volatility case study the cliquet option

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1、CHAPTER56volatilitycasestudy:thecliquetoptionInthisChapter...•howthecliquetoptiondependsonvolatility•pricingwithvolatilityranges56.1INTRODUCTIONInChapter29Ilookedatthepricingofcliquetoptionsinadeterministicvolatilitysetting,includingtheinterpretationofthepathdependenc

2、yintermsofpartialdifferentialequations.Inthischapter,Iwillshowhowthecontractvaluedependsonthetreatmentofvolatility.Inparticular,Ishallshowresultsfortheuncertainvolatilitymodel.56.2THESUBTLENATUREOFTHECLIQUETOPTIONTraditionallyonemeasuressensitivitytovolatilityviatheve

3、ga.Thisisdefinedasthederivativeoftheoptionvaluewithrespecttoa(usuallyconstant)volatility.Thisnumberisthenusedtodeterminehowaccurateapricemightbeshouldvolatilitychange.Aspartofone’sriskmanagement,perhapsonewillvegahedgetoreducesuchsensitivity.Thisisentirelyreasonablewhe

4、nthecontractinquestionisanexchange-tradedvanillacontractandoneismeasuringsensitivitytothemarket’s(implied)volatility.However,whenitcomestotheriskmanagementofexoticoptionsthesensitivitytoaconstantvolatilityisatbestirrelevantandatworsttotallymisleading.Bynowthisiscommon

5、knowledgeandIdon’tneedtodwellonthedetails.Itsufficestosaythatwheneveracontracthasagammawhichchangessign(asdoesany‘interesting’exotic)vegamaybesmallatpreciselythoseplaceswheresensitivitytoactualvolatilityisverylarge.Confused?Asaruleofthumbifyouincreasevolatilitywhengamm

6、aispositiveyouwillincreaseacontract’svalue.Atpointsofinflectionintheoptionvalue(wheregammaiszero)theoptionvaluemayhardlymove.Butthisissensitivitytoaparameterthattakesthesamevalueeverywhere.Whatifyouincreasevolatilitywhengammaispositiveanddecreaseitwhengammaisnegative?T

7、heneteffectisanincreaseinoptionvalueevenatpointsofinflection.916PartFiveadvancedtopicsFigure56.1Exampletermsheetforacliquetoption.Skewsandsmilescanmakesmattersevenworse,unlessyouarefortunateenoughthatyourskew/smilemodelforecastsactualvolatilitybehavioraccurately.Thecla

8、ssicalreferencestothisphenomenonareAvellaneda,Levy&Paras(1995)andLyons´(1995).Andtherelevancetocliquetoptions?Takealookatthe