Statistical Models and Methods for Financial Markets

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Statistical Models and Methods for Financial Markets
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ISBN:978-0-387-77826-6e-ISBN:978-0-387-77827-3
DOI:10.1007/978-0-387-77827-3
LibraryofCongressControlNumber:2008930111
2008 Springer Science+BusinessMedia,LLC

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关键词：Statistical statistica financial statistic inancial Methods Statistical financial models Markets

Preface ........................................................vii
PartIBasicStatisticalMethodsandFinancialApplications
1LinearRegressionModels .................................3
1.1Ordinaryleastsquares(OLS)............................4
1.1.1Residualsandtheirsumofsquares..................4
1.1.2Propertiesofprojectionmatrices....................5
1.1.3Propertiesofnonnegativede?nitematrices...........6
1.1.4StatisticalpropertiesofOLSestimates...............7
1.2Statisticalinference......................................8
1.2.1Con?denceintervals...............................8
1.2.2ANOVA(analysisofvariance)tests..................10
1.3Variableselection........................................12
1.3.1Test-basedandothervariable selectioncriteria........12
1.3.2Stepwisevariableselection..........................15
1.4Regressiondiagnostics...................................16
1.4.1Analysisofresiduals...............................17
1.4.2In?uencediagnostics...............................18
1.5Extensiontostochasticregressors.........................19
1.5.1Minimum-variancelinearpredictors..................19
1.5.2Futuresmarketsandhedgingwithfuturescontracts...20
1.5.3Inferenceinthecaseofstochasticregressors..........21
1.6Bootstrappinginregression...............................22
1.6.1Theplug-inprincipleandbootstrapresampling.......22
1.6.2Bootstrappingregressionmodels....................24
1.6.3Bootstrapcon?denceintervals.......................25
1.7Generalizedleastsquares.................................25

xivContents
1.8Implementationandillustration...........................26
Exercises...................................................32
2MultivariateAnalysisandLikelihoodInference ............37
2.1Jointdistributionofrandomvariables......................38
2.1.1Changeofvariables................................39
2.1.2Meanandcovariancematrix........................39
2.2Principalcomponentanalysis(PCA).......................41
2.2.1Basicde?nitions...................................41
2.2.2Propertiesofprincipalcomponents..................42
2.2.3Anexample:PCAofU.S.Treasury-LIBOR
swaprates........................................44
2.3Multivariatenormaldistribution...........................48
2.3.1De?nitionanddensityfunction......................48
2.3.2Marginalandconditionaldistributions...............50
2.3.3Orthogonalityandindependence,withapplications
toregression......................................50
2.3.4SamplecovariancematrixandWishartdistribution....52
2.4Likelihoodinference.....................................55
2.4.1Methodofmaximumlikelihood.....................55
2.4.2Asymptoticinference..............................58
2.4.3Parametricbootstrap..............................59
Exercises...................................................60
3BasicInvestmentModelsandTheirStatisticalAnalysis ...63
3.1Assetreturns...........................................64
3.1.1De?nitions.......................................64
3.1.2Statisticalmodelsforassetpricesandreturns.........66
3.2Markowitz!ˉsportfoliotheory..............................6
3.2.1Portfolioweights..................................67
3.2.2Geometryofe?cientsets...........................68
3.2.3Computationofe?cientportfolios...................69
3.2.4Estimationof and andanexample..............71
3.3Capitalassetpricingmodel(CAPM).......................72
3.3.1Themodel........................................72
3.3.2Investmentimplications............................77
3.3.3Estimationandtesting.............................77
3.3.4EmpiricalstudiesofCAPM.........................79
3.4Multifactorpricingmodels................................81
3.4.1Arbitragepricingtheory............................81
3.4.2Factoranalysis....................................82
3.4.3ThePCAapproach................................85
3.4.4TheFama-Frenchthree-factormodel.................86

Contentsxv
3.5Applicationsofresamplingtoportfoliomanagement.........87
3.5.1Michaud!ˉsresamplede?cientfrontier................8
3.5.2Bootstrapestimatesofperformance..................88
Exercises...................................................89
4ParametricModelsandBayesianMethods.................93
4.1Maximumlikelihoodandgeneralizedlinearmodels...........94
4.1.1NumericalmethodsforcomputingMLE..............94
4.1.2Generalizedlinearmodels..........................95
4.2Nonlinearregressionmodels...............................97
4.2.1TheGauss-Newtonalgorithm.......................98
4.2.2Statisticalinference................................100
4.2.3Implementationandanexample.....................101
4.3Bayesianinference.......................................103
4.3.1Priorandposteriordistributions....................103
4.3.2Bayesprocedures..................................104
4.3.3Bayesestimatorsofmultivariatenormalmean
andcovariancematrix.............................105
4.3.4BayesestimatorsinGaussianregressionmodels.......107
4.3.5EmpiricalBayesandshrinkageestimators............108
4.4Investmentapplicationsofshrinkageestimatorsand
Bayesianmethods.......................................109
4.4.1Shrinkageestimatorsof and fortheplug-in
e?cientfrontier...................................110
4.4.2AnalternativeBayesianapproach...................111
Exercises...................................................113
5TimeSeriesModelingandForecasting .....................115
5.1Stationarytimeseriesanalysis............................115
5.1.1Weakstationarity.................................115
5.1.2Testsofindependence..............................117
5.1.3WolddecompositionandMA,AR,andARMAmodels.119
5.1.4ForecastinginARMAmodels.......................121
5.1.5Parameterestimationandorderdetermination........122
5.2Analysisofnonstationarytimeseries.......................123
5.2.1Detrending.......................................123
5.2.2Anempiricalexample..............................124
5.2.3Transformationanddi?erencing.....................128
5.2.4Unit-rootnonstationarityandARIMAmodels........129
5.3Linearstate-spacemodelsandKalman?ltering..............130
5.3.1Recursiveformulasfor P
,x ,and x .........131
t|t?1 t|t?1 t|t
5.3.2Dynamiclinearmodelsandtime-varyingbetas
inCAPM........................................133

xviContents
Exercises...................................................135
6DynamicModelsofAsserReturnsandTheirVolatilities ..139
6.1Stylizedfactsontimeseriesofassetreturns.................140
6.2Movingaverageestimatorsoftime-varyingvolatilities........144
6.3Conditionalheteroskedasticmodels........................146
6.3.1TheARCHmodel.................................146
6.3.2TheGARCHmodel...............................147
6.3.3TheintegratedGARCHmodel......................152
6.3.4TheexponentialGARCHmodel.....................152
6.4TheARMA-GARCHandARMA-EGARCHmodels.........155
6.4.1Forecastingfuturereturnsandvolatilities.............156
6.4.2Implementationandillustration.....................156
Exercises...................................................157
PartIIAdvancedTopicsinQuantitativeFinance
7NonparametricRegressionandSubstantive-Empirical
Modeling ..................................................163
7.1Regressionfunctionsandminimum-varianceprediction.......164
7.2Univariatepredictors.....................................165
7.2.1Running-mean/running-linesmoothersandlocal
polynomialregression..............................165
7.2.2Kernelsmoothers..................................166
7.2.3Regressionsplines.................................166
7.2.4Smoothingcubicsplines............................169
7.3Selectionofsmoothingparameter..........................170
7.3.1Thebias-variancetrade-o?.........................170
7.3.2Cross-validation...................................171
7.4Multivariatepredictors...................................172
7.4.1Tensorproductbasisandmultivariateadaptive
regressionsplines..................................172
7.4.2Additiveregressionmodels.........................173
7.4.3Projectionpursuitregression........................174
7.4.4Neuralnetworks...................................174
7.5Amodelingapproachthatcombinesdomainknowledgewith
nonparametricregression.................................176
7.5.1Penalizedsplinemodelsandestimationof
forwardrates.....................................177
7.5.2Asemiparametricpenalizedsplinemodelforthe
forwardratecurveofcorporatedebt.................178
Exercises...................................................179

Contentsxvii
8OptionPricingandMarketData ..........................181
8.1Optionpricesandpricingtheory..........................182
8.1.1Optionsdataandput¨Ccallparity....................18
8.1.2TheBlack-ScholesformulasforEuropeanoptions......183
8.1.3OptimalstoppingandAmericanoptions..............187
8.2Impliedvolatility........................................188
8.3Alternativestoandmodi?cationsoftheBlack-Scholesmodel
andpricingtheory.......................................192
8.3.1Theimpliedvolatilityfunction(IVF)model..........192
8.3.2Theconstantelasticityofvariance(CEV)model......192
8.3.3Thestochasticvolatility(SV)model.................193
8.3.4Nonparametricmethods............................194
8.3.5Acombinedsubstantive-empiricalapproach...........195
Exercises...................................................197
9AdvancedMultivariateandTimeSeriesMethods
inFinancialEconometrics .................................199
9.1Canonicalcorrelationanalysis.............................200
9.1.1Cross-covarianceandcorrelationmatrices.............200
9.1.2Canonicalcorrelations.............................201
9.2Multivariateregressionanalysis...........................203
9.2.1Leastsquaresestimatesinmultivariateregression.....203
9.2.2Reduced-rankregression............................203
9.3Modi?edCholeskydecompositionandhigh-dimensional
covariancematrices......................................205
9.4Multivariatetimeseries..................................206
9.4.1Stationarityandcross-correlation....................206
9.4.2DimensionreductionviaPCA.......................206
9.4.3Linearregressionwithstochasticregressors...........207
9.4.4Unit-roottests....................................211
9.4.5CointegratedVAR.................................213
9.5Long-memorymodelsandregime
switching/structuralchange...............................217
9.5.1Longmemoryinintegratedmodels..................217
9.5.2Change-pointAR-GARCHmodels...................219
9.5.3Regime-switchingmodels...........................224
9.6Stochasticvolatilityandmultivariatevolatilitymodels.......225
9.6.1Stochasticvolatilitymodels.........................225
9.6.2Multivariatevolatilitymodels.......................228
9.7Generalizedmethodofmoments(GMM)...................229
9.7.1Instrumentalvariablesforlinearrelationships.........229
9.7.2Generalizedmomentrestrictionsand
GMMestimation..................................231

xviiiContents
9.7.3Anexample:Comparisonofdi?erentshort-term
interestratemodels................................233
Exercises...................................................234
10InterestRateMarkets .....................................239
10.1Elementsofinterestratemarkets..........................240
10.1.1Bankaccount(moneymarketaccount)
andshortrates....................................241
10.1.2Zero-couponbondsandspotrates...................241
10.1.3Forwardrates.....................................244
10.1.4Swapratesandinterestrateswaps..................245
10.1.5Caps,?oors,andswaptions.........................247
10.2Yieldcurveestimation...................................247
10.2.1Nonparametricregressionusingspline
basisfunctions....................................248
10.2.2Parametricmodels.................................248
10.3Multivariatetimeseriesofbondyieldsandother
interestrates............................................252
10.4Stochasticinterestratesandshort-ratemodels..............255
10.4.1Vasicek,Cox-Ingersoll-Ross,andHull-Whitemodels...258
10.4.2Bondoptionprices................................259
10.4.3Black-Karasinskimodel............................260
10.4.4Multifactora?neyieldmodels......................261
10.5StochasticforwardratedynamicsandpricingofLIBORand
swapratederivatives.....................................261
10.5.1StandardmarketformulasbasedonBlack!ˉsmode
offorwardprices..................................262
10.5.2Arbitrage-freepricing:martingalesandnumeraires.....263
10.5.3LIBORandswapmarketmodels....................264
10.5.4TheHJMmodelsoftheinstantaneousforwardrate....266
10.6Parameterestimationandmodelselection..................267
10.6.1Calibratinginterestratemodelsinthe
?nancialindustry..................................267
10.6.2Econometricapproachto?tting
term-structuremodels.............................270
10.6.3Volatilitysmilesandasubstantive-empiricalapproach..271
Exercises...................................................272
11StatisticalTradingStrategies ..............................275
11.1Technicalanalysis,tradingstrategies,
anddata-snoopingchecks.................................277
11.1.1Technicalanalysis.................................277
11.1.2Momentumandcontrarianstrategies................279

Contentsxix
11.1.3Pairstradingstrategies.............................279
11.1.4Empiricaltestingofthepro?tability
oftradingstrategies...............................282
11.1.5Valueinvestingandknowledge-based
tradingstrategies..................................285
11.2High-frequencydata,marketmicrostructure,andassociated
tradingstrategies........................................286
11.2.1Institutionalbackgroundandstylizedfactsabout
transactiondata...................................287
11.2.2Bid¨Caskbounceandnonsynchronoustradingmodels...29
11.2.3Modelingtimeintervalsbetweentrades..............292
11.2.4Inferenceonunderlyingpriceprocess................297
11.2.5Real-timetradingsystems..........................299
11.3Transactioncostsanddynamictrading.....................300
11.3.1Estimationandanalysisoftransactioncosts..........300
11.3.2Heterogeneoustradingobjectivesandstrategies.......300
11.3.3Multiperiodtradinganddynamicstrategies...........301
Exercises...................................................302
12StatisticalMethodsinRiskManagement ..................305
12.1Financialrisksandmeasuresofmarketrisk.................306
12.1.1Typesof?nancialrisks.............................306
12.1.2Internalmodelsforcapitalrequirements..............307
12.1.3VaRandothermeasuresofmarketrisk..............307
12.2StatisticalmodelsforVaRandES.........................309
12.2.1TheGaussianconventionandthe t-modi?cation.......309
12.2.2ApplicationsofPCAandanexample................310
12.2.3Timeseriesmodels................................311
12.2.4BacktestingVaRmodels...........................311
12.3Measuringriskfornonlinearportfolios.....................312
12.3.1LocalvaluationviaTaylorexpansions................312
12.3.2FullvaluationviaMonteCarlo......................314
12.3.3Multivariatecopulafunctions.......................314
12.3.4Variancereductiontechniques.......................316
12.4Stresstestingandextremevaluetheory....................318
12.4.1Stresstesting.....................................318
12.4.2Extraordinarylossesandextremevaluetheory........318
12.4.3ScenarioanalysisandMonteCarlosimulations........321
Exercises...................................................321

xxContents
AppendixA.MartingaleTheoryandCentralLimitTheorems .325
AppendixB.LimitTheoremsforStationaryProcesses.........331
AppendixC.LimitTheoremsUnderlyingUnit-RootTests
andCointegration .........................................333
References.....................................................337
Index ..........................................................349