stochastic volatility and mean-variance analysis

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1、CHAPTER54stochasticvolatilityandmean-varianceanalysisInthisChapter...•howtoanalyzeriskwhenvolatilityisstochastic•mean-varianceanalysis54.1INTRODUCTIONItendnottolikeanymodelthatrequirestheinputofamar-ketpriceofrisk.Themainreasonisthatthisquantityisnotdirectlyobservable.Atbestitcan

2、bededucedfromthepricesofderivatives,socalled‘fitting.’Butthisisfarfromadequate,sincethefittingwillonlyworkifthosewhosetthepricesofderivativesareusingthesamemodelandtheyareconsistentinthatthefittedmarketpriceofriskdoesnotchangewhenthemodelisrefittedafewdayslater.ToseewhatImean,thinkba

3、cktoChapter50.InthatchapterIshowedhowtofindthelocaldeterministicvolatilitysurfacethatisconsistentwithallquotedpricesofvanillaoptions.Ifafewdayslaterwefindthatthissurfacehaschanged,stillbeingchosentomatchmarketprices,thenthemodelwaswrong.Exactlythesameproblemoccurswhenwehavestochast

4、icvolatilityandwehavefittedtofindthemarketpriceofvolatilityrisk.1Whetherwehaveadeterministicvolatilitysurfaceorastochasticvolatilitymodelwithprescribedorfittedmarketpriceofrisk,wewillalwaysbefacedwithhowtointerpretrefitting.Wasthemarketwrongbeforebutisnowright,orwasthemarketcorrectin

5、itiallyandnowtherearearbitrageopportunities?Wewon’tbefacedwithawkwardquestionslikethisifwedon’texpectourmodel,whateveritmaybe,togiveuniqueandcorrectvalues.Inthischapterwe’llseehowtoestimateprobabilitiesforpricesbeingcorrect.Wedothisbyonlydeltahedgingandnotdynamicallyvegahedging.I

6、nsteadwelookatmeansandvariancesforoptionvalues.54.2THEMODELFORTHEASSETANDITSVOLATILITYWearegoingtoworkwiththeveryclassicalmodeldS=µSdt+σSdX11Icallthisthemarket-price-of-riskrisk.890PartFiveadvancedtopicsanddσ=p(S,σ)dt+q(S,σ)dX2withacorrelationofρ(S,σ).We’llonlyconsideranon-divide

7、nd-payingasset;themodificationsneededtoallowfordividendsaretheusual.Wearegoingtoexaminethestatisticalpropertiesofaportfoliothattriestoreplicateascloselyaspossibletheoriginaloptionposition.Wewillnothedgetheportfoliodynamicallywithotheroptionssoourportfoliowillnotberiskfree.Insteadw

8、ewillexaminethemeanandvarianceofthevalue