ARCH Models for Financial Applications

558 Pages·2010·5.3 MB·English

by  Evdokia Xekalaki

ARCH Models for
Financial Applications
ARCH Models for Financial Applications Evdokia Xekalaki and Stavros Degiannakis
© 2010 John Wiley & Sons Ltd. ISBN: 978-0-470-06630-0
ARCH Models for
Financial Applications
.
Evdokia Xekalaki Stavros Degiannakis
Department of Statistics
Athens University ofEconomics andBusiness, Greece
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Xekalaki,Evdokia.
ARCHmodelsforfinancialapplications/EvdokiaXekalaki,StavrosDegiannakis.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-470-06630-0(cloth)
1. Finance–Mathematicalmodels.2. Autoregression(Statistics) I.Degiannakis,Stavros.II.Title.
HG106.X452010
332.01’519536–dc22
2009052104
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ISBN978-0-470-06630-0(H/B)
Setin10/12ptTimesbyThomsonDigital,Noida,India
PrintedandboundinGreatBritainbyTJInternational,Padstow,Cornwall
To my husband and my son, a wonderful family
Evdokia Xekalaki
To thememory of the most importantperson in mylife, my father Antonis,
and to my mother and my brother
Stavros Degiannakis
Contents
Preface xi
Notation xv
1 What is an ARCH process? 1
1.1 Introduction 1
1.2 The autoregressiveconditionally heteroscedasticprocess 8
1.3 The leverage effect 13
1.4 The non-trading period effect 15
1.5 The non-synchronous tradingeffect 15
1.6 The relationshipbetween conditional variance and
conditional mean 16
1.6.1 The ARCH in mean model 16
1.6.2 Volatility and serial correlation 18
2 ARCHvolatilityspecifications 19
2.1 Model specifications 19
2.2 Methodsof estimation 23
2.2.1 Maximumlikelihood estimation 23
2.2.2 Numerical estimation algorithms 25
2.2.3 Quasi-maximum likelihood estimation 28
2.2.4 Other estimation methods 29
2.3 Estimating the GARCH model withEViews 6:
an empiricalexample 31
2.4 Asymmetric conditional volatility specifications 42
2.5 Simulating ARCH models using EViews 49
2.6 Estimating asymmetric ARCH models with [email protected]
OxMetrics: an empiricalexample 55
2.7 Misspecificationtests 66
2.7.1 The Box–Pierce and Ljung–Box Qstatistics 66
2.7.2 Tse’s residualbased diagnostic test for
conditional heteroscedasticity 67
2.7.3 Engle’sLagrangemultiplier test 67
2.7.4 Engle and Ng’s sign bias tests 68
2.7.5 The Breusch–Pagan,Godfrey,Glejser,Harvey andWhite tests 69
viii CONTENTS
2.7.6 TheWald, likelihood ratio andLagrange multipliertests 69
2.8 Other ARCH volatilityspecifications 70
2.8.1 Regime-switching ARCH models 70
2.8.2 Extended ARCH models 72
2.9 Other methods of volatilitymodelling 76
2.10 Interpretation ofthe ARCH process 82
Appendix 86
3 Fractionally integratedARCHmodels 107
3.1 Fractionally integrated ARCH modelspecifications 107
3.2 Estimating fractionally integratedARCHmodels using
[email protected] OxMetrics: an empirical example 111
3.3 A more detailed investigation of the normalityof the
standardizedresiduals: goodness-of-fittests 122
3.3.1 EDF tests 123
3.3.2 Chi-square tests 124
3.3.3 QQ plots 125
3.3.4 Goodness-of-fittests using EViews and G@RCH 126
Appendix 129
4 Volatility forecasting: an empirical example using EViews 6 143
4.1 One-step-ahead volatilityforecasting 143
4.2 Ten-step-ahead volatility forecasting 150
Appendix 154
5 Otherdistributionalassumptions 163
5.1 Non-normally distributedstandardized innovations 163
5.2 Estimating ARCH models with non-normally distributed
standardizedinnovations using [email protected] OxMetrics:
an empirical example 168
5.3 Estimating ARCH models with non-normally distributed
standardizedinnovations using EViews 6:an empirical example 174
5.4 Estimating ARCH models with non-normally distributed
standardizedinnovations using EViews 6:the logl object 176
Appendix 182
6 Volatility forecasting: an empirical example using G@RCH Ox 185
Appendix 195
7 Intraday realizedvolatility models 217
7.1 Realized volatility 217
7.2 Intraday volatilitymodels 220
7.3 Intraday realized volatilityandARFIMAX models
in G@RCH 4.2OxMetrics: anempirical example 223
7.3.1 Descriptivestatistics 223
CONTENTS ix
7.3.2 In-sample analysis 228
7.3.3 Out-of-sample analysis 232
8 Applications invalue-at-risk,expected shortfalland optionspricing 239
8.1 One-day-ahead value-at-risk forecasting 239
8.1.1 Value-at-risk 239
8.1.2 Parametric value-at-risk modelling 240
8.1.3 Intraday data andvalue-at-riskmodelling 242
8.1.4 Non-parametric andsemi-parametric value-at-risk modelling 244
8.1.5 Back-testing value-at-risk 245
8.1.6 Value-at-risk loss functions 248
8.2 One-day-aheadexpectedshortfall forecasting 248
8.2.1 Historical simulationand filtered historical simulation
for expectedshortfall 251
8.2.2 Loss functions for expectedshortfall 251
8.3 FTSE100 index:one-step-ahead value-at-risk and expected
shortfall forecasting 252
8.4 Multi-period value-at-riskand expectedshortfall forecasting 258
8.5 ARCH volatility forecasts inBlack–Scholes option pricing 260
8.5.1 Options 261
8.5.2 Assessingthe performance of volatility forecastingmethods 269
8.5.3 Black–Scholes option pricing usinga set ofARCHprocesses 270
8.5.4 Trading straddles based on aset of ARCH processes 271
8.5.5 Discussion 279
8.6 ARCH option pricing formulas 281
8.6.1 Computation of Duan’sARCHoption prices: an example 286
Appendix 288
9 Implied volatilityindices and ARCHmodels 341
9.1 Implied volatility 341
9.2 The VIX index 342
9.3 The implied volatility indexas anexplanatoryvariable 344
9.4 ARFIMAX model for implied volatilityindex 349
Appendix 352
10 ARCHmodel evaluation and selection 357
10.1 Evaluation ofARCH models 358
10.1.1 Model evaluationviewed interms ofinformation criteria 359
10.1.2 Model evaluationviewed interms ofstatistical
loss functions 360
10.1.3 Consistentranking 367
10.1.4 Simulation, estimation andevaluation 377
10.1.5 Point, interval and density forecasts 383
10.1.6 Model evaluationviewed interms ofloss functionsbased
on the use of volatilityforecasts 384
x CONTENTS
10.2 Selection of ARCH models 386
10.2.1 TheDiebold–Marianotest 386
10.2.2 TheHarvey–Leybourne–Newbold test 389
10.2.3 TheMorgan–Granger–Newboldtest 389
10.2.4 White’srealitycheck for datasnooping 390
10.2.5 Hansen’s superior predictiveabilitytest 390
10.2.6 Thestandardized predictionerror criterion 393
10.2.7 Forecastencompassing tests 400
10.3 Application ofloss functionsas methods of model selection 401
10.3.1 Applying the SPEC model selection method 401
10.3.2 Applying loss functionsas methods of model selection 402
10.3.3 Medianvalues ofloss functions asmethods
of model selection 407
10.4 TheSPAtest for VaR and expectedshortfall 408
Appendix 410
11 Multivariate ARCHmodels 445
11.1 Model Specifications 446
11.1.1 Symmetric model specifications 446
11.1.2 Asymmetric and long-memory modelspecifications 453
11.2 Maximumlikelihood estimation 454
11.3 Estimating multivariate ARCH models using EViews 6 456
11.4 Estimating multivariate ARCH models using [email protected] 465
11.5 Evaluation ofmultivariateARCHmodels 473
Appendix 475
References 479
Author Index 521
Subject Index 533
Preface
There has been wide interest throughout the financial literature on theoretical and
applied problems in the context of ARCH modelling. While a plethora of articles
existsinvariousinternationaljournals,theliteraturehasbeenrathersparsewhenit
comes to books with an exclusive focus on ARCH models. As a result, students,
academics in the area offinance and economics, and professional economists with
only a superficial grounding in the theoretical aspects of econometric modelling,
whileabletounderstandthebasictheoriesaboutmodelconstruction,estimationand
forecasting, often fail toget a grasp of howthesecan beused inpractice.
The present book addresses precisely these issues by interweaving practical
questions with approaches hinging on financial and statistical theory: we have
adopted an interactional exposition of the ARCH theory and its implementation
throughout. This is a book of practical orientation and applied nature intended for
readerswithabasicknowledgeoftimeseriesanalysiswishingtogainanaptitudein
theapplicationsoffinancialeconometricmodelling.Balancingstatisticalmethodol-
ogyandstructuraldescriptivemodelling,itaimstointroducereaderstotheareaof
discretetimeappliedstochasticvolatilitymodelsandtohelpthemacquiretheability
to deal with applied economic problems. It provides background on the theory of
ARCHmodels,butwithafocusonpracticalimplementationviaapplicationstoreal
data (the accompanying CD-ROM provides programs and data) and via examples
worked with econometricspackages (EViews and the G@RCHmodule for the Ox
package)withstep-by-stepexplanationsoftheiruse.Readersarefamiliarizedwith
theoreticalissuesofARCHmodelsfrommodelconstruction,fittingandforecasting
throughtomodelevaluationandselection,andwillgainfacilityinemployingthese
models in the context of financial applications: volatility forecasting, value-at-risk
forecasting,expectedshortfallestimation,andvolatilityforecastsforpricingoptions.
Chapter 1 introduces the concept of an autoregressive conditionally heterosce-
dastic(ARCH)processanddiscussestheeffectsthatvariousfactorshaveonfinancial
time series such as the leverage effect, the non-trading period effect, and the non-
synchronous trading effect. Chapter 2 provides an anthology of representations of
ARCHmodelsthathavebeenconsideredintheliterature.Estimationandsimulation
ofthemodelsisdiscussed,andseveralmisspecificationtestsareprovided.Chapter3
deals with fractionally integrated ARCH models and discusses a series of tests for
testing the hypothesis of normality of the standardized residuals. Chapter 4 famil-
iarizes readers with the use of EViews in obtaining volatility forecasts. Chapter 5
treats the case of ARCH models with non-normally distributed standardized