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1、ExtendingFiglewski’soptionpricingformula0VickyHenderson1PrincetonUniversityDavidHobson2UniversityofBathTinoKluge3UniversityofOxfordSeptember20040WewouldliketothankGurdipBakshi,NikunjKapadiaandRobertTompkinsforkindlysharingtheirdatasets.WealsothankseminarparticipantsatStanfordUniversityandSteveFiglewskiforcommentsonapreviousversionofthispaper.ThesecondauthorissupportedbyanAdvancedFellowshipfromtheEPSRC.ThethirdauthoracknowledgespartialfinancialsupportfromDAAD,EPSRCandKWI.1ORFEandBendheimCenterfor。
2、Finance,E-Quad,PrincetonUniversity,Princeton,NJ.08544.USA.Email:vhenders@princeton.edu2DepartmentofMathematics,UniversityofBath,Bath.BA27AY.UK.Email:dgh@maths.bath.ac.uk3OCIAM,MathematicalInstitute,24-29StGiles’,Oxford.OX13LB.UK.Email:kluge@maths.ox.ac.ukExtendingFiglewski’soptionpricingformulaOneoftheusesofanoptionpricingmodelistoinferthepriceofanoptionfromthemarketpriceofa“nearby”option.Forexample,giventheBlack-ScholesoptionpricingformulaandthemarketpriceofanoptionitispossibletocalculatetheBla。
3、ck-Scholesimpliedvolatility.ThisvolatilitycanbesubstitutedbackintotheBlack-Scholespricingformulatogivethepriceofanyotherderivative.AsFiglewski(2002)haspointedout,iftheoptionpricingmodelistobeusedinthiswaythenthereisnothingspecialabouttheBlack-Scholesequationandanyfunctionwiththerightshape,couldinprin-ciplebeusedinstead.Figlewskisuggestsasimplealternativefunction.Unfortunatelyhisproposedfunctionviolatesstaticarbitrage.Wesuggestasimplemodificationwhichcorrectsforthisdeficiency.Wealsoshowhowtoincorpo。
4、ratematurityintothepricingmodel.Oncematurityisincludedinthemodelitispossibletoinferthedynamicsoftheunderlyingwhichareconsistentwiththepricingequations.WealsoundertakeanumericalinvestigationofthefitofboththeFiglewskimodelandourmodifiedversion.Indoingso,weoftenreachthesameconclusionsasFiglewski,butinterestingly,wealsosometimesfindtheoppositeresults.1TheBlackandScholes(1973)modelforoptionpricingistheindustrystandardandwonitsinventorsaNobelprize.Despiteitswidespreaduse,thetheoreticalunderpinningsofthem。
5、odelareoftenviolatedinpractice.Volatilityisnotconstant,andiswidelydocumentedtoexhibitsmilesandskews,seeRubinstein(1985).Oneoftheusesofanoptionpricingmodelistoinferanoptionpricefrommarketpricesof“nearby”options,perhapsinvolvingasimilarstrikeortimetomaturity.IntheBlackandScholes(1973)modelthisisaccomplishedviaimpliedvolatility.Forexampleyesterday’simpliedvolatilitymightbeusedtocomputeanoptionpricetodayoranoptionpricemightbecalculatedfrominterpolationbetweenimpliedvolatilitiesoftwooptionswithstrike。
6、sspanningthestrikeofinterest.TherecentpaperofFiglewski(2002)recognizesthatthisusageoftheBlack-Scholesoptionpricingformuladoesnotrelyonitspreciseform.Infactanyfunctionoftherightshapecouldbeusedinitsplace.FiglewskicomparestheBlack-Scholesformulawithan“informationallypassive”al-ternativemodel,which,followingFiglewski,werefertoastheFIGmodel1.ThepointisthattheFIGmodelisnotchosentoprovideabestfit,butratherisasimplestattemptatfindingapricingfunctionofapproximatelytherightshape.Therearetwodistinctusagesof。
7、theBlack-Scholesmodel.Inthefirstusageatradercalculatestheimpliedvolatilityfromasingleoptionandusesthatvolatilitytocalculatethepriceofarelatedsecurity.(Foreachdifferentsecuritythetraderwishestopricehemightcalibratewithadifferentoption.)Inthesecondusagethetradercalculatesthebest-fitimpliedvolatilityfrom1HereFIGcaneitherbetakenasanacronymforflexibleimpliedGoranabbreviationoftheauthor’ssurname.2asetoftradedoptionsofdifferentstrikes,andusesthatvolatilitytogivepricesforeachofadifferentsetofoptions.Thefirstoft。
8、heseapproachesrecognizesthatmarketdataadmitssmilesandskewsandallowsthetradertoaccountforthis.However,indoingthisthetraderisbeinginconsistentinhisuseofBlack-Scholes.Ontheonehandheisassumingthatvolatilityisconstant(whenapplyingtheBlack-Scholespricingfunction)andontheotherheisassumingthatdifferentvolatilitiescanbeappliedindifferentcases.Thesecondsituationsuffersnosuchinconsistency,butthenthetradercannotmatchhismodeltomarketdata,hecanonlygiveabestfit.Figlewski(2002)testshisalternativemodelagainstBlack-S。
9、cholesinbothoftheseusages.Inthefirstcaseheusestoday’soptionvaluetopredictthepricetomorrowofanoptionwiththesamestrikeandmaturity.Inthesecondcaseheusestoday’spricesofalltheoptionsofagivenmaturitytocalculateabestfitvolatility,whichisthenusedtopredicttomorrow’spricesforthosesameoptions.FiglewskifindsthathismodelprovidesroughlyasgoodafittothedataastheBlack-Scholesmodel.Inthefirstcase,whentheBlack-Scholesmodelisusedinconsistently,ittendstooutperformthepassivealternative,whereaswhenBlack-Scholesisusedconsis。
10、tently,theFIGmodelprovidesabetterfit.UnfortunatelythemodelFIGadmitssimplearbitrage.Formarketpa-rameters(basedonthedatausedbothbyFiglewskiandinthispaper)theFiglewskimodelwouldgiveapricerangingfrom50centsto$2foraputoptionwithzerostrike,whichmustbynecessitybeworthless.Withthisinmind,ourpapermakesatleastfourcontributionstotheliterature.Firstlyweproposeamodified“informationallypassive”model3MFIG,satisfyingstaticarbitrageconstraints.2Secondly,weshowhowtoin-corporateatimeparameterintoboththeFIGandMFIGmode。
本文标题:Extending Figlewski’s option pricing formula 0
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