Risk Estimation on High Frequency Financial Data: Empirical Analysis of the DAX 30

78 Pages·2015·1.38 MB·English

by  Florian Jacob (auth.)

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Florian Jacob
Risk Estimation on High
Frequency Financial
Data
Empirical Analysis of the DAX 30
Florian Jacob
Karlsruhe, Germany
BestMasters
ISBN 978-3-658-09388-4 ISBN 978-3-658-09389-1 (eBook)
DOI 10.1007/978-3-658-09389-1
Library of Congress Control Number: 2015936902
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Acknowledgements
This work would not have been possible without the support, encouragement, and advice of
several people. Foremost, I would like to thank Dr. habil. Young Shin (Aaron) Kim for the
introduction to the topic during my time as a master student at the Karlsruher Institute of
Technology,aswellasfortheguidanceandsupportthroughoutthisthesis. Iwouldalsowantto
expressmygratitudetoProf. Dr. Wolf-DieterHellerandProf. Dr. Svetlozar(Zari)Rachevfor
giving me the opportunity to pursue my interests in Germany and during my two month stay
atStonyBrookUniversity,USA.Thistimeaswellashelpingmetodevelopaprofessionalskill
set,letmedevelopasaperson.
FlorianJacob
Contents
1. Introduction 1
2. Theory of Time Series Modeling and Risk Estimation 5
2.1. FinancialEconometrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.1. GARCHmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2. FIGARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2. ModelChoiceandValidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1. DetectingtheAutocorrelation. . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2. TestingforLong-RangeDependency . . . . . . . . . . . . . . . . . . . . . 11
2.3. Theinnovationprocess. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1. L´evyProcesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.2. Subordination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.3. Time-changedStochasticProcess . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.4. Time-changedBrownianMotion . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.5. Theα-stableProcess. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.6. TemperedStableDistribution . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.7. CTSSubordinator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.8. UnivariateNormalTemperedStableDistribution . . . . . . . . . . . . . . 23
2.3.9. StandardUnivariateNTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.10.MultivariateStandardNormalTemperedStableDistribution . . . . . . . 26
2.4. GoodnessofFit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4.1. Kolmogorov-Smirnov-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
viii Contents
2.4.2. Anderson-DarlingTest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5. RiskManagement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5.1. Value-at-Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5.2. CoherentRiskMeasures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5.3. AverageValue-at-Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5.4. Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.5.5. Backtesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3. Data and Methodology 37
3.1. DataSelection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2. DataTransformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3. AutocorrelationandDependence . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4. EmpiricalEstimationandGoFTests . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4.1. IndexReturns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4.2. StockReturns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4.3. ARMA-GARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5. ComparisonofRiskMeasuresforARMA-GARCHModels . . . . . . . . . . . . . 57
3.5.1. RiskPrediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.5.2. Backtestings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4. Conclusion 61
A. DAX30Stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Bibliography 67
List of Figures
2.1. IllustrationofautocorrelationoftheS&P500of01.01.2002-22.10.2012 . . . . . 14
2.2. Illustrationoftheroleofdifferentparametersintheα-stabledistribution. . . . . 19
2.3. CTSSubordinator-Influenceofθ,α . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4. IllustrationoftheroleofdifferentparametersintheNTSdistribution . . . . . . 25
3.1. IllustrationoftheDAX30IndexduringtheanalyzedtimeperiodbetweenMarch,
22009andDecember,292009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2. Intra-daily volatility profile for the DAX 30 index on 5 min log-returns for the
analyzed time period between March, 2 2009 and December, 29 2009. The sum
ofsquaredreturnsserveasaproxyfortheunobservedvolatilityinthischart. . 40
3.3. SampleAutocorrelationFunctionoftheabsolutelog-returnsfortheanalyzedtime
periodbetweenMarch,22009andDecember,292009. . . . . . . . . . . . . . . 41
3.4. SACFandSPACFforlog-returnsaswellastheSACFforabsoluteandsquared
log-returns for DAX 30 time series for 5 min high frequency data for the ana-
lyzedtimeperiodbetweenMarch,22009andDecember,292009. Theyexhibit
significantautocorrelationaswellassignificantandpersistentdependencyeffects. 43
3.5. R/SStatisticsforlog-returnsaswellasforabsolutelog-returnsforDAX30time
seriesfor5minhighfrequencydatafortheanalyzedtimeperiodbetweenMarch,
22009andDecember,292009. . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
x ListofFigures
3.6. KerneldensitiesofstandardizedresidualsofARMA(1,1)-GARCH(1,1)incompar-
ison with the fitted standard Normal Distribution and the fitted standard NTS
DistributionandQQ-PlotsforDAX30timeseriesfor5minhighfrequencydata
fortheanalyzedtimeperiodbetweenMarch,22009andDecember,292009. . . 46
3.7. Kernel densities of standardized residuals of ARMA(1,1)-FIGARCH(1,0.24, 1),
compared to fitted standard NTS Distribution and QQ-Plots for DAX 30 time
seriesfor5minhighfrequencydatafortheanalyzedtimeperiodbetweenMarch,
22009andDecember,292009. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.8. Backtesting of MNTS ARMA(1,1)-FIGARCH(1,d,1) and MNTS ARMA(1,1)-
GARCH(1,1)forbacktestingwindowof250.. . . . . . . . . . . . . . . . . . . . . 59
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