Extending Figlewski’s option pricing formula 0

ExtendingFiglewski’soptionpricingformula0VickyHenderson1PrincetonUniversityDavidHobson2UniversityofBathTinoKluge3UniversityofOxfordSeptember20040WewouldliketothankGurdipBakshi,NikunjKapadiaandRobertTompkinsforkindlysharingtheirdatasets.WealsothankseminarparticipantsatStanfordUniversityandSteveFiglewskiforcommentsonapreviousversionofthispaper.ThesecondauthorissupportedbyanAdvancedFellowshipfromtheEPSRC.ThethirdauthoracknowledgespartialfinancialsupportfromDAAD,EPSRCandKWI.1ORFEandBendheimCenterforFinance,E-Quad,PrincetonUniversity,Princeton,NJ.08544.USA.Email:[e-mail]2DepartmentofMathematics,UniversityofBath,Bath.BA27AY.UK.Email:[e-mail]3OCIAM,MathematicalInstitute,24-29StGiles’,Oxford.OX13LB.UK.Email:[e-mail]’soptionpricingformulaOneoftheusesofanoptionpricingmodelistoinferthepriceofanoptionfromthemarketpriceofa“nearby”option.Forexample,giventheBlack-ScholesoptionpricingformulaandthemarketpriceofanoptionitispossibletocalculatetheBlack-Scholesimpliedvolatility.ThisvolatilitycanbesubstitutedbackintotheBlack-Scholespricingformulatogivethepriceofanyotherderivative.AsFiglewski(2002)haspointedout,iftheoptionpricingmodelistobeusedinthiswaythenthereisnothingspecialabouttheBlack-Scholesequationandanyfunctionwiththerightshape,couldinprin-ciplebeusedinstead.Figlewskisuggestsasimplealternativefunction.Unfortunatelyhisproposedfunctionviolatesstaticarbitrage.Wesuggestasimplemodificationwhichcorrectsforthisdeficiency.Wealsoshowhowtoincorporatematurityintothepricingmodel.Oncematurityisincludedinthemodelitispossibletoinferthedynamicsoftheunderlyingwhichareconsistentwiththepricingequations.WealsoundertakeanumericalinvestigationofthefitofboththeFiglewskimodelandourmodifiedversion.Indoingso,weoftenreachthesameconclusionsasFiglewski,butinterestingly,wealsosometimesfindtheoppositeresults.1TheBlackandScholes(1973)modelforoptionpricingistheindustrystandardandwonitsinventorsaNobelprize.Despiteitswidespreaduse,thetheoreticalunderpinningsofthemodelareoftenviolatedinpractice.Volatilityisnotconstant,andiswidelydocumentedtoexhibitsmilesandskews,seeRubinstein(1985).Oneoftheusesofanoptionpricingmodelistoinferanoptionpricefrommarketpricesof“nearby”options,perhapsinvolvingasimilarstrikeortimetomaturity.IntheBlackandScholes(1973)modelthisisaccomplishedviaimpliedvolatility.Forexampleyesterday’simpliedvolatilitymightbeusedtocomputeanoptionpricetodayoranoptionpricemightbecalculatedfrominterpolationbetweenimpliedvolatilitiesoftwooptionswithstrikesspanningthestrikeofinterest.TherecentpaperofFiglewski(2002)recognizesthatthisusageoftheBlack-Scholesoptionpricingformuladoesnotrelyonitspreciseform.Infactanyfunctionoftherightshapecouldbeusedinitsplace.FiglewskicomparestheBlack-Scholesformulawithan“informationallypassive”al-ternativemodel,which,followingFiglewski,werefertoastheFIGmodel1.ThepointisthattheFIGmodelisnotchosentoprovideabestfit,butratherisasimplestattemptatfindingapricingfunctionofapproximatelytherightshape.TherearetwodistinctusagesoftheBlack-Scholesmodel.Inthefirstusageatradercalculatestheimpliedvolatilityfromasingleoptionandusesthatvolatilitytocalculatethepriceofarelatedsecurity.(Foreachdifferentsecuritythetraderwishestopricehemightcalibratewithadifferentoption.)Inthesecondusagethetradercalculatesthebest-fitimpliedvolatilityfrom1HereFIGcaneitherbetakenasanacronymforflexibleimpliedGoranabbreviationoftheauthor’ssurname.2asetoftradedoptionsofdifferentstrikes,andusesthatvolatilitytogivepricesforeachofadifferentsetofoptions.Thefirstoftheseapproachesrecognizesthatmarketdataadmitssmilesandskewsandallowsthetradertoaccountforthis.However,indoingthisthetraderisbeinginconsistentinhisuseofBlack-Scholes.Ontheonehandheisassumingthatvolatilityisconstant(whenapplyingtheBlack-Scholespricingfunction)andontheotherheisassumingthatdifferentvolatilitiescanbeappliedindifferentcases.Thesecondsituationsuffersnosuchinconsistency,butthenthetradercannotmatchhismodeltomarketdata,hecanonlygiveabestfit.Figlewski(2002)testshisalternativemodelagainstBlack-Scholesinbothoftheseusages.Inthefirstcaseheusestoday’soptionvaluetopredictthepricetomorrowofanoptionwiththesamestrikeandmaturity.Inthesecondcaseheusestoday’spricesofalltheoptionsofagivenmaturitytocalculateabestfitvolatility,whichisthenusedtopredicttomorrow’spricesforthosesameoptions.FiglewskifindsthathismodelprovidesroughlyasgoodafittothedataastheBlack-Scholesmodel.Inthefirstcase,whentheBlack-Scholesmodelisusedinconsistently,ittendstooutperformthepassivealternative,whereaswhenBlack-Scholesisusedconsistently,theFIGmodelprovidesabetterfit.UnfortunatelythemodelFIGadmitssimplearbitrage.Formarketpa-rameters(basedonthedatausedbothbyFiglewskiandinthispaper)theFiglewskimodelwouldgiveapricerangingfrom50centsto$2foraputoptionwithzerostrike,whichmustbynecessitybeworthless.Withthisinmind,ourpapermakesatleastfourcontributionstotheliterature.Firstlyweproposeamodified“informationallypassive”model3MFIG,satisfyingstaticarbitrageconstraints.2Secondly,weshowhowtoin-corporateatimeparameterintoboththeFIGandMFIGmode