paul wilmott on quantitative finance stochastic volatility

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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