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1、CHAPTER54stochasticvolatilityandmean-varianceanalysisInthisChapter...?howtoanalyzeriskwhenvolatilityisstochastic?mean-varianceanalysis54.1INTRODUCTIONItendnottolikeanymodelthatrequirestheinputofamar-ketpriceofrisk.Themainreasonisthatthisquantityisnotdirectlyobservable.Atbestitcan
2、bededucedfromthepricesofderivatives,socalled‘?tting.’Butthisisfarfromadequate,sincethe?ttingwillonlyworkifthosewhosetthepricesofderivativesareusingthesamemodelandtheyareconsistentinthatthe?ttedmarketpriceofriskdoesnotchangewhenthemodelisre?ttedafewdayslater.ToseewhatImean,thinkba
3、cktoChapter50.InthatchapterIshowedhowto?ndthelocaldeterministicvolatilitysurfacethatisconsistentwithallquotedpricesofvanillaoptions.Ifafewdayslaterwe?ndthatthissurfacehaschanged,stillbeingchosentomatchmarketprices,thenthemodelwaswrong.Exactlythesameproblemoccurswhenwehavestochast
4、icvolatilityandwehave?ttedto?ndthemarketpriceofvolatilityrisk.1Whetherwehaveadeterministicvolatilitysurfaceorastochasticvolatilitymodelwithprescribedor?ttedmarketpriceofrisk,wewillalwaysbefacedwithhowtointerpretre?tting.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;themodi?cationsneededtoallowfordividendsaretheusual.Wearegoingtoexaminethestatisticalpropertiesofaportfoliothattriestoreplicateascloselyaspossibletheoriginaloptionposition.Wewillnothedgetheportfoliodynamicallywithotheroptionssoourportfoliowillnotberiskfree.Insteadw
8、ewillexaminethemeanandvarianceofthevalue
