Mean Reverting Stochastic Volatility Models

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1、MeanRevertingStochasticVolatilityModelsforEquityIndicesSubmitted:August2009Candidate:CliveHillSupervisor:Prof.JohnHatgioannidesThisdissertationissubmittedaspartoftherequirementsfortheawardoftheMScinMathematicalTradingandFinance.MathematicalTradingandFinanc

2、ePT2007-20091CliveHillAbstractThepurposeofthisprojectistoinvestigatethemeanrevertingnatureofequityvolatilities,andtoshowthattheassumptionofconstantvolatilityisinvalid.Fourspecificstochasticvolatilitymodelsarechosenwiththeaimofprovingthesecharacteristics.Tw

3、overydifferentmethodsarethenusedtofindtheunknownparametersofthemodels:Hansen’sGMMusinghistoricalimpliedvolatilities,andcalibrationagainstoptionmarketdataonaspecificday.Theresultsshowthatvolatilityisstronglymeanreverting,andhashighlysignificantvolofvolparam

4、etervaluesforallstochasticvolatilitymodelsconsidered,thusdemonstratingthatvolatilityisnotconstant.MonteCarlosimulationsarethenappliedtotestthestabilityoftheresults.Thisfindsthatthebiasissmallforallthemodelsconsidered,buthasrelativelyhighvaluesofstandarddev

5、iation,meaningthattheparametervaluesobtainedarequiteunstable.Thisisparticularlytrueofthemeanreversionparameter.AcknowledgementsIwouldliketothankmywifeforhersupportandpatience,specificallyoverthefinalmonthsofthisMSc;tomyparents,withouttheirsupportthisdisser

6、tationwouldnothavebeenpossible;and,finally,tomysupervisorJohnHatgioannides,forhisvaluablefeedbackandguidancethroughout.MathematicalTradingandFinancePT2007-20092CliveHillTableofContentsExecutiveSummary........................................................

7、................................51Introduction...................................................................................................71.1Notation..................................................................................................9

8、2LiteratureReview.......................................................................................103StochasticVolatilityModels......................................................................224Pa