Econometric Analysis of Volatility in Financial Additive Noise Models

252 Pages·2014·2.43 MB·English

2014-12
Rasmus Tangsgaard Varneskov
PhD Thesis
Econometric Analysis of Volatility in Financial
Additive Noise Models
DEPARTMENT OF ECONOMICS AND BUSINESS
AARHUS UNIVERSITY (cid:2) DENMARK
ECONOMETRIC ANALYSIS OF VOLATILITY IN
FINANCIAL ADDITIVE NOISE MODELS
By Rasmus Tangsgaard Varneskov
APhDthesissubmittedto
SchoolofBusinessandSocialSciences,AarhusUniversity,
inpartialfulfilmentoftherequirementsof
thePhDdegreein
EconomicsandBusiness
August2014
CREATES
Center for Research in Econometric Analysis of Time Series

PREFACE
ThisthesiswaswrittenintheperiodSeptember2010throughAugust2014during
mygraduatestudiesattheDepartmentofEconomicsandBusinessatAarhusUni-
versity,theDepartmentofEconomicsatBostonUniversity,theFinanceDepartment
atKelloggSchoolofManagement,NorthwesternUniversity,andattheOxford-Man
Institute,UniversityofOxford.Iwouldliketothankallfourinstitutionsforproviding
mewithexcellentresearchenvironments.Iamalsogratefulforthefinancialsupport
throughoutmystudiesprovidedbytheDepartmentofEconomicsandBusiness,
AarhusUniversity,theCenterforResearchinEconometricAnalysisofTimeSeries
(CREATES),fundedbytheDanishNationalResearchFoundation,KøbmandFerdi-
nandSallingsMindefond,OticonFonden,KnudHøjgaardsFond,Statsautoriseret
revisorOlufChristianOlsenoghustruJulieRasmineOlsensMindefond,andFonden
JuliusSkrikesStiftelse.
I would like to thank my supervisor Bent Jesper Christensenfor hisvaluable
guidance,hishelpwithapplications,andhisinsightfulcontributionstoourjoint
researchproject.Iwouldalsoliketothankmyco-supervisorAsgerLundeforhelping
meoutwithdataissuesandNielsHaldrupforhisencouragement.Inaddition,Iam
gratefultoValeriVoevforintroducingmetohigh-frequencyfinancialeconometrics,
alwayshavinganopendoorpolicy,andhiscontinuingsupport.
FromAugust2012throughJanuary2013,IhadthepleasureofvisitingNeilShep-
hardattheOxford-ManInstitute,UniversityofOxford.IwishtothankNeilforinviting
meandforsomeverystimulatingdiscussionsthathave,amongothers,improved
thefirsttwochaptersofthisthesis.Furthermore,IamthankfultoOMIfortheir
hospitalityandtoBentNielsenforinvitingmetoattendsomeinterestingseminars
anddinnersatNuffieldCollege.
IhavebeenveryfortunatetovisittheDepartmentofEconomicsatBostonUni-
versitytwice.Firstduringthe2010fallsemesterandsubsequentlyfromApril2013
throughmid-June2013.IhavealwaysenjoyedthewelcomingatmosphereatBU.I
amgratefultoPierrePerronforhisteachings,ongoingadvice,andhisdedicationto
ourjointresearchpaper,whichIampleasedtohaveaspartofthisthesis.Working
togetherhasbeenaninspiringlearningexperience.
i
ii
Forthepastthreeyears,IhavehadthegreatpleasureofworkingwithTorbenG.
Andersenontworesearchpapers,thefirstofwhichcomprisesthefourthchapter
ofthisthesis.OurcollaborationpromptedmetovisittheFinanceDepartmentat
KelloggSchoolofManagement,NorthwesternUniversity,fromOctober2013through
December2013,whosehospitalityIamgratefulfor.Whetheritwasduringoneof
ourweeklySkypemeetingsorourworkingsessionsinFlorence,London,Montreal
andSanDiego,Ihavealwayslearnedalotfromourdiscussions.Icansafelysaythat
mythesiswouldnothavebeenthesamewithoutthem.Iamtrulygratefulforour
collaboration,andIhopethatwewillcontinueitintheyearstocome.
NomatterwhereIhavebeen,Ihavehadtheprivilegetobesurroundedbygreat
PhDcolleagues,andIhaveverymuchenjoyedyourcompanyduringthisendeavor.
Thankyou.Tomyvariousofficemates,AndreasEmmertsen,FedericoCarlini,Diaa
Noureldin,andDorLe,Iappreciateyourendurancewhenlisteningtometalkabout
financialeconometrics,butmostlyforremindingmetotakeabreakonceinawhile.
Inaddition,ManuelLukas,KasperV.Olesen,MartinSchultz-Nielsen,JesperWulff,
YaseminSatır,AdamMcCloskey,JamesWolter,MathiasKruttli,andSigridKällblad
deserveaspecialthanksformanyacademicandnon-academicdiscussions.
Lastly,Iwishtothankmyfriends,myparents,CarstenandLone,andmybrother
Tobiasforyourpatienceandsupportovertheyears,andforbearingoverwithme
beingslightlyabsentmindedattimes.
RasmusTangsgaardVarneskov
Aarhus,August2014
UPDATED PREFACE
ThepredefencemeetingwithanassessmentcommitteeconsistingofAndersRah-
bek,UniversityofCopenhagen,OliverLinton,UniversityofCambridge,andKim
Christensen,AarhusUniversity,tookplaceonOctober9th,2014.Iamthankfultothe
membersofthecommitteefortheircarefulreadingofmythesisalongwithanumber
ofconstructivecommentsandsuggestionsthatwillproveusefulintryingtopublish
itsfourchapters,inadditiontoopenupinterestingavenuesforfutureresearch.
RasmusTangsgaardVarneskov
Aarhus,November2014
iii

CONTENTS
Summary ix
DanskResumé xiii
1 EstimatingtheQuadraticVariationSpectrumofNoisyAssetPricesusing
GeneralizedFlat-topRealizedKernels 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 ASemimartingaleSetupandAssumptions . . . . . . . . . . . . . . . 6
1.2.1 TheEfficientPriceProcess . . . . . . . . . . . . . . . . . . . . 7
1.2.2 TheNoiseProcess . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 TheRealizedKernelApproach . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 Flat-TopRealizedKernels . . . . . . . . . . . . . . . . . . . . . 13
1.3.2 AsymptoticVarianceandOptimalBandwidthSelection . . . 16
1.3.3 SelectingtheFlat-topShrinkage . . . . . . . . . . . . . . . . . 17
1.3.4 RelationtoJack-KnifeKernels . . . . . . . . . . . . . . . . . . 18
1.3.5 RelationtothePre-AveragingApproach . . . . . . . . . . . . 19
1.4 Flat-topRealizedKernelsandJumps . . . . . . . . . . . . . . . . . . . 21
1.4.1 TheObservablePriceProcesswithJumps . . . . . . . . . . . 21
1.4.2 BlockSamplingandJump-robustEstimation . . . . . . . . . 23
1.5 SimulationStudy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.5.1 SimulationDesign . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.5.2 SelectingBandwidthandFlat-topShrinkage. . . . . . . . . . 26
1.5.3 RelativeFiniteSamplePerformanceofRealizedEstimators . 27
∗ ∗
1.5.4 FiniteSampleBehaviorofBRK andMBRK . . . . . . . . . 29
1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.7 Appendix:FiguresandTables . . . . . . . . . . . . . . . . . . . . . . . 31
1.8 Appendix:ABias-CorrectedPre-AveragingEstimator . . . . . . . . . 37
1.9 Appendix:Proofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.9.1 ProofofTheorem1. . . . . . . . . . . . . . . . . . . . . . . . . 39
1.9.2 ProofofProposition1 . . . . . . . . . . . . . . . . . . . . . . . 45
v
vi CONTENTS
1.9.3 ProofofProposition2 . . . . . . . . . . . . . . . . . . . . . . . 46
1.9.4 ProofofLemma2. . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.9.5 ProofofTheorem2. . . . . . . . . . . . . . . . . . . . . . . . . 47
1.9.6 ProofofLemma3. . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.9.7 ProofsofTheorems3and4 . . . . . . . . . . . . . . . . . . . . 52
1.10 Appendix:TechnicalResultsandDefinitions . . . . . . . . . . . . . . 56
2 Flat-TopRealizedKernelEstimationofQuadraticCovariationwithNon-
SynchronousandNoisyAssetPrices 67
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.2 TheoreticalSetup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.2.1 TheSemimartingaleProcess . . . . . . . . . . . . . . . . . . . 71
2.2.2 SynchronizationSchemes. . . . . . . . . . . . . . . . . . . . . 73
2.2.3 AnEncompassingAdditiveNoiseModel . . . . . . . . . . . . 74
2.3 Flat-TopRealizedKernelEstimation . . . . . . . . . . . . . . . . . . . 76
2.3.1 MotivationforFlat-TopKernels . . . . . . . . . . . . . . . . . 78
2.3.2 CentralLimitTheory. . . . . . . . . . . . . . . . . . . . . . . . 80
2.4 FiniteSampleAdjustments,ApplicationsandImplementation . . . 81
2.4.1 APositiveDefiniteProjection. . . . . . . . . . . . . . . . . . . 82
2.4.2 Non-LinearTransformations . . . . . . . . . . . . . . . . . . . 83
2.4.3 TheChoiceofKernelandTuningParameters . . . . . . . . . 83
2.5 AnInitialGaugeatTick-by-TickTradeData . . . . . . . . . . . . . . . 84
2.6 SimulationStudy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.6.1 SimulationDesign . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.6.2 SynchronizationEffectsandReturnAutocorrelation . . . . . 87
2.6.3 TheImpactofTruncatingEigenvalues . . . . . . . . . . . . . 89
2.6.4 RelatedandCompetingEstimators . . . . . . . . . . . . . . . 89
2.6.5 EstimatesofQuadraticCovariationandNon-linearTransfor-
mations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
2.7 EmpiricalAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.7.1 AnalyzingIn-SampleInformation . . . . . . . . . . . . . . . . 93
2.7.2 Out-of-SampleForecasting . . . . . . . . . . . . . . . . . . . . 95
2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
2.9 Appendix:TablesandFigures . . . . . . . . . . . . . . . . . . . . . . . 97
2.10 Appendix:AdditionalTheory . . . . . . . . . . . . . . . . . . . . . . . 105
2.10.1 MatrixConcepts . . . . . . . . . . . . . . . . . . . . . . . . . . 105
2.10.2 AsymptoticVarianceofNon-linearTransformations . . . . . 105
2.11 Appendix:Proofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
CONTENTS vii
2.11.1 ProofsofLemmas1-2andTheorem1. . . . . . . . . . . . . . 105
2.11.2 ProofofTheorem2. . . . . . . . . . . . . . . . . . . . . . . . . 110
3 CombiningLongMemoryandLevelShiftsinModelingandForecasting
theVolatilityofAssetReturns 113
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.2 TheStochasticVolatilityModel:MotivationandSpecification . . . . 117
3.2.1 Discrete-andContinuous-timeSVModels. . . . . . . . . . . 117
3.2.2 TheStochasticVolatilityProcess . . . . . . . . . . . . . . . . . 119
3.2.3 TheDataandConstructionoftheVolatilitySeries . . . . . . 121
3.2.4 AnInitialGaugeattheVolatilityDynamics. . . . . . . . . . . 122
3.3 EconometricMethodology . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.3.1 StateSpaceRepresentation . . . . . . . . . . . . . . . . . . . . 128
3.3.2 MaximumLikelihoodEstimation . . . . . . . . . . . . . . . . 129
3.3.3 ForecastingwiththeRLS-ARFIMAModel . . . . . . . . . . . 131
3.4 HandlingMeasurementErrors . . . . . . . . . . . . . . . . . . . . . . 132
3.4.1 MeasurementErrorsandtheARFIMArepresentation . . . . 132
3.4.2 MotivationalEvidencefromanRLS-LMSV(1,d)Model . . . 133
3.5 SimulationStudy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3.5.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.5.2 SimulationResults . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.6 EmpiricalAnalysisofAssetReturnVolatility. . . . . . . . . . . . . . . 138
3.6.1 Full-SampleParameterEstimates . . . . . . . . . . . . . . . . 139
3.6.2 ForecastingPerformanceEvaluation . . . . . . . . . . . . . . 141
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
3.8 Appendix:FiguresandTables . . . . . . . . . . . . . . . . . . . . . . . 148
3.9 Appendix:AdditionalTheory . . . . . . . . . . . . . . . . . . . . . . . 160
3.9.1 TheFlat-topRealizedKernelApproach . . . . . . . . . . . . . 160
3.9.2 TheModelConfidenceSet . . . . . . . . . . . . . . . . . . . . 161
3.10 Appendix:Proofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
3.10.1 ProofofProposition1 . . . . . . . . . . . . . . . . . . . . . . . 162
3.10.2 ProofofProposition2 . . . . . . . . . . . . . . . . . . . . . . . 163
4 OntheInformationalEfficiencyofOption-ImpliedandTimeSeriesFore-
castsofRealizedVolatility 165
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
4.2 TraditionalVolatilityForecastAnalysis . . . . . . . . . . . . . . . . . . 168
4.2.1 TheQuadraticReturnVariation . . . . . . . . . . . . . . . . . 168
4.2.2 RealizedVarianceMeasurementandForecasting . . . . . . . 169