesformulaeandthe‘greeks’Chapter71330.30.250.2Out-of-the-moneyputsOut-of-the-moneycalls0.15Impliedvolatility0.10.0506080100120140StrikeFigure7.17Impliedvolatilityversusstrike.0.30.250.20.15Out-of-the-moneycallsOut-of-the-moneyputsImpliedvolatility0.10.0500%10%20%30%40%50%60%70%80%90%100%DeltaFigure7.18Impliedvolatilityversusdelta. 134PartOnemathematicalandfinancialfoundations7.10ACLASSIFICATIONOFHEDGINGTYPES7.10.1WhyHedge?‘Hedging’initsbroadestsensemeansthereductionofriskbyexploitingrelationshipsorcorrelationbetweenvariousriskyinvestments(orbets).Theconceptisusedwidelyinhorserac-ing,othersportsbettingand,ofcourse,highfinance.Thereasonforhedgingisthatitcanleadtoanimprovedrisk/return.IntheclassicalModernPortfolioThe-oryframework(Chapter18),forexample,itisusuallypossibletoconstructmanyportfolioshav-ingthesameexpectedreturnbutwithdifferentvarianceofreturns(‘risk’).Clearly,ifyouhavetwoportfolioswiththesameexpectedreturntheonewiththelowerriskisthebetterinvestment.7.10.2TheTwoMainClassificationsProbablythemostimportantdistinctionbetweentypesofhedgingisbetweenmodel-independentandmodel-dependenthedgingstrategies.Model-independenthedging:AnexampleofsuchhedgingisPut-callParity.Thereisasimplerelationshipbetweencallsandputsonanasset(whentheyarebothEuropeanandwiththesamestrikesandexpiries),theunderlyingstockandazero-couponbondwiththesamematurity.Thisrelationshipiscompletelyindependentofhowtheunderlyingassetchangesinvalue.AnotherexampleisSpot-forwardParity.Inneithercasedowehavetospecifythedynamicsoftheasset,notevenitsvolatility,tofindapossiblehedge.Suchmodel-independenthedgesarefewandfarbetween.Model-dependenthedging:Mostsophisticatedfinancehedgingstrategiesdependonamodelfortheunderlyingasset.TheobviousexampleisthehedgingusedintheBlack–Scholesanalysisthatleadstoawholetheoryforthevalueofderivatives.Inpricingderivativeswetypicallyneedtoknowatleastthevolatilityoftheunderlyingasset.Ifthemodeliswrongthentheoptionvalueandanyhedgingstrategywillalsobewrong.7.10.3DeltaHedgingOneofthebuildingblocksofderivativestheoryisdeltahedging.Thisisthetheoreticallyperfecteliminationofallriskbyusingaverycleverhedgebetweentheoptionanditsunderlying.Deltahedgingexploitstheperfectcorrelationbetweenthechangesintheoptionvalueandthechangesinthestockprice.Thisisanexampleof‘dynamic’hedging;thehedgemustbecontinuallymonitoredandfrequentlyadjustedbythesaleorpurchaseoftheunderlyingasset.Becauseofthefrequentrehedging,anydynamichedgingstrategyisgoingtoresultinlossesduetotransactioncosts.Insomemarketsthiscanbeveryimportant.7.10.4GammaHedgingToreducethesizeofeachrehedgeand/ortoincreasethetimebetweenrehedges,andthusreducecosts,thetechniqueofgammahedgingisoftenemployed.Aportfoliothatisdeltahedgedis theBlack–Scholesformulaeandthe‘greeks’Chapter7135insensitivetomovementsintheunderlyingaslongasthosemovementsarequitesmall.Thereisasmallerrorinthisduetotheconvexityoftheportfoliowithrespecttotheunderlying.Gammahedgingisamoreaccurateformofhedgingthattheoreticallyeliminatesthesesecond-ordereffects.Typically,onehedgesone,exotic,say,contractwithavanillacontractandtheunderlying.Thequantitiesofthevanillaandtheunderlyingarechosensoastomakeboththeportfoliodeltaandtheportfoliogammainstantaneouslyzero.7.10.5VegaHedgingAsIsaidabove,thepricesandhedgingstrategiesareonlyasgoodasthemodelfortheunderlying.Thekeyparameterthatdeterminesthevalueofacontractisthevolatilityoftheunderlyingasset.Unfortunately,thisisaverydifficultparametertomeasureorevenestimate.Norisitusuallyaconstantasassumedinthesimpletheories.Obviously,thevalueofacontractdependsonthisparameter,andsotoensurethatourportfoliovalueisinsensitivetothisparameterwecanvegahedge.Thismeansthatwehedgeoneoptionwithboththeunderlyingandanotheroptioninsuchawaythatboththedeltaandthevega,thesensitivityoftheportfoliovaluetovolatility,arezero.Thisisoftenquitesatisfactoryinpracticebutisusuallytheoreticallyinconsistent;weshouldnotuseaconstantvolatility(basicBlack–Scholes)modeltocalculatesensitivitiestoparametersthatareassumednottovary.Thedistinctionbetweenvariables(underlyingassetpriceandtime)andparameters(volatility,dividendyield,interestrate)isextremelyimportanthere.Itisjustifiabletorelyonsensitivitiesofpricestovariables,butusuallynotsensitivitytoparameters.Togetaroundthisproblemitispossibletomodelindependentlyvolatilityetc.asvariablesthemselves.Insuchawayitispossibletobuildupaconsistenttheory.7.10.6StaticHedgingTherearequiteafewproblemswithdeltahedging,onboththepracticalandthetheoreticalside.Inpractice,hedgingmustbedoneatdiscretetimesandiscostly.Sometimesonehastobuyorsellaprohibitivelylargenumberoftheunderlyinginordertofollowthetheory.Thisisaproblemwithbarrieroptionsandoptionswithdiscontinuouspayoff.Onthetheoreticalside,themodelfortheunderlyingisnotperfect,attheveryleastwedonotknowparametervaluesaccurately.Deltahedgingaloneleavesusveryexposedtothemodel;thisismodelrisk.Manyoftheseproblemscanbereducedoreliminatedifwefollowastrategyofstatichedgingaswellasdeltahedging:buyorsellmoreliquidtradedcontractstoreducethecashflowsintheoriginalcontract.Thestatichedgeisputintoplacenow,andleftuntilexpiry.Intheextremecasewhereanexoticcontracthasallofitscashflowsmatchedbycashflowsfromtradedoptionsthenitsvalueisgivenbythecostofsettingupthestatichedge;amodelisnotneeded.(Butthentheoptionwasn’texoticinthefirstplace.)7.10.7MarginHedgingOftenwhatcausesbanks,andotherinstitutions,tosufferduringvolatilemarketsisnotthechangeinthepapervalueoftheirassetsbuttherequirementtocomeupsuddenlywithalargeamountofcashtocoveranunexpectedmargincall.RecentexampleswheremarginhascausedsignificantdamageareMetallgesellschaftandLongTermCapitalManagement.Writingoptionsisveryrisky.Thedownsideofbuyinganoptionisjusttheinitialpremium,whiletheupsidemaybeunlimited.Theupsideofwritinganoptionislimited,butthedownsidecouldbe 136PartOnemathematicalandfinancialfoundationshuge.Forthisreason,tocovertheriskofdefaultintheeventofanunfavorableoutcome,theclearinghousesthatregisterandsettleoptionsinsistonthedepositofamarginbythewritersofoptions.Margincomesintwoforms,theinitialmarginandthemaintenancemargin.Theinitialmarginistheamountdepositedattheinitiationofthecontract.Thetotalamountheldasmarginmuststayaboveaprescribedmaintenancemargin.Ifiteverfallsbelowthislevelthenmoremoney(orequivalentinbonds,stocksetc.)mustbedeposited.Theamountofmarginthatmustbedepositeddependsontheparticularcontract.Adramaticmarketmovecouldresultinasuddenlargemargincallthatmaybedifficulttomeet.Topreventthissituationitispossibletomarginhedge.Thatis,setupaportfoliosuchthatmargincallsononepartoftheportfolioarebalancedbyrefundsfromotherparts.Usuallyover-the-countercontractshavenoassociatedmarginrequirementsandsowon’tappearinthecalculation.7.10.8Crash(Platinum)HedgingThefinalvarietyofhedgingthatwediscussisspecifictoextrememarkets.Marketcrasheshaveatleasttwoobviouseffectsonourhedging.Firstofall,themovesaresolargeandrapidthattheycannotbetraditionallydeltahedged.Theconvexityeffectisnotsmall.Second,normalmarketcorrelationsbecomemeaningless.Typicallyallcorrelationsbecomeone(orminusone).CrashorPlatinumhedgingexploitsthelattereffectinsuchawayastominimizetheworstpossibleoutcomefortheportfolio.Themethod,calledCrashMetrics(Chapter43),doesnotrelyondifficulttomeasureparameterssuchasvolatilitiesandsoisaveryrobusthedge.Platinumhedgingcomesintwotypes:hedgingthepapervalueoftheportfolioandhedgingthemargincalls.7.11SUMMARYInthischapterwewentthroughthederivationofsomeofthemostimportantformulae.Wealsosawthedefinitionsanddescriptionsofthehedgeratios.Tradinginderivativeswouldbenomorethangamblingifyoutookawaytheabilitytohedge.Hedgingisallaboutmanagingriskandreducinguncertainty.FURTHERREADING•SeeTaleb(1997)foralotofdetailedanalysisofvega.•SeePressetal.(1992)formoreroutinesforfindingroots,i.e.forfindingimpliedvolatilities.•Therearemany‘virtual’optionpricersontheinternet.See,forexample,www.cboe.com.•I’mnotgoingtospendmuchtimeonderivingorevenpresentingformulae.Thereare1001booksthatcontainoptionformulae,thereisevenonebookwith1001formulae(Haug,1997).•SeetheseriesofarticlesbyThorp(2002)onhowhederivedthecorrectoptionpricingformulaeinthelate1960s.•Haug(2003)discussesthesophisticatedtraderuseofthesimpleequations. theBlack–Scholesformulaeandthe‘greeks’Chapter71372))x)212)(tN(d2−e))D−−(d)2π2−Td2t1(N(d12))×)r√1−t)t)√2))N(d2σ)×dT×)−−N2t2(d2t)−2d2σr(Tt)=N(d(d(dt)−−(d1√(dr(T−−−−1(N−σ+−e(x)BinaryPut−NTN1t)t)t)N+Ntt)etr(T−1(t)−d(T−−1−t)(T1t)−−eNt)√t)2r(Td−3Sd−−Tσt−r(TσS−2Sr(T−(Tr(T22r(TT−and−r(Tσ−e2−σ−−√(T√σr(T−e−re−ee−−Te−e√dξ2ξ)12()−t)2)e2(d2D−(dx−T×2tN(dN−∞)2×)r√)d1−×)t)t)−Nπ)2σ2t)dT22−t)12)(dt)2(d(d−(ddσr(T−)2N−−−1√r(T−r(T√2(dt1N(dNNσN+−te−=−d(Tt)t)t)t)(T+t)tt)eeBinaryCallN(dNt)2−−1−−3S−−−tt)t)T−Sdr(T2d1−σ−−−√r(T2σr(Tr(T−(T−σ2r(T−TTσr(T−−e2e−e√(T√Tr(T−−σSere+−−−+√t,N(x)ee−−−)Td2)√))1σd1t))1−d)1)−2(d−−dd2)1−(d)×N(1−−tD(T−1Nt)N(d−N−−(dt)tt)−t)=N()1)1t))eN(−−−−t)N(t−T1t)Nr(T−t)Put−t)−(d√d−t)(TTD(TD(T(N(dN−D(T2−r(T−2√−t)e−−D(Tr(Tt)t)T−−2Sσte−−−−−√D(Tσ+−t)Se)(T−2SeEeD(TσSσSeDSN(rEeed1T−σ−+D(T−−−−+−
√E(T12teeS−(T−−DT))1−√)1)σ2(r1t))×(dN(d)(d−2)t)+)2t1NN(d−1ND(T(dt)tt)t)N(d))1t)−−N(d−−−−N(dt)1t−T)et)ND(T(S/E)Call−(d1−t)(TTD(Tr(T−t)−D(T√−2√−−−N(dN−2r(TStelogr(T−t)t)T−2σt)et)SeD(T−−√D(T−σ+−−−=−EeσSeDSN(drEee1T2Se−D(TD(TσS−+−−
d(T−−e√eSE(T−,dt)−)(T2σ12t+−DT−√(rσ+V22∂∂SV3∂V∂r∂V∂D∂V∂t3∂∂S)(S/E)V∂V∂S∂V∂σ)log(r(D=ValueBlack–ScholesvalueDeltaSensitivitytounderlyingGammaSensitivityofdeltatounderlyingThetaSensitivitytotimeSpeedSensitivityofgammatounderlyingVegaSensitivitytovolatilityRhoSensitivitytointerestrateRhoSensitivitytodividendyield1d CHAPTER8simplegeneralizationsoftheBlack–ScholesworldInthisChapter...•complexdividendstructures•jumpconditions•time-dependentvolatility,interestrateanddividendyield8.1INTRODUCTIONThischapterisanintroductiontosomeofthepossiblegeneralizationsofthe‘Black–Scholesworld.’Inparticular,Iwilldiscusstheeffectofdividendpaymentsontheunderlyingassetandhowtoincorporatetime-dependentparametersintotheframework.Thesesubjectsleadtosomeinterestingandimportantmathematicalandfinancialconclusions.Thegeneralizationsareverystraightforward.However,later,inPartFive,IdescribeothermodelsofthefinancialworldthattakeusalongwayfromBlack–Scholes.8.2DIVIDENDS,FOREIGNINTERESTANDCOSTOFCARRYInChapter5IshowedhowtoincorporatecertaintypesofdividendstructuresintotheBlack–Scholesoptionpricingframework,andtheninChapter7Igavesomeformulaeforthevaluesofsomecommonvanillacontracts,againwithdividendsontheunderlying.ThedividendstructurethatIdealtwithwastheverysimplestfromamathematicalpointofview.Iassumedthatanamountwaspaidtotheholderoftheassetthatwasproportionaltothevalueoftheassetandthatitwaspaidcontinuously.Inotherwords,theownerofoneassetreceivedadividendofDSdtinatimestepdt.Thisdividendstructureisrealisticiftheunderlyingisanindexonalargenumberofindividualassetseachreceivingalumpsumdividendbutwithallthesedividendsspreadoutthroughtheyear.Itisalsoagoodmodeliftheunderlyingisacurrencyinwhichcasewesimplytakethe‘dividendyield’tobetheforeigninterestrate.Similarly,iftheunderlyingisacommoditywithacostofcarrythatisproportionaltoitsvalue, 140PartOnemathematicalandfinancialfoundationsthenthe‘dividendyield’isjustthecostofcarry(withaminussign,webenefitfromdividendsbutmustpayoutthecostofcarry).Torecap,iftheunderlyingreceivesadividendofDSdtinatimestepdtwhentheassetpriceisSthen∂V∂2V∂V+1σ2S2+(r−D)S−rV=0.∂t2∂S2∂SHowever,iftheunderlyingisastock,thentheassumptionofconstantandcontinuously-paiddividendyieldisnotagoodone.8.3DIVIDENDSTRUCTURESTypically,dividendsarepaidoutquarterlyintheUSandsemi-annuallyorquarterlyintheUK.Thedividendissetbytheboardofdirectorsofthecompanysometimebeforeitispaidoutandtheamountofthepaymentismadepublic.Theamountisoftenchosentobesimilartopreviouspayments,butwillobviouslyreflectthesuccessorotherwiseofthecompany.Theamountspecifiedisadollaramount,itisnotapercentageofthestockpriceonthedaythatthepaymentismade.Sorealitydiffersfromtheabovesimplemodelinthreerespects:•theamountofthedividendisnotknownuntilshortlybeforeitispaid•thepaymentisagivendollaramount,independentofthestockprice•thedividendispaiddiscretely,andnotcontinuouslythroughouttheyear.InwhatfollowsIamgoingtomakesomeassumptionsaboutthedividend.Iwillassumethat•theamountofthedividendisaknownamount,possiblywithsomefunctionaldependenceontheassetvalueatthepaymentdate•thedividendispaiddiscretelyonaknowndate.OtherassumptionsthatIcould,butwon’t,makebecauseofthesubsequentcomplexityofthemodelingarethatthedividendamountand/ordatearerandom,thatthedividendamountisafunctionofthestockpriceonthedaythatthedividendisset,thatthedividenddependsonhowwellthestockhasdoneinthepreviousquarter...8.4DIVIDENDPAYMENTSANDNOARBITRAGEHowdoesthestockreacttothepaymentofadividend?Toputthequestionanotherway,ifyouhaveachoicewhethertobuyastockjustbeforeorjustafteritgoesex-dividend,whichshouldyouchoose?Letmeintroducesomenotation.ThedatesofdividendswillbetiandtheamountofthedividendpaidonthatdaywillbeDi.Thismaybeafunctionoftheunderlyingasset,butitthenmustbeadeterministicfunction.Themomentjustbeforethestockgoesex-dividendwillbedenotedbyt−andthemomentjustafterwillbet+.iiThepersonwhobuysthestockonorbeforet−willalsogettherightstothedividend.Theipersonwhobuysitatt+orlaterwillnotreceivethedividend.Itlookslikethereisanadvantagei simplegeneralizationsoftheBlack–ScholesworldChapter81411401201008060Stockprice4020000.20.40.60.81TimeFigure8.1Astockpricepathacrossadividenddate.inbuyingthestockjustbeforethedividenddate.Ofcourse,thisadvantageisbalancedbyafallinthestockpriceasitgoesex-dividend.Acrossadividenddatethestockfallsbytheamountofthedividend.Ifitdidnot,thentherewouldbearbitrageopportunities.WecanwriteS(t+)=S(t−)−D.(8.1)iiiInFigure8.1isshownanassetpricepathshowingthefallintheassetpriceasitgoesex-dividend;thedrophasbeenexaggerated.Thisjumpinthestockpricewillpresumablyhavesomeeffectonthevalueofanoption.Wewilldiscussthisnext.8.5THEBEHAVIOROFANOPTIONVALUEACROSSADIVIDENDDATEWehavejustseenhowtheunderlyingassetjumpsinvalue,inacompletelypredictableway,acrossadividenddate.Jumpconditionstellusaboutthevalueofadependentvariable,anoptionprice,whenthereisadiscontinuouschangeinoneoftheindependentvariables.Inthepresentcase,thereisadiscontinuouschangeintheassetpriceduetothepaymentofadividendbuthowdoesthisaffecttheoptionprice?Doestheoptionpricealsojump?Thejumpconditionrelatesthevaluesoftheoptionacrossthejump,fromtimest−tot+.Thejumpconditionwillbederivedbyaiisimpleno-arbitrageargument.Toseewhatthejumpconditionshouldbe,askthequestion:‘ByhowmuchdoIprofitorlosewhenthestockpricejumps?’Ifyouholdtheoptionthenyoudonotseeanyofthedividend, 142PartOnemathematicalandfinancialfoundations120B10080A60Stockprice4020000.10.20.30.40.50.60.70.80.91Time3025C2015Optionvalue105000.20.40.60.81Time70OptionValueAfterOptionValueBefore605040V30C2010AB0020406080100120140SFigure8.2Toppicture,arealizationofthestockpriceshowingafallacrossthedividenddate.Middlepicture,thecorrespondingrealizationoftheoptionprice(inthisexampleacall).Bottompicture,theoptionvalueasafunctionofthestockpricejustbeforeandjustafterthedividenddate. simplegeneralizationsoftheBlack–ScholesworldChapter8143thatgoestotheholderofthestocknotyou,theholderoftheoption.Ifthedividendamountanddateareknowninadvancethenthereisnosurpriseinthefallinthestockprice.Theconclusionmustbethattheoptiondoesnotchangeinvalueacrossthedividenddate,itspathiscontinuous.Continuityoftheoptionvalueacrossadividenddatecanbewrittenas

VS(t−),t−=VS(t+),t+(8.2)iiiior,intermsoftheamountofthedividend,

VS,t−=VS−D,t+.(8.3)iiiThejumpconditionanditseffectontheoptionvaluecanbeexplainedbyreferencetoFigure8.2.Inthisfigure,thetoppictureshowsarealizationofthestockpricewithafallacrossthedividenddate.Themiddlepictureshowsthecorrespondingrealizationofacalloptionprice.Thebottompictureshowstheoptionvalueasafunctionofthestockpricejustbeforeandjustafterthedividenddate.Observethepoints‘A’and‘B’onthesepictures.‘A’isthestockpriceafterthedividendhasbeenpaidand‘B’isthepricebefore.Onthebottompictureweseethevaluesoftheoptionassociatedwiththesebeforeandafterassetprices.Theseoptionvaluesarethesameandaredenotedby‘C.’Eventhoughthereisafallintheassetvalue,theoptionvalueisunchangedbecausethewholeVversusSplotchanges.Therelationshipbetweenthebeforeandaftervaluesoftheoptionarerelatedby(8.3).Iwillgivetwoexamples.Supposethatthedividendpaidoutisproportionaltotheassetvalue,Di=DS.InthiscaseS(t+)=(1−D)S(t−).iiEquation(8.3)isthenjust

VS,t−=V(1−D)S,t+.iiThetwooptionpricecurvesareidenticalifonestretchestheaftercurvebyafactorof(1−D)−1inthehorizontaldirection.Thus,eventhoughtheoptionvalueiscontinuousacrossadividenddate,thedeltachangesdiscontinuously.IfthedividendisindependentofthestockpricethenS(t+)=S(t−)−D,iiiwhereDiisindependentoftheassetvalue.Thebeforecurveisthenidenticaltotheaftercurve,butshiftedbyanamountDi.8.6COMMODITIESConvenienceyieldisthebenefitorpremiumassociatedwithholdinganunderlyingproductorphysicalgood,ratherthanafuturepositioninthatproduct.Forexample,thereisanobviousbenefittotheactualholdingofbarrelsofoil.Thisnaturallyleadsustothinkingofcommoditiesasbeingoftwotypes. 144PartOnemathematicalandfinancialfoundations•Investmentcommodities:Commoditiesheldforinvestment,suchasgold.•Consumptioncommodities:Commoditiesheldforconsumption,suchasoilorwheat.Sincetheyareheldforconsumption,andhaveavalueassociatedwiththis,theymaynotbesoldwhenthepricerises.Crucially,thiscanmakearbitrageargumentsonesided.8.6.1FuturesPricesandArbitrageThestandardargumentfortherelationshipbetweenspotandfuturespricesdoesnotholdforconsumptioncommodities.Onlyanupperboundcanbefound.Thisisbecausetheremaybeconstraintsonthesellingofconsumptioncommodities;theoilisneededforheating,transportetc.andcannotbereasonablysold.Theno-arbitrageargumentcanstillbeappliedtoinvestmentcommodityfutures.8.6.2StorageCostsNostoragecostsIftherearenostoragecoststhentherelationshipbetweenspotandforwardpricesisF=Ser(T−t).Storagecostsproportionaltospotprice,forinvestmentcommoditiesIftherearestoragecosts,andtheyareproportionaltothepriceofthecommoditythenF=Se(r+u)(T−t).Thisisequivalenttotherebeinganegativedividendyield.Storagecostsproportionaltospotprice,forconsumptioncommoditiesForconsumptioncommoditiesallwecansayisF≤Se(r+u)(T−t).Thisisbecauseholdersofcommoditieswillbereluctanttosellthecommodity(atthespotprice)whichtheyarekeepingforconsumption.Theymaybuymoreoilifthespotpriceislow,buttheywon’tsellitifthespotpriceishighsincethentheywouldhavenofuel.8.6.3ConvenienceYieldSinceF≤Se(r+u)(T−t)weintroducetheconvenienceyieldysuchthatF=Se(r+u−y)(T−t),withy≥0.Forinvestmentcommoditiesy=0,ofcourse. simplegeneralizationsoftheBlack–ScholesworldChapter81458.6.4CostofCarryCostofcarryisthesumofstoragecost,interestpaidtofinancetheasset,lessanyincomefromtheasset.Examples:UsuallythedriftcoefficientintheBlack–Scholesequationisjustr,butthiswillchangeifthereisacostofcarryasfollows.1.Nodividendsetc.:r2.Dividendyield:r−D3.Foreignexchange:r−rf4.Commodity:r+uTheconvenienceyieldisnotincludedinthis.8.6.5EffectonOptionsIfSisthespotpricethenoptionsonthespotsatisfy∂V∂2V∂V+1σ2S2+(r+u)S−rV=0.∂t2∂S2∂SChangevariablestoF=Se(r+u−y)(T−t),H(F,t)=V(S,t)toget∂H∂2H∂H+1σ2F2+yF−rH=0.∂t2∂F2∂FThisistherelevantequationincorporatingcostofcarryandtheconvenienceyield.8.7STOCKBORROWINGANDREPOIhavemanytimesreferredtosellingstockforhedgingpurposes,goingshortthestock.Butinpracticehowcanonesellaquantitythatonedoesnotownandwhichisnaturallysomethingyouwouldbuy,in‘positive’quantitiesasopposedtonegativequantities?Tokeepitreal,let’simagineyouwanttogoshortalawnmower.Youdon’townalawnmowertosell,sowhatcanyoudo?Easy,justborrowonefromyourneighborandsellthat!Whenyourneighborwantshislawnmowerbackyouhavetogooutandbuyanotheronetogivehim.Iflawnmowerpriceshavemeanwhilefallenyouwillmakeaprofit.Intheworldofstocksandsharesthesameideaapplies.Ifyouwanttogoshortastockyoumustfirstborrowit.Butthisisnotgoingtobecostless,usuallythereissomepaymenttobemade,likeaninterestchargeontheamountyouborrow.Thisshouldreallybefactoredintoanyoptionpricingmodel.Toquantifythis,let’ssupposethatyouhavetopayinterestatarateofRonthevalueofthestockthatyouhaveborrowed.NowgothroughtheBlack–Scholesargumentandincludethiscostintheanalysis. 146PartOnemathematicalandfinancialfoundationsWebeginwiththelognormalrandomwalkforthestock,dS=µSdt+σSdX.Andwesetupaportfolio,longtheoptionandshortthestock,=V(S,t)−S.Thisthenchangesbyd=dV−dS,whichcanbewritten,usingIto’slemma,asˆ∂V∂V∂2Vd=dt+dS+1σ2S2dt−dS.∂t∂S2∂S2Exceptthatthisisnotquiteright.Thevalueofourportfoliochangesbecauseoftheinterestrate,thereporate;wemustpayfortheborrowedstock.Soscratchthatequation,itshouldbe∂V∂V∂2Vd=dt+dS+1σ2S2dt−dS−Rmax(,0)Sdt.∂t∂S2∂S2Letmeexplainthenewterm.First,theinterestpaymentisonthevalueoftheshortposition,thatisS.Second,becauseitisaninterestratetheactualpaymentisproportionaltothetimestep,dt.Finally,andthisistheinterestingpart,thepaymentisonlywhenispositive,sothatweareshortthestock(remembertheminussigninfrontoftheintheportfolio).Hencethemaximumfunctionabove.Toeliminatetherandomtermswestillchoose∂V=,∂Sleavinguswith∂V∂2Vd=+1σ2S2dt−Rmax(,0)Sdt.∂t2∂S2Becausethisisdeterministicwecansetthereturnontheportfolioequaltotherisk-freerated=rdt.Theendresultisthepartialdifferentialequation∂V∂2V∂V∂V+1σ2S2+rS−rV−RSmax,0=0.∂t2∂S2∂S∂SThisistheequationforpricingderivativesinthepresenceofinterestpaymentsforshortingstocks.Notethatitisactuallyanon-linearequation.Theconsequencesofnon-linearitywilldiscussedindepthlateroninthebook. simplegeneralizationsoftheBlack–ScholesworldChapter81478.8TIME-DEPENDENTPARAMETERSThenextgeneralizationconcernsthetermstructureofparameters.InthissectionIshowhowtoderiveformulaeforoptionswhentheinterestrate,volatilityanddividendyield/foreigninterestratearetimedependent.TheBlack–Scholespartialdifferentialequationisvalidaslongastheparametersr,Dandσareknownfunctionsoftime;inpracticeoneoftenhasaviewonthefuturebehavioroftheseparameters.Forinstance,youmaywanttoincorporatethemarket’sviewonthedirectionofinterestrates.Assumethatyouwanttopriceoptionsknowingr(t),D(t)andσ(t).NotethatwhenIwrite‘D(t)’Iamspecificallyassumingatime-dependentdividendyield,thatis,theamountofthedividendisD(t)Sdtinatimestepdt.Theequationthatwemustsolveisnow∂V∂2V∂V+1σ2(t)S2+(r(t)−D(t))S−r(t)V=0,(8.4)∂t2∂S2∂Swherethedependenceontisshownexplicitly.Introducenewvariablesasfollows:S=Seα(t),V=Veβ(t),t=γ(t).Wearefreetochoosethefunctionsα,βandγandsowewillchoosethemsoastoeliminatealltime-dependentcoefficientsfrom(8.4).Afterchangingvariables(8.4)becomes∂V2∂2V
∂V
γ(t)˙+1σ(t)2S+r(t)−D(t)+˙α(t)S−r(t)+β(t)˙V=0,(8.5)∂t22∂S∂Swhere˙=d/dt.BychoosingTβ(t)=r(τ)dτtwemakethecoefficientofVzeroandthenbychoosingTα(t)=(r(τ)−D(τ))dτ,twemakethecoefficientof∂V/∂Salsozero.Finally,theremainingtimedependence,inthevolatilityterm,canbeeliminatedbychoosingTγ(t)=σ2(τ)dτ.tNow(8.5)becomesthemuchsimplerequation∂V2V12∂=S.(8.6)∂t22∂S 148PartOnemathematicalandfinancialfoundationsTheimportantpointaboutthisequationisthatithascoefficientswhichareindependentoftime,andthereisnomentionofr,Dorσ.IfweuseV(S,t)todenoteanysolutionof(8.6),thenthecorrespondingsolutionof(8.5),intheoriginalvariables,is
V=e−β(t)VSeα(t),γ(t).(8.7)NowuseVBStomeananysolutionoftheBlack–Scholesequationforconstantinterestraterc,dividendyieldDcandvolatilityσc.Thissolutioncanbewrittenintheform
V=e−rc(T−t)VSe−(rc−Dc)(T−t),σ2(T−t)(8.8)BSBScforsomefunctionVBS.Bycomparing(8.7)and(8.8)itfollowsthatthesolutionofthetime-dependentparameterproblemisthesameasthesolutionoftheconstantparameterproblemifweusethefollowingsubstitutions:T1rc=r(τ)dτT−ttT1Dc=D(τ)dτT−ttT212σc=σ(τ)dτT−ttTheseformulaegivetheaverage,overtheremaininglifetimeoftheoption,oftheinterestrate,thedividendyieldandthesquaredvolatility.Justtomakethingsabsolutelyclear,hereistheformulaforaEuropeancalloptionwithtime-dependentparameters:TT−D(τ)dτ−r(τ)dτSetN(d1)−EetN(d2)whereT1T2log(S/E)+(r(τ)−D(τ))dτ+σ(τ)dτd=t2t1Tσ2(τ)dτtandT1T2log(S/E)+(r(τ)−D(τ))dτ−σ(τ)dτd=t2t.2Tσ2(τ)dτtTherearesomeconditionsthatImustattachtotheuseoftheseformulae.Theyaregenerallynotcorrectifthereisearlyexercise,orforcertaintypesofexoticoption.Thequestiontoasktodecidewhethertheyarecorrectis:‘Arealltheconditions,finalandboundary,preservedbythetransformations?’ simplegeneralizationsoftheBlack–ScholesworldChapter81498.9FORMULAEFORPOWEROPTIONSAnoptionwithapayoffthatdependsontheassetpriceatexpiryraisedtosomepoweriscalledapoweroption.SupposethatithasapayoffPayoff(Sα)wecanfindasimpleformulaforthevalueoftheoptionifwehaveasimpleformulaforanoptionwithpayoffgivenbyPayoff(S).(8.9)Thisisbecauseofthelognormalityoftheunderlyingasset.WritingS=SαtheBlack–Scholesequationbecomes,inthenewvariableS,∂V∂2V
∂V+1α2σ2S2+α1σ2(α−1)+rS−rV=0.∂t2∂S22∂SThuswhatevertheformulafortheoptionvaluewithsimplepayoff(8.9),theformulaforthepowerversionhasSαinsteadofSandadjustmentmadetoσ,randD.8.10THElogCONTRACTThelogcontracthasthepayofflog(S/E).Thetheoreticalfairvalueforthiscontractisoftheforma(t)+b(t)log(S/E).SubstitutingthisexpressionintotheBlack–Scholesequationresultsina˙+b˙log(S/E)−1σ2b+(r−D)b−ra−rblog(S/E)=0,2where˙denotesd/dt.Equatingtermsinlog(S/E)andthoseindependentofSresultsin
b(t)=e−r(T−t)anda(t)=r−D−1σ2(T−t)e−r(T−t).2Thetwoarbitraryconstantsofintegrationhavebeenchosentomatchthesolutionwiththepayoffatexpiry.Thisvalueisratherspecialinthatthedependenceoftheoptionpriceontheunderlyingasset,S,andthevolatility,σ,uncouples.OnetermcontainsSandnoσandtheothercontainsσandnoS.WebrieflysawinChapter7theconceptofvegahedgingtoeliminatevolatilityrisk.Itisconceivable,eventhoughnotentirelyjustifiably,thatthesimplicityofthelogcontractvaluemakesitausefulweaponforhedgingothercontractsagainstfluctuationsinvolatility.Havingsaidthat,it’snotexactlyahighlyliquidcontract. 150PartOnemathematicalandfinancialfoundationsThelogcontractpayoffcanbepositiveornegativedependingonwhetherS>EorS0.Thisisarisklessprofit.Ifwebelievethattherearenoarbitrageopportunitiesthenwemustbelieve(9.1).Whiletheoptionvalueisstrictlygreaterthanthepayoff,itmustsatisfytheBlack–Scholesequation;Ireturntothispointinthenextsection.Recallingthattheoptionisperpetualandthereforethatthevalueisindependentoft,itmustsatisfyd2VdV1σ2S2+rS−rV=0.2dS2dSThisistheordinarydifferentialequationyougetwhentheoptionvalueisafunctionofSonly.Thegeneralsolutionofthissecond-orderordinarydifferentialequationis−2r/σ2V(S)=AS+BS,whereAandBarearbitraryconstants.Thefirstpartofthissolution(thatwithcoefficientA)issimplytheasset:theassetitselfsatisfiestheBlack–Scholesequation.IfwecanfindAandBwehavefoundthesolutionfortheperpetualAmericanput.Clearly,fortheperpetualAmericanputthecoefficientAmustbezero;asS→∞thevalueoftheoptionmusttendtozero.WhataboutB?Letuspostulatethatwhiletheassetvalueis‘high’wewon’texercisetheoption.Butifitfallstoolowweimmediatelyexercisetheoption,receivingE−S.(Commonsensetellsuswedon’texercisewhenS>E.)SupposethatwedecidethatS=S∗isthevalueatwhichweexercise,i.e.assoonasSreachesthisvaluefromaboveweexercise.HowdowechooseS∗?WhenS=S∗theoptionvaluemustbethesameastheexercisepayoff:V(S∗)=E−S∗.Itcannotbeless,thatwouldresultinanarbitrageopportunity,anditcannotbemoreorwewouldn’texercise.Continuityoftheoptionvaluewiththepayoffgivesusoneequation:∗∗−2r/σ2∗V(S)=B(S)=E−S. earlyexerciseandAmericanoptionsChapter9153ButsincebothBandS∗areunknown,weneedonemoreequation.Let’slookatthevalueoftheoptionasafunctionofS∗,eliminatingBusingtheabove.WefindthatforS>S∗
−2r/σ2∗SV(S)=(E−S).(9.2)S∗WearegoingtochooseS∗tomaximizetheoption’svalueatanytimebeforeexercise.Inotherwords,whatchoiceofS∗makesVgivenby(9.2)aslargeaspossible?Thereasonforthisisobvious,ifwecanexercisewheneverwelikethenwedosoinsuchawaytomaximizeourworth.Wefindthisvaluebydifferentiating(9.2)withrespecttoS∗andsettingtheresultingexpressionequaltozero:
−2r/σ2
−2r/σ2
∂∗S1S∗2r∗(E−S)=−S+(E−S)=0.∂S∗S∗S∗S∗σ2Wefindthat∗ES=.1+σ2/2rThischoicemaximizesV(S)forallS≥S∗.ThesolutionwiththischoiceforS∗andwiththecorrespondingBgivenby2
1+2r/σ2σE2r1+σ2/2risshowninFigure9.1.1.210.8PayoffV0.6Optionvalue0.40.2S*000.511.5SFigure9.1ThesolutionfortheperpetualAmericanput. 154PartOnemathematicalandfinancialfoundationsTheobservantreaderwillnoticesomethingspecialaboutthisfunction:TheslopeoftheoptionvalueandtheslopeofthepayofffunctionarethesameatS=S∗.ToseethatthisfollowsfromthechoiceofS∗letusexaminethedifferencebetweentheoptionvalueandthepayofffunction:
−2r/σ2∗S(E−S)−(E−S).S∗DifferentiatethiswithrespecttoSandyouwillfindthattheexpressioniszeroatS=S∗.Thisdemonstrates,inacompletelynon-rigorousway,thatifwewanttomaximizeouroption’svaluebyacarefulchoiceofexercisestrategy,thenthisisequivalenttosolvingtheBlack–Scholesequationwithcontinuityofoptionvalueandoptiondelta,theslope.Thisiscalledthehigh-contactorsmooth-pastingcondition.TheAmericanoptionvalueismaximizedbyanexercisestrategythatmakestheoptionvalueandoptiondeltacontinuousWeexercisetheoptionassoonastheassetpricereachesthelevelatwhichtheoptionpriceandthepayoffmeet.Thisposition,S∗,iscalledtheoptimalexercisepoint.Anotherwayoflookingattheconditionofcontinuityofdeltaistoconsiderwhathappensifthedeltaisnotcontinuousattheexercisepoint.ThetwopossibilitiesareshowninFigure9.2.Inthisfigurethecurve(a)correspondstoexercisethatisnotoptimalbecauseitispremature;theoptionvalueislowerthanitcouldbe.Incase(b)thereisclearlyanarbitrageopportunity.1.210.8(b)0.6V0.40.2(a)000.511.5SFigure9.2Optionpricewhenexerciseis(a)toosoonor(b)toolate. earlyexerciseandAmericanoptionsChapter9155Ifwetakecase(a)butprogressivelydelayexercisebyloweringtheexercisepoint,wewillmaximizetheoptionvalueeverywherewhenthedeltaiscontinuous.Whenthereisacontinuouslypaidandconstantdividendyieldontheasset,ortheassetisaforeigncurrency,therelevantordinarydifferentialequationfortheperpetualoptionisd2VdV1σ2S2+(r−D)S−rV=0.2dS2dSThegeneralsolutionisnowα+α−AS+BS,where
±1121222α=−r−D−σ±r−D−σ+2rσ,σ222withα−<0<α+.TheperpetualAmericanputnowhasvalueα−BS,where
1−α−1EB=−.α−1−1/α−ItisoptimaltoexercisewhenSreachesthevalueE.1−1/α−BeforeconsideringtheformulationofthegeneralAmericanoptionproblem,weconsideronemorespecialcase.9.3PERPETUALAMERICANCALLWITHDIVIDENDSThesolutionfortheAmericanperpetualcallisα+AS,where
1−α+1EA=α+1−1/α+anditisoptimaltoexerciseassoonasSreaches∗ES=1−1/α+frombelow.AninterestingspecialcaseiswhenD=0.ThenthesolutionisV=SandS∗becomesinfinite.ThusitisneveroptimaltoexercisetheAmericanperpetualcallwhentherearenodividendsontheunderlying;itsvalueisthesameastheunderlying.Asweseeinamoment,itisalsoneveroptimaltoexercisetheordinarynon-perpetualAmericancallintheabsenceofdividends. 156PartOnemathematicalandfinancialfoundations9.4MATHEMATICALFORMULATIONFORGENERALPAYOFFNowIbuildupthetheoryforAmerican-stylecontractswitharbitrarypayoffandexpiry,fol-lowingthestandardBlack–Scholesargumentwithminormodifications.ThecontractwillnolongerbeperpetualandtheoptionvaluewillnowbeafunctionofbothSandt.ConstructaportfolioofoneAmericanoptionwithvalueV(S,t)andshortanumberoftheunderlying:=V−S.Thechangeinvalueofthisportfolioinexcessoftherisk-freerateisgivenby

∂V∂2V∂Vd−rdt=+1σ2S2−r(V−S)dt+−dS.∂t2∂S2∂SWiththechoice∂V=∂Sthisbecomes
∂V∂2V∂V+1σ2S2+rS−rVdt.∂t2∂S2∂SIntheBlack–ScholesargumentforEuropeanoptionswesetthisexpressionequaltozero,sincethisprecludesarbitrage.Butitprecludesarbitragewhetherwearebuyingorsellingthecontract.WhenthecontractisAmericanthelong/shortrelationshipisasymmetrical,itistheholderoftheexerciserightswhocontrolstheearly-exercisefeature.Thewriteroftheoptioncandonomorethansitbackandenjoytheview.IfVisthevalueofalongpositioninanAmericanoptionthenallwecansayisthatwecanearnnomorethantherisk-freerateonourportfolio.Thuswearriveattheinequality∂V∂2V∂V+1σ2S2+rS−rV≤0.(9.3)∂t2∂S2∂SThewriteroftheAmericanoptioncanmakemorethantherisk-freerateiftheholderdoesnotexerciseoptimally.Healsomakesmoreprofitiftheholderoftheoptionhasapoorestimateofthevolatilityoftheunderlyingandexercisesinaccordancewiththatestimate.Equation(9.3)caneasilybemodifiedtoaccommodatedividendsontheunderlying.IfthepayoffforearlyexerciseisP(S,t),possiblytime-dependent,thentheno-arbitrageconstraintV(S,t)≥P(S,t),(9.4)mustapplyeverywhere.AtexpirywehavethefinalconditionV(S,T)=P(S,T).(9.5) earlyexerciseandAmericanoptionsChapter9157Theoptionvalueismaximizediftheowneroftheoptionexercisessuchthat∂V(9.6)=iscontinuous∂STheAmericanoptionvaluationproblemconsistsof(9.3),(9.4),(9.5)and(9.6).IfwesubstitutetheBlack–ScholesEuropeancallsolution,intheabsenceofdividends,intotheinequality(9.3)thenitisclearlysatisfied;itactuallysatisfiestheequality.Ifwesubstitutetheexpressionintotheconstraint(9.4)withP(S,t)=max(S−E,0)thenthistooissatisfied.TheconclusionisthatthevalueofanAmericancalloptionisthesameasthevalueofaEuropeancalloptionwhentheunderlyingpaysnodividends.Comparethiswithouraboveresultthattheperpetualcalloptionshouldnotbeexercised—theAmericancalloptionwithafinitetimetoexpiryshouldalsonotbeexercisedbeforeexpiry.Toexercisebeforeexpirywouldbe‘sub-optimal.’Noneofthisistrueiftherearedividendsontheunderlying.Again,toseethissimplysubstitutetheexpressionsfromChapter8intotheconstraint.SincethecalloptionhasavaluewhichapproachesSe−D(T−t)asS→∞thereisclearlyapointatwhichtheEuropeanvaluefailstosatisfytheconstraint(9.4).Iftheconstraintisnotsatisfiedsomewherethentheproblemhasnotbeensolvedanywhere.Thisisveryimportant,our‘solution’mustsatisfytheinequalitieseverywhereorthe‘solution’isinvalid.Thisisduetothediffusivenatureofthedifferentialequation;anerrorinthesolutionatanypointisimmediatelypropagatedeverywhere.TheproblemfortheAmericanoptioniswhatisknownasafreeboundaryproblem.IntheEuropeanoptionproblemweknowthatwemustsolveforallvaluesofSfromzerotoinfinity.WhentheoptionisAmericanwedonotknowaprioriwheretheBlack–Scholesequationistobesatisfied;thismustbefoundaspartofthesolution.Thismeansthatwedonotknowthepositionoftheearlyexerciseboundary.Moreover,exceptinspecialandtrivialcases,thispositionistime-dependent.Forexample,weshouldexercisetheAmericanputiftheassetvaluefallsbelowS∗(t),buthowdowefindS∗(t)?Notonlyisthisproblemmuchharderthanthefixedboundaryproblem(forexample,whereweknowthatwesolveforSbetweenzeroandinfinity),butthisalsomakestheproblemnonlinear.Thatis,ifwehavetwosolutionsoftheproblemwedonotgetanothersolutionifweaddthemtogether.ThisiseasilyshownbyconsideringtheperpetualAmericanstraddleonadividend-payingstock.Ifthisisdefinedasasinglecontractthatmayatanytimebeexercisedforanamountmax(S−E,0)+max(E−S,0)=|S−E|thenitsvalueisnotthesameasthesumofaperpetualAmericanputandaperpetualAmericancall.Itssolutionisagainoftheformα+α−V(S)=AS+BS,andIsuggestthatthereaderfindthesolutionforhimself.ThereasonthatthiscontractisnotthesumoftwootherAmericanoptionsisthatthereisonlyoneexerciseopportunity;thetwo-optioncontracthasoneexerciseopportunitypercontract.IfthecontractswerebothEuropeanthenthesumofthetwoseparatesolutionswouldgivethecorrectanswer;theEuropeanvaluationproblemislinear.Thiscontractcanalsobeusedtodemonstratethattherecaneasilybemore 158PartOnemathematicalandfinancialfoundationsthanoneoptimalexerciseboundary.WiththeperpetualAmericanstraddle,asdefinedhere,oneshouldexerciseeitheriftheassetgetstoolowortoohigh.Theexactpositionsoftheboundariescanbedeterminedbymakingtheoptionanditsdeltaeverywherecontinuous.Onecanimaginethatifacontracthasareallystrangepayoff,thattherecouldbeanynumberoffreeboundaries.Sincewedon’tknowapriorihowmanyfreeboundariestherearegoingto120100AmericanoptionEuropeanoption80V604020S0050100150S*Figure9.3ValuesofaEuropeanandanAmericanput,seetextforparametervalues.Theoptimalexercisepointismarked.120100EuropeanoptionAmericanoption8060V40S*20S0050100150200Figure9.4ValuesofaEuropeanandanAmericancall,seetextforparametervalues.Theoptimalexercisepointismarked. earlyexerciseandAmericanoptionsChapter9159Figure9.5Bloombergoptionvaluation.Source:BloombergL.P.be(althoughcommonsensegivesusaclue)itisusefultohaveanumericalmethodthatcanfindtheseboundarieswithouthavingtobetoldhowmanytolookfor.IdiscusstheseissuesinChapter78.InFigure9.3areshownthevaluesofaEuropeanandAmericanputwithstrike100,volatility20%,interestrate5%andwithoneyeartoexpiry.Thepositionofthefreeboundary,theoptimalexercisepoint,ismarked.Rememberthatthispointmovesintime.InFigure9.4areshownthevaluesofaEuropeanandAmericancallwithstrike100,volatility20%,interestrate5%andwithoneyeartoexpiry.Thereisaconstantdividendyieldof5%ontheunderlying.(Iftherewerenodividendpaymentthenthetwocurveswouldbeidentical.)Thepositionofthefreeboundary,theoptimalexercisepoint,ismarked.Figure9.5showstheBloombergoptionvaluationcalculator,appliedtoanAmericanoption.9.5LOCALSOLUTIONFORCALLWITHCONSTANTDIVIDENDYIELDIfwecannotfindfullsolutionstonon-trivialproblems,wecanatleastfindlocalsolutions,solutionsthataregoodapproximationsforsomevaluesoftheassetatsometimes.WehaveseenthesolutionfortheAmericancallwithdividendswhenthereisalongtimetoexpiry, 160PartOnemathematicalandfinancialfoundationswhataboutclosetoexpiry?Iwillstatetheresultswithoutanyproof.Theproofsaresimplebuttedious;therelevantliteratureiscitedattheendofthechapter.Firstlet’sconsiderthecaser>D.Thisisusuallytrueforoptionsonequities,forwhichthedividendissmall.Closetoexpirytheoptimalexerciseboundaryis
∗rE1S(t)∼1+0.9034...σ(T−t)+···.D2Thecallshouldbeexercisediftheassetrisesabovethisvalue.NotethatasT−t→∞wehavefromtheperpetualcallanalysisthatthefreeboundarytendstoE.1−1/α+Iftheassetvaluerisesabovethefreeboundaryitisbettertoexercisetheoptiontoreceivethedividendsthantocontinueholdingit.Nearthepointt=T,S=Er/Dtheoptionpriceisapproximately
3/2log(SD/Er)V∼S−E+E(T−t)f√,T−twhere
2r12x122−x12−sf(x)=−x+0.075...(x+4)e4+(x+6x)e4ds.σ22−∞WhenD=0thereisnofreeboundary;itisneveroptimaltoexerciseearly.Whenrt.Thisequationistobeusedifthereissomespecialstatenowandyouwanttoknowwhatcouldhappenlater.Forexample,youknowthecurrentvalueofyandwanttoknowthedistributionofvaluesatsomelaterdate. probabilitydensityfunctionsandfirst-exittimesChapter101730.0060.0050.0040.003PDF0.0020.0010050100150200250SFigure10.3Theprobabilitydensityfunctionforthelognormalrandomwalk.ExampleThemostimportantexampletousisthatofthedistributionofequitypricesinthefuture.IfwehavetherandomwalkdS=µSdt+σSdXthentheforwardequationbecomes∂p∂2∂=1(σ2S
2p)−(µS
p).∂t
2∂S
2∂S
Aspecialsolutionofthisistheonehavingadeltafunctioninitialconditionp(S,t;S
,t)=δ(S
−S),representingavariablethatbeginswithcertaintywithvalueSattimet.Thesolutionofthisproblemisp(S,t;S
,t
)=√1e−(log(S/S
)+(µ−1/2σ2)(t
−t))2/2σ2(t
−t).σS
2π(t
−t)(10.3)ThisisplottedasafunctionofS
inFigure10.3andasafunctionofbothS
andt
inFigure10.4.10.5THESTEADY-STATEDISTRIBUTIONSomerandomwalkshaveasteady-statedistribution.Thatis,inthelongrunast
→∞thedistributionp(y,t;y
,t
)asafunctionofy
settlesdowntobeindependentofthestartingstateyandtimet.Looselyspeaking,thisrequiresatleastthattherandomwalkistimehomogeneous,i.e.thatAandBareindependentoft,asymptotically.Somerandomwalkshavenosuchsteadystateeventhoughtheyhaveatime-independentequation;thelognormalrandomwalkeithergrowswithoutboundordecaystozero. 174PartOnemathematicalandfinancialfoundations0.0140.0120.010.0080.006PDF0.00410.0020.7500.5Time2302500.251902101501707090110130305010AssetFigure10.4Theprobabilitydensityfunctionforthelognormalrandomwalkevolvingthroughtime.Ifthereisasteady-statedistributionp∞(y
)thenitsatisfiesd2d1(B2p)−(Ap)=0.2dy
2∞∞dy
∞∞InthisequationA∞andB∞arethefunctionsinthelimitt→∞.We’llseethisequationusedseveraltimesinlaterchapters,sometimestocalculatep(y
)knowingAandBand∞sometimestocalculateAknowingp(y
)andB.∞10.6THEBACKWARDEQUATIONNowwecometofindthebackwardequation.Thiswillbeusefulifwewanttocalculateprobabilitiesofreachingaspecifiedfinalstatefromvariousinitialstates.Itwillbeabackwardparabolicpartialdifferentialequationrequiringconditionsimposedinthefuture,andsolvedbackwardsintime.TheequationIamabouttoderiveisverysimilartotheBlack–Scholesequationforthefairvalueofanoption,indeed,thevalueofanoptioncanbeinterpretedasanexpectationoverpossiblefuturestates;muchmoreofthislater.Hereisthederivation.Themathsmaybeeasierforthebackwardequation,however,thejustificationfortherelation-shipbetweentheprobabilitiesofbeingatthefour‘nodes’ismoresubtlethaninthederivationoftheforwardequation.So,let’slookataconcreteexample(seeFigure10.5).At5pmyouareintheoffice.(Thisisthepoint(y,t).)At6pmyouwillbeatoneofthreeplaces:ThePub;Stillattheoffice,workinglate;MadameJojo’s.(Thesearethepoints(y+δy,t+δt),(y,t+δt)and(y−δy,t+δt).)Wearegoingtolookattheprobabilitythatatmidnightyouaretuckedupinbed.(Thisisthepoint(y
,t
).)Rememberthatp(y,t;y
,t
)representstheprobabilityofbeingatthefuturepoint(y
,t
),bedatmidnight,giventhatyoustartedat(y,t),theofficeat5pm.Youcanonlygettothebedatmidnightviaeitherthepub,theofficeorMadameJojo’sat6pm.Whathappensafter6pmdoesn’tmatter(youmaynotevenremember!),weareonlyconcernedwiththeprobabilitythatyouareinbedatmidnight,nothowyougotthere. probabilitydensityfunctionsandfirst-exittimesChapter10175y+dyBed,Midnight(Pub,6pm)ay1−2ay(Office,workinglate,6pm)(Office,a5pm)y−dy(MadameJojo's,6pm)dtFigure10.5Thepubcrawlexampleusedinderivingthebackwardequation.Inwords:Theprobabilityofbeinginbedatmidnight(giventhatyouwereintheofficeat5pm)istheprobabilityofgoingtothepubandbeingthereat6pm(giventhatyouwereintheofficeat5pm)plustheprobabilityofworkinglateattheofficeandbeingthereat6pm(giventhatyouwereintheofficeat5pm)plustheprobabilityofgoingtoMadameJojo’sandbeingthereat6pm(giventhatyouwereintheofficeat5pm).The‘giventhatyouwereintheofficeat5pm’ismathematicallythe‘;y,t)’bitofthetransitionprobabilitydensityfunction.Insymbolswecanwritethisasp(y,t;y
,t
)=φ+(y,t)p(y+δy,t+δt;y
,t
)+(1−φ+(y,t)−φ−(y,t))p(y,t+δt;y
,t
)+φ−(y,t)p(y−δy,t+δt;y
,t
).TheTaylorseriesexpansionleadstothebackwardKolmogorovequation∂p∂2p∂p+1B(y,t)2+A(y,t)=0.(10.4)∂t2∂y2∂yExampleThetransitionprobabilitydensityfunction(10.3)forthelognormalrandomwalksatisfiesthisequation,butnotethedifferentindependentvariables.10.7FIRST-EXITTIMESThefirst-exittimeisthetimeatwhichtherandomvariablereachesagivenboundary.PerhapswewanttoknowhowlongbeforeacertainlevelisreachedorperhapswewanttoknowhowlongbeforeanAmericanoptionshouldbeoptimallyexercised.Anexampleofafirst-exittimeisgiveninFigure10.6. 176PartOnemathematicalandfinancialfoundations3.02.52.01.5FirsttimethatassetStockpricereaches2.51.00.50.024-Feb-9812-Sep-9831-Mar-9917-Oct-9904-May-00TimeFigure10.6Anexampleofafirst-exittime.Questionstoaskaboutfirst-exittimesare‘Whatistheprobabilityofanassetlevelbeingreachedbeforeacertaintime?’,‘Howlongdoweexpectittotakeforaninterestratetofalltoagivenlevel?’Iwilladdresstheseproblemsnow.10.8CUMULATIVEDISTRIBUTIONFUNCTIONSFORFIRST-EXITTIMESWhatistheprobabilityofyourfavoriteassetdoublingorhalvinginvalueinthenextyear?Thisisaquestionthatcanbeansweredbythesolutionofasimplediffusionequation.Itisanexampleofthemoregeneralquestion,‘Whatistheprobabilityofarandomvariableleavingagivenrangebeforeagiventime?’ThisquestionisillustratedinFigure10.7.LetmeintroducethefunctionC(y,t;t
)astheprobabilityofthevariableyleavingtheregionbeforetimet
.Thisfunctioncanbethoughtofasacumulativedistributionfunction.Thisfunctionalsosatisfiesthebackwardequation∂C∂2C∂C+1B(y,t)2+A(y,t)=0.∂t2∂y2∂yWhatmakestheproblemdifferentfromthatforthetransitionprobabilitydensityfunctionaretheboundaryandfinalconditions.Ifthevariableyisactuallyontheboundaryoftheregionthenclearlytheprobabilityofexitingisone:C(y,t,t
)=1ontheedgeof.Ontheotherhandifweareinsidetheregionattimet
thenthereisnotimeleftforthevariabletoleavetheregionandsotheprobabilityiszero.ThuswehaveC(y,t
,t
)=0. probabilitydensityfunctionsandfirst-exittimesChapter101773.0Assetdoubles2.52.0One-year1.5horizonStockprice1.00.5AssethalvesTime0.005-Apr-22-Oct-10-May-26-Nov-13-Jun-30-Dec-18-Jul-98989999000001Figure10.7Whatistheprobabilityoftheassetleavingtheregionbeforethegiventime?10.9EXPECTEDFIRST-EXITTIMESIntheprevioussectionIshowedhowtocalculatetheprobabilityofleavingagivenregion.Wecanusethisfunctiontofindtheexpectedtimetoexit.OncewehavefoundCthenitissimpletofindtheexpectedfirst-exittime.Letmecalltheexpectedfirst-exittimeu(y,t).Itisafunctionofwherewestartout,yandt.SinceCisacumulativedistributionfunctiontheexpectedfirst-exittimecanbewrittenas
∞
∂C
u(y,t)=(t−t)dt.∂t
tAfteranintegrationbypartsweget
∞u(y,t)=1−C(y,t;t
)dt
.tThefunctionCsatisfiesthebackwardequationinyandtsothat,afterdifferentiatingundertheintegralsign,wefindthatusatisfiestheequation∂u∂2u∂u+1B(y,t)2+A(y,t)=−1.(10.5)∂t2∂y2∂ySinceCisoneontheboundaryof,umustbezeroaroundtheboundaryoftheregion.Whataboutthefinalcondition?Typicallyonesolvesoveraregionthatisboundedintime,forexampleasshowninFigure10.8. 178PartOnemathematicalandfinancialfoundationsSu=0tFigure10.8Thefirst-exittimeproblem.ExampleWhenthestochasticdifferentialequationisindependentoftime,thatis,bothAandBarefunctionsofyonly,andtheregionisalsotimehomogeneous,thentheremaybeasteady-statesolutionof(10.5).Returningtothelogarithmicassetproblem,whatistheexpectedtimefortheassettoleavetherange(S0,S1)?Theanswertothisquestionisthesolutionofd2udu1σ2S2+µS=−1,2dS2dSwithu(S0)=u(S1)=0,andis11−(S/S)1−2µ/σ20u(S)=12log(S/S0)−1−2µ/σ2log(S1/S0).2σ−µ1−(S1/S0)10.10ANOTHEREXAMPLEOFOPTIMALSTOPPINGYouholdsomeinvestment,itgoesupinvalue,itgoesdowninvalue.Generallyspeaking,it’sgoingnowherefast.Whenshouldyousellit?Let’smaketwobigassumptions.Firstofall,let’ssaythatyouknowthestatistical/stochasticpropertiesofyourinvestment’svaluesothatyoucanwritedS=µ(S)dt+σ(S)dXforsomeknownµ(S)andσ(S)(independentoftimetokeepitsimple).Second,let’sassumethatyouwanttosellatthetimewhichmaximizestheexpectedvalueofyourinvestment,withsuitableallowancebeingtakenforthetimevalueofmoney.We’lluseVtodenotethismaximum. probabilitydensityfunctionsandfirst-exittimesChapter10179Sincewewanttocalculateanexpectation,therelevantequationtobesolvedistheback-wardKolmogorovequation.AndbecausethegoverningstochasticdifferentialequationistimehomogeneousandwehavenofinitetimehorizonwemustsolveforV(S)withnotdependence:d2VdV1σ(S)2+µ(S)−rV=0.2dS2dSThelasttermontheleftistheusualtime-value-of-moneyterm.ThismustbesolvedsubjecttoV≥SwithcontinuityofVanddV/dS.Thisconstraintensuresthatwemaximizeourexpectedvalue.ExampleWhenwehavealogarithmicassetsothatµ(S)=µSandσ(S)=σSwegetsomeinterestingresults.ThegeneralsolutionofthesecondorderordinarydifferentialequationisthenASα++BSα−whereAandBarearbitraryconstantsand±1121222α=−µ+σ±µ−σ+2rσ.σ222Ifµ>rthenthereisnofinitesolutionforV(withafinitetimehorizon)andoneshouldneverselltheasset.IfµE2.TopricethiscontractwemustfindV(S1,S2,t)where∂V22∂2V2∂V+1σσρSS+rS−rV=0,2ijijiji∂t∂Si∂Sj∂Sii=1j=1i=1withV(S1,S2,T)=P(S1)subjecttoV(S1,S2,t)≥P(S1)forS2>E2,andcontinuityofVanditsfirstderivatives.Ifwecanpriceandhedgeanoptiononasingleassetwithasetofcharacteristics,thentheoreticallywecanpriceandhedgeamulti-assetversionaswell.ThepracticeofpricingandhedgingmaybemuchharderasImentionbelow. 194PartOnemathematicalandfinancialfoundations11.10REALITIESOFPRICINGBASKETOPTIONSThefactorsthatdeterminetheeaseordifficultyofpricingandhedgingmulti-assetoptionsare•existenceofaclosed-formsolution•numberofunderlyingassets,thedimensionality•pathdependency•earlyexerciseWehaveseenalloftheseexceptpathdependency,whichisoneofthesubjectsofPartTwo.Thesolutiontechniquethatweusewillgenerallybeoneof•finite-differencesolutionofapartialdifferentialequation•numericalintegration•MonteCarlosimulationThesemethodsarethesubjectsofPartSix.11.10.1EasyProblemsIfwehaveaclosed-formsolutionthenourworkisdone;wecaneasilyfindvaluesandhedgeratios.Thisisprovidedthatthesolutionisintermsofsufficientlysimplefunctionsforwhichtherearespreadsheetfunctionsorotherlibraries.IfthecontractisEuropeanwithnopath-dependencythenthesolutionmaybeoftheform(11.2).Ifthisisthecase,thenweoftenhavetodotheintegrationnumerically.Thisisnotdifficult.SeveralmethodsaredescribedinChapter81,includingMonteCarlointegrationandtheuseoflow-discrepancysequences.11.10.2MediumProblemsIfwehavelowdimensionality,lessthanthreeorfour,say,thefinite-differencemethodsaretheobviouschoice.Theycopewellwithearlyexerciseandmanypath-dependentfeaturescanbeincorporated,thoughusuallyatthecostofanextradimension.Forhigherdimensions,MonteCarlosimulationsaregood.Theycopewithallpath-dependentfeatures.Unfortunately,theyarenotveryefficientforAmerican-styleearlyexercise.11.10.3HardProblemsThehardestproblemstosolvearethosewithbothhighdimensionality,forwhichwewouldliketouseMonteCarlosimulation,andwithearlyexercise,forwhichwewouldliketousefinite-differencemethods.Thereiscurrentlynonumericalmethodthatcopeswellwithsuchaproblem.11.11REALITIESOFHEDGINGBASKETOPTIONSEvenifwecanfindoptionvaluesandthegreeks,theyareoftenverysensitivetothelevelofthecorrelation.ButasIhavesaid,thecorrelationisaverydifficultquantitytomeasure.So multi-assetoptionsChapter11195thehedgeratiosareverylikelytobeinaccurate.Ifwearedeltahedgingthenweneedaccurateestimatesofthedeltas.Thismakesbasketoptionsverydifficulttodeltahedgesuccessfully.Whenwehaveacontractthatisdifficulttodeltahedgewecantrytoreducesensitivitytoparameters,andthemodel,byhedgingwithotherderivatives.Thiswasthebasisofvegahedging,mentionedinChapter7.Wecouldtrytousethesameideatoreducesensitivitytothecorrelation.Unfortunately,thatisalsodifficultbecausetherejustaren’tenoughcontractstradedthatdependontherightcorrelations.11.12CORRELATIONVERSUSCOINTEGRATIONThecorrelationsbetweenfinancialquantitiesarenotoriouslyunstable.Onecouldeasilyarguethatatheoryshouldnotbebuiltupusingparametersthataresounpredictable.Iwouldtendtoagreewiththispointofview.Onecouldproposeastochasticcorrelationmodel,butthatapproachhasitsownproblems.Analternativestatisticalmeasuretocorrelationiscointegration.Verylooselyspeaking,twotimeseriesarecointegratedifalinearcombinationhasconstantmeanandstandarddeviation.Inotherwords,thetwoseriesneverstraytoofarfromoneanother.Thisisprobablyamorerobustmeasureofthelinkagebetweentwofinancialquantitiesbutasyetthereislittlederivativestheorybasedontheconcept.11.13SUMMARYThenewideasinthischapterwerethemultifactor,correlatedrandomwalksforassets,andIto’slemmainhigherdimensions.Thesearebothsimpleconcepts,andwewillusethemoften,ˆespeciallyininterest-rate-relatedtopics.FURTHERREADING•SeeHamilton(1994)forfurtherdetailsofthemeasurementofcorrelationandcointegration.•ThefirstsolutionoftheexchangeoptionproblemwasbyMargrabe(1978).•Foranalyticalresults,formulaeornumericalalgorithmsforthepricingofsomeothermulti-factoroptionsseeStulz(1982),Johnson(1987),Boyle,Evnine&Gibbs(1989),Boyle&Tse(1990),Rubinstein(1991)andRich&Chance(1993).•SeeEmanuelDerman’sautobiographyfordiscussionofquantos(Derman,2004).•Fordetailsofcointegration,whatitmeansandhowitworksseethepapersbyAlexander&Johnson(1992,1994).•Krekeletal.(2004)comparedifferentpricingmethodsforbasketoptions. CHAPTER12howtodeltahedgeInthisChapter...•howtomakemoneyifyourvolatilityforecastismoreaccuratethanthemarket’s•differentwaysofdeltahedging•howmuchprofitshouldyouexpecttomake12.1INTRODUCTIONInthischapterwearegoingtoassertboldlythatthereismoneytobemadefromoptions,becauseoptionsmaybemispricedbythemarket.Insimpleterms,therearearbitrageopportunities.Shock,horror!IknowthatthewholeofChicagoUniversityhasjustthrowndownthisbookindisgust.However,Ihopetherestofyouwillenjoythecontentsofthischapterforitputsintoconcretemathematicssomeideasthataremostimportant,andfrighteninglyunder-explainedintheliterature.Thisisthesubjectofhowtodeltahedgewhenyourestimateoffutureactualvolatilitydiffersfromthatofthemarketasmeasuredbytheimpliedvolatility.AsIhintedabove,tosomepeople,sayingthatactualvolatilityandimpliedvolatilitycanbeinconsistentwitheachotherisaheresy,foritimpliesarbitrageandhencefreemoney.Well,it’sonlyasfreeasthemodelisaccurate,thatis,notatall.Butevenso,ifthereissuchadifference(andvolarbhedgefundscertainlythinkthereis)thenwecanonlygetatthatmoneybyhedging,andifwehavetwoestimatesofvolatilitywhichonegoesintothefamousBlack–Scholesdeltaformula?We’llseehowyoucanhedgeusingadeltabasedoneitheractualvolatilityoronimpliedvolatility.Neitheriswrong,theyjusthavedifferentrisk/returnprofiles.IfyoudodoubtthatimpliedvolatilityandactualvolatilitycanbeindisagreementthentakealookatFigure12.1.ThisissimplyaplotofthedistributionsofthelogarithmsoftheVIXandoftherollingrealizedSPXvolatility.TheVIXisanimpliedvolatilitymeasurebasedontheSPXindexandsoyouwouldexpectitandtherealizedSPXvolatilitytobearsomeresemblance.Notquite,ascanbeseeninthefigure.TheimpliedvolatilityVIXseemstobehigherthantherealizedvolatility.Bothofthesevolatilitiesareapproximatelylognormallydistributed(sincetheirlogarithmsappeartobeNormal),especiallytherealizedvolatility.TheVIXdistributionissomewhattruncatedontheleft.Themeanoftherealizedvolatility,about15%,issignificantlylessthanthemeanoftheVIX,about20%,butitsstandarddeviationisgreater. 198PartOnemathematicalandfinancialfoundations1.41.2Ln(VIX)Ln(SPXvol.)1Normal(VIX)Normal(SPXvol.)0.8PDF0.60.40.2011.522.533.54Ln('Vol')Figure12.1DistributionsofthelogarithmsoftheVIXandtherollingrealizedSPXvolatility,andtheNormaldistributionsforcomparison.12.2WHATIFIMPLIEDANDACTUALVOLATILITIESAREDIFFERENT?Actualvolatilityistheamountof‘noise’inthestockprice;itisthecoefficientoftheWienerprocessinthestockreturnsmodel;itistheamountofrandomnessthat‘actually’transpires.Impliedvolatilityishowthemarketispricingtheoptioncur-rently.Sincethemarketdoesnothaveperfectknowledgeaboutthefuturethesetwonumberscanandwillbedifferent.Actualvolatilitybeingdifferentfromimpliedvolatilityistheheartofthischapter.Let’slookatthesimplecaseofexploitingsuchadifferencebybuyingorsellingoptions,butnotdeltahedgingthem.Imaginethatwehaveaforecastforvolatilityovertheremaininglifeofanoption,thisvolatilityisforecasttobeconstant,and,crucially,ourforecastturnsouttobecorrect.Ifyoubelievethatactualvolatilityishigherthanimpliedyoumightwanttobuyastraddlebecausethereisthenagoodchancethatthestockwillmovesofarbeforeexpirythatyouwillgetapayoffofmorethanthepremiumyoupaid,evenafterallowingforthetimevalueofmoney.Thisisaverysimplestrategy,requiringnomaintenance.Thereisonebigproblemwiththishowever.Itisrisky.Sometimesyou’llwin,sometimesyou’lllose.Unlessyoucandothisstrategymany,manytimesyoucouldenduplosingagreatdeal.Evenifyouarerightabouttheactualvolatilitybeinglargethestockmightendupatthemoney,andyouloseout.Therelationshipbetweenactualvolatilityandtherangeoverwhichanassetmovesisaprobabilisticone,therearenoguaranteesthatahighvolatilityresultsinlargemoves. howtodeltahedgeChapter12199Ifyoubuyanat-the-moneystraddleclosetoexpiry,theprofityouexpecttomakefromthisstrategyisapproximately
2(T−t)(σ−˜σ)S.πTheexpressionusestheclosetoexpiryandATMapproximationwesawinChapter7.Thenotationisobvious:σistheactualvolatility,assumedconstant,andσ˜istheimpliedvolatility.Notethatthisisanexpectation.Itisalsoarealexpectation,howevertherealdriftdoesn’tappearinthisexpressionbecauseitisanapproximationvalidonlywhenclosetoexpiration.Thestandarddeviationoftheprofit(therisk)isapproximately
2√1−σST−t.πObservehowthisdependsontheactualvolatilityandnotontheimpliedvolatility.Thisstandarddeviationisofthesameorderofmagnitudeastheexpectedprofit.Thatisalotofrisk.Wecanimprovetherisk-rewardprofilebydeltahedgingasweshallseenext.Themainpurposeofwritingdowntheaboveexpressionsistoshowhowtheyarelinearinthetwovolatilities.12.3IMPLIEDVERSUSACTUAL;DELTAHEDGINGBUTUSINGWHICHVOLATILITY?Let’sbuyanunderpricedoption,orportfolioasabove,butnow,toimproveriskandreward,wewilldeltahedgetoexpiry.Thisisalessriskystrategy.Butwhichdeltadoyouchoose?Deltabasedonactualorimpliedvolatility?Thisisoneofthosequestionsthatpeoplealwaysask,andonethatnooneseemstoknowthefullanswerto.Scenario:Impliedvolatilityforanoptionis20%,butwebelievethatactualvolatilityis30%.Question:Howcanwemakemoneyifourforecastiscorrect?Answer:Buytheoptionanddeltahedge.Butwhichdeltadoweuse?Weknowthat=N(d1)wherex1s2−N(x)=√e2ds2π−∞andln(S/E)+r+1σ2(T−t)d=√2.1σT−tWecanallagreeonS,E,T−tandr(almost),butnotonσ,soshouldweuseσ=0.2or0.3,impliedvolatilityoractual?Inthisexample,σ=actualvolatility,30%andσ˜=impliedvolatility,20%.whichofthesegoesintothed1? 200PartOnemathematicalandfinancialfoundations12.4CASE1:HEDGEWITHACTUALVOLATILITY,σByhedgingwithactualvolatilitywearereplicatingashortpositioninacorrectlypricedoption.Thepayoffsforourlongoptionandourshortreplicatedoptionwillexactlycancel.TheprofitwemakewillbeexactlythedifferenceintheBlack–Scholespricesofanoptionwith30%volatilityandonewith20%volatility.(AssumingthattheBlack–Scholesassumptionshold.)IfV(S,t;σ)istheBlack–ScholesformulathentheguaranteedprofitisV(S,t;σ)−V(S,t;˜σ).Buthowisthisguaranteedprofitrealized?Let’sdothemathonamark-to-marketbasis.Inthefollowing,superscript‘a’meansactualand‘i’meansimplied;thesecanbeappliedtodeltasandoptionvalues.Forexample,aisthedeltausingtheactualvolatilityintheformula.Viisthetheoreticaloptionvalueusingtheimpliedvolatilityintheformula.NotealsothatV,,andareallsimple,known,Black–Scholesformulae.ThemodelisasusualdS=µSdt+σSdX.Now,setupaportfoliobybuyingtheoptionforViandhedgewithaofthestock.ThevaluesofeachofthecomponentsofourportfolioareshowninTable12.1.Leavethishedgedportfolioovernight,andcomebacktoitthenextday.ThenewvaluesareshowninTable12.2.(Ihaveincludedacontinuousdividendyieldinthis.)Thereforewehavemade,marktomarket,dVi−adS−r(Vi−aS)dt−aDSdt.SincetheoptionwouldbecorrectlyvaluedatVa,wehavedVa−adS−r(Va−aS)dt−aDSdt=0.Table12.1Portfoliocompositionandvalues,today.ComponentValueOptionViStock−aSCash−Vi+aSTable12.2Portfoliocompositionandvalues,tomorrow.ComponentValueOptionVi+dViStock−aS−adSCash(−Vi+aS)(1+rdt)−aDSdt howtodeltahedgeChapter12201Sowecanwritethemark-to-marketprofitoveronetimestepasdVi−dVa+r(Va−aS)dt−r(Vi−aS)dt=dVi−dVa−r(Vi−Va)dt=ertd(e−rt(Vi−Va)).Thatistheprofitfromtimettot+dt.Thepresentvalueofthisprofitattimet0ise−r(t−t0)ertd(e−rt(Vi−Va))=ert0d(e−rt(Vi−Va)).Sothetotalprofitfromt0toexpirationisTert0d(e−rt(Vi−Va))=Va−Vi.t0ThisconfirmswhatIsaidearlierabouttheguaranteedtotalprofitbyexpiration.Wecanalsowritethatonetimestepmark-to-marketprofit(usingIto’slemma)asˆidt+idS+1σ2S2idt−adS−r(Vi−aS)dt−aDSdt2=idt+µS(i−a)dt+1σ2S2idt−r(Vi−Va)dt+(i−a)σSdX−aDSdt2=(i−a)σSdX+(µ+D)S(i−a)dt+1(σ2−˜σ2)S2idt2(usingBlack–Scholeswithσ=˜σ)=1(σ2−˜σ2)S2idt+(i−a)((µ−r+D)Sdt+σSdX).254.543.532.521.5Mark-to-marketP&L10.5000.10.20.30.40.50.60.70.80.91−0.5TimeFigure12.2P&Lforadelta-hedgedoptiononamark-to-marketbasis,hedgedusingactualvolatility. 202PartOnemathematicalandfinancialfoundationsTheconclusionisthatthefinalprofitisguaranteed(thedifferencebetweenthetheoreticaloptionvalueswiththetwovolatilities)buthowthatisachievedisrandom,becauseofthedXtermintheabove.Onamark-to-marketbasisyoucouldlosebeforeyougain.Moreover,themark-to-marketprofitdependsontherealdriftofthestock,µ.ThisisillustratedinFigure12.2,whichshowsseveralrealizationsofthesamedelta-hedgedposition.NotethatthefinalP&Lisnotexactlythesameineachcasebecauseoftheeffectofhedgingdiscretely,atopicdiscussedinChapter47.WhenSchanges,sowillV.Butthesechangesdonotcanceleachotherout.Fromariskmanagementpointofviewthisisnotideal.Thereisasimpleanalogyforthisbehavior.Itissimilartoowningabond.Forabondthereisaguaranteedoutcome,butwemayloseonamark-to-marketbasisinthemeantime.12.5CASE2:HEDGEWITHIMPLIEDVOLATILITY,σ˜Compareandcontrastnowwiththecaseofhedgingusingadeltabasedonimpliedvolatility.Byhedgingwithimpliedvolatilitywearebalancingtherandomfluctuationsinthemark-to-marketoptionvaluewiththefluctuationsinthestockprice.Theevolutionoftheportfoliovalueisthen‘deterministic’asweshallsee.Buytheoptiontoday,hedgeusingtheimplieddelta,andputanycashinthebankearningr.Themark-to-marketprofitfromtodaytotomorrowisdVi−idS−r(Vi−iS)dt−iDSdt=idt+1σ2S2idt−r(Vi−iS)dt−iDSdt2=1(σ2−˜σ2)S2idt.(12.1)2Thisisafarnicerwaytomakemoney.Observehowthedailyprofitisdeterministic,therearen’tanydXterms.Fromariskmanagementperspectivethisismuchbetterbehaved.Thereisanother,ratherwonderful,advantageofhedgingusingimpliedvolatility...wedon’tevenneedtoknowwhatactualvolatilityis.Andtomakeaprofitallweneedtoknowisthatactualisalwaysgoingtobegreaterthanimplied(ifwearebuying)oralwaysless(ifweareselling).Thistakessomeofthepressureoffforecastingvolatilityaccuratelyinthefirstplace.AddupthepresentvalueofalloftheseprofitstogetatotalprofitofT1(σ2−˜σ2)e−r(t−t0)S2idt.2t0Thisisalwayspositive,buthighlypathdependent.Beingpathdependentitwilldependonthedriftµ.Ifwestartoffatthemoneyandthedriftisverylarge(positiveornegative)wewillfindourselvesquicklymovingintoterritorywheregammaandhence(12.1)issmall,sothattherewillbenotmuchprofittobemade.Thebestthatcouldhappenwouldbeforthestocktoendupclosetothestrikeatexpiration,thiswouldmaximizethetotalprofit.ThispathdependencyisshowninFigure12.3.Thefigureshowsseveralrealizationsofthesamedelta-hedgedposition.Notethatthelinesarenotperfectlysmooth,againbecauseoftheeffectofhedgingdiscretely.Thesimpleanalogyisnowjustputtingmoneyinthebank.TheP&Lisalwaysincreasinginvaluebuttheendresultisrandom. howtodeltahedgeChapter1220376543Mark-to-marketP&L21000.10.20.30.40.50.60.70.80.91TimeFigure12.3P&Lforadelta-hedgedoptiononamark-to-marketbasis,hedgedusingimpliedvolatility.PeterCarr(2005)andHenrard(2001)showthatifyouhedgeusingadeltabasedonavolatilityσhthenthePVofthetotalprofitisgivenbyTV(S,t;σ)−V(S,t;˜σ)+1σ2−σ2e−r(t−t0)S2hdt,(12.2)h2ht0wherethesuperscriptonthegammameansthatitusestheBlack–Scholesformulawithavolatilityofσh.12.5.1TheExpectedProfitafterHedgingusingImpliedVolatilityWhenyouhedgeusingdeltabasedonimpliedvolatilitytheprofiteach‘day’isdeterministicbutthepresentvalueoftotalprofitbyexpirationispathdependent,andgivenbyT1(σ2−˜σ2)e−r(s−t0)S2ids.2t0IntroducetI=1(σ2−˜σ2)e−r(s−t0)S2ids.2t0SincethereforedI=1(σ2−˜σ2)e−r(t−t0)S2idt2 204PartOnemathematicalandfinancialfoundationswecanwritedownthefollowingpartialdifferentialequationfortherealexpectedvalue,P(S,I,t),ofI:∂P∂2P∂P∂P+1σ2S2+µS+1(σ2−˜σ2)e−r(t−t0)S2i=0,∂t2∂S2∂S2∂IwithP(S,I,T)=I.LookforasolutionofthisequationoftheformP(S,I,t)=I+H(S,t)sothat∂H∂2H∂H+1σ2S2+µS+1(σ2−˜σ2)e−r(t−t0)S2i=0.∂t2∂S2∂S2Thesourcetermcanbesimplifiedto22−r(T−t0)−d2/2E(σ−˜σ)ee2√.2σ˜2π(T−t)Changevariablestox=log(S/E)+µ−1σ2τandτ=T−t2andwriteH=w(x,τ).Theresultingpartialdifferentialequationisthenabitnicer.DetailscanbefoundintheappendixtothischapterAftersomemanipulationsweendupwiththeexpectedprofitinitially(t=t0,I=0)beingthesingleintegralEe−r(T−t0)(σ2−˜σ2)T1√22πt0σ2(s−t0)+˜σ2(T−s)2log(S/E)+µ−1σ2(s−t)+r−D−1σ˜2(T−s)202×exp−ds.2(σ2(s−t)+˜σ2(T−s))0Resultsareshowninthefollowingfigures.InFigure12.4isshowntheexpectedprofitversusthegrowthrateµ.ParametersareS=100,σ=0.4,r=0.05,D=0,E=110,T=1,σ˜=0.2.Observethattheexpectedprofithasamaximum.Thiswillbeatthegrowthratethatensures,roughlyspeaking,thatthestockendsupclosetoatthemoneyatexpiration,wheregammaislargest.Inthefigureisalsoshowntheprofittobemadewhenhedgingwithactualvolatility.Formostrealisticparametersregimesthemaximumexpectedprofithedgingwithimpliedissimilartotheguaranteedprofithedgingwithactual.InFigure12.5isshownexpectedprofitversusEandµ.Youcanseehowthehigherthegrowthratethelargerthestrikepriceatthemaximum.ThecontourmapisshowninFigure12.6. howtodeltahedgeChapter122059876543ExpectedProfit210−1.5−1−0.500.511.5GrowthFigure12.4Expectedprofit,hedgingusingimpliedvolatility,versusgrowthrateµ;S=100,σ=0.4,r=0.05,D=0,E=110,T=1,σ˜=0.2.Thedashedlineistheprofittobemadewhenhedgingwithactualvolatility.98765ExpectedProfit40.230.120.042Growth−0.041−0.120−0.212011511010510095908580StrikeFigure12.5Expectedprofit,hedgingusingimpliedvolatility,versusgrowthrateµandstrikeE;S=100,σ=0.4,r=0.05,D=0,T=1,σ˜=0.2. 206PartOnemathematicalandfinancialfoundationsExpectedProfit120115110105100Strike9590858000.20.2−0.160.120.080.040.040.080.120.16−−−−GrowthFigure12.6Contourmapofexpectedprofit,hedgingusingimpliedvolatility,versusgrowthrateµandstrikeE;S=100,σ=0.4,r=0.05,D=0,T=1,σ˜=0.2.TheeffectofskewisshowninFigure12.7.HereIhaveusedalinearnegativeskew,from22.5%atastrikeof75,fallingto17.5%atthe125strike.Theat-the-moneyimpliedvolatilityis20%whichinthiscaseistheactualvolatility.Thispicturechangeswhenyoudividetheexpectedprofitbythepriceoftheoption(putsforlowerstrikes,callforhigher),seeFigure12.8.Thereisnomaximum,profitabilityincreaseswithdistanceawayfromthemoney.Ofcourse,thisdoesn’ttakeintoaccounttherisk,thestandarddeviationassociatedwithsuchtrades.12.5.2TheVarianceofProfitafterHedgingusingImpliedVolatilityOncewehavecalculatedtheexpectedprofitfromhedgingusingimpliedvolatilitywecancalculatethevarianceinthefinalprofit.Usingtheabovenotation,thevariancewillbetheexpectedvalueofI2lessthesquareoftheaverageofI.Sowewillneedtocalculatev(S,I,t)where∂v∂2v∂v∂v+1σ2S2+µS+1(σ2−˜σ2)e−r(t−t0)S2i=0,∂t2∂S2∂S2∂Iwithv(S,I,T)=I2.Thedetailsoffindingthisfunctionvarerathermessy,butasolutioncanbefoundoftheformv(S,I,t)=I2+2IH(S,t)+G(S,t).TheinitialvarianceisG(S,t)−F(S,t)2,where0000E2(σ2−˜σ2)2e−2r(T−t0)TTG(S0,t0)=4πσσ˜t0sep(u,s;S0,t0)×√duds√s−tT−sσ2(u−s)+˜σ2(T−u)1+1+10σ2(s−t0)σ˜2(T−s)σ2(u−s)+˜σ2(T−u)(12.3) howtodeltahedgeChapter1220725%0.7VolatilityImpliedExpectedProfit0.620%0.515%0.40.310%ImpliedvolatilityExpectedprofit0.25%0.10%07580859095100105110115120125StrikeFigure12.7Effectofskew,expectedprofit,hedgingusingimpliedvolatility,versusstrikeE;S=100,µ=0,σ=0.2,r=0.05,D=0,T=1.25%0.6VolatilityImpliedExpectedProfit/Price0.520%0.415%0.310%Impliedvolatility0.2Expectedprofit/price5%0.10%07580859095100105110115120125StrikeFigure12.8Effectofskew,ratioofexpectedprofittoprice,hedgingusingimpliedvolatility,versusstrikeE;S=100,µ=0,σ=0.2,r=0.05,D=0,T=1. 208PartOnemathematicalandfinancialfoundationswhere(x+α(T−s))2(x+α(T−u))2p(u,s;S,t)=−1−10022222σ˜(T−s)σ(u−s)+˜σ(T−u)2x+α(T−s)x+α(T−u)+σ˜2(T−s)σ2(u−s)+˜σ2(T−u)+12111++σ2(s−t0)σ˜2(T−s)σ2(u−s)+˜σ2(T−u)andx=ln(S/E)+µ−1σ2(T−t),andα=µ−1σ2−r+D+1σ˜2.02022Thederivationofthiscanbefoundintheappendixtothischapter.InFigure12.9isshownthestandarddeviationofprofitversusgrowthrate,S=100,σ=0.4,r=0.05,D=0,E=110,T=1,σ˜=0.2.Figure12.10showsthestandarddeviationofprofitversusstrike,S=100,σ=0.4,r=0.05,D=0,µ=0.1,T=1,σ˜=0.2.Notethatintheseplotstheexpectationsandstandarddeviationshavenotbeenscaledwiththecostoftheoptions.InFigure12.11isshownexpectedprofitdividedbycostversusstandarddeviationdividedbycost,asbothstrikeandexpirationvary.IntheseplotsS=100,σ=0.4,r=0.05,D=0,µ=0.1,σ˜=0.2.Tosomeextent,althoughweemphasizeonlysome,thesediagramscanbeinterpretedinaclassicalmean-variancemanner,seeChapter18.Themaincriticismis,ofcourse,thatwearenotworkingwithNormaldistributions,and,furthermore,thereisnodownside,nopossibilityofanylosses.Figure12.12completestheearlierpicturefortheskew,sinceitnowcontainsthestandarddeviation.987ExpectedprofitStandarddeviationofprofit6543210−1.5−1−0.500.511.5GrowthFigure12.9Standarddeviationofprofit,hedgingusingimpliedvolatility,versusgrowthrateµ;S=100,σ=0.4,r=0.05,D=0,E=110,T=1,σ˜=0.2.(Theexpectedprofitisalsoshown.) howtodeltahedgeChapter122099Expectedprofit8Standarddeviationofprofit76543210708090100110120130StrikeFigure12.10Standarddeviationofprofit,hedgingusingimpliedvolatility,versusstrikeE;S=100,σ=0.4,r=0.05,D=0,µ=0,T=1,σ˜=0.2.(Theexpectedprofitisalsoshown.)257620T=0.25T=0.55154E=1251032ExpectedreturnExpectedreturn5E=7510001020304002468StandarddeviationStandarddeviation4.53.5433.5T=0.75T=112.532.5221.51.5ExpectedreturnExpectedreturn110.50.5000123400.511.522.5StandarddeviationStandarddeviationFigure12.11Scaledexpectedprofitversusscaledstandarddeviation;S=100,σ=0.4,r=0.05,D=0,µ=0.1,σ˜=0.2.Fourdifferentexpirations,varyingstrike. 210PartOnemathematicalandfinancialfoundations0.60.5RatioofstandarddeviationtopriceRatioofexpectedprofittoprice0.40.30.20.10708090100110120130Figure12.12Effectofskew,ratioofexpectedprofittoprice,andratioofstandarddeviationtoprice,versusstrikeE;S=100,µ=0,σ=0.2,r=0.05,D=0,T=1.12.5.3HedgingwithDifferentVolatilitiesWewillbrieflyexaminehedgingusingvolatilitiesotherthanactualorimplied,usingthegeneralexpressionforprofitgivenby(12.2).TheexpressionsfortheexpectedprofitandstandarddeviationsnowmustallowfortheV(S,t;σh)−V(S,t;˜σ),sincetheintegralofgammatermcanbetreatedasbeforeifonereplacesσ˜withσhinthisterm.Resultsarepresentedinthenexttwofigures.InFigure12.13isshowntheexpectedprofitandstandarddeviationofprofitwhenhedgingwithvariousvolatilities.Thethin,dottedlines,continuingonfromtheboldlines,representhedgingwithvolatilitiesoutsidetheimplied-actualrange.Thechartalsoshowsstandarddevi-ationofprofit,andminimumandmaximum.ParametersareE=90,S=100,µ=−0.1,σ=0.4,r=0.1,D=0,T=1,andσ˜=0.2.Notethatitispossibletolosemoneyifyouhedgeatbelowimplied,buthedgingwithahighervolatilityyouwillnotbeabletoloseuntilhedgingwithavolatilityofapproximately70%.Inthisexample,theexpectedprofitdecreaseswithincreasinghedgingvolatility.Figure12.14showsthesamequantitiesbutnowforanoptionwithastrikepriceof110.Theupperhedgingvolatility,beyondwhichitispossibletomakealoss,isnowslightlyhigher.Theexpectedprofitnowincreaseswithincreasinghedgingvolatility.Inpracticewhichvolatilityoneusesisoftendeterminedbywhetheroneisconstrainedtomarktomarketormarktomodel.Ifoneisabletomarktomodelthenoneisnotnecessarilyconcernedwiththeday-to-dayfluctuationsinthemark-to-marketprofitandlossandsoitisnaturaltohedgeusingactualvolatility.Thisisusuallynotfarfromoptimalinthesenseofpossibleexpectedtotalprofit,andithasnostandarddeviationoffinalprofit.However,itiscommontohavetoreportprofitandlossbasedonmarketvalues.Thisconstraintmaybeimposedbyariskmanagementdepartment,byprimebrokers,orbyinvestorswhomaymonitor howtodeltahedgeChapter12211Expectedprofit18StandarddeviationofprofitMinimumprofitMaximumprofit138300.10.20.30.40.50.60.70.80.91−2HedgingvolatilityFigure12.13Expectedprofitandstandarddeviationofprofithedgingwithvariousvolatilities.E=90,S=100,µ=−0.1,σ=0.4,r=0.1,D=0,T=1,σ˜=0.2.ExpectedprofitStandarddeviationofprofitMinimumprofitMaximumprofit18138300.10.20.30.40.50.60.70.80.91−2−7HedgingvolatilityFigure12.14Expectedprofitandstandarddeviationofprofithedgingwithvariousvolatilities.E=110,S=100,µ=−0.1,σ=0.4,r=0.1,D=0,T=1,σ˜=0.2. 212PartOnemathematicalandfinancialfoundationsthemark-to-marketprofitonaregularbasis.Inthiscaseitismoreusualtohedgebasedonimpliedvolatilitytoavoidthedailyfluctuationsintheprofitandloss.Fortheremainderofthischapterwewillonlyconsiderthecaseofhedgingusingadeltabasedonimpliedvolatility.12.6PORTFOLIOSWHENHEDGINGWITHIMPLIEDVOLATILITYAnaturalextensiontotheaboveanalysisistolookatportfoliosofoptions,optionswithdifferentstrikesandexpirations.Sinceonlyanoption’sgammamatterswhenwearehedgingusingimpliedvolatility,callsandputsareeffectivelythesamesincetheyhavethesamegamma.TheprofitfromaportfolioisnowTk1q(σ2−˜σ2)e−r(s−t0)S2ids,2kkkkt0wherekistheindexforanoption,andqkisthequantityofthatoption.IntroducetI=1q(σ2−˜σ2)e−r(s−t0)S2ids,(12.4)2kkkkt0asanewstatevariable,andtheanalysiscanproceedasbefore.Notethatsincetheremaybemorethanoneexpirationdatesincewehaveseveraldifferentoptions,itmustbeunderstoodinEquation(12.4)thatiiszerofortimesbeyondtheexpirationoftheoption.kThegoverningdifferentialoperatorforexpectation,variance,etc.isthen∂∂2∂∂+1σ2S2+µS+1(σ2−˜σ2)e−r(t−t0)S2i=0,∂t2∂S2∂S2kk∂Ikwithfinalconditionrepresentingexpectation,variance,etc.12.6.1ExpectationThesolutionforthepresentvalueoftheexpectedprofit(t=t0,S=S0,I=0)issimplythesumofindividualprofitsforeachoption,Ee−r(Tk−t0)(σ2−˜σ2)Tk1kkF(S0,t0)=qk√k22πt0σ2(s−t)+˜σ2(T−s)0kk2ln(S/E)+µ−1σ2(s−t)+r−D−1σ˜2(T−s)0k202kk×exp−ds.2(σ2(s−t)+˜σ2(T−s))0kkThederivationcanbefoundinthischapter’sappendix. howtodeltahedgeChapter1221312.6.2VarianceThevarianceismorecomplicated,obviously,becauseofthecorrelationbetweenalloftheoptionsintheportfolio.Nevertheless,wecanfindanexpressionfortheinitialvarianceasG(S,t)−F(S,t)2where0000G(S0,t0)=qjqkGjk(S0,t0)jkwhereEjEk(σ2−˜σ2)(σ2−˜σ2)e−r(Tj−t0)−r(Tk−t0)min(Tj,Tk)TjjkGjk(S0,t0)=4πσσ˜kt0sep(u,s;S0,t0)×
duds√√s−tT−sσ2(u−s)+˜σ2(T−u)1+1+10kjjσ2(s−t0)σ˜2σ2(u−s)+˜σ2k(Tk−s)j(Tj−u)(12.5)where(ln(S/E)+µ(s−t)+r(T−s))210k0kkp(u,s;S0,t0)=−22σ˜k(Tk−s)(ln(S/E)+µ(u−t)+r(T−u))210j0jj−2σ2(u−s)+˜σ2(T−u)jj2ln(S0/Ek)+µ(s−t0)+rk(Tk−s)+ln(S0/Ej)+µ(u−t0)+rj(Tj−u)σ˜2σ2(u−s)+˜σ21k(Tk−s)j(Tj−u)+21+1+1σ2(s−t0)σ˜2σ2(u−s)+˜σ2k(Tk−s)j(Tj−u)andµ=µ−1σ2,r=r−D−1σ˜2andr=r−D−1σ˜2.2j2jk2kThederivationcanbefoundinthischapter’sappendix.12.6.3PortfolioOptimizationPossibilitiesThereisclearlyplentyofscopeforusingtheaboveformulaeinportfoliooptimizationproblems.HereIgiveoneexample.Thestockiscurrentlyat100.Thegrowthrateiszero,actualvolatilityis20%,zerodivi-dendyieldandtheinterestrateis5%.Table12.3showstheavailableoptions,andassociatedparameters.Observethenegativeskew.Theout-of-the-moneyputsareovervaluedandtheout-of-the-moneycallsareundervalued.(The‘ProfitTotalExpected’rowassumesthatwebuyasingleoneofthatoption.)Usingtheaboveformulaewecanfindtheportfoliothatmaximizesorminimizestargetquantities(expectedprofit,standarddeviation,ratioofprofittostandarddeviation).Letusconsiderthesimplecaseofmaximizingtheexpectedreturn,whileconstrainingthestandard 214PartOnemathematicalandfinancialfoundationsTable12.3Availableoptions.ABCDETypePutPutCallCallCallStrike8090100110120Expiration11111Volatility,Implied0.2500.2250.2000.1750.150OptionPrice,Market1.5113.01210.4515.0541.660OptionValue,Theory0.6872.31010.4516.0403.247ProfitTotalExpected−0.933−0.7520.0000.9361.410Table12.4Anoptimalportfolio.ABCDETypePutPutCallCallCallStrike8090100110120Quantity−2.10−2.2501.461.2815Payofffunction(withinitialdeltahedge)105Asset060708090100110120130140−5−10−15−20−25−30Figure12.15Payoffwithinitialdeltahedgeforoptimalportfolio;S=100,µ=0,σ=0.2,r=0.05,D=0,T=1.Seetextforadditionalparametersandinformation.deviationtobeone.Thisisaverynaturalstrategywhentryingtomakeaprofitfromvolatilityarbitragewhilemeetingconstraintsimposedbyregulators,brokers,investorsetc.TheresultisgiveninTable12.4.Thepayofffunction(withitsinitialdeltahedge)isshowninFigure12.15.Thisoptimizationhaseffectivelyfoundanidealriskreversaltrade.Thisportfoliowouldcost−$0.46tosetup,i.e.itwouldbringinpremium.Theexpectedprofitis$6.83. howtodeltahedgeChapter1221512.7HEDGINGWHENIMPLIEDVOLATILITYISSTOCHASTICItisdemonstrablytruethatimpliedvolatilitytendstovary,whereasitrequiresquiteadvancedstatisticstomakethesameobservationaboutactualvolatility.So,let’srepeatsomeoftheanalysesaboveassumingthatdσ˜=adt+bdX2withthestochasticdifferentialequationforSbeingtheusual,withrandomcomponentdX1andactualvolatilityconstant.TherewillbeacorrelationcoefficientbetweendX1anddX2ofρ.SeealsoCarr&Verma(2005).12.7.1Case1:HedgewithActualVolatility,σTheargumentthatthefinalprofitisguaranteedisnotaffectedbyhavingimpliedvolatilitystochastic,exceptinsofarasyoumaygettheopportunitytoclosethepositionearlyifimpliedvolatilityreachesthelevelofactual.Themark-to-marketprofitnowbecomes1(σ2−˜σ2)S2idt+(i−a)((µ−r+D)Sdt+σSdX)21∂Vi∂2Vi∂2Vi+dσ˜+1b2dt+ρσbSdt.∂σ˜2∂σ˜2∂S∂σ˜RememberthatViistheBlack–Scholesformulafortheoption.12.7.2Case2:HedgewithImpliedVolatility,σ˜?Thereisaquestionmarkinthetitleofthissectionbecauseitisnotclearthatusingthedeltabasedonimpliedvolatilityismeaningfulwhenimpliedvolatilityisstochastic.Soforthemomentwewilljusthedgeusing‘,’tobechosen.Themark-to-marketprofitisdVi−dS−r(Vi−S)dt−DSdt∂Vi∂2Vi=idt+idS+1σ2S2idt+dσ˜+1b2dt2∂σ˜2∂σ˜2∂2Vi+ρσbSdt−dS−r(Vi−S)dt−DSdt.∂S∂σ˜Thevarianceofthisis2∂Vi∂Vib2+σ2S2(i−)2+2ρσbS(i−)+···.∂σ˜∂σ˜(multipliedbydt).Thisisminimizedbythechoiceρb∂Vi=i+.σS∂σ 216PartOnemathematicalandfinancialfoundationsThisconfirmsthattominimizeriskonamark-to-marketbasisyoumustadjustthedeltabythevega.Withthischoiceofdeltawefindthatthemark-to-marketprofitcontainsadeterministicandarandomterm.Therandomtermisρb∂Vi−dS+dσ˜.σS∂σ˜Thishasstandarddeviation∂Vib1−ρ2dt1/2.∂σ˜UsingtheBlack–Scholesformulaethedeterministicpartbecomes√e−D(T−t)T−tSN(d)1σ2−˜σ2∂dρbσdρb11211×2−2bd1+ρbσ−√+a+(r−µ−D)(12.6)σ(T˜−t)∂σ˜σ˜T−tσ(multipliedbydt).12.8HOWDOESIMPLIEDVOLATILITYBEHAVE?Nowisthenaturaltimetotalkalittlebitabouthowimpliedvolatilitybehavesinpractice.Asthestockpricegoesupanddownrandomlyweoftenseethattheimpliedvolatilityofeachoptionwillalsovary.Thismayormaynotbeconsistentwithcertainmodels,andmayormaynotbeconsistentwithnoarbitrage.Butmoreimportantly,whatdoesitmeanformakingmoneyifwethinkthatthemarketiswrong?Belowareacoupleof‘models’forhowimpliedvolatilitymightchangeasthemarketmoves.12.8.1StickyStrikeInthismodelimpliedvolatilityremainsconstantforeachoption(i.e.eachstrikeandexpiration).EffectivelyeachoptioninhabitsitsownlittleBlack–Scholesworldofconstantvolatility.Thisbehaviorseemstobemostcommonintheequitymarkets.Asfarasmakingaprofitiftheimpliedvolatilityisdifferentfromactualvolatilitythenthefirsthalfofthischapterisclearlyveryrelevant.12.8.2StickyDeltaSincethedeltaofanoptionisafunctionofitsmoneyness,S/E,thestickydeltabehaviorcouldalsobecalledthestickymoneynessrule.ThisbehavioriscommonlyseenintheFXmarkets,possiblybecausethereitisusualtoquoteoptionprices/volatilitiesforoptionswithspecificdeltasratherthanspecificstrike.(Thereis,ofcourse,aone-to-onecorrespondenceforvanillas.)Inthismodelwehaveσ˜=g(S/E,t). howtodeltahedgeChapter12217Therefore∂gS∂gS2∂2gS∂gdσ˜=+µ+1σ2dt+σdX,∂tE∂ξ2E22E∂ξ1∂ξwhereξ=S/E.ThemostimportantpointaboutthisexpressionisthatitisperfectlycorrelatedwithdS,ρ=1,sothatperfecthedging(inthemark-to-marketsense)ispossible.WecansubstituteforaandbintoEquation(12.6)togetthe‘daily’mark-to-marketprofit.AvariationonthisthemeistohaveimpliedvolatilityatdifferentstrikesbeingproportionaltotheATMvolatilityandafunctionofthemoneyness,suchaslog(S/E)σ˜=σATMg√.T−tSeeNatenberg(1994)fordetails.Ofcourse,thisthenrequiresamodelforthebehavioroftheATMvolatility.12.8.3Time-periodicBehaviorJusttomakemattersevenmoreinteresting,thereappearstobeaday-of-the-weekeffectinimpliedvolatility.Figures12.16–12.19showhowtheVIXvolatilityindex(ameasureoftheimpliedvolatilityoftheATMSPXadjustedtoalwayshaveanexpirationof30days)changes0.5AverageofVIXChanges0.40.30.20.10MondayTuesdayWednesdayThursdayFriday−0.1−0.2−0.3Figure12.16AveragechangeinlevelofVIXversusdayofweek. 218PartOnemathematicalandfinancialfoundations1.4StdDev.ofVIXChanges1.210.80.60.40.20MondayTuesdayWednesdayThursdayFridayFigure12.17StandarddeviationofchangeinlevelofVIXversusdayofweek.0.8AverageofVIXChanges0.60.40.2086420323028262422201816141210−−−−−−−−−−−−−−−−−0.2−0.4−0.6Figure12.18AveragechangeinlevelofVIXversusdaysbeforenextexpiration. howtodeltahedgeChapter122191.8StdDev.ofVIXChanges1.61.41.210.80.60.40.2086420323028262422201816141210−−−−−−−−−−−−−−−−Figure12.19StandarddeviationofchangeinlevelofVIXversusdaysbeforenextexpiration.withdayoftheweekandnumberofdaystonextexpiration.Bothaveragechangesandstandarddeviationareshown.112.9SUMMARYInthischapterwehaveseensomehintsofhowwecanstarttomoveawayfromtheBlack–Scholesworld,andperhapsevenprofitfromoptions.FURTHERREADING•SeeDerman(1999)foradescriptionofstickystrikeanddelta,andothervolatilityregimes.•Natenberg’sbook(Natenberg,1994)isstilltheclassicreferenceforvolatilitytrading.•SeeCarr’s(Carr,2005)excellentFAQspaperforfurtherinsightintowhichvolatilitytouseforhedging.AlsoHenrard(2001),whoexaminedtheroleoftherealdriftrate.•Ahmad&Wilmott(2005)delveevendeeperintothesubjectofhedgingwithdifferentvolatilities.1Ofcourse,someofthisisnodoubtrelatedtotheroleofweekendsinthecalculationofvolatility.Thereisalotof‘potential’forvolatilityoverweekendsinthesensethatthereisplentyofnewsthatcomesoutthatwillimpactonmarketpriceswhenmarketsopenontheMonday. 220PartOnemathematicalandfinancialfoundationsAPPENDIX:DERIVATIONOFRESULTSPreliminaryResultsInthefollowingderivationsweoftenrequirethefollowingsimpleresults.First,
∞π−ax2edx=.(12.7)−∞aSecond,thesolutionof∂w∂2w=+f(x,τ)∂τ∂x2thatisinitiallyzeroandiszeroatplusandminusinfinityis∞τ1f(x,τ)−(x−x)2/4(τ−τ)√√edτdx.(12.8)2π−∞0τ−τFinally,thetransformations2σ2x=ln(S/E)+µ−1σ2τandτ=(T−t),σ222turntheoperator∂∂2∂+1σ2+µS∂t2∂S2∂Sinto∂∂21σ2−+.(12.9)2∂τ∂x2Expectation,SingleOptionTheequationtobesolvedforF(S,t)is∂F∂2F∂F+1σ2S2+µS+1(σ2−˜σ2)e−r(t−t0)S2i=0,∂t2∂S2∂S2withzerofinalandboundaryconditions.UsingtheabovechangesofvariablesthisbecomesF(S,t)=w(x,τ)where222−r(T−t0)−d2/2∂w∂wE(σ−˜σ)ee2=+√∂τ∂x2σσ˜πτwhere22x−µ−1σ2τ+r−D−1σ˜2τσσ22σ22d2=√.σ˜2τ howtodeltahedgeChapter12221Thesolutionofthisproblemis,using(12.8),1E(σ2−˜σ2)e−r(T−t0)∞τ11√√2πσσ˜−∞0ττ−τ(x−x)2σ2222×exp−−x−µ−1σ2τ+r−D−1σ˜2τdτdx.4(τ−τ)4σ˜2τσ22σ22Ifwewritetheargumentoftheexponentialfunctionas−a(x+b)2+cwehavethesolution1E(σ2−˜σ2)e−r(T−t0)τ11∞√√exp−a(x+b)2+cdxdτ2πσσ˜0ττ−τ−∞1E(σ2−˜σ2)e−r(T−t0)τ111=√√√√exp(c)dτ.2πσσ˜0ττ−τaItiseasytoshowthat1σ2a=+4(τ−τ)4σ˜2τand22τ1212x−µ−σ−r+D+σ˜σ2σ222c=−.4σ˜2τ(τ−τ)1σ2+τ−τσ˜2τWith2s−t=τσ2wehave2ln(S/E)+µ−1σ2(s−t)+r−D−1σ˜2(T−s)22c=−.2(σ2(s−t)+˜σ2(T−s))Fromthisfollows,thattheexpectedprofitinitially(t=t0,S=S0,I=0)isEe−r(T−t0)(σ2−˜σ2)T1√22πt0σ2(s−t0)+˜σ2(T−s)2ln(S/E)+µ−1σ2(s−t)+r−D−1σ˜2(T−s)0202×exp−ds.2(σ2(s−t)+˜σ2(T−s))0 222PartOnemathematicalandfinancialfoundationsVariance,SingleOptionTheproblemfortheexpectationofthesquareoftheprofitis∂v∂2v∂v∂v+1σ2S2+µS+1(σ2−˜σ2)e−r(t−t0)S2i=0,(12.10)∂t2∂S2∂S2∂Iwithv(S,I,T)=I2.Asolutioncanbefoundoftheformv(S,I,t)=I2+2IH(S,t)+G(S,t).SubstitutingthisintoEquation(12.10)leadstothefollowingequationsforHandG(bothtohavezerofinalandboundaryconditions):∂H∂2H∂H+1σ2S2+µS+1(σ2−˜σ2)e−r(t−t0)S2i=0;∂t2∂S2∂S2∂G∂2G∂G+1σ2S2+µS+(σ2−˜σ2)e−r(t−t0)S2iH=0.∂t2∂S2∂SComparingtheequationsforHandtheearlierFwecanseethatEe−r(T−t0)(σ2−˜σ2)T1H=F=√22πtσ2(s−t)+˜σ2(T−s)2ln(S/E)+µ−1σ2(s−t)+r−D−1σ˜2(T−s)22×exp−ds.2(σ2(s−t)+˜σ2(T−s))Noticeinthisthattheexpressionisapresentvalueattimet=t,hencethee−r(T−t0)termat0thefront.TherestofthetermsinthismustbekeptastherunningvariablesSandt.Returningtovariablesxandτ,thegoverningequationforG(S,t)=w(x,τ)is222−r(T−t0)−d2/2∂w∂w2Eσ(σ−˜σ)ee2=+√∂τ∂x2σ24σ˜πτE(σ2−˜σ2)e−r(T−t0)τ111×√√√√exp(c)dτ(12.11)2σσ˜π0ττ−τawhere21212x−µ−σ+r−D−σ˜τσσ222d2=√,σ˜2τandaandcareasabove. howtodeltahedgeChapter12223Thesolutionistherefore12Eσ(σ2−˜σ2)e−r(T−t0)E(σ2−˜σ2)e−r(T−t0)∞τf(x,τ)e−d22/2√√√√2πσ24σ˜π2σσ˜π−∞0τ−τ−(x−x)2/4(τ−τ)×edτdx.Wherenowτ1111f(x,τ)=√√√√exp(c)dττ0ττ−τaandinaandcallτsbecomeτsandallτsbecomeτs,andind2allτsbecomeτsandallxsbecomexs.Thecoefficientinfrontoftheintegralsignssimplifiesto1E2(σ2−˜σ2)2e−2r(T−t0).8π3/2σ2σ˜2Theintegraltermisoftheform∞ττ···dτdτdx,−∞00withtheintegrandbeingtheproductofanalgebraicterm1√√√√√τττ−ττ−τaandanexponentialterm(x−x)2exp−1d2−+c.224(τ−τ)Thisexponentis,infull,2221σ212212(x−x)−x−µ−στ+r−D−σ˜τ−4τσ˜2σ22σ224(τ−τ)22τ1212x−µ−σ−r+D+σ˜σ2σ222−.4σ˜2τ(τ−τ)1σ2+τ−τσ˜2τThiscanbewrittenintheform−d(x+f)2+g,where1σ21111σ2d=++4σ˜2τ4τ−τ4σ2(τ−τ)+˜σ2τ 224PartOnemathematicalandfinancialfoundationsand2222σ2ατσ2ατg=−x−−x−4σ˜2τσ24(σ2(τ−τ)+˜σ2τ)σ22σ22ατσ22ατx−+x−1σ˜2τσ2(σ2(τ−τ)+˜σ2τ)σ2+,4σ211σ2++σ˜2ττ−τσ2(τ−τ)+˜σ2τwhereα=µ−1σ2−r+D+1σ˜2.22UsingEquation(12.7)weendupwith1E2(σ2−˜σ2)2e−2r(T−t0)4π3/2σ2σ˜2ττ
1π√√√√√exp(g)dτdτ.00τττ−ττ−τadChangingvariablestoσ2σ2σ2τ=(T−t),τ=(T−s),andτ=(T−u),222andevaluatingatS=S0,t=t0,givestherequiredresultforthevariance.Expectation,PortfolioofOptionsThisexpressionfollowsfromtheadditivityofexpectations.Variance,PortfolioofOptionsThemanipulationsandcalculationsrequiredfortheanalysisoftheportfoliovariancearesimilartothatforasinglecontract.Thereisagainasolutionoftheformv(S,I,t)=I2+2IH(S,t)+G(S,t).Themaindifferencesarethatwehavetocarryaroundtwoimpliedvolatilities,σ˜jandσ˜k,andtwoexpirations,TjandTk.WewillfindthatthesolutionforthevarianceisthesumoftermssatisfyingdiffusionequationswithsourcetermslikeinEquation(12.11).Thesubscript‘k’isthenassociatedwiththegammaterm,andsoappearsoutsidetheintegralintheequivalentof(12.11),andthesubscript‘j’isassociatedwiththeintegralandsoappearsintheintegrand.Thereisoneadditionalsubtletyinthederivationsandthatconcernstheexpirations.WemustconsiderthegeneralcaseTj=Tk.Theintegrationsin(12.5)mustonlybetakenovertheintervalsupuntiltheoptionshaveexpired.Theeasiestwaytoapplythisistousetheconventionthatthegammasarezeroafterexpiration.Forthisreasonthesintegralisovert0tomin(Tj,Tk). CHAPTER13fixed-incomeproductsandanalysis:yield,durationandconvexityInthisChapter...•thenamesandpropertiesofthebasicandmostimportantfixed-incomeproducts•thedefinitionsoffeaturescommonlyfoundinfixed-incomeproducts•simplewaystoanalyzethemarketvalueoftheinstruments:yield,durationandconvexity•howtoconstructyieldcurvesandforwardrates13.1INTRODUCTIONThischapterisanintroductiontosomebasicinstrumentsandconceptsintheworldoffixedincome,thatis,theworldofcashflowsthatareinthesimplestcasesindependentofanystocks,commoditiesetc.Iwilldescribethemostelementaryoffixed-incomeinstruments,thecoupon-bearingbond,andshowhowtodeterminevariouspropertiesofsuchbondstohelpintheiranalysis.Thischapterisselfcontained,anddoesnotrequireanyknowledgefromearlierchapters.Alotofitisalsonotreallynecessaryreadingforanythingthatfollows.Thereasonforthisisthat,althoughtheconceptsandtechniquesIdescribehereareusedinpracticeandareusefulinpractice,itisdifficulttopresentacompletelycoherenttheoryformoresophisticatedproductsinthisframework.PartThreeisallaboutsuchcoherentframeworks.13.2SIMPLEFIXED-INCOMECONTRACTSANDFEATURES13.2.1TheZero-couponBondThezero-couponbondisacontractpayingaknownfixedamount,theprincipal,atsomegivendateinthefuture,thematuritydateT.Forexample,thebondpays$100in10years’time,seeFigure13.1.We’regoingtoscalethispayoff,sothatinfutureallprincipalswillbe$1. 226PartOnemathematicalandfinancialfoundations$100tTFigure13.1Thezero-couponbond.Thispromiseoffuturewealthisworthsomethingnow:itcannothavezeroornegativevalue.Furthermore,exceptinextremecircumstances,theamountyoupayinitiallywillbesmallerthantheamountyoureceiveatmaturity.WediscussedtheideaoftimevalueofmoneyinChapter1.Thisisclearlyrelevanthereandwewillreturntothisinamoment.13.2.2TheCoupon-bearingBondAcoupon-bearingbondissimilartotheaboveexceptthataswellaspayingtheprincipalatmaturity,itpayssmallerquantities,thecoupons,atintervalsuptoandincludingthematuritydate(seeFigure13.2).Thesecouponsareusuallyprespecifiedfractionsoftheprincipal.Forexample,thebondpays$1in10yearsand2%,i.e.2cents,everysixmonths.Thiswouldbecalleda4%coupon.Thisbondisclearlymorevaluablethanthebondinthepreviousexamplebecauseofthecouponpayments.Wecanthinkofthecoupon-bearingbondasaportfolioofzero-couponbearingbonds;onezero-couponbearingbondforeachcoupondatewithaprincipalbeingthesameastheoriginalbond’scoupon,andthenafinalzero-couponbondwiththesamematurityastheoriginal.Figure13.3isanexcerptfromTheWallStreetJournalEuropeof14thApril2005showingUSTreasuryBonds,NotesandBills.Observethattherearemanydifferent‘rates’orcoupons,anddifferentmaturities.Thevaluesofthedifferentbondswilldependonthesizeofthecoupon,thematurityandthemarket’sviewofthefuturebehaviorofinterestrates.$100tTFigure13.2Thecoupon-bearingbond. fixed-incomeproductsandanalysis:yield,durationandconvexityChapter13227Publisher'sNote:Permissiontoreproducethisimageonlinewasnotgrantedbythecopyrightholder.Readersarekindlyrequestedtorefertotheprintedversionofthischapter.Figure13.3TheWallStreetJournalEuropeof14thApril2005TreasuryBonds,NotesandBills.ReproducedbypermissionofDowJones&Company,Inc. 228PartOnemathematicalandfinancialfoundations13.2.3TheMoneyMarketAccountEveryonewhohasabankaccounthasamoneymarketaccount.Thisisanaccountthataccumulatesinterestcompoundedataratethatvariesfromtimetotime.Therateatwhichinterestaccumulatesisusuallyashort-termandunpredictablerate.Inthesensethatmoneyheldinamoneymarketaccountwillgrowatanunpredictablerate,suchanaccountisriskywhencomparedwithaone-yearzero-couponbond.Ontheotherhand,themoneymarketaccountcanbeclosedatanytimebutifthebondissoldbeforematuritythereisnoguaranteehowmuchitwillbeworthatthetimeofthesale.13.2.4FloatingRateBondsInitssimplestformafloatinginterestrateistheamountthatyougetonyourbankaccount(seeFigure13.4).Thisamountvariesfromtimetotime,reflectingthestateoftheeconomyandinresponsetopressurefromotherbanksforyourbusiness.Thisuncertaintyabouttheinterestrateyoureceiveiscompensatedbytheflexibilityofyourdeposit;itcanbewithdrawnatanytime.ThemostcommonmeasureofinterestisLondonInterbankOfferRateorLIBOR.LIBORcomesinvariousmaturities,onemonth,threemonth,sixmonthetc.,andistherateofinterestofferedbetweenEurocurrencybanksforfixed-termdeposits.Sometimesthecouponpaymentonabondisnotaprescribeddollaramountbutdependsonthelevelofsome‘index,’measuredatthetimeofthepaymentorbefore.Typically,wecannotknowatthestartofthecontractwhatlevelthisindexwillbeatwhenthepaymentismade.Wewillseeexamplesofsuchcontractsinlaterchapters.13.2.5ForwardRateAgreementsAForwardRateAgreement(FRA)isanagreementbetweentwopartiesthataprescribedinterestratewillapplytoaprescribedprincipaloversomespecifiedperiodinthefuture.Thecashflowsinthisagreementareasfollows:partyApayspartyBtheprincipalattimeT1andBpaysAtheprincipalplusagreedinterestattimeT2>T1.Thevalueofthisexchangeatthetimethecontractisenteredintoisgenerallynotzeroandsotherewillbeatransferofcashfromonepartytotheotheratthestartdate.tFigure13.4Thefloating-ratebond. fixed-incomeproductsandanalysis:yield,durationandconvexityChapter1322913.2.6ReposArepoisarepurchaseagreement.Itisanagreementtosellsomesecuritytoanotherpartyandbuyitbackatafixeddateandforafixedamount.Thepriceatwhichthesecurityisboughtbackisgreaterthanthesellingpriceandthedifferenceimpliesaninterestratecalledthereporate.Thecommonestrepoistheovernightrepoinwhichtheagreementisrenegotiateddaily.Iftherepoagreementextendsfor30daysitiscalledatermrepo.Areverserepoistheborrowingofasecurityforashortperiodatanagreedinterestrate.Reposcanbeusedtolockinfutureinterestrates.Forexample,buyasix-monthTreasurybilltodayandrepoitoutforthreemonths.Thereisnocashflowtodaysincethebondhasbeenpaidfor(moneyout)andthenrepoed(sameamountin).Inthreemonths’timeyouwillhavetorepurchasethebillattheagreedprice;thisisanoutflowofcash.Insixmonthsyoureceivetheprincipal.Moneyoutinthreemonths,moneyinsixmonths;fortheretobenoarbitragetheequivalentinterestrateshouldbethatcurrentlyprevailingbetweenthreeandsixmonths’time.13.2.7STRIPSSTRIPSstandsfor‘SeparateTradingofRegisteredInterestandPrincipalofSecurities’.Thecouponsandprincipalofnormalbondsaresplitup,creatingartificialzero-couponbondsoflongermaturitythanwouldotherwisebeavailable.13.2.8AmortizationInalloftheaboveproductsIhaveassumedthattheprincipalremainsfixedatitsinitiallevel.Sometimesthisisnotthecase;theprincipalcanamortizeordecreaseduringthelifeofthecontract.Theprincipalisthuspaidbackgraduallyandinterestispaidontheamountoftheprincipaloutstanding.Suchamortizationisarrangedattheinitiationofthecontractandmaybefixed,sothattherateofdecreaseoftheprincipalisknownbeforehand,orcandependonthelevelofsomeindex:iftheindexishightheprincipalamortizesfasterforexample.WeseeanexampleofacomplexamortizingstructureinChapter32.13.2.9CallProvisionSomebondshaveacallprovision.Theissuercancallbackthebondoncertaindatesoratcertainperiodsforaprescribed,possiblytime-dependent,amount.Thislowersthevalueofthebond.ThemathematicalconsequencesofthisarediscussedinChapter30.13.3INTERNATIONALBONDMARKETS13.3.1UnitedStatesofAmericaIntheUS,bondsofmaturitylessthanoneyeararecalledbillsandareusuallyzerocoupon.Bondswithmaturity2–10yearsarecallednotes.Theyarecouponbearingwithcouponseverysixmonths.Bondswithmaturitygreaterthan10yearsarecalledbonds.Againtheyarecouponbearing.InthisbookItendtocallallofthese‘bonds,’merelyspecifyingwhetherornottheyhavecoupons. 230PartOnemathematicalandfinancialfoundationsBondstradedintheUnitedStatesforeignbondmarketbutwhichareissuedbynon-USinstitutionsarecalledYankeebonds.Sincethebeginningof1997theUSgovernmenthasalsoissuedbondslinkedtotherateofinflation.13.3.2UnitedKingdomBondsissuedbytheUKgovernmentarecalledgilts.Someofthesebondsarecallable,someareirredeemable,meaningthattheyareperpetualbondshavingacouponbutnorepaymentofprin-cipal.Thegovernmentalsoissuesconvertiblebondswhichmaybeconvertedintoanotherbondissue,typicallyoflongermaturity.Finally,thereareindex-linkedbondshavingtheamountofthecouponandprincipalpaymentslinkedtoameasureofinflation,theRetailPriceIndex(RPI).13.3.3JapanJapaneseGovernmentBonds(JGBs)comeasshort-termtreasurybills,medium-term,long-term(10-yearmaturity)andsuperlong-term(20-yearmaturity).Thelong-andsuperlong-termbondshavecouponseverysixmonths.Theshort-termbondshavenocouponsandthemedium-termbondscanbeeithercoupon-bearingorzero-couponbonds.Yendenominatedbondsissuedbynon-JapaneseinstitutionsarecalledSamuraibonds.13.4ACCRUEDINTERESTThemarketpriceofbondsquotedinthenewspapersarecleanprices.Thatis,theyarequotedwithoutanyaccruedinterest.Theaccruedinterestistheamountofinterestthathasbuiltupsincethelastcouponpayment:accruedinterest=interestdueinfullperiodnumberofdayssincelastcoupondate×.numberofdaysinperiodbetweencouponpaymentsTheactualpaymentiscalledthedirtypriceandisthesumofthequotedcleanpriceandtheaccruedinterest.13.5DAY-COUNTCONVENTIONSBecauseofsuchmattersastheaccrualofinterestbetweencoupondatestherenaturallyarisesthequestionofhowtoaccrueinterestovershorterperiods.Interestisaccruedbetweentwodatesaccordingtotheformulanumberofdaysbetweenthetwodates×interestearnedinreferenceperiod.numberofdaysinperiodTherearethreemainwaysofcalculatingthe‘numberofdays’intheaboveexpression.•Actual/ActualSimplycountthenumberofcalendardays•30/360Assumethereare30daysinamonthand360daysinayear•Actual/360Eachmonthhastherightnumberofdaysbutthereareonly360daysinayear fixed-incomeproductsandanalysis:yield,durationandconvexityChapter1323113.6CONTINUOUSLYANDDISCRETELYCOMPOUNDEDINTERESTTobeabletocomparefixed-incomeproductswemustdecideonaconventionforthemeasure-mentofinterestrates.Sofar,wehaveusedacontinuouslycompoundedrate,meaningthatthepresentvalueof$1paidattimeTinthefutureise−rT×$1forsomer.Wehaveseenhowthisfollowsfromthecash-in-the-bankormoneymarketaccountequationdM=rMdt.Thisistheconventionusedintheoptionsworld.Anothercommonconventionistousetheformula1×$1,(1+r
)Tforpresentvalue,wherer
issomeinterestrate.ThisrepresentsdiscretelycompoundedinterestandassumesthatinterestisaccumulatedannuallyforTyears.Theformulaisderivedfromcalculatingthepresentvaluefromasingle-periodpayment,andthencompoundingthisforeachyear.Thisformulaiscommonlyusedforthesimplertypeofinstrumentssuchascoupon-bearingbonds.Thetwoformulaeareidentical,ofcourse,whenr=log(1+r
).Thisgivestherelationshipbetweenthecontinuouslycompoundedinterestraterandthediscreteversionr
.Whatwouldtheformulabeifinterestwasdiscretelycompoundedtwiceperyear?Inthisbookwetendtousethecontinuousdefinitionofinterestrates.13.7MEASURESOFYIELDThereissuchavarietyoffixed-incomeproducts,withdifferentcouponstructures,amortiza-tion,fixedand/orfloatingrates,thatitisnecessarytobeabletocomparedifferentproductsconsistently.Supposeyouhavetochoosebetweenaten-yearzero-couponbondanda20-yearcoupon-bearingbond.Onehasnoincomefortenyearsbutthengetsabiglumpsum,theotherhasatrickleofincomebutyouhavetowaitmuchlongerforthebigamount.Onewaytodothisisthroughmeasuresofhowmucheachcontractearns;thereareseveralmeasuresofthisallcomingunderthenameyield.13.7.1CurrentYieldThesimplestmeasurementofhowmuchacontractearnsisthecurrentyield.Thismeasureisdefinedbyannual$couponincomecurrentyield=.bondprice 232PartOnemathematicalandfinancialfoundationsForexample,considerthe10-yearbondthatpays2centseverysixmonthsand$1atmaturity.Thisbondhasatotalincomeperannumof4cents.Supposethatthequotedmarketpriceofthisbondis88cents.Thecurrentyieldissimply0.04=4.5%.0.88Thismeasurementoftheyieldofthebondmakesnoallowanceforthepaymentoftheprincipalatmaturity,norforthetimevalueofmoneyifthecouponpaymentisreinvested,norforanycapitalgainorlossthatmaybemadeifthebondissoldbeforematurity.Itisarelativelyunsophisticatedmeasure,concentratingverymuchonshort-termpropertiesofthebond.13.7.2TheYieldtoMaturity(YTM)orInternalRateofReturn(IRR)Supposethatwehaveazero-couponbondmaturingattimeTwhenitpaysonedollar.AttimetithasavalueZ(t;T).ApplyingaconstantrateofreturnofybetweentandT,thenonedollarreceivedattimeThasapresentvalueofZ(t;T)attimet,whereZ(t;T)=e−y(T−t).ItfollowsthatlogZy=−.T−tLetusgeneralizethis.Supposethatwehaveacoupon-bearingbond.Discountallcouponsandtheprincipaltothepresentbyusingsomeinterestratey.Thepresentvalueofthebond,attimet,isthen
NV=Pe−y(T−t)+Ce−y(ti−t),(13.1)ii=1wherePistheprincipal,Nthenumberofcoupons,andCithecouponpaidondateti.Ifthebondisatradedsecuritythenweknowthepriceatwhichthebondcanbebought.IfthisisthecasethenwecancalculatetheyieldtomaturityorinternalrateofreturnasthevalueythatwemustputintoEquation(13.1)tomakeVequaltothetradedpriceofthebond.Thiscalculationmustbeperformedbysometrialanderror/iterativeprocedure.Forexample,inthebondinTable13.1wehaveaprincipalof$1paidinfiveyearsandcouponsofthreecents(threepercent)paideverysixmonths.Supposethatthemarketvalueofthisbondis96cents.Weask‘Whatistheinternalrateofreturnwemustusetogivethesecashflowsatotalpresentvalueof96cents?’Thisvalueistheyieldtomaturity.Inthefourthcolumninthistableisthepresentvalue(PV)ofeachofthecashflowsusingarateof6.8406%:sincethesumofthesepresentvaluesis96centstheYTMorIRRis6.8406%.Thisyieldtomaturityisavalidmeasureofthereturnonabondifweintendtoholdittomaturity.Tocalculatetheyieldtomaturityofaportfolioofbondssimplytreatallthecashflowsasiftheywerefromtheonebondandcalculatethevalueofthewholeportfoliobyaddingupthemarketvaluesofthecomponentbonds. fixed-incomeproductsandanalysis:yield,durationandconvexityChapter13233Table13.1Anexampleofacoupon-bearingbond.TimeCouponPrincipalPV(discountingrepaymentat6.8406%)000.5.030.02901.0.030.02801.5.030.02702.0.030.02622.5.030.02533.0.030.02443.5.030.02364.0.030.02284.5.030.02205.0.031.000.7316Total0.960013.8THEYIELDCURVETheplotofyieldtomaturityagainsttimetomaturityiscalledtheyieldcurve.Forthemomentassumethatthishasbeencalculatedfromzero-couponbondsandthatthesebondshavebeenissuedbyaperfectlycreditworthysource.Ifthebondshavecouponsthenthecalculationoftheyieldcurveismorecomplicatedandthe‘forwardcurve,’describedbelow,isabettermeasureoftheinterestratepertainingatsometimeinthefuture.Figure13.5showstheyieldcurveforUSTreasuriesasitwason9thSeptember1999.13.9PRICE/YIELDRELATIONSHIPFromEquation(13.1)wecaneasilyseethattherelationshipbetweenthepriceofabondanditsyieldisoftheformshowninFigure13.6(assumingthatallcashflowsarepositive).Onthisfigureismarkedthecurrentmarketpriceandthecurrentyieldtomaturity.Sinceweareofteninterestedinthesensitivityofinstrumentstothemovementofcertainunderlyingfactorsitisnaturaltoaskhowdoesthepriceofabondvarywiththeyield,orviceversa.Toafirstapproximationthisvariationcanbequantifiedbyameasurecalledtheduration.Figure13.7showsthePrice/Yieldrelationshipforaspecificfive-yearUSTreasury.13.10DURATIONFromEquation(13.1)wefindthat
NdV=−(T−t)Pe−y(T−t)−C(t−t)e−y(ti−t).iidyi=1 234PartOnemathematicalandfinancialfoundationsFigure13.5YieldcurveforUSTreasuries.Source:BloombergL.P.10.90.80.70.6today'sprice0.5Price0.40.30.2today'syield0.100%2%4%6%8%10%12%14%16%YieldFigure13.6ThePrice/Yieldrelationship. fixed-incomeproductsandanalysis:yield,durationandconvexityChapter13235Figure13.7Bloomberg’sPrice/Yieldgraph.Source:BloombergL.P.ThisistheslopeofthePrice/Yieldcurve.Thequantity1dV−VdyiscalledtheMacaulayduration.(Themodifieddurationissimilarbutusesthediscretelycompoundedrate.)Intheexpressionforthedurationthetimeofeachcouponpaymentisweightedbyitspresentvalue.Thehigherthevalueofthepresentvalueofthecouponthemoreitcontributestothedura-tion.Also,sinceyismeasuredinunitsofinversetime,theunitsofthedurationaretime.Thedurationisameasureoftheaveragelifeofthebond.ItiseasilyshownthattheMacaulaydurationforazero-couponbondisthesameasitsmaturity.Let’stakealookattheideaofaveragetime.Supposeweaskedtowhatzero-couponbondisour(coupon-bearing)bondequivalent?Thatis,whatmaturitywouldan‘equivalent’bondhave?Taketheactualbond’svalueandequateittoazero-couponbond,havingthesameyieldbutanunknownmaturity(andunknownquantity!):
NV=Pe−y(T−t)+Ce−y(ti−t)=Xe−y(T−t).ii=1 236PartOnemathematicalandfinancialfoundationsDifferentiatebothsideswithrespecttoy:
NdV=−(T−t)Pe−y(T−t)−C(t−t)e−y(ti−t)=−X(T−t)e−y(T−t).iidyi=1Finally,dividebothsidesby−V:1dV−=···=T−t.VdyHencethestatementaboutthebond’saveragelife,oreffectivematurity.Forsmallmovementsintheyield,thedurationgivesagoodmeasureofthechangeinvaluewithachangeintheyield.ForlargermovementsweneedtolookathigherordertermsintheTaylorseriesexpansionofV(y).Oneofthemostcommonusesofthedurationisinplotsofyieldversusdurationforavarietyofinstruments.AnexampleisshowninFigure13.8.Lookatthebondmarked‘CPU.’Thisbondhasacouponof4.75%paidtwiceperyear,callablefromJune1998andmaturinginJune2000.Wecanusethisplottogrouptogetherinstrumentswiththesameorsimilardurationsandmakecomparisonsbetweentheiryields.Twobondshavingthesamedurationbutwithonebondhavingahigheryieldmightbesuggestiveofvalueformoneyinthehigher-yieldingbond,orofcreditriskissues.However,suchindicatorsofrelativevaluemustbeusedwithcare.Itispossiblefortwobondstohavevastlydifferentcashflowprofilesyethavethesameduration;onemayhaveamaturityof30yearsbutanaveragelifeandhenceadurationofsevenyears,whereasanothermaybeaseven-yearzero-couponbond.Clearly,theformerhas23yearsmoreriskthanthelatter.12FNEN5/1/05-9810HCNA10.75010/15/03-98FNRI9.7510/1/06-01NVR114/15/03-988CPU9.56/15/00-98Yield642001234567DurationFigure13.8Yieldversusduration;measuringtherelativevalueofbonds. fixed-incomeproductsandanalysis:yield,durationandconvexityChapter1323713.11CONVEXITYTheTaylorseriesexpansionofVgivesdV1dV1d2V=δy+(δy)2+···,VVdy2Vdy2whereδyisachangeinyield.Forverysmallmovementsintheyield,thechangeinthepriceofabondcanbemeasuredbytheduration.ForlargermovementswemusttakeaccountofthecurvatureinthePrice/Yieldrelationship.Thedollarconvexityisdefinedasd2V
N=(T−t)2Pe−y(T−t)+C(t−t)2e−y(ti−t)2iidyi=1andtheconvexityis1d2V.Vdy2Toseehowthesecanbeused,examineFigure13.9.InthisfigureweseethePrice/Yieldrelationshipfortwobondshavingthesamevalueanddurationwhentheyieldisaround8%,butthentheyhavedifferentconvexities.BondAhasagreaterconvexitythanbondB.ThisfiguresuggeststhatbondAisbettervaluethanBbecauseasmallchangeintheyieldsresultsinahighervalueforA.Whenwedevelopaconsistenttheoryforpricingbondswheninterestratesarestochasticwewillseehowtheabsenceofarbitragewillleadtorelationshipsbetweensuchquantitiesasyield,durationandconvexity,notunliketheBlack–Scholesequation.43.5BondABondB32.52Price1.510.500%5%10%15%YieldFigure13.9Twobondswiththesamepriceanddurationbutdifferentconvexities. 238PartOnemathematicalandfinancialfoundationsACBDEGIKFHJ1DateCouponPrincipalPVsTimeTime^220wtdwtd3YTM4.95%0.52%0.01950.00980.00494Mktprice0.92112%0.01900.01900.01905Th.Price0.9211.52%0.01860.02790.04186Error1.4E-0822%0.01810.03620.07257Duration8.25442.52%0.01770.04420.11048Convexity76.872832%0.01720.05170.155193.52%0.01680.05890.2060=C4-C510=SUM(H3:H22)42%0.01640.06560.2625114.52%0.01600.07200.32411252%0.01560.07810.3903135.52%0.01520.08380.46071462%0.01490.08920.5349156.52%0.01450.09420.61241672%0.01410.09900.6929=SUM(I3:I22)/C5177.52%0.01380.10350.77601882%0.01350.10770.8613198.52%0.01310.11160.948520=SUM(J3:J22)/C592%0.01280.11531.0374219.52%0.01250.11871.127622102%10.62166.216162.16142324=F20*EXP(-E20*$C$3)25=E20*H2026=(G22+F22)*EXP(-E22*$C$3)2728=E20*I20293031323334353637Figure13.10Aspreadsheetshowingthecalculationofyield,durationandconvexity. fixed-incomeproductsandanalysis:yield,durationandconvexityChapter13239Thecalculationofyieldtomaturity,durationandconvexityisshowninFigure13.10.Inputsareinthegreyboxes.13.12ANEXAMPLEFigure13.11showstheyieldanalysisscreenfromBloomberg.Theyield,durationandconvexityhavebeencalculatedforaspecificUSTreasury.Figures13.12and13.13showtimeseriesofthepriceandyieldrespectively.13.13HEDGINGInmeasuringandusingyieldstomaturity,itmustberememberedthattheyieldistherateofdiscountingthatmakesthepresentvalueofabondthesameasitsmarketvalue.Ayieldisthusidentifiedwitheachindividualinstrument.Itisperfectlypossiblefortheyieldononeinstrumenttorisewhileanotherfalls,especiallyiftheyhavesignificantlydifferentmaturitiesordurations.Nevertheless,oneoftenwantstohedgemovementsinonebondwithmovementsinanother.Thisiscommonlyachievedbymakingonebigassumptionabouttherelativemovementsofyieldsonthetwobonds.BondAhasayieldofyA,bondBhasayieldofyB,theyhavedifferentFigure13.11Yieldanalysis.Source:BloombergL.P. 240PartOnemathematicalandfinancialfoundationsFigure13.12Pricetimeseries.Source:BloombergL.P.Figure13.13Yieldtimeseries.Source:BloombergL.P. fixed-incomeproductsandanalysis:yield,durationandconvexityChapter13241maturitiesanddurationsbutwewillassumethatamoveofx%inA’syieldisaccompaniedbyamoveofx%inB’syield.Thisistheassumptionofparallelshiftsintheyieldcurve.Ifthisisthecase,thenifweholdAbondsandBbondsintheinverseratiooftheirdurations(withonelongpositionandoneshort)wewillbeleading-orderhedged:=VA(yA)−VB(yB),withtheobviousnotationforthevalueandyieldofthetwobonds.Thechangeinthevalueofthisportfoliois∂VA∂VBδ=x−x+higher-orderterms.∂yA∂yBChoose∂VA∂VB=∂yA∂yBtoeliminatetheleading-orderrisk.Thehigher-ordertermsdependontheconvexityofthetwoinstruments.Ofcourse,thisisasimplificationoftherealsituation;theremaybelittlerelationshipbetweentheyieldsonthetwoinstruments,especiallyifthecashflowsaresignificantlydifferent.Inthiscasetheremaybetwistingorarchingoftheyieldcurve.13.14TIME-DEPENDENTINTERESTRATEInthissectionweexaminebondpricingwhenwehaveaninterestratethatisaknownfunctionoftime.Theinterestrateweconsiderwillbewhatisknownasashort-terminterestrateorspotinterestrater(t).Thismeansthattherater(t)istoapplyattimet:interestiscompoundedatthisrateateachmomentintimebutthisratemaychange;generallyweassumeittobetimedependent.Ifthespotinterestrater(t)isaknownfunctionoftime,thenthebondpriceisalsoafunctionoftimeonly:V=V(t).(Thebondpriceis,ofcourse,alsoafunctionofmaturitydateT,butIsuppressthatdependenceexceptwhenitisimportant.)Webeginwithazero-couponbondexample.Becausewereceive1attimet=TweknowthatV(T)=1.Inowderiveanequationforthevalueofthebondatatimebeforematurity,tKThenPayoff=S-KEndFunction2Thisparameterizationofthebinomialmethodistheoneexplainedintheappendixofthischapter. thebinomialmodelChapter152876.366.346.326.36.286.266.46.38Value6.246.366.226.346.326.26.3951051151256.186.166.14050100150200NumberoftimestepsFigure15.33Optionpriceasafunctionofnumberoftimesteps.SinceIneverusetheassetnodesotherthanatexpiryIcouldhaveusedonlytheonearrayintheabove,withthesamearraybeingusedforbothSandV.Ihavekeptthemseparatetomaketheprogrammoretransparent.Also,IcouldhavesavedthevaluesofVatallofthenodes,whereasintheaboveIhaveonlysavedthenodeatthepresenttime.Savingallthevalueswillbeimportantifyouwanttoseehowtheoptionvaluechangeswiththeassetpriceandtime,ifyouwanttocalculategreeksforexample.InFigure15.33Ishowaplotofthecalculatedoptionpriceagainstthenumberoftimestepsusingthisalgorithm.Theinsetfigureisacloseup.Observetheoscillation.Inthisexample,anoddnumberoftimestepsgivesananswerthatistoohighandanevenananswerthatistoolow.15.19THEGREEKSThegreeksaredefinedasderivativesoftheoptionvaluewithrespecttovariousvariablesandparameters.Itisimportanttodistinguishwhetherthedifferentiationiswithrespecttoavariableoraparameter(itcould,ofcourse,bewithrespecttoboth).Ifthedifferentiationisonlywithrespecttotheassetpriceand/ortimethenthereissufficientinformationinourbinomialtreetoestimatethederivative.Itmaynotbeanaccurateestimate,butitwillbeanestimate.Theoption’sdelta,gammaandthetacanallbeestimatedfromthetree.Ontheotherhand,ifyouwanttoexaminethesensitivityoftheoptionwithrespecttooneoftheparameters,thenyoumustperformanotherbinomialcalculation.Thisappliestotheoption’svegaandrhoforexample.Letmetakethesetwocasesinturn.Fromthebinomialmodeltheoption’sdeltaisdefinedbyV+−V−.(u−v)S 288PartOnemathematicalandfinancialfoundationsGDGDGFigure15.34Calculatingthedeltaandgamma.Wecancalculatethisquantitydirectlyfromthetree.ReferringtoFigure15.34,thedeltausestheoptionvalueatthetwopointsmarked‘D,’togetherwithtoday’sassetpriceandtheparametersuandv.Thisisasimplecalculation.Inthelimitasthetimestepapproacheszero,thedeltabecomes∂V.∂SThegammaoftheoptionisalsodefinedasaderivativeoftheoptionwithrespecttotheunderlying:∂2V.∂S2Toestimatethisquantityusingourtreeisnotsoclear.Itwillbemucheasierwhenweuseafinite-differencegrid.However,gammaisameasureofhowmuchwemustrehedgeatthenexttimestep.ButwecancalculatethedeltaatpointsmarkedwithaDinFigure15.34fromtheoptionvalueonetimestepfurtherinthefuture.Thegammaisthenjustthechangeinthedeltafromoneofthesetotheotherdividedbythedistancebetweenthem.Thiscalculationusesthepointsmarked‘G’inFigure15.34.Thethetaoftheoptionisthesensitivityoftheoptionpricetotime,assumingthattheassetpricedoesnotchange.Again,thisiseasiertocalculatefromafinite-differencegrid.Anobviouschoiceforthediscrete-timedefinitionofthetaistointerpolatebetweenV+andV−tofindatheoreticaloptionvaluehadtheassetnotchangedandusethistoestimate∂V.∂tThisresultsin1(V++V−)−V2.δtAsthetimestepgetssmallerandsmallerthesegreeksapproachtheBlack–Scholescon-tinuous-timevalues. thebinomialmodelChapter15289Estimatingtheothertypeofgreeks,theonesinvolvingdifferentiationwithrespecttoparam-eters,isslightlyharder.Theyarehardertocalculateinthesensethatyoumustperformasecondbinomialcalculation.Iwillillustratethiswiththecalculationoftheoption’svega.Thevegaisthesensitivityoftheoptionvaluetothevolatility∂V.∂σSupposewewanttofindtheoptionvalueandvegawhenthevolatilityis20%.Themostefficientwaytodothisistocalculatetheoptionpricetwice,usingabinomialtree,withtwodifferentvaluesofσ.Calculatetheoptionvalueusingavolatilityofσ±,forasmallnumber;callthevaluesyoufindV±.TheoptionvalueisapproximatedbytheaveragevalueV=1(V+V)2+−andthevegaisapproximatedbyV+−V−.2Theideacanbeappliedtoothergreeks.15.20EARLYEXERCISEAmerican-styleexerciseiseasytoimplementinabinomialsetting.ThealgorithmisidenticaltothatforEuropeanexercisewithoneexception.Weusethesamebinomialtree,withthesameu,vandp,butthereisaslightdifferenceintheformulaforV.Wemustensurethattherearenoarbitrageopportunitiesatanyofthenodes.Forreasonswhichwillbecomeapparent,I’mgoingtochangemynotationnow,makingitmorecomplexbutmoreinformative.IntroducethenotationSntomeanjtheassetpriceatthenthtimestep,atthenodejfromthebottom,0≤j≤n.Thisnotationisconsistentwiththecodeabove.InourlognormalworldwehaveSn=Sujvn−j,jwhereSisthecurrentassetprice.AlsointroduceVnastheoptionvalueatthesamenode.OurjultimategoalistofindV0knowingthepayoff,i.e.knowingVMforall0≤j≤MwhereM0jisthenumberoftimesteps.ReturningtotheAmericanoptionproblem,arbitrageispossibleiftheoptionvaluegoesbelowthepayoffatanytime.Ifourtheoreticalvaluefallsbelowthepayoffthenitistimetoexercise.Ifwedothenexercisetheoption,itsvalueandthepayoffmustbethesame.IfwefindthatVn+1−Vn+1uVn+1−vVn+1j+1j1jj+1n+≥Payoff(Sj)u−v1+rδtu−vthenweusethisasournewvalue.ButifVn+1−Vn+1uVn+1−vVn+1j+1j1jj+1n+0thenf>0andwestandtomakeaprofit,inthelongterm.Ifµ<0,asitisforrouletteorifyoufollowanaiveblackjackstrategy,thenyoushouldinvestanegativeamount,i.e.ownthecasino.(Ifyoumustplayroulette,putallyourmoneyyouwouldgambleinyourlifetimeonacolor,andplayonce.Notonlydoyoustandanalmost50%chanceofdoublingyourmoney,youwillgainaninvaluablereputationasaseriousplayer.)Thelong-rungrowthratemaximizationandtheoptimalamounttoinvestiscalledtheKellycriterion.InFigure17.5isshownthefunctiongiveninEquation(17.1)fortheexpectedlong-termgrowthrate.Thisexampleusesµ=0.01andσ=1.Inthisfigureyouwillseetheoptimalbettingfractionisthevalueforfwhichmaximizesthefunction;thisistheKellycriterion.Totheleftofthisisconservativebetting,totherightiscrazy.Isaycrazybecausegoingbeyondtheoptimalfractionincreasesyourvolatilityofwinnings,anddecreasestheirexpectation.At‘twiceKelly’yourexpectedwinningsarezero,andbeyondthattheybecomenegative.Giventhatinpracticeyourarelyknowtheoddsaccuratelyitmakessensetobetconserva-tively,incaseyouaccidentallystrayintothecrazyzone.Forthatreasonmanypeopleuse‘halfExpectedlong-termgrowthrate0.00006TwiceKelly0.000040.00002f000.0050.010.0150.020.0250.03−0.00002−0.00004ConservativeKelly−0.00006CrazySuicidal−0.00008−0.0001−0.00012Figure17.5Theexpectedlong-termgrowthrateasafunctionofinvestmentfraction. investmentlessonsfromblackjackandgamblingChapter17309Kelly,’thatisafractionthatishalfoftheKellyfraction.Thishalvesyourvolatility,keepsyounicelyawayfromthecrazyzone,yetonlydecreasesyourexpectedgrowthrateby25%.IfyoucanplayMtimesinaneveningyouwouldexpectatotalgrowthofµ2M,(17.2)2σ2usingfullKelly.Thisillustratesonepossiblewayofchoosingaportfolio,whichassettoinvestin(blackjack)andhowmuchtoinvest(f∗).Facedwithotherpossibleinvestments,thenyoucouldargueinfavorofchoosingtheonewiththehighest(17.2),dependingonthemeanofthereturn,itsstandarddeviationandhowoftentheinvestmentopportunitycomesyourway.Theseideasareparticularlyimportanttothetechnicalana-lystorchartistwhotradesonthebasisofsignalssuchasgoldencrosses,saucerbottoms,andheadandshoulderpat-terns.Notonlydotheriskandreturnofthesesignalsmatter,butsodoestheirfrequencyofoccurrence.TheKellycriterionisaboutmaximizingexpectedlong-termgrowth.Manypeoplebelievethistobeaquiteaggressivestrategyleadingtopossiblelargedownturns.Therearemanyotherquantitiestooptimize,ofcourse,sotheKellycriterionmightnotbetherightchoiceforyou.Youcould,forexample,choosetominimizedownturn,ormaximizesomerisk-adjustedreturn,asdiscussedinthenextchapter.17.6CANYOUWINATROULETTE?Let’sgobacktoEdThorp,butafewyearsbeforehisworkonblackjack.Inspring1955EdThorpwasinhissecondyearofgraduatephysicsatUCLA.AtteatimeoneSundayhegottochattingwithcolleaguesabouthowtomake‘easymoney.’Theconversationturnedtogambling,androuletteinparticular.Wasitpossibletopredict,atleastwithsomeexploitabledegreeofaccuracy,theoutcomeofaspinofthewheel?Someofhiscolleagues,theonesintheknow,werecertainthattheroulettewheelsweremanufacturedsopreciselythattherewerenoimperfectionsthatcouldbediscerned,nevermindexploited.ButEd’scountertothatwassimple,ifthewheelsaresoperfectyoushouldbeabletopredict,usingsimpleNewtonianprinciples,thepathoftheballanditsfinalrestingplace.Edgottoworkonthisprobleminthelate1950s,playingaroundwithacheapminiatureroulettewheel,filmingandtimingtherevolutions.HemetupwithClaudeShannon,thefatherofinformationtheoryin1959,originallytodiscusshisblackjackresults,buttheconversationsoonturnedtoothergamesandrouletteinparticular.Shannonwasfascinated.ShortlyafterwardstheymetupatShannon’shouse,thebasementofwhichwaspackedwithmechanicalandengineeringgadgets,theperfectplaygroundforfurtherrouletteexperiments.EdandShannontogethertooktherouletteanalysistogreaterheights,investing$1,500inafull-sizeprofessionalwheel.Theycalibratedasimplemathematicalmodeltotheexperiments,totrytopredictthemomentwhenthespinningballwouldfallintothewaitingpockets.Fromtheirmodeltheywereabletopredictanysinglenumberwithastandarddeviationof10pockets.Thisconvertstoa44percentedgeonabetonasinglenumber.Bettingonaspecificoctantgavethema43percentadvantage. 310PartOnemathematicalandfinancialfoundationsFromNovember1960untilJune1961EdandShannondesignedandbuilttheworld’sfirstwearablecomputer.Thetwelvetransistors,cigarette-packsizedcomputerwasfeddatabyswitchesoperatedbytheirbigtoes.Oneswitchinitializedthecomputerandtheotherwasfortimingtherotationoftheballandrotor.Thecomputerpredictionswereheardbythecomputerwearerasoneofeighttonesviaanearpiece.(EdandShannondecidedthatthebestbetwasonoctantsratherthansinglenumberssincethefatherofinformationtheoryknewthatfacedwithnoptionsindividualstakeatimea+blog(n)tomakeadecision.)ThiscomputerwastestedoutinLasVegasinthesummerof1961.Butforproblemswithbrokenwiresandearpiecesfallingout,thetripwasasuccess.SimilarsystemswerelaterbuiltfortheWheelofFortunewhichhadanevengreateredge,anoutstanding200percent.On30thMay1985Nevadaoutlawedtheuseofanydeviceforpredictingoutcomesoranalyzingprobabilitiesorstrategies.17.7HORSERACEBETTINGANDNOARBITRAGESeveraltimeswehaveseenhowtheabsenceofarbitrageopportunitiesleadstotheideaofrisk-neutralpricing.Thevalueofanoptioncanbeinterpretedasthepresentvalueoftheexpectedpayoff,withtheexpectationbeingwithrespecttotherisk-neutralassetpricepath.Inthiscontextriskneutraljustmeansthattheassetpriceincreaseswithagrowthratethatisthesameastherisk-freeinterestrate.Inotherwords,whatwereallybelievethattheassetpriceisgoingtodointhefuture(intermsofitsgrowthrate)isirrelevant.Wedon’tevenneedtoknowthegrowthrateofanassettopriceitsoptions,onlyitsvolatility.Somethingrelatedhappensintheworldofsportsbetting.17.7.1SettingtheOddsinaSportingGameInahorserace,footballorbaseballgametheoddsaresetnottoreflecttherealprobabilitiesofahorseorateamwinningbuttoreflectthebettingthathasoccurred.Dependingonhowthebettinggoes,theoddswillbesetsothatthehouse/bookiecannotlose.Forexample,inasoccermatchbetweenEnglandandGermanytheGermansaremorelikelytowin,butthepatrioticEnglishwillbetmoreheavilyonEngland(presumably).TheoddsgivenbythebookieswillreflectthisbettingandmakeitlooklikeEnglandismorelikelytowin.Ofcourse,inGermanythesituationisreversed.ThebestbetwouldbeonGermany,butplacedinEngland,andoneonEnglandplacedinGermany.Inpractice,however,bookiesinonecountrywouldlayofftheirbetsonbookiesinothercountriessoallbookieshaveroughlythesameodds.Otherwisetherewouldbestraightforwardarbitrageopportunities.Thereforeit’sunlikelyfortheretobeasure-firebet(unlessthebookiehasmadeamistake,theraceisfixed,oryoucanfindtwoormorebookiesthataren’tdirectlyorindirectlylayingofftheirbetsoneachother).Butyoucanwin,onaverage.Byexploitingthedifferencebetweentherealprobabilityofahorsewinningandtheoddsyoucanget.(Therearedifferencesbetweenrealoddsandwhatyougetpaidinallcasinogames,butit’sonlyinblackjackthatthiscanbeexploited.)17.7.2TheMathematicsSupposethatthereareNhorsesinarace,withanamountWibetontheithhorse.Theoddssetbythebookieareqi:1.Thismeansthatifyoubet1onhorseiyouwilllosethe1ifthe investmentlessonsfromblackjackandgamblingChapter17311horseloses,butwilltakehomeqi+1ifthehorsewins,youroriginal1plusafurtherqi.Howdoesthebookiesettheoddstoensureheneverloses?ThetotaltakingsbeforetheraceisNWi.i=1Ifhorsejwinsthebookiehastopayout(qj+1)Wj.AllthatthebookiehastodoistoensurethatNWi≥(qj+1)Wj,i=1orequivalentlyNi=1Wiqj≤−1forallj.WjNothingtoocomplicated.Butseehowtheoddshavebeenchosentoreflectthebetting.Nowherewasthereanymentionofthelikelihoodofhorsejwinning.17.8ARBITRAGESupposethebookiemadeanerrorwhensettingtheodds.Howcouldyoudeterminewhethertherewasanarbitrageopportunity?(Don’tforgetthatonlypositivebetsareallowed,there’snogoingshorthere.)Let’sintroducesomemorenotation.Thewiarethebetsthatyouplace.(Wecanforgetaboutthewagersmadebyeveryoneelse,theWs.)Let’sassumethatyourtotalwageris1,sothatNwi=1.(17.3)i=1Theamountyouwinis(qj+1)wj(17.4)ifhorsejisthewinner.Canyoufindawiforallisuchthattheyaddupto1,areallpositiveandthatexpression(17.4)ispositiveforallj?Ifyoucanthereisanarbitrageopportunity.Therequirementthat(17.4)ispositivecanbewrittenas1wj≥.(17.5)qj+1Canwefindpositivewssuchthat(17.3)and(17.5)hold?Thisisveryeasytovisualize,atleastwhentherearetwoorthreehorses.Let’slookatthetwo-horserace. 312PartOnemathematicalandfinancialfoundations1.210.8Normalsituation0.6w_20.4Arbitragepossible0.2000.20.40.60.811.2w_1Figure17.6Arbitrageinatwo-horserace.InFigure17.6theaxesrepresenttheamountofthewageroneachofthetwohorses.Thelineshowstheconstraint(17.3).Thewagersmustlieonthisline.Thetwodotsmarkthepoint11,(17.6)q+1q+112ineachoftwosituations.Onedotisthetypicalsituationwherethereisnoarbitrageopportunityandtheotherdotdoeshaveanassociatedarbitrageopportunity.Let’sseethedetails.Tofindanarbitrageopportunitywemustfindapair(w1,w2)lyingonthelinesuchthateachcoordinateisgreaterthanacertainquantity,dependingontheqs.Plotthepoint(17.6)anddrawalineverticallyup,andanotherlinehorizontallytotheright,asshowninthefigure,emanatingfromthedot.Doesthequadrantdefinedbythesetwolinesincludeanyoftheline?Ifnot,aswouldbethecasewiththehigherdot,thenthereisnoarbitragepossible.Ifsomeofthelineisincludedthenarbitrageispossible.17.8.1HowBesttoProfitfromtheOpportunity?There’sasimpletesttoseewhetherweareinanarbitragesituation.Ingeneral,ifN1≥1qi+1i=1thenthereisnoarbitrage.Ifthesumislessthan1,thereisanarbitrage. investmentlessonsfromblackjackandgamblingChapter17313Youcanbenefitfromthearbitragebyplacingwagerswisuchthattheylieonthepartofthelineencompassedbythequadrant.Whichpartoftheline,though,isuptoyou.BythatImeanthatyoumustmakesomestatementaboutwhatyouaretryingtoachieveoroptimizeinthearbitrage.Onepossibilityistolookattheworst-casescenarioandmaximizethepaybackinthatcase.Alternatively,specifyrealprobabilitiesforeachofthehorseswinning.17.9HOWTOBETWesawhowoddsareestablishedbybookies.Weevensawhowtospotarbitrageopportunities.Inpractice,ofcourse,youcouldspendalifetimelookingforarbitrageopportunitiesthatrarelyoccurinreallife.Nowwearegoingtoseeifwecanexploitthedifferencebetweentheoddsassetbythebookieandtheoddsthatyouestimate.Remember,theoddssetbythebookiearereallydeterminedbythewagersplaced,whicharemoretodowithirrationalsentiment(‘I’mgoingtobetonthishorse’cosit’sgotthesamenameasthepetratIhadwhenIwasachild’)thanwithacold-heartedestimationoftheprobabilities.Weneedsomemorenotation.Let’susepiastheprobabilityoftheithhorsewinningtherace.Thisissupposedtobetherealprobability,notthebookie’sprobability.Obviously,theoddsmustsumto1:Npi=1.i=1IfwewagerwiontheithhorsethenweexpecttomakeNm=piwi(qi+1)−1(17.7)i=1Thisisundertheassumptionthatthetotalwager,thesumofallthews,is1.Anobviousgoalistomakethisquantitypositive;wewanttogetapositivereturnonaverage.Buttheremaybemanywaystomakethispositive.Howdowedecidewhichwayisbest?Anotherquantitywemightwanttolookatisthestandarddeviationofwinnings.ThisisgivenbyNp(w(q+1)−1−m)2(17.8)iiii=1Thismeasuresthedispersionofwinningsabouttheaverage,andisofteninterpretedasameasureofrisk.Ifthiswerezeroourprofitorlosswouldbeasurething.SeeTable17.1foranexample.Howshouldyoubet?Thefollowingcalculationsareeasilydoneonaspreadsheet.Scenario1:MaximizeexpectedreturnSinceyouplacenopremiumonreducingriskyoushouldbeteverythingonthehorsethatmaximizespi(qi+1). 314PartOnemathematicalandfinancialfoundationsTable17.1Oddsandprobabilitiesinahorserace.HorseBookie’soddsYourestimateofprobabilityWagerNijinsky50.2RedRum60.2Oxo10.1RedMarauder10.1GayLad20.1Roquefort20.1RedAlligator20.1Shergar20.1Table17.2Maximizingexpectation.HorseBookie’soddsYourestimateofprobabilityWagerNijinsky50.20RedRum60.21Oxo10.10RedMarauder10.10GayLad20.10Roquefort20.10RedAlligator20.10Shergar20.10Table17.3Minimizingstandarddeviation.HorseBookie’soddsYourestimateofprobabilityWagerNijinsky50.20.063062RedRum60.20.054068Oxo10.10.189203RedMarauder10.10.189246GayLad20.10.126108Roquefort20.10.126108RedAlligator20.10.126108Shergar20.10.126108Inthiscase,thatisRedRum.Theexpectedreturnis40%withastandarddeviationof280%.Averyriskybet(seeTable17.2).Scenario2:MinimizestandarddeviationAninterestingstrategy.Isay‘interesting’becausethisstrategyresultsinzerostandarddeviation,andareturnof−62%.Inotherwords,aguaranteedloss!(SeeTable17.3). investmentlessonsfromblackjackandgamblingChapter17315Table17.4Maximizereturndividedbystandarddeviation.HorseBookie’soddsYourestimateofprobabilityWagerNijinsky50.20.459016RedRum60.20.540984Oxo10.10RedMarauder10.10GayLad20.10Roquefort20.10RedAlligator20.10Shergar20.10Scenario3:MaximizereturndividedbystandarddeviationAstrategythatseekstobenefitfromapositiveexpectationbutwithasmallerrisk.Formathe-maticalreasons(theCentralLimitTheorem)thisisanaturalstrategy.ThesolutionisgiveninTable17.4.Theexpectedreturnisnow31%withastandarddeviationof164%.17.10SUMMARYThemathematicsofgamblingisalmostidenticaltothemathematicsof‘investing.’Themaindifferencebetweengamblingandinvestingisthattheparametersareusuallyeasiertomeasurewithgamblinggames.Ifyoucan’tcopewiththemathematics(andtheemotionalrollercoasterride)ofgamblingthenyoushouldn’tbeworkinginabank;-)FURTHERREADING•SeeKelly’soriginal1956paper.•TheclassicreferencetextsonblackjackarebyThorp(1962)andWong(1981).•ThegrippingstoryofJohnKelly,ClaudeShannon,EdThorpandacastofmanyintriguingcharacterscanbefoundinPoundstone(2005). CHAPTER18portfoliomanagementInthisChapter...•ModernPortfolioTheoryandtheCapitalAssetPricingModel•optimizingyourportfolio•alternativemethodologiessuchascointegration•howtoanalyzeportfolioperformance18.1INTRODUCTIONThetheoryofderivativepricingisatheoryofdeterministicreturns:Wehedgeourderivativewiththeunderlyingtoeliminaterisk,andourresultingrisk-freeportfoliothenearnstherisk-freerateofinterest.Banksmakemoneyfromthishedgingprocess;theysellsomethingforabitmorethanit’sworthandhedgeawaytherisktomakeaguaranteedprofit.Butnoteveryoneishedging.Fundmanagersbuyandsellassets(includingderivatives)withtheaimofbeatingthebank’srateofreturn.Insodoingtheytakerisk.InthischapterIexplainsomeofthetheoriesbehindtheriskandrewardofinvestment.AlongthewayIshowthebenefitsofdiversification,howthereturnandriskonaportfolioofassetsisrelatedtothereturnandriskontheindividualassets,andhowtooptimizeaportfoliotogetthebestvalueformoney.Forthemostpart,theassumptionsareasfollows.•Weholdaportfoliofor‘asingleperiod,’examiningthebehaviorafterthistime.•DuringthisperiodreturnsonassetsareNormallydistributed.•Thereturnonassetscanbemeasuredbyanexpectedreturn(thedrift)foreachasset,astandarddeviationofreturn(thevolatility)foreachassetandcorrelationsbetweentheassetreturns. 318PartOnemathematicalandfinancialfoundations18.2DIVERSIFICATIONInthissectionIintroducesomemorenotation,andshowtheeffectsofdiversificationonthereturnoftheportfolio.WeholdaportfolioofNassets.ThevaluetodayoftheithassetisSianditsrandomreturnisRioverourtimehorizonT.TheRsareNormallydistributedwithmean√µiTandstandarddeviationσiT.Thecorrelationbetweenthereturnsontheithandjthassetsisρij(withρii=1).Theparametersµ,σandρcorrespondtothedrift,volatilityandcorrelationthatweareusedto.Notethescalingwiththetimehorizon.Ifweholdwioftheithasset,thenourportfoliohasvalue
N=wiSi.i=1Attheendofourtimehorizonthevalueis
N+δ=wiSi(1+Ri).i=1Wecanwritetherelativechangeinportfoliovalueas
Nδ=WiRi,(18.1)i=1wherewiSiWi=.Ni=1wiSiTheweightsWisumtoone.From(18.1)itissimpletocalculatetheexpectedreturnontheportfolioN1δ
µ=E=Wiµi(18.2)Ti=1andthestandarddeviationofthereturn
N
N1δσ=√var=WiWjρijσiσj.(18.3)Ti=1j=1Inthese,wehaverelatedtheparametersfortheindividualassetstotheexpectedreturnandthestandarddeviationoftheentireportfolio. portfoliomanagementChapter1831918.2.1UncorrelatedAssetsSupposethatwehaveassetsinourportfoliothatareuncorrelated,ρij=0,i=j.TomakethingssimpleassumethattheyareequallyweightedsothatWi=1/N.Theexpectedreturnontheportfolioisrepresentedby
N1µ=µi,Ni=1theaverageoftheexpectedreturnsonalltheassets,andthevolatilitybecomes1
Nσ=σ2.N2ii=1ThisvolatilityisO(N−1/2)sincethereareNtermsinthesum.Asweincreasethenumberofassetsintheportfolio,thestandarddeviationofthereturnstendstozero.ItisratherextremetoassumethatallassetsareuncorrelatedbutwewillseesomethingsimilarwhenIdescribetheCapitalAssetPricingModelbelow;diversificationreducesvolatilitywithouthurtingexpectedreturn.Iamnowgoingtorefertovolatilityorstandarddeviationasrisk,abadthingtobeavoided(withinreason),andtheexpectedreturnasreward,agoodthingthatwewantasmuchofaspossible.18.3MODERNPORTFOLIOTHEORYWecanusetheaboveframeworktodiscussthe‘best’portfolio.Thedefinitionof‘best’wasaddressedverysuccessfullybyNobelLaureateHarryMarkowitz.Hismodelprovidesawayofdefiningportfoliosthatareefficient.Anefficientportfolioisonethathasthehighestrewardforagivenlevelofrisk,orthelowestriskforagivenreward.Toseehowthisworksimaginethattherearefourassetsintheworld,A,B,CandDwithrewardandriskasshowninFigure18.1(ignoreEforthemoment).Ifyoucouldbuyanyoneofthese(butasyetyouarenotallowedmorethanone),whichwouldyoubuy?WouldyouchooseD?No,becauseithasthesameriskasBbutlessreward.IthasthesamerewardasCbutforahigherrisk.WecanruleoutD.WhataboutBorC?TheyarebothappealingwhensetagainstD,butagainsteachotheritisnotsoclear.Bhasahigherrisk,butgetsahigherreward.However,comparingthembothwithAweseethatthereisnocontest.Aisthepreferredchoice.IfweintroduceassetEwiththesameriskasBandahigherrewardthanA,thenwecannotobjectivelysaywhichoutofAandEisthebetter;thisisasubjectivechoiceanddependsonaninvestor’sriskpreferences.NowsupposethatIhavethetwoassetsAandEofFigure18.2,andIamallowedtocombinetheminmyportfolio,whateffectdoesthishaveonmyrisk/reward? 320PartOnemathematicalandfinancialfoundations25%20%E15%ABReturn10%CD5%0%0%5%10%15%20%25%30%35%40%RiskFigure18.1Riskandrewardforfiveassets.30%25%20%E15%Return10%A5%0%0%10%20%30%40%50%60%RiskFigure18.2Twoassetsandanycombination. portfoliomanagementChapter18321From(18.2)and(18.3)wehaveµ=WµA+(1−W)µEandσ2=W2σ2+2W(1−W)ρσσ+(1−W)2σ2.AAEEHereWistheweightofassetAand,rememberingthattheweightsmustadduptoone,theweightofassetEis1−W.AswevaryW,sotheriskandtherewardchange.Thelineinrisk/rewardspacethatisparameterizedbyWisahyperbola,asshowninFigure18.2.Thepartofthiscurveinboldisefficient,andispreferabletotherestofthecurve.Again,anindividual’sriskpreferenceswillsaywherehewantstobeontheboldcurve.Whenoneofthevolatilitiesiszerothelinebecomesstraight.Anywhereonthecurvebetweenthetwodotsrequiresalongpositionineachasset.Outsidethisregion,oneoftheassetsissoldshorttofinancethepurchaseoftheother.Everythingthatfollowsassumesthatwecansellshortasmuchofanassetaswewant.Theresultschangeslightlywhentherearerestrictions.Ifwehavemanyassetsinourportfoliowenolongerhaveasimplehyperbolaforourpossiblerisk/rewardprofiles;insteadwegetsomethinglikethatshowninFigure18.3.ThisfigurenowusesallofA,B,C,DandE,notjusttheAandE.EventhoughB,CandDarenotindividuallyappealingtheymaywellbeusefulinaportfolio,dependinghowtheycorrelate,ornot,withotherinvestments.Inthisfigurewecanseetheefficientfrontiermarkedinbold.Givenanychoiceofportfoliowewouldchoosetoholdonethatliesonthisefficientfrontier.50%45%40%35%30%marketportfolio25%Return20%15%10%F5%0%0%10%20%30%40%50%60%70%80%RiskFigure18.3Portfoliopossibilitiesandtheefficientfrontier. 322PartOnemathematicalandfinancialfoundationsABCDEFGHIJ1Portfolioreturn30.0%ABCD2Portfoliorisk45.6%Return10.0%15.0%18.0%20.0%3Volatility20.0%30.0%25.0%35.0%4Targetreturn30.0%Weights−1.463710.3936381.3344330.735637156Correlations720.0%−1.46371A10.20.30.1=SUM(D4:G4)830.0%0.393638B0.210.050.1925.0%1.334433C0.30.0510.051035.0%0.735637D0.10.10.0511112=SUMPRODUCT(D2:G2,D4:G4)130.08570−0.00691−0.0293−0.0075414−0.006910.0139460.001970.003041=SQRT(SUM(D13:G16))=E715−0.02930.001970.1112940.00429516-0.007540.0030410.0042950.06629217{=TRANSPOSE(D4:G4)}1819{=TRANSPOSE(D3:G3)}=$A10*$B10*E10*E$4*E$320212223242526272829303132333435363738394041424344Figure18.4Spreadsheetforcalculatingonepointontheefficientfrontier.ThecalculationoftheriskforagivenreturnisdemonstratedinthespreadsheetinFigure18.4.Thisspreadsheetcanbeusedtofindtheefficientfrontierifitisusedmanytimesfordifferenttargetreturns.18.3.1IncludingaRisk-freeInvestmentArisk-freeinvestmentearningaguaranteedrateofreturnrwouldbethepointFinFigure18.3.Ifweareallowedtoholdthisassetinourportfolio,thensincethevolatilityofthisassetiszero,wegetthenewefficientfrontierwhichisthestraightlineinthefigure.Theportfolioforwhichthestraightlinetouchestheoriginalefficientfrontieriscalledthemarketportfolio.Thestraightlineitselfiscalledthecapitalmarketline.11Intherisk-neutralworldtheythinkthatallinvestmentslieonthehorizontallinegoingthroughthepoint(0,r). portfoliomanagementChapter1832318.4WHEREDOIWANTTOBEONTHEEFFICIENTFRONTIER?Havingfoundtheefficientfrontierwewanttoknowwhereaboutsonitweshouldbe.Thisisapersonalchoice,theefficientfrontierisobjective,giventhedata,butthe‘best’positiononitissubjective.Thefollowingisawayofinterpretingtherisk/rewarddiagramthatmaybeusefulinchoosingthebestportfolio.ThereturnonportfolioisNormallydistributedbecauseitiscomprisedofassetswhicharethemselvesNormallydistributed.Ithasmeanµandstandarddeviationσ(IhaveignoredthedependenceonthehorizonT).Theslopeofthelinejoiningtheportfoliototherisk-freeassetisµ−rs=.σThisisanimportantquantity;itisameasureofthelikelihoodofhavingareturnthatexceedsr.IfC(·)isthecumulativedistributionfunctionforthestandardizedNormaldistributionthenC(s)istheprobabilitythatthereturnonisatleastr.Moregenerallyµ−r∗Cσistheprobabilitythatthereturnexceedsr∗.Thissuggeststhatifwewanttominimizethechanceofareturnoflessthanr∗weshouldchoosetheportfoliofromtheefficientfrontierseteffwiththelargestvalueoftheslopeµ−r∗eff.σeffConversely,ifwekeeptheslopeofthislinefixedatsthenwecansaythatwithaconfidenceofC(s)wewilllosenomorethanµeff−sσeff.Ourportfoliochoicecouldbedeterminedbymaximizingthisquantity.ThesetwostrategiesareshownschematicallyinFigure18.5.Neitherofthesemethodsgivesatisfactoryresultswhenthereisarisk-freeinvestmentamongtheassetsandthereareunrestrictedshortsales,sincetheyresultininfiniteborrowing.Anotherwayofchoosingtheoptimalportfolioiswiththeaidofautilityfunction.Thisapproachispopularwitheconomists.InFigure18.6Ishowindifferencecurvesandtheefficientfrontier.Thecurvesarecalledbythisnamebecausetheyaremeanttorepresentlinesalongwhichtheinvestorisindifferenttotherisk/rewardtrade-off.Aninvestorwantshighreturn,andlowrisk.FacedwithportfoliosAandBinthefigure,heseesAwithlowreturnandlowrisk,butBhasabetterrewardatthecostofgreaterrisk.Theinvestorisindifferentbetweenthesetwo.However,Cisbetterthanbothofthem,beingonapreferredcurve. 324PartOnemathematicalandfinancialfoundations30%25%20%15%Returnslopes10%5%r*0%0%10%20%30%40%50%60%RiskFigure18.5Twosimplewaysforchoosingthebestefficientportfolio.35%30%25%B20%CReturn15%A10%5%0%0%10%20%30%40%50%60%RiskFigure18.6Theefficientfrontierandindifferencecurves. portfoliomanagementChapter1832518.5MARKOWITZINPRACTICETheinputstotheMarkowitzmodelareexpectedreturns,volatil-itiesandcorrelations.WithNassetsthismeansN+N+N(N−1)/2parameters.Mostofthesecannotbeknownaccu-rately(dotheyevenexist?);onlythevolatilitiesareatallreliable.Havinginputtheseparameters,wemustoptimizeoverallweightsofassetsintheportfolio:Chooseaportfolioriskandfindtheweightsthatmakethereturnontheportfolioamaximumsubjecttothisvolatility.Thisisaverytime-consumingprocesscomputationallyunlessoneonlyhasasmallnumberofassets.Theproblemwiththepracticalimplementationofthismodelwasoneofthereasonsfordevelopmentofthesimplermodelofthenextsection.18.6CAPITALASSETPRICINGMODELBeforediscussingtheCapitalAssetPricingModelorCAPMwemustintroducetheideaofasecurity’sbeta.Thebeta,βi,ofanassetrelativetoaportfolioMistheratioofthecovariancebetweenthereturnonthesecurityandthereturnontheportfoliotothevarianceoftheportfolio.ThusCov[RiRM]βi=.Var[RM]18.6.1TheSingle-indexModelIwillnowbuildupasingle-indexmodelanddescribeextensionslater.Iwillrelatethereturnonallassetstothereturnonarepresentativeindex,M.Thisindexisusuallytakentobeawide-rangingstockmarketindexinthesingle-indexmodel.WewritethereturnontheithassetasRi=αi+βiRM+i.Usingthisrepresentationwecanseethatthereturnonanassetcanbedecomposedintothreeparts:Aconstantdrift,arandompartcommonwiththeindexMandarandompartuncorrelatedwiththeindex,i.Therandompartiisuniquetotheithasset,andhasmeanzero.NoticehowalltheassetsarerelatedtotheindexMbutareotherwisecompletelyuncorrelated.InFigure18.7isshownaplotofreturnsonWaltDisneystockagainstreturnsontheS&P500;αandβcanbedeterminedfromalinearregressionanalysis.ThedatausedinthisplotranfromJanuary1985untilalmosttheendof1997.TheexpectedreturnontheindexwillbedenotedbyµManditsstandarddeviationbyσM.Theexpectedreturnontheithassetisthenµi=αi+βiµMandthestandarddeviationσ=β2σ2+e2iiMiwhereeiisthestandarddeviationofi. 326PartOnemathematicalandfinancialfoundations30.0%20.0%y=1.1773x+0.00042=0.4038R10.0%0.0%0.0%−25.0%−20.0%−15.0%−10.0%−5.0%5.0%10.0%15.0%−10.0%−20.0%−30.0%−40.0%Figure18.7ReturnsonWaltDisneystockagainstreturnsontheS&P500.Ifwehaveaportfolioofsuchassetsthenthereturnisgivenbyδ
N
N
N
N=WiRi=Wiαi+RMWiβi+Wii.i=1i=1i=1i=1Fromthisitfollowsthat
N
Nµ=Wiαi+E[RM]Wiβi.i=1i=1Letuswrite
N
Nα=Wiαiandβ=Wiβi,i=1i=1 portfoliomanagementChapter18327sothatµ=α+βE[RM]=α+βµM.Similarlytheriskinismeasuredby
N
N
Nσ=WWββσ2+W2e2.ijijMiii=1j=1i=1Iftheweightsareallaboutthesame,N−1,thenthefinaltermsinsidethesquarerootarealsoO(N−1).Thusthisexpressionis,toleadingorderasN→∞,
Nσ=WiβiσM=|β|σM.i=1Observethatthecontributionfromtheuncorrelatedstotheportfoliovanishesasweincreasethenumberofassetsintheportfolio:Theriskassociatedwiththesiscalleddiversifiablerisk.Theremainingrisk,whichiscorrelatedwiththeindex,iscalledsystematicrisk.18.6.2ChoosingtheOptimalPortfolioTheprincipalisthesameastheMarkowitzmodelforoptimalportfoliochoice.Theonlydifferenceisthattherearealotfewerparameterstobeinput,andthecomputationisalotfaster.Theprocedureisasfollows.Chooseavaluefortheportfolioreturnµ.Subjecttothisconstraint,minimizeσ.Repeatthisminimizationfordifferentportfolioreturnstoobtaintheefficientfrontier.Thepositiononthiscurveisthenasubjectivechoice.18.7THEMULTI-INDEXMODELThemodelpresentedaboveisasingle-indexmodel.Theideacanbeextendedtoincludefurtherrepresentativeindices.Forexample,aswellasanindexrepresentingthestockmarketonemightincludeanindexrepresentingbondmarkets,anindexrepresentingcurrencymarketsorevenaneconomicindexifitisbelievedtoberelevantinlinkingassets.Inthemulti-indexmodelwewriteeachasset’sreturnas
nRi=αi+βjiRj+i,j=1wheretherearenindiceswithreturnRj.Theindicescanbecorrelatedtoeachother.Similarresultstothesingle-indexmodelfollow.Itisusuallynotworthhavingmorethanthreeorfourindices.Thefewertheparameters,themorerobustwillbethemodel.AttheotherextremeistheMarkowitzmodelwithoneindexperasset. 328PartOnemathematicalandfinancialfoundations18.8COINTEGRATIONWhetheryouuseMPTorCAPMyouwillalwaysworryabouttheaccuracyofyourparameters.Bothofthesemethodsareonlyasaccurateastheinputdata,CAPMbeingmorereliablethanMPTgenerallyspeaking,becauseithasfewerparameters.Thereisanothermethodthatisgainingpopularity,andwhichIwilldescribeherebriefly.Itisunfortunatelyacomplextechniquerequiringsophisticatedstatisticalanalysis(todoitproperly)butwhichatitscoremakesalotofsense.Insteadofaskingwhethertwoseriesarecorrelatedweaskwhethertheyarecointegrated.Twostocksmaybeperfectlycorrelatedovershorttimescalesyetdivergeinthelongrun,withonegrowingandtheotherdecaying.Conversely,twostocksmayfolloweachother,neverbeingmorethanacertaindistanceapart,butwithanycorrelation,positive,negativeorvarying.Ifwearedeltahedgingthenmaybetheshorttimescalecorrelationmatters,butnotifweareholdingstocksforalongtimeinanunhedgedportfolio.Toseewhethertwostocksstayclosetogetherweneedadefinitionofstationarity.Atimeseriesisstationaryifithasfiniteandconstantmean,standarddeviationandautocorrelationfunction.Stocks,whichtendtogrow,arenotstationary.Inasense,stationaryseriesdonotwandertoofarfromtheirmean.Wecanseethedifferencebetweenstationaryandnon-stationarywithourfirstcoin-tossingexperiment.Thetimeseriesgivenby1everytimewethrowaheadand−1everytimewethrowatailisstationary.Ithasameanofzero,astandarddeviationof1andanautocorrelationfunctionthatiszeroforanynon-zerolag.Butwhatifweadduptheresults,aswemightdoifwearebettingoneachtoss?Thistimeseriesisnon-stationary.Thisisbecausethestandarddeviationofthesumgrowslikethesquarerootofthenumberofthrows.Themeanmaybezerobutthesumiswanderingfurtherandfurtherawayfromthatmean.TestingforthestationarityofatimeseriesXtinvolvesalinearregressiontofindthecoeffi-cientsa,bandcinXt=aXt−1+b+ct.Ifitisfoundthat|a|>1thentheseriesisunstable.If−1≤a<1thentheseriesisstationary.Ifa=1thentheseriesisnon-stationary.Aswithallthingsstatistical,wecanonlysaythatourvalueforaisaccuratewithacertaindegreeofconfidence.Todecidewhetherwehavegotastationaryornon-stationaryseriesrequiresustolookattheDickey–Fullerstatistictoestimatethedegreeofconfidenceinourresult.Fromthispointonthesubjectofcointegrationgetscomplicated.Howisthisusefulinfinance?Eventhoughindividualstockpricesmightbenon-stationaryitispossibleforalinearcombination(i.e.aportfolio)tobestationary.Canwefindλi,withNi=1λi=1,suchthat
NλiSii=1isstationary?Ifwecan,thenwesaythatthestocksarecointegrated.Forexample,supposewefindthattheS&P500iscointegratedwithaportfolioof15stocks.Wecanthenusethese15stockstotracktheindex.Theerrorinthistrackingportfoliowillhaveconstantmeanandstandarddeviation,andsoshouldnotwandertoofarfromitsaverage.Thisisclearlyeasierthanusingall500stocksforthetracking(when,ofcourse,thetrackingerrorwouldbezero). portfoliomanagementChapter18329Wedon’thavetotracktheindex,wecouldtrackanythingwewant,suchase0.2ttochooseaportfoliothatgetsa20%return.Wecouldanalyzethecointegrationpropertiesoftworelatedstocks,NikeandReebok,forexample,tolookforrelationships.Thiswouldbepairstrading.ClearlytherearesimilaritieswithMPTandCAPMinconceptssuchasmeansandstandarddeviations.Theimportantdifferenceisthatcointegrationassumesfarfewerpropertiesfortheindividualtimeseries.Mostimportantly,volatilityandcorrelationdonotappearexplicitly.18.9PERFORMANCEMEASUREMENTIfonehasfollowedoneoftheassetallocationstrategiesoutlinedabove,orjusttradedongutinstinct,canonetellhowwellonehasdone?Weretheoutstandingresultsbecauseofanuncannynaturalinstinct,orweretheawfulresultssimplybadluck?Theidealperformancewouldbeoneforwhichreturnsoutperformedtherisk-freerate,butinaconsistentfashion.Notonlyisitimportanttogetahighreturnfromportfoliomanagement,butonemustachievethiswithaslittlerandomnessaspossible.Thetwocommonestmeasuresof‘returnperunitrisk’aretheSharperatioof‘rewardtovariability’andtheTreynorratioof‘rewardtovolatility’.Thesearedefinedasfollows:µ−rSharperatio=σandµ−rTreynorratio=.β70605040Profit302010t0Figure18.8Agoodandabadmanager;samereturns,differentvariability. 330PartOnemathematicalandfinancialfoundationsIntheseµandσaretherealizedreturnandstandarddeviationfortheportfolioovertheperiod.Theβisameasureoftheportfolio’svolatility.TheSharperatioisusuallyusedwhentheportfolioisthewholeofone’sinvestmentandtheTreynorratiowhenoneisexaminingtheperformanceofonecomponentofthewholefirm’sportfolio,say.Whentheportfoliounderexaminationishighlydiversifiedthetwomeasuresarethesame(uptoafactorofthemarketstandarddeviation).InFigure18.8weseetheportfoliovalueagainsttimeforagoodmanagerandabadmanager.18.10SUMMARYPortfoliomanagementandassetallocationareabouttakingrisksinreturnforareward.Thequestionsarehowtodecidehowmuchrisktotake,andhowtogetthebestreturn.Butderivativestheoryisbasedonnottakinganyriskatall,andsoIhavespentlittletimeonportfoliomanagementinthebook.Ontheotherhand,asIhavestressed,thereissomuchuncertaintyinthesubjectoffinancethateliminationofriskiswell-nighimpossibleandtheideasbehindportfoliomanagementshouldbeappreciatedbyanyoneinvolvedinderivativestheoryorpractice.Ihavetriedtogivetheflavorofthesubjectwithonlytheeasiest-to-explainmathematics;thefollowingsourceswillproveusefultoanyonewantingtopursuethesubjectfurther.FURTHERREADING•SeeMarkowitz’soriginalbookforallthedetailsofMPT,Markowitz(1959).•Oneofthebesttextsoninvestments,includingchaptersonportfoliomanagement,isSharpe(1985).•ForadescriptionofcointegrationandothertechniquesineconometricsseeHamilton(1994)andHendry(1995).•SeeFarrell(1997)forfurtherdiscussionofportfolioperformance.•Ihavenotdiscussedthesubjectofcontinuous-timeassetallocation(yet),buttheelegantsubjectisexplainednicelyinthecollectionofRobertMerton’spapers,Merton(1992).•Transactioncostscanhaveabigeffectonportfoliosthataresupposedtobecontinuouslyrebalanced.SeeMorton&Pliska(1995)foramodelwithcosts,andAtkinson&Wilmott(1995),Atkinson,Pliska&Wilmott(1997)andAtkinson&Al–Ali(1997)forasymptoticresults.•Foradescriptionofchaos-basedmethodsinfinance,andhowtheywontheFirstInter-nationalNon-linearFinancialForecastingCompetition,seeAlexander&Giblin(1997).•ForareviewofcurrentthinkinginriskmanagementseeAlexander(1998). CHAPTER19ValueatRiskInthisChapter...•themeaningofVaR•howVaRiscalculatedinpractice•someofthedifficultiesassociatedwithVaRforportfolioscontainingderivatives19.1INTRODUCTIONItisthemarkofaprudentinvestor,betheyamajorbankwithbillionsofdollars’worthofassetsorapensionerwithjustafewhundred,thattheyhavesomeideaofthepossiblelossesthatmayresultfromthetypicalmovementsofthefinancialmarkets.Havingsaidthat,therehavebeenwell-publicizedexampleswheretheinstitutionhadnoideawhatmightresultfromsomeoftheirmoreexotictransactions,ofteninvolvingderivatives.Aspartofthesearchformoretransparencyininvestments,therehasgrownuptheconceptofValueatRiskasameasureofthepossibledownsidefromaninvestmentorportfolio.19.2DEFINITIONOFVALUEATRISKOneofthedefinitionsofValueatRisk(VaR),andthedefinitionnowcommonlyintended,isthefollowing.ValueatRiskisanestimate,withagivendegreeofconfidence,ofhowmuchonecanlosefromone’sportfoliooveragiventimehorizon.Theportfoliocanbethatofasingletrader,withVaRmeasuringtheriskthatheistakingwiththefirm’smoney,oritcanbetheportfoliooftheentirefirm.Theformermeasurewillbeofinterestincalculatingthetrader’sefficiencyandthelatterwillbeofinteresttotheownersofthefirmwhowillwanttoknowthelikelyeffectofstockmarketmovementsonthebottomline. 332PartOnemathematicalandfinancialfoundationsThedegreeofconfidenceistypicallysetat95%,97.5%,99%etc.Thetimehorizonofinterestmaybeoneday,say,fortradingactivities,ormonthsforportfoliomanagement.Itissupposedtobethetimescaleassociatedwiththeorderlyliquidationoftheportfolio,meaningthesaleofassetsatasufficientlylowrateforthesaletohavelittleeffectonthemarket.ThustheVaRisanestimateofalossthatcanberealized,notjusta‘paper’loss.AsanexampleofVaR,wemaycalculate(bythemethodstobedescribedhere)thatoverthenextweekthereisa95%probabilitythatwewilllosenomorethan$10m.Wecanwritethisas
ProbδV≤−$10m=0.05,whereδVisthechangeintheportfolio’svalue.(Iuseδ·for‘thechangein’toemphasizethatweareconsideringchangesoverafinitetime.)Insymbols,Prob{δV≤−VaR}=1−c,wherethedegreeofconfidenceisc,95%intheaboveexample.VaRiscalculatedassumingnormalmarketcircumstances,meaningthatextrememarketconditionssuchascrashesarenotconsidered,orareexaminedseparately.Thus,effectively,VaRmeasureswhatcanbeexpectedtohappenduringtheday-to-dayoperationofaninstitu-tion.ThecalculationofVaRrequiresatleasthavingthefollowingdata:Thecurrentpricesofallassetsintheportfolioandtheirvolatilitiesandthecorrelationsbetweenthem.Iftheassetsaretradedwecantakethepricesfromthemarket(markingtomarket).ForOTCcontractswemustusesome‘approved’modelfortheprices,suchasaBlack–Scholes-typemodel;thisisthenmarkingtomodel.Usually,oneassumesthatthemovementofthecomponentsoftheportfolioarerandomanddrawnfromNormaldistributions.Imakethatassumptionhere,butmakeafewgeneralcommentslateron.19.3VaRFORASINGLEASSETLetusbeginbyestimatingtheVaRforaportfolioconsistingofasingleasset.WeholdaquantityofastockwithpriceSandvolatilityσ.Wewanttoknowwith99%certaintywhatisthemaximumwecanloseoverthenextweek.Iamdeliberatelyusingnotationsimilartothatfromthederivativesworld.InFigure19.1isthedistributionofpossiblereturnsoverthetimehorizonofoneweek.HowdowecalculatetheVaR?FirstofallweareassumingthatthedistributionisNormal.Sincethetimehorizonissosmall,wecanreasonablyassumethatthemeaniszero.Thestandarddeviationofthestockpriceoverthistimehorizonis1/21σS,52sincethetimestepis1/52ofayear.Finally,wemustcalculatethepositionoftheextremeleft-handtailofthisdistributioncorrespondingto1%=(100−99)%oftheevents.WeonlyneeddothisforthestandardizedNormaldistributionbecausewecangettoanyotherNormaldistributionbyscaling.ReferringtoTable19.1,weseethatthe99%confidenceintervalcorrespondsto ValueatRiskChapter19333161412108PDF6420−0.15−0.1−0.0500.050.10.15ReturnFigure19.1Thedistributionoffuturestockreturns.Table19.1Degreeofconfidenceandtherelationshipwithdeviationfromthemean.DegreeofNumberofstandardconfidencedeviationsfrom(%)themean992.326342982.053748971.88079961.750686951.644853901.2815512.33standarddeviationsfromthemean.Sinceweholdanumberofthestock,theVaRisgivenby2.33σS(1/52)1/2.Moregenerally,ifthetimehorizonisδtandtherequireddegreeofconfidenceisc,wehaveVaR=−σS(δt)1/2α(1−c),(19.1)whereα(·)istheinversecumulativedistributionfunctionforthestandardizedNormaldistri-bution,showninFigure19.2.In(19.1)wehaveassumedthatthereturnontheassetisNormallydistributedwithameanofzero.Theassumptionofzeromeanisvalidforshorttimehorizons:Thestandarddeviationofthereturnscaleswiththesquarerootoftimebutthemeanscaleswithtimeitself.Forlongertimehorizons,thereturnisshiftedtotheright(onehopes)byanamountproportionaltothetimehorizon.Thusforlongertimescales,expression(19.1)shouldbemodifiedtoaccountfor 334PartOnemathematicalandfinancialfoundations543210Return00.10.20.30.40.50.60.70.80.91−1CDF−2−3−4−5Figure19.2TheinversecumulativedistributionfunctionforthestandardizedNormaldistribution.thedriftoftheassetvalue.Iftherateofthisdriftisµthen(19.1)becomesVaR=Sµδt−σδt1/2α(1−c).NotethatIusetherealdriftrateandnottherisk-neutral.Ishallnotworryaboutthisdriftadjustmentfortherestofthischapter.19.4VaRFORAPORTFOLIOIfweknowthevolatilitiesofalltheassetsinourportfolioandthecorrelationsbetweenthemthenwecancalculatetheVaRforthewholeportfolio.Ifthevolatilityoftheithassetisσiandthecorrelationbetweentheithandjthassetsisρij(withρii=1),thentheVaRfortheportfolioconsistingofMassetswithaholdingofioftheithassetisMM−α(1−c)δt1/2σσρSS.ijijijijj=1i=1Youwillrecognizethis,apartfromthemultiplicativefactoratthefront,asbeingthesameasthestandarddeviationofastockportfolio’sreturnfromChapter18.SeveralobviouscriticismscanbemadeofthisdefinitionofVaR:1ReturnsarenotNormal,volatilitiesandcorrelationsarenotoriouslydifficulttomeasure,anditdoes1VaRislikeyourstarsign.Youhavetotellyourinvestors/riskpeoplewhatyourVaRis,justlikeyouhavetotellthepersonyouarechattingupyourstarsign.Youdon’tnecessarilybelievethereismeaningineither. ValueatRiskChapter19335notallowforderivativesintheportfolio.Wediscussthefirstcriticismlater;Inowdescribeinsomedetailwaysofincorporatingderivativesintothecalculation.19.5VaRFORDERIVATIVESThekeypointaboutestimatingVaRforaportfoliocontainingderivativesisthat,evenifthechangeintheunderlyingisNormal,theessentialnon-linearityinderivativesmeansthatthechangeinthederivativecanbefarfromNormal.Nevertheless,ifweareconcernedwithverysmallmovementsintheunderlying,forexampleoveraveryshorttimehorizon,wemaybeabletoapproximateforthesensitivityoftheportfoliotochangesintheunderlyingbytheoption’sdelta.Forlargermovementswemayneedtotakeahigher-orderapproximation.Weseetheseapproachesandpitfallsnext.19.5.1TheDeltaApproximationConsiderforamomentaportfolioofderivativeswithasingleunderlying,S.Thesensitivityofanoption,oraportfolioofoptions,totheunderlyingisthedelta,.IfthestandarddeviationofthedistributionoftheunderlyingisσSδt1/2thenthestandarddeviationofthedistributionoftheoptionpositionisσSδt1/2.mustherebethedeltaofthewholeposition,thesensitivityofalloftherelevantoptionstotheparticularunderlying,i.e.thesumofthedeltasofalltheoptionpositionsonthesameunderlying.Itisbutasmall,andobvious,steptothefollowingestimatefortheVaRofaportfoliocontainingoptions:MM−α(1−c)δt1/2σσρSS.ijijijijj=1i=1Hereiistherateofchangeoftheportfoliowithrespecttotheithasset.19.5.2WhichVolatilityDoIUse?Forasingleunderlying,thedeltaapproximationtoVaRdependsonthestandarddeviationσSδt1/2.(19.2)Aswe’veseenmanytimes,therearedifferenttypesofvolatility.2Sowhichvolatilitygoesintothisformula?Supposeyouhaveanestimateforactualvolatility,σ,anditdiffersfromimpliedvolatility,σ˜.Whichgoesintoexpression(19.2)?Theanswerisboth.Thedeltarepresentshowtheoptionvaluewillvaryasthestockpricevaries,andthisisgovernedbythemarket’spricingoftheoptions.Thereforethedeltamustbetheusualformulausingσ˜.However,theσinfrontrepresentsthemovementinthestock,itsrealmovement,andshouldthereforebetheactualvolatility.Subtle.2...fortunatelyorunfortunatelydependingonwhetheryoumakemoneyfromthedifferences. 336PartOnemathematicalandfinancialfoundations19.5.3TheDelta-GammaApproximationThedeltaapproximationissatisfactoryforsmallmovementsintheunderlying.Abetterapprox-imationmaybeachievedbygoingtohigherorderandincorporatingthegammaorconvexityeffect.Idemonstratethisbyexample.Supposethatourportfolioconsistsofanoptiononastock.Therelationshipbetweenthechangeintheunderlying,δS,andthechangeinthevalueoftheoption,δV,is∂V∂2V∂VδV=δS+1(δS)2+δt+···.∂S2∂S2∂tSinceweareassumingthatδS=µSδt+σSδt1/2φ,whereφisdrawnfromastandardizedNormaldistribution,wecanwrite∂V∂V∂2V∂VδV=σSδt1/2φ+δtµS+1σ2S2φ2++···.∂S∂S2∂S2∂tThiscanberewrittenasδV=σSδt1/2φ+δtµS+1σ2S2φ2++···.(19.3)2Toleadingorder,therandomnessintheoptionvalueissimplyproportionaltothatintheunderlying.TothenextorderthereisadeterministicshiftinδVduetothedeterministicdriftofSandthethetaoftheoption.Moreimportantly,however,theeffectofthegammaistointroduceatermthatisnon-linearintherandomcomponentofδS.InFigure19.3areshownthreepictures.First,thereistheassumeddistributionforthechangeintheunderlying.ThisisaNormaldistributionwithstandarddeviationσSδt1/2,drawninboldinthefigure.Second,isshownthedistributionforthechangeintheoptionassumingthedeltaapproximationonly.ThisisaNormaldistributionwithstandarddeviationσSδt1/2.Finally,thereisthedistributionforthechangeintheunderlyingassumingthedelta/gammaapproximation.Fromthisfigurewecanseethatthedistributionforthedelta/gammaapproximationisfarfromNormal.Infact,becauseexpression(19.3)isquadraticinφ,δVmustsatisfythefollowingconstraint2δV≥−if>02or2δV≤−if<0.2Theextremevalueisattainedwhenφ=−.σSδt1/2 ValueatRiskChapter1933735distributionof30optiondistributionofportfolio,deltaoptionapproximationportfolio,delta/gamma25approximationdistributionofunderlying20151050−0.15−0.1−0.0500.050.10.15Figure19.3ANormaldistributionforthechangeintheunderlying(bold),thedistributionforthechangeintheoptionassumingthedeltaapproximation(anotherNormaldistribution)andthedistributionforthechangeintheoptionassumingthedelta/gammaapproximation(definitelynotaNormaldistribution).Thequestiontoaskisthen:‘Isthiscriticalvalueforφinthepartofthetailinwhichweareinterested?’Ifitisnotthenthedeltaapproximationmaybesatisfactory,otherwiseitwillnotbe.Ifwecannotuseanapproximationwemayhavetorunsimulationsusingvaluationformulae.Oneobviousconclusiontobedrawnisthatpositivegammaisgoodforaportfolioandnegativegammaisbad.Withapositivegammathedownsideislimited,butwithanegativegammaitistheupsidethatislimited.19.5.4UseofValuationModelsTheobviouswayaroundtheproblemsassociatedwithnon-linearinstrumentsistouseasimu-lationfortherandombehavioroftheunderlyingsandthenusevaluationformulaeoralgorithmstodeducethedistributionofthechangesinthewholeportfolio.Thisistheultimatesolutiontotheproblembuthasthedisadvantagethatitcanbeveryslow.Afterall,wemaywanttoruntensofthousandsofsimulationsbutifwemustsolveamultifactorpartialdifferentialequationeachtimethenwefindthatitwilltakefartoolongtocalculatetheVaR.19.5.5Fixed-incomePortfoliosWhentheassetorportfoliohasinterestratedependencethenitisusualtotreattheyieldtomaturityoneachinstrumentastheNormallydistributedvariable.Yieldsondifferentinstruments 338PartOnemathematicalandfinancialfoundationsarethensuitablycorrelated.Therelationshipofpricetochangeinyieldisviaduration(andconvexityathigherorder).Soourfixed-incomeassetcanbethoughtofasaderivativeoftheyield.TheVaRisthenestimatedusingdurationinplaceofdelta(andconvexityinplaceofgamma)intheobviousway.19.6SIMULATIONSThetwosimulationmethodsdescribedinthisbookareMonteCarlo,basedonthegenerationofNormallydistributedrandomnumbers,andbootstrappingusingactualassetpricemovementstakenfromhistoricaldata.Withinthesetwosimulationmethods,therearetwowaystogeneratefuturescenarios,depend-ingonthetimescaleofinterestandthetimescaleforone’smodelordata.Ifoneisinterestedinahorizonofoneyearandonehasamodelordataforreturnswiththissamehorizon,thenthisiseasilyusedtogenerateadistributionoffuturescenarios.Ontheotherhand,ifthemodelordataisforashortertimescale,astochasticdifferentialequationordailydata,say,andthehorizonisoneyear,thenthemodelmustbeusedtobuildupaone-yeardistributionbygeneratingwholeyear-longpathsoftheasset.Thisismoretimeconsumingbutisimportantforpath-dependentcontractswhenthewholepathtakenmustobviouslybemodeled.Remember,thesimulationmustuserealreturnsandnotrisk-neutral.19.6.1MonteCarloMonteCarlosimulationisthegenerationofadistributionofreturnsand/orassetpricepathsbytheuseofrandomnumbers.ThissubjectisdiscussedingreatdepthinChapter80.ThetechniquecanbeappliedtoVaRusingnumbers,φ,drawnfromaNormaldistribution,tobuildupadistributionoffuturescenarios.Foreachofthesescenariosusesomepricingmethodologytocalculatethevalueofaportfolio(oftheunderlyingassetanditsoptions)andthusdirectlyestimateitsVaR.19.6.2BootstrappingAnothermethodforgeneratingaseriesofrandommovementsinassetsistousehistoricaldata.Again,therearetwopossiblewaysofgeneratingfuturescenarios:Aone-stepprocedureifyouhaveamodelforthedistributionofreturnsovertherequiredtimehorizon,oramulti-stepprocedureifyouonlyhavedata/modelforshortperiodsandwanttomodelalongertimehorizon.Thedatathatweusewillconsistofdailyreturns,say,forallunderlyingassetsgoingbackseveralyears.Thedataforeachdayarerecordedasavector,withoneentryperasset.Supposewehaverealtime-seriesdataforNassetsandthatourdataaredailydatastretchingbackfouryears,resultinginapproximately1000dailyreturnsforeachasset.Wearegoingtousethesereturnsforsimulationpurposes.Thisisdoneasfollows.Assignan‘index’toeachdailychange.Thatis,weassign1000numbers,oneforeachvectorofreturns.Tovisualizethis,imaginewritingthereturnsforalloftheNassetsontheblankpagesofanotebook.Onpage1wewritethechangesinassetvaluesthatoccurredfrom8thJuly1998to9thJuly1998.Onpage2wedothesame,butforthechangesfrom9thJulyto10thJuly1998.Onpage3...from10thto11thJulyetc.Wewillfill1000pagesifwehave ValueatRiskChapter193391000datasets.Now,drawanumberfrom1to1000,uniformlydistributed;itis534.Gotopage534inthenotebook.Changeallassetsfromtoday’svaluebythevectorofreturnsgivenonthepage.Nowdrawanothernumberbetween1and1000atrandomandrepeattheprocess.Incrementthisnewvalueagainusingoneofthevectors.Continuethisprocessuntiltherequiredtimehorizonhasbeenreached.Thisisonerealizationofthepathoftheassets.Repeatthissimulationtogeneratemany,manypossiblerealizationstogetanaccuratedistributionofallfutureprices.Bythismethodwegenerateadistributionofpossiblefuturescenariosbasedonhistoricaldata.Notehowwekeeptogetherallassetchangesthathappenonacertaindate.Bydoingthisweensurethatwecaptureanycorrelationthattheremaybebetweenassets.Thismethodofbootstrappingisverysimpletoimplement.Theadvantagesofthismethodarethatitnaturallyincorporatesanycorrelationbetweenassets,andanynon-Normalityinassetpricechanges.Itdoesnotcap-tureanyautocorrelationinthedata,butthenneitherdoesaMonteCarlosimulationinitsbasicform.Themaindisadvan-tageisthatitrequiresalotofhistoricaldatathatmaycor-respondtocompletelydifferenteconomiccircumstancesthanthosethatcurrentlyapply.InFigure19.4isshownthedailyhistoricalreturnsforseveralstocksandthe‘index’usedintherandomchoice.19.7USEOFVaRASAPERFORMANCEMEASUREOneoftheusesofVaRisinthemeasurementofperformanceofbanks,desksorindividualtraders.Inthepast,‘tradingtalent’hasbeenmeasuredpurelyintermsofprofit;atrader’sbonusisrelatedtothatprofit.Thisencouragestraderstotakerisks;thinkoftossingacoinwithyoureceivingapercentageoftheprofitbutwithoutthedownside(whichistakenbythebank),howmuchwouldyoubet?Abettermeasureoftradingtalentmighttakeintoaccounttheriskinsuchabet,andrewardagoodreturn-to-riskratio.TheratioReturninexcessofrisk-freeµ−r=,volatilityσtheSharperatio,issuchameasure.Alternatively,useVaRasthemeasureofriskandprofit/lossasthemeasureofreturn:dailyP&L.dailyVaR19.8INTRODUCTORYEXTREMEVALUETHEORYMoremoderntechniquesforestimatingtailriskuseExtremeValueTheory.Theideaistorepresentmoreaccuratelytheouterlimitsofreturnsdistributionssincethisiswherethemostimportantriskis.ThrowNormaldistributionsaway;theirtailsarefartoothintocapturethefre-quentmarketcrashes(andrallies).Iwon’tgointothedetailshere;awholebookcouldbewrittenonthissubject(andhasbeen,Embrechts,KluppelbergandMikosch1997,averygoodbook).¨ 340PartOnemathematicalandfinancialfoundations0.01538490.01449300.00651470.02298950.01052640.02681130.03423320.02331110.04329680.00443460.02777960.01843370.12429770.06001800.00975620.01481510.0150379−−−−−−−−−−−−−−−−−0.02353050.01392130.0284379-0.03717900.02677020.01169600.00579710.00809720.00000000.00757580.02697660.05927660.00732600.00738010.02803920.01096900.00368320.01151640.00769230.02343860.10487960.03200270.00816330.01025650.00396040.01197620.01619470.0050378−−−−−−−−−−−−−−−−−−−−−0.01821900.00263930.00675060.03278980.01445200.07466150.00202990.00196380.04391920.00000000.01662940.03846630.01700730.01719240.00637620.00391390.00000000.02285810.00355370.01123610.04191090.04750230.00805550.00000000.00379010.00884960.00000000.02805180.00750470.00966080.00394480.05470120.00975620.00651980.0109291−−−−−−−−−−−−−−−−−−0.02409760.00962380.01142870.00000000.00000000.00000000.01232050.01129960.01104980.03045920.01520950.00995030.02020270.00966190.0196085−−−−−−−−−0.01069530.02540970.01197620.00000000.01694960.00858370.03807190.03352270.00574710.00000000.03001380.00527710.01369880.00000000.00932410.01011760.02850550.00000000.02686360.01105540.00000000.00053240.00526320.00000000.02403320.00000000.00439560.01086970.01844920.02210030.00000000.03409420.0202027-0.01005030.01096900.01980270.01941810.02298950.0310527−−−−−−−−−−−−−−−−−−−0.01219530.01234580.02083410.00000000.01005030.00506330.02570840.01503790.00459770.00527710.00000000.00071870.01912100.00385360.01165060.00698490.01178800.02400120.00000000.00341040.00400800.00247220.03509130.01348340.03960910.03687050.00906630.01895790.00913250.02790880.0094787−−−−−−−−−−−−−−−−−−−−−0.00682600.00687290.00619200.00366970.00619200.00623050.01000010.01162800.00574710.01092910.01739170.01441830.01652930.04115810.07453310.02230580.00547950.00000000.02323600.01671350.00360360.00560230.01941810.00000000.00249070.02424360.00000000.00615390.03135050.01282070.0180185−−−−−−−−−−−−−−0.01117330.02262440.03559090.01156080.02076200.06021000.01857640.00682600.01219530.00719430.01069530.02347590.01212140.00000000.00400800.01140300.00000000.02262480.01268350.00000000.02541460.00082560.01299140.04154900.02675070.02777960.02312240.03093030.00917440.04082200.00143750.02770260.00878680.05049620.00000000.04403960.00020430.04927100.02854560.01734390.01111120.01866610.01208310.01455340.04607660.02666820.05638280.0000000−−−−−−−−−−−−−−−−−−−−−−0.02409750.02469260.10013230.00229090.00041550.00673400.00472820.04520540.00485410.01129960.02409750.04385190.00000000.01129960.01156080.00041340.01005030.04441060.00000000.01408630.01126770.04216080.01129960.00219470.09325750.00813040.04286780.02216460.00946970.00205000.00583090.03795530.01222010.01459880.03922070.00024230.00753150.01393530.00868590.00560230.08498820.02020270.03257850.00496280.02040890.00396040.01477920.02666820.00382410.05377850.01801850.0277796−−−−−−−−−−−−−−−−−−−0.00461300.03243530.00010390.03585690.02550430.03463930.01610400.01169610.01222500.00233920.00000000.02840210.02597550.00010350.02510590.01769960.02481870.01142870.02644210.00785410.00030840.01228040.01582100.00086140.00994820.03428910.01606460.02641210.03447810.02257510.02040890.01510710.02082620.01990910.06795070.03865220.04649760.02643320.04275170.0000000−−−−−−−−−−−−−−−−−−−−0.03978830.04947120.02040130.01124130.03836260.02641840.00365630.01143880.04033770.01486390.01425200.00693060.00722250.02298960.00422300.01386200.00638980.02427920.01015240.01920600.02058230.00853490.01142870.00368100.00147130.00540030.02891440.00697700.00659560.00674090.00841740.02847930.01748390.00369820.02040630.04709360.01833910.02617910.00962700.00696860.00431970.04648350.06280090.0690578−−−−−−−−−−−−−−−−−−−−−−−0.01522100.01801850.04985460.03577620.01183450.01398010.00522190.01306930.01066560.00000000.04753770.00370450.02083410.00911090.03099870.00451470.06156720.01103410.00445440.04173950.00445440.01769950.05069320.01781630.02052190.00851080.00921670.00000000.02569770.01728060.03239860.01130790.07695000.04792360.0374621−−−−−−−−−−−−−−−−−−−−−−Spreadsheetshowingbootstrapdata.10.00120.00130.00140.00916040.00150.00160.00722140.00170.00330580.00180.04243060.00190.02793170.0010.02601110.00062160.00062160.0006216100.001110.001120.001130.001140.00790350.001150.00000000.001160.01841670.00000000.001170.001180.02974280.001190.001200.01026520.001210.05688740.01144450.001220.06559110.001230.08282760.07149500.001240.02091540.02314700.10525770.001250.01579980.02099220.001260.04771500.001270.04204560.02702870.06126610.00128-0.00892860.02247560.001290.02750120.02710190.01336320.00683620.001300.05837110.07534940.001310.02321830.00386860.001320.00010220.07464350.00445440.05129330.02489680.001330.03614210.03883980.001340.00010220.02020270.00886850.001350.02801300.05997620.04367500.001360.0010.0498324370.001380.04380260.001390.03031670.001400.00000000.001410.03984800.0010.01781850.01889540.00030500.02040670.01227350.02785470.00935480.06252040.00000000.04216080.00913250.03966530.00829880.00000000.02985300.0471675ABCDEFGHIJKLMNIndexProb.TELEBRASELETROBRASPETROBRASCVDRUSIMINASYPFTARTEOTGSPEREZTELMEXTELEVISA123456789101112131415161718192021222324252627282930313233343536373839404142Figure19.4 ValueatRiskChapter1934119.8.1SomeEVTResultsDistributionofmaxima/minimaIfXiareindependent,identicallydistributedrandomvariablesandx=max(X1,X2,...,Xn)thenthedistributionofxconvergesto−1/ξξ(x−µ)exp−1+.σWhenξ=0thisisaGumbeldistribution,whenξ<0itisaWeibullandwhenξ>0aFrechet.Frechetistheoneofinterestinfinancebecauseitisassociatedwithfattails.PeaksoverthresholdConsidertheprobabilitythatlossexceedsubyanamounty(giventhatthethresholduhasbeenexceeded):Fu(y)=P(X−u≤y|X>u).ThiscanbeapproximatedbyaGeneralizedParetoDistribution:−1/ξξX1−1+βForheavytailswehaveξ>0inwhichcasenotallmomentsexist:E[Xk]=∞fork≥1/ξ.TheparametersinthemodelsarefittedbyMaximumLikelihoodEstimation,usinghistoricaldataforexample,andfromthatwecanextrapolatetothefuture.Example(FromAlexanderMcNeil,1998).FitaFrechetdistributiontothe28annualmaximafrom1960toOctober16th1987,thebusinessdaybeforethebigone.Nowcalculateprobabilityofvariousreturns.Forexample,50-yearreturnlevelbeingthelevelwhichonaverageshouldonlybeexceededinoneyearevery50years.Result:24%.Andthenthenextday,October19th:20.4%.Notethatinthedatasetthelargestfallhadbeen‘just’6.7%.19.9COHERENCESomemeasuresofriskmakemoresensethanothers.Artzner,Delbaen,Eber&Heath(1997)havestatedsomepropertiesthatsensibleriskmeasuresoughttohave;theycallsuchsensiblemeasures‘coherent.’Ifweuseρ(X)todenotethisriskmeasureforasetofoutcomesXthenthenecessarypropertiesareasfollows. 342PartOnemathematicalandfinancialfoundations1.Sub-additivity:ρ(X+Y)≤ρ(X)+ρ(Y).Thisjustsaysthatifyouaddtwoportfoliostogetherthetotalriskcan’tgetanyworsethanaddingthetworisksseparately.Indeed,theremaybecancellationeffectsoreconomiesofscalethatwillmaketheriskbetter.2.Monotonicity:IfX≤Yforeachscenariothenρ(X)≤ρ(Y).Prettyobviously,ifoneportfoliohasbettervaluesthananotherunderallscenariosthenitsriskwillbebetter.3.Positivehomogeneity:Forallλ>0,ρ(λX)=λρ(X).Doubleyourportfoliothenyoudoubleyourrisk.4.Translationinvariance:Forallconstantc,ρ(X+c)=ρ(X)−c.Thinkofjustaddingcashtoaportfolio.Dothecommonmeasuresofrisksatisfythesereasonablecriteria?Generallyspeaking,sur-prisinglyandratherunfortunatelynot!Forexample,theclassicalVaRviolatessub-additivity.19.10SUMMARYTheestimationofdownsidepotentialinanyportfolioisclearlyveryimportant.Nothavinganideaofthiscouldleadtothedisappearanceofabank,andhas.Inpractice,itismoreimportanttothemanagersinbanks,andnotthetraders.Whatdotheycareiftheirbankcollapsesaslongastheycanwalkintoanewjob?IhaveshownthesimplestwaysofestimatingValueatRisk,butthesubjectgetsmuchmorecomplicated.‘Complicated’isnottherightword,‘messy’and‘time-consuming’arebetter.Andcurrentlytherearemanysoftwarevendors,banksandacademicstoutingtheirownversionsinthehopeofbecomingthemarketstandard.InChapter42we’llseeafewoftheseinmoredetail.FURTHERREADING•SeeLawrence(1996)fortheapplicationoftheValueatRiskmethodologytoderivatives.•SeeChew(1996)forawiderangingdiscussionofriskmanagementissuesanddetailsofimportantreal-lifeVaR‘crises.’•SeeJorion(1997)forfurtherinformationaboutthemathematicsofValueatRisk.•TheallocationofbankcapitalisaddressedinMatten(1996).•Alexander&Leigh(1997)andAlexander(1997a)discusstheestimationofcovariancematricesforVaR.•Artzner,Delbaen,Eber&Heath(1997)discussthepropertiesthatasensibleVaRmodelmustpossess.•Lillo,Mantegna,Bouchaud&Potters(2002)introducetheconceptof‘variety’asameasureofthedispersionofstocks. CHAPTER20forecastingthemarkets?InthisChapter...•someofthecommonlyusedtechnicalmethodsforpredictingmarketdirection•somemodernapproachestomodelingmarketsandtheirmicrostructure20.1INTRODUCTIONPeoplehavebeenmakingpredictionsaboutthefuturesincethedawnoftime.Andpredictingthefutureofthefinancialmar-ketshasbeenespeciallypopular.Despitetheclaimsofmany‘legendary’investorsitisnotclearwhetherthereisanyvalid-ityinanyofthemethodstheyuse,orwhethertheclaimsareexamplesofsurvivorbias.Thebigloserstendtokeepquiet.Humansjustseemtolikedeterminism,andhavedifficulty,especiallyatanearlyage,handlingtherandomandmeaningless.Itgoesbacktocavemantimes.UgandhisfriendOgwerestandingintheentrancetotheircavewhenalongcameasaber-toothedtiger.ThetigerwentforOganddraggedhimoffforhissupper.ItjustsohappenedthatUgwasscratchinghisleftearwhenthishappened.NoweverytimethatUgseesasaber-toothedtigerhescratcheshisleftear,justtobeonthesafeside.NowmaybetherewasnoconnectionbetweenwhatUgwasdoingwhenOgwasdraggedoff,butperhapstherewas,andthere’snoharminplayingitsafeinthefuture.Inthischapterwelookatsomeofthetraditionalmethodsfordeterminingtrends,technicalanalysis,andalsosomeofthemorerecentmethods,oftenemanatingfromphysics.Iwon’tbedescribingsomeofthemoredubiousideas,suchasastrology,butthenweScorpiostendtobesceptical.120.2TECHNICALANALYSISTechnicalanalysisisawayofpredictingfuturepricemovementsbasedonlyonobservingthepasthistoryofprices.Thispricehistorymayalsoincludeotherquantitiessuchasvolume1Ididknowatraderoncewhoalwayshadastrologicalchartsonhiscomputer.Iassumedtheywerejustascreensaver,usedtohidethetrader’sactualstrategywhilehewasawayfromhisdesk.Shortlyafterhewasfiredwealldiscoveredthatactually... 344PartOnemathematicalandfinancialfoundationsoftrade.Thesemethodscontrastwithfundamentalanalysisinwhichpredictionismadebasedonanexaminationofthefactorsunderlyingthestockorotherinstrument.Thismayincludegeneraleconomicorpoliticalanalysis,oranalysisoffactorsspecifictothestock,suchastheeffectofglobalwarmingonsnowfallintheAlps,ifoneisconcernedwithatravelcompany.Inpractice,mosttraderswilluseacombinationofbothtechnicalandfundamentalanalysis.Technicalanalysisisalsocalledchartingbecausethegraphicalrepresentationofpricesetc.playsanimportantpart.Technicalanalysisisthoughttobeparticularlygoodfortimingmarketmoves;fundamentalanalysismaygetthedirectionright,butnotnecessarilywhenthemovewillhappen.20.2.1PlottingThesimplestcharttypesjustjointogetherthepricesfromonedaytothenext,withtimealongthehorizontalaxis.Thesearethesortofplotswehaveseenthroughoutthisbook.Sometimesalogarithmicscaleisusedfortheverticalpriceaxistorepresentreturnratherthanabsolutelevel.Lateronwe’llseesomemorecomplicatedtypesofplotting.Sometimesyouwillseetradingvolumeonthesamegraph;thisisalsousedforpredictionbutIwon’tgointoanydetailshere,seeFigure20.1.Figure20.1Priceandvolume.Source:BloombergL.P. forecastingthemarkets?Chapter2034520.2.2SupportandResistanceResistanceisapricelevelwhichanassetseemstohavedifficultyrisingabove.Thismaybeapreviouslyrealizedhighestvalue,oritmaybeapsychologicallyimportant(round)number.Supportisalevelbelowwhichanassetpriceseemstobereluctanttofall.Theremaybesufficientdemandatthislowpricetostopitfallinganyfurther.ExamplesofsupportandresistanceareshowninFigure20.2.Whenasupportorresistancelevelfinallybreaksitissaidtodosoquitedramatically.20.2.3TrendlinesSimilartosupportandresistancearetrendlines.Theseareformedbyjoiningtogethersuccessivepeaksand/ortroughsinthepricehistorytoformarisingorfallingsupportorresistancelevel.AnexampleisshowninFigure20.3.20.2.4MovingAveragesMovingaveragesarecalculatedinmanyways.Differenttimewindowscanbeused,orevenexponentially-weightedaveragescanbecalculated.Movingaveragesaresup-posedtodistilloutthebasictrendinapricebysmoothingtherandomnoise.Sometimestwomovingaveragesarecalculated,sayaten-dayanda250-dayaver-age.Thecrossingofthesetwowouldsignifyachangeintheunderlyingtrendandatimetobuyorsell.AlthoughI’mnotthegreatestfanoftechnicalanalysis,thereissomeevidencethattheremaybepredictivepowerinmovingaverages.Figure20.4showsaBloombergscreenwithMicrosoftshareprice,five-and15-daymovingaverages.20.2.5RelativeStrengthTherelativestrengthindexisthepercentageofupmovesinthelastNdays.Anumberhigherthan70%issaidtobeoverboughtandarethereforelikelytofallandbelow30%issaidtobeoversoldandshouldrise.20.2.6OscillatorsAnoscillatorisanotherindicatorofover/underboughtconditions.Onewayofcalcu-latingitisasfollows.DefinekbyCurrentclose−lowestovernperiods100×.Highestovernperiods−lowestovernperiodsNowtakeamovingaverageofthelastthreedays,say.Thisaverageisplottedagainsttimeandanymoveoutsidetherange30–70%couldbeanindicationofamoveintheasset.SeeFigure20.5. 346PartOnemathematicalandfinancialfoundations16014012010080Stockprice6040200012468012Time1201008060Stockprice40200051234678TimeFigure20.2Supportandresistance. forecastingthemarkets?Chapter203477060504030Stockprice2010006248TimeFigure20.3Atrendingstock.Figure20.4Twomovingaverages.Source:BloombergL.P. 348PartOnemathematicalandfinancialfoundationsFigure20.5Oscillator.Source:BloombergL.P.20.2.7BollingerBandsBollingerBandsareplotsofaspecifiednumberofstandarddeviationsaboveandbelowaspecifiedmovingaverage,seeFigure20.6.20.2.8MiscellaneousPatternsAswellasthe‘quantitative’sideofchartingthereisalsothe‘artistic’side.Practitionerssaythatcertainpatternsanticipatecertainfuturemoves.It’sratherlikeyourgrandmotherreadingtealeaves.HeadandshouldersisacommonpatternandisbestdescribedwithreferencetoFigure20.7.Therearealeftandarightshoulderwiththeheadrisingabove.Followingonfromtherightshouldershouldbeadramaticdeclineintheassetprice.Thispatternissupposedtobeoneofthemostreliablepredictors.Itisalsoseeninanupside-downformation.Saucertopsandbottomsarealsoknownasroundingtopsandbottoms.Theyaretheresultofagradualchangeinsupplyanddemand.Theshapeisgenerallyfairlysymmetricalasthepricesrisesandfalls.Thesepatternsarequiterare.Theycontainnoinformationaboutthestrengthofthenewtrend.SeeFigure20.8. forecastingthemarkets?Chapter20349Figure20.6BollingerBands.Source:BloombergL.P.Doubleandtripletopsandbottomsarequiterarepatterns,thetriplebeingevenrarerthanthedouble.Thedoubletoplookslikean‘M’andadoublebottomlikea‘W.’Thetripletopissimilarbutwiththreepeaks,asshowninFigure20.9.Thekeypointaboutthepeaksandtroughsisthattheyshouldallbeatapproximatelythesamelevel.20.2.9JapaneseCandlesticksJapanesecandlestickscontainmoreinformationthanthesimpleplotsdescribedsofar.Theyrecordtheopeningandclosingpricesaswellastheday’shighandlow.Arectangleisdrawnextendingfromtheclosetotheopen,andiscoloredwhiteifcloseisaboveopenandblackifcloseisbelowopen(seeFigure20.10).Thehigh-lowrangeismarkedbyacontinuousline.Certaincombinationsofcandlesticks,appearingconsecutivelyhavespecialmean-ingsandnameslike‘HangingMan’and‘UpsideGapTwoCrows.’SeeFigure20.11forcandlesticksinaction.OnthisLetterchartareshown‘HR’=BearishHarami,‘D’=Doji(representingindecision),‘BH’BullishHarami,‘EL’=BearishEngulfingLine,and‘H’=HangingMan(representingreversalafteratrend).Figure20.12showssomeofthepossiblecandlestickshapesandtheirinterpretation. 350PartOnemathematicalandfinancialfoundations16014012010080Stockprice6040200024681012TimeFigure20.7Headandshoulders.600500400300Stockprice2001000026481012TimeFigure20.8Saucerbottom. forecastingthemarkets?Chapter20351250Tripletop200150Stockprice1005000246810TimeFigure20.9Atripletop.HighOpenCloseCloseOpenLowFigure20.10Japanesecandlesticks. 352PartOnemathematicalandfinancialfoundationsFigure20.11Acandlestickchart.Source:BloombergL.P.1234567891011121314151Extremelybearish9Indowntrend,bullish;inuptrend,bearish2Extremelybullish10Indowntrend,bullish;inuptrend,bearish3Bearish11Aturningperiod4Bearish12Aturningperiod5Bullish13Endofdowntrend6Bullish14Aturningperiod7Neutral15Possibleturningperiod8NeutralFigure20.12Themeaningsofthevariouscandlesticks. forecastingthemarkets?Chapter20353Figure20.13ApointandfigurechartofMicrosoft.Source:BloombergL.P.20.2.10PointandFigureChartsPointandfigurechartsaredifferentfromthechartsdescribedaboveinthattheydonothaveanyexplicittimescaleonthehorizontalaxis.Figure20.13isanexampleofapointandfigurechart.Eachboxonthechartrepresentsaprespecifiedassetpricemove.Theboxesareawayofdiscretizingassetpricemoves,insteadofdiscretizingintime.Foreachconsecutiveassetpriceriseoftheboxsizedrawan‘X’inthebox,inarisingcolumn,oneabovetheother.Whenthisuptrendfinishes,andtheassetfalls,startputting‘O’inadescendingcolumn,totherightofthepreviousrisingXs.•AlongcolumnofXsdenotesdemandexceedingsupply.•AlongcolumnofOsdenotessupplyexceedingdemand.•Shortupanddowncolumnsdenoteabalanceofsupplyanddemand.20.3WAVETHEORYAswellasplottingandspottingtrendsinpricemovementstherehavebeensometheoriesforpricepredictionbasedonmarketcyclesorwaves.Below,Ibrieflymentionacouple. 354PartOnemathematicalandfinancialfoundationsStockpriceTimeFigure20.14Elliotwaves.Figure20.15Fibonaccilines.Source:BloombergL.P. forecastingthemarkets?Chapter203551201001/48060Stockprice1/24012000246810TimeFigure20.16Ganncharts.20.3.1ElliottWavesandFibonacciNumbersRalphN.Elliottobservedrepetitivepatterns,wavesorcyclesinprices.Roughlyspeaking,therearesupposedtobefivepointsinabullishwaveandthenthreeinabearishone(seeFigure20.14).2WithinthisElliottwavetheorythereisalsosupposedtobesomepredictiveabilityintermsofthesizesofthepeaksineachwave.Forsomereason,theratioofpeaksinatrendaresupposedtobefairlyconstant;theratioofsecondpeaktofirstshouldbeapproximately1.618andofthethirdtothesecond2.618.Unfortunately,thenumber1.618isapproximately√theGoldenratiooftheancientGreeks;1(5+1).Itisalsotheratioofsuccessivenumbers2intheFibonacciseriesgivenbyan=an−1+an−2forlargen.Isay,unfortunately,becausepeopleextrapolatewildlyfromthis.Andifit’sacoincidencethen...Figure20.15showsthekeylevelscomingfromtheFibonacciseries.20.3.2GannChartsFigure20.16showsaGannchart.Thelinesallhaveslopeswhicharefractionsoftheslopeofthelowestline.NeedIsaymore?20.4OTHERANALYTICSThere’sanalmostendlessnumberofwaysthatchartistsanalyzedata.I’llmentionjustacouplemorebeforemovingon.2InBrownianMotionthereare,ofcourse,aninfinitenumberofpeaksandtroughsinanyperiod. 356PartOnemathematicalandfinancialfoundationsVolumeissimplythenumberofcontractstradedinagivenperiod.Arisingpriceandhighvolumemeansastrong,upwardlytrendingmarket.Butarisingpricewithlowvolumecouldbeasignthatthemarketisabouttoturn.Openinterestisthenumberofstilloutstandingfuturescontracts,thosewhichhavenotbeenclosedout.Becausethereareequalnumbersofbuyersandsellers,openinterestdoesnotnecessarilygiveanydirectionalinfo,butanincreaseinopeninterestcanmeanthatanexistingtrendisstrong.20.5MARKETMICROSTRUCTUREMODELINGThefinancialmarketsaremadeupofmanytypesofplayers.Therearethe‘producers’whomanufactureorproduceorsellvariousgoodsandwhomaybeinvolvedinthefinancialmarketsforhedging.Therearethe‘speculators’whotryandspottrendsinthemarket,toexploitthemandmakemoney.Thesespeculatorsmaybeusingtechnicalanalysismethods,suchasthosedescribedabove,orfundamentalanalysis,wherebytheyexaminetherecordsandfutureplansoffirmstodeterminewhetherstocksareunder-oroverpriced.Almostalltradersusetechnicalanalysisatsometime.Thentherearethemarketmakerswhobuyandsellfinancialinstruments,holdingthemforaveryshorttime,oftenseconds,andprofitonbid-offerspreads.Therehavebeenmanyattemptstomodeltheinteractionoftheseagents,sometimesinagametheoreticway,totryandmodeltheassetpricemovementsthatinthisbookwehavetakenforgranted.Forexample,canthedynamicsinducedbytheactionsofacombinationofthesethreetypesofagentresultinBrownianmotionandlognormalrandomwalks?Belowarejustaveryfewexamplesofworkinthisarea.20.5.1EffectofDemandonPriceBuyingandsellingassetsmovestheirprices.Marketmakersrespondtodemandbyincreasingprice,andreducepriceswhenthemarketisselling.Ifonecanmodeltherelationshipbetweendemandandpricethenitshouldbepossibletoanalyzetheeffectthatvarioustypesoftechnicaltradingrulehaveontheevolutionofprices,andeventuallytomodelthedynamicsofprices.Acommonstartingpointistoassumethattherearetwotypesoftraderandonemarketmaker.Onetraderfollowsatechnicaltradingrulesuchaswatchingamovingaverageandtheotherisanoisetraderwhorandomlybuysorsells.Interestingresultsfollowfromsuchmodels.Forexample•trendfollowerscaninducepatternsinassetpricetimeseries;•theseartificiallyinducedpatternscanonlybeexploitedforgainbysomeonefollowingasuitablydifferenttrend;•themorepeoplefollowingthesametrendasyou,themoremoneyyouwilllose.Therearegoodreasonsfortherebeinggenuinetrendsinthemarket:Thereisaslowdiffusionofinformationfromtheknowledgeabletothelessknowledgeable;Thepiece-by-piecesecretacquisitionofacompanywillgraduallymoveastockpriceupwards.Ontheotherhand,ifthereisnogenuinereasonforatrend,ifitissimplyacaseoftrendfollowersbegettingatrend,thenitmaybebeneficialtobeacontrarian. forecastingthemarkets?Chapter2035720.5.2CombiningMarketMicrostructureandOptionTheoryArbitragedoesexist;manypeoplemakemoneyfromitsexistence.Yettheactionofarbitragerswill,viaademand/pricerelationship,removethearbitrage.Buttherewillbeatimescaleasso-ciatedwiththisremoval.Whatistheoptimalwaytoexploitthearbitrageopportunitywhileknowingthatyouractionswilltosomeextentbeself-defeating?20.5.3ImitationAnotherapproachtomarketmicrostructuremodelingisbasedonthetrueobservationthatpeoplecopyeachother.Inthesemodelsthereareanumberoftraderswhoactpartlyinresponsetoprivateinformationaboutastock,partlyrandomlyasnoisetraders,andpartlytoimitatetheirnearestneighbors.Thesemodelscanresultinmarketbubblesormarketcrashes.20.6CRISISPREDICTIONTherehasbeensomeworkonanalyzingdataovervarioustimescalestodeterminethelikelihoodofamarketcrash.Someideasfromearthquakemodelinghavebeenusedtoderivea‘Richter’-likemeasureofmarketmoves.Ofcourse,aneffectivepredictorofmarketcrashescouldeither:•increasethechanceorsizeofacrashaseveryonepanics,or•reducethechanceorsizeofthecrashsinceeveryonegetsadvancewarningandcancalmlyandlogicallyactaccordingly.20.7SUMMARYIstartedoutinfinancemanyyearsagoplottingallofthetechnicalindicators.Iwasnotverysuccessfulatit.Icouldonlygetdirectionsrightforthoseassetswithobviousseasonalityeffects,suchassomecommodities.ThereisonlyonetechnicalindicatorthatIbelievein.Thereisdefinitelyastrongcorrelationbetweenhemlinesandthestateoftheeconomy.Theshortertheskirts,thebettertheeconomy.FURTHERREADING•ThebookontechnicalanalysiswrittenbythenewsagencyReuters(1999)isexcellent,asisMeyers(1994).•Farmer&Joshi(2000)discussandmodeltrendfollowing,andthecreationoftrends.Theyalsodemonstratepropertiesoftherelationshipbetweendemandandpricethatpreventarbitrage.•Bhamra(2000)hasworkedonimitationinfinancialmarkets.•Olsen&Associates(www.olsen.ch)arecurrentlyworkingintheareaofcrisismodelingandprediction. 358PartOnemathematicalandfinancialfoundations•Johnson,Hui&Lo(1999)modelsself-organizedsegregationoftraders,andconcludesthatcautioustradersperformpoorly.•Theaboveisonlyabriefdescriptionofaveryfewexamplesfromanexpandingfield.SeeO’Hara(1995)forawide-rangingdiscussionofmarketmicrostructuremodels.•Bernstein(1998)hasawholechapterontheGoldenRatio.•Elton&Gruber(1995)describetheefficientmarkethypothesisandcriticizetechnicalanalysis.•Prast(2000a,b)discusses‘herding’inthefinancialmarkets.•AmathematicalmodelleadingtosupportandresistancecanbefoundinOsband(2004). CHAPTER21atradinggameInthisChapter...•Asimplegamesimulatingtradinginoptions21.1INTRODUCTIONAlotofpeoplereadingthisbookwillneverhavetradedoptionsorevenstocks.InthischapterIdescribeaverysimpletradinggamesothatagroupofpeoplecantryouttheirskillwithoutlosingtheirshirts.Thegameisbasedonthatbyoneofmyex-students,DavidEpstein.21.2AIMSTheaimsofthisgamearetofamiliarizestudentswiththebasicmarket-tradedderivativecontractsandtopromoteanunderstandingoftheconceptsinvolvedintrading,suchasbid,offer,arbitrageandliquidity.21.3OBJECTOFTHEGAMETomakemoremoneythanyouropponents.Afterthefinalroundoftrading,eachplayersumsuptheirprofitsandlosses.Theplayerwhohasmadethemostprofitisthewinner.21.4RULESOFTHEGAME1.Oneperson(possiblyalecturer)isthegameorganizerandinchargeofchoosingthetypesofcontractsavailablefortrading,thenumberandlengthofthetradingrounds,judginganydisputesandjollyingthegamealongduringslackperiods.2.Thetradinggametakesplaceoveranumberofrounds.Attheendofeachround,asix-sideddieisthrown.Afterthelastround,the‘shareprice’isdeemedtobethesumofallofthedierolls.3.Tradedcontractsmayincludesomeorallofforwards,callsandputsatthediscretionoftheorganizer(Figure21.1).Theorganizermustalsodecidewhatexercisepricesareavailableforcallorputoptions. 360PartOnemathematicalandfinancialfoundations1510Forward50051015202530−5−10−15Asset1510Call50051015202530Asset1510Put50051015202530AssetFigure21.1Availablecontracts.4.Allcontractsexpireattheendofthefinalround.Thesettlementvalueofeachtradedcontractcanthenbedeterminedbysubstitutingthesharepriceintotheappropriateformula.Aplayer’sprofitsandlossesoneachtradecanthenbecalculatedandtheresultantprofit/lossistheirfinalscore.5.Duringaround,playerscanoffertobuyorsellanyofthetradedcontracts.Ifanotherplayerchoosestotakethemupontheiroffer,thenthedealisagreedandbothpartiesmustrecordthetransactionontheirtradingsheet. atradinggameChapter213616.Adealonacontractmustincludethefollowinginformation:•Forward:Forwardpriceandquantity•CallorPut:Typeofoption(callorput),exerciseprice,costandquantityTheorganizerchoosesthetypesofcontractsavailableandthestrikeprices.Theforwardpriceoroptioncostandthequantityinadealarechosenbytheplayers.Forbeginners,playthreegamesinsuccession,withthefollowingstructures:1.Playwithjusttheforwardcontract.2.Playwiththeforwardcontractandthecalloptionwithexerciseprice15.3.Playwiththeforwardandthecallandputoptionswithexerciseprice15.Allthreegamestakeplaceoverfiverounds,eachfiveminutesinlength.21.5NOTES1.Dependingonthelevelofpriorknowledgeoftheplayers,theorganizermayneedtoexplainthecharacteristicsofthevarioustradedcontracts.Itwillbeinstructivetoemphasizethattheforwardcontracthasnocostinitially.2.Therewillprobablybetimeswhentheorganizerhastoactasa‘market-maker’andpromotetrading,forinstance,askingthegroupifanyonewantstobuysharesoratwhatpricesomeoneiswillingtodoso.3.Formoreadvancedstudents,considerintroducingsomeofthefollowingideas:•Increasethenumberofrounds.•Decreasethelengthofeachround.•Includeextracallsandputswithdifferentexercisepricesorwhicheithercomeintoexistenceorexpireatdifferenttimes.Youmustfixthedetailsoftheseextracontractsinadvanceofthegame.•Includeothercontractse.g.AsianoptionsorBarriers.•Includeaseconddieforasecondunderlyingshareprice.4.Includingfutureswith‘daily’markingtomarketcanbetried,butslowsdownthegame.Nevertheless,itdoesillustratetheimportanceofmargin,especiallyifthestudentshavealimitonhowmuch‘indebt’theyareallowedtobecome.21.6HOWTOFILLINYOURTRADINGSHEET21.6.1DuringaTradingRoundInthe‘Contract’columnofFigure21.2,fillinthespecificationsoftheinstrumentthatyouhavebought/sold.Specifytheforwardpriceorexercisepriceifapplicable(e.g.ifthereismorethanonecontractofthistypeinthegame).Inthe‘Buy/Sell’column,fillinwhetheryouhaveboughtorsoldthecontractandthequantity.Inthe‘Costpercontract’column,fillinthecostofasinglecontract. 362PartOnemathematicalandfinancialfoundationsTradingsheetCostperContractBuy/SellTotalcostSettlementvalueProfit/losscontractFigure21.2TheTradingGame—designedbyDavidEpstein,1999.21.6.2AttheEndoftheGameInthe‘SettlementValue’column,fillinthevalueofasinglecontractwiththefinalshareprice.Inthe‘Profit/Losspercontract’column,fillintheprofit/lossforasinglecontract.Inthe‘TotalProfit/Loss’fillinthetotalprofit/lossforthetrade(=profit/loss×quantity).ExampleDuringaround,yourtransactionsare:Buy10calloptions,withexerciseprice20,atacostof$2each.Sell1putoption,withexerciseprice15,atacostof$1.Buy5forwards,withforwardprice19.Yourtradingsheetshouldbefilledinasbelow:ContractBuy/SellCostperTotalcostSettlementvalueProfit/losscontractCall20Buy102Put15Sell11Forward19Buy5— atradinggameChapter21363Attheendofthegame,thefinalsharepriceis21.Consequently,thetradingsheetiscompletedasfollows:ContractBuy/SellCostperTotalcostSettlementvalueProfit/losscontractCall20Buy10221−20=11−2=−1−1×10=−10Put15Sell1101−0=+1+1×1=+1Forward19Buy5—21−19=2+2+2×5=+10Thetotalprofitandlossforthetraderistherefore−10+1+10=+1.Rememberthat:•Ifyoubuyacontract,yourprofit/loss=settlementvalue−costpercontract•Ifyousellacontract,yourprofit/loss=costpercontract−settlementvalue