Extending Figlewski’s option pricing formula 0

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。