PRICING DIGITAL OUTPERFORMANCE OPTIONS WITH UNCERTAIN CORRELATION

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1、August8,20119:56WSPC/S0219-0249104-IJTAFSPI-J071S0219024911006425InternationalJournalofTheoreticalandAppliedFinanceVol.14,No.5(2011)709–722
cWorldScientificPublishingCompanyDOI:10.1142/S0219024911006425PRICINGDIGITALOUTPERFORMANCEOPTIONSWITHUNCERTAINCORRELATIONJACIN

2、TOMARABELROMOBBVA,V´ıadelosPobladoss/n28033Madrid,SpainandUniversityInstituteforEconomicandSocialAnalysisUniversityofAlcal´a,PlazadelaVictoria228802Alcal´adeHenares,Spainjacinto.marabel@grupobbva.comReceived28March2010Accepted20December2010Multi-assetoptionsexhibit

3、sensitivitytothecorrelationsbetweentheunderlyingassetsandthesecorrelationsarenotoriouslyunstable.Moreover,someoftheseoptionssuchasthedigitaloutperformanceoptions,haveacross-gammathatchangessigndependingontherelativeevolutionoftheunderlyingassets.Inthispaper,Ipresen

4、tamodeltopricedigitaloutperformanceoptionswhenthereisuncertaintyaboutcorrelation,butitisassumedtoliewithinacertainrange.UndertheassumptionthatassetspricesfollowaGeometricBrownianmotionwithconstantinstantaneousvolatilitiesIpresentananalyticexpressionforthepriceofthe

5、digitaloutperformanceoptionunderthecon-stantcorrelationassumption,aswellasthepartialdifferentialequationcorrespondingtotheuncertaincorrelationmodel.Thecomparisonofthepricesobtainedusingbothmodelsshowsthatthereisnoconstantcorrelationwhichallowsattainingthepriceobtain

6、edundertheuncertaincorrelationmodel.Thisfactshowsthatitcanbedanger-oustoassumeaconstantinstantaneouscorrelationforproductswithacross-gammathatchangessign.Keywords:Digitaloutperformanceoptions;uncertaincorrelation;cross-gamma.1.IntroductionThereareanumberofmodelstop

7、riceexoticoptions.Thesemodelsareincludedinthreemaincategories:stochasticvolatility,localvolatilityandjumpsmodels.StochasticvolatilitymodelsleavetheconstantinstantaneousvolatilityassumptionoftheBlack-Scholes[3]model,andassumethatvolatilityfollowsastochasticpro-cessp

8、ossiblycorrelatedwiththeprocessforthestockprice.WithinthisgroupareHullandWhite[13]andHeston[12]stochasticvolatilitymodels.Merton[17],amongothers,