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页数:16页
时间:2018-02-11
《paul wilmott on quantitative finance stochastic volatility》由会员上传分享,免费在线阅读,更多相关内容在工程资料-天天文库。
1、CHAPTER51stochasticvolatilityInthisChapter...•modelingvolatilityasastochasticvariable•discontinuousvolatility•howtopricecontractswhenvolatilityisstochastic51.1INTRODUCTIONVolatilitydoesnotbehavehowtheBlack–Scholesequationwouldlikeittobehave;itisnotconstant
2、,itisnotpredictable,itisnotevendirectlyobservable.Thismakesitaprimecandidateformodelingasarandomvariable.Thereisplentyofevidencethatreturnsonequities,currenciesandcommoditiesarenotNormallydistributed;theyhavehigherpeaksandfattertailsthanpredictedbyaNormald
3、istribution.Wehaveseenthisinseveralplacesinthisbook.Thishasalsobeencitedasevidencefornon-constantvolatility.51.2RANDOMVOLATILITYBeforesayingwhatwemeanbystochasticvolatility,let’sbeclearaboutwhatrandomvolatilityisnotabout.Ifwedrawanumberatrandomfromadistrib
4、utionwitheithera10%volatility,20%volatilityor30%volatilitywithequalprobability,thentheresultingdistributionwouldhavethehigherpeak-fattertailspropertieswehaveseeninthedata.However,aslongasthestandarddeviationofthisdistributionisfinite(andscaleswiththesquarer
5、ootofthetimestep)wecanpriceoptionsusingthestandarddeviationinplaceofthevolatilityinBlack–Scholes(seealsoChapter47).Sothisisnotwhatwemeanwhenwetalkaboutstochasticvolatility.Thekeypointistimescale.Pricingonlybecomesaproblemwithrandomvolatilitywhenthetimescal
6、eoftheevolu-tionofvolatilityisofthesameorderastheevolutionoftheunderlying.Forexample,ifwehaveastochasticdifferentialequationmodelforthevolatilitythenwemustmovebeyondtheBlack–Scholesworld.854PartFiveadvancedtopics51.3ASTOCHASTICDIFFERENTIALEQUATIONFORVOLATI
7、LITYFigure51.1showsfourestimatesofvolatilityoftheDowJonesIndex,usingweightedaveragesofthedailychangesintheindexovertheprevious10,30,50and100days.Firstofall,thisgraphshowsthatvolatilityisnotconstant.Moreover,itisnotatallclearhowtomeasureit.Eachestimategives
8、adifferentanswer.Modelingvolatilityasastochasticprocessrequirestheretobesuchaquantityasvolatility...butmaybeobservabilityisn’tthatimportant.WecontinuetoassumethatSsatisfiesdS=µSdt+σSdX1,butwefurtherassumethatv
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