Financial Statistics and Mathematical Finance: Methods, Models and Applications

422 Pages·2012·2.48 MB·English

by  Ansgar Steland(auth.)

Financial Statistics and
Mathematical Finance
Financial Statistics and
Mathematical Finance
Methods, Models and Applications
Ansgar Steland
InstituteforStatisticsandEconomics
RWTHAachenUniversity,Germany
Thiseditionfirstpublished2012
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LibraryofCongressCataloging-in-PublicationData
Steland,Ansgar.
Financialstatisticsandmathematicalfinance:methods,modelsand
applications/AnsgarSteland.
Includesbibliographicalreferencesandindex.
ISBN978-0-470-71058-6
1. Businessmathematics. 2. Calculus. I. Title.
HF5691.S65852012
332.01’5195–dc23
2012007001
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
ISBN:978-0-470-71058-6
Setin10/12ptTimesbyThomsonDigital,Noida,India.
Contents
Preface xi
Acknowledgements xv
1 Elementaryfinancialcalculus 1
1.1 Motivatingexamples 1
1.2 Cashflows,interestrates,pricesandreturns 2
1.2.1 Bondsandthetermstructureofinterestrates 5
1.2.2 Assetreturns 6
1.2.3 Somebasicmodelsforassetprices 8
1.3 Elementarystatisticalanalysisofreturns 11
1.3.1 Measuringlocation 13
1.3.2 Measuringdispersionandrisk 16
1.3.3 Measuringskewnessandkurtosis 20
1.3.4 Estimationofthedistribution 21
1.3.5 Testingfornormality 27
1.4 Financialinstruments 28
1.4.1 Contingentclaims 28
1.4.2 Spotcontractsandforwards 29
1.4.3 Futurescontracts 29
1.4.4 Options 30
1.4.5 Barrieroptions 31
1.4.6 Financialengineering 32
1.5 Aprimeronoptionpricing 32
1.5.1 Theno-arbitrageprinciple 32
1.5.2 Risk-neutralevaluation 33
1.5.3 Hedgingandreplication 36
1.5.4 Nonexistenceofarisk-neutralmeasure 37
1.5.5 TheBlack–Scholespricingformula 37
1.5.6 TheGreeks 39
1.5.7 Calibration,impliedvolatilityandthesmile 41
1.5.8 Optionpricesandtherisk-neutraldensity 41
1.6 Notesandfurtherreading 43
References 43
2 Arbitragetheoryfortheone-periodmodel 45
2.1 Definitionsandpreliminaries 45
2.2 Linearpricingmeasures 47
2.3 Moreonarbitrage 50
vi CONTENTS
2.4 SeparationtheoremsinRn 53
2.5 No-arbitrageandmartingalemeasures 56
2.6 Arbitrage-freepricingofcontingentclaims 65
2.7 Constructionofmartingalemeasures:generalcase 70
2.8 Completefinancialmarkets 73
2.9 Notesandfurtherreading 76
References 76
3 Financialmodelsindiscretetime 79
3.1 Adaptedstochasticprocessesindiscretetime 81
3.2 Martingalesandmartingaledifferences 85
3.2.1 Themartingaletransformation 91
3.2.2 Stoppingtimes,optionalsamplingandamaximalinequality 93
3.2.3 ExtensionstoRd 101
3.3 Stationarity 102
3.3.1 Weakandstrictstationarity 102
3.4 LinearprocessesandARMAmodels 111
3.4.1 Linearprocessesandthelagoperator 111
3.4.2 Inversion 116
3.4.3 AR(p)andAR(∞)processes 119
3.4.4 ARMAprocesses 122
3.5 Thefrequencydomain 124
3.5.1 Thespectrum 124
3.5.2 Theperiodogram 126
3.6 EstimationofARMAprocesses 132
3.7 (G)ARCHmodels 133
3.8 Long-memoryseries 139
3.8.1 Fractionaldifferences 139
3.8.2 Fractionallyintegratedprocesses 144
3.9 Notesandfurtherreading 144
References 145
4 Arbitragetheoryforthemultiperiodmodel 147
4.1 Definitionsandpreliminaries 148
4.2 Self-financingtradingstrategies 148
4.3 No-arbitrageandmartingalemeasures 152
4.4 Europeanclaimsonarbitrage-freemarkets 154
4.5 Themartingalerepresentationtheoremindiscretetime 159
4.6 TheCox–Ross–Rubinsteinbinomialmodel 160
4.7 TheBlack–Scholesformula 165
4.8 Americanoptionsandcontingentclaims 171
4.8.1 Arbitrage-freepricingandtheoptimalexercisestrategy 171
4.8.2 Pricingamericanoptionsusingbinomialtrees 174
4.9 Notesandfurtherreading 175
References 175
CONTENTS vii
5 Brownianmotionandrelatedprocessesincontinuoustime 177
5.1 Preliminaries 177
5.2 Brownianmotion 181
5.2.1 Definitionandbasicproperties 181
5.2.2 Brownianmotionandthecentrallimittheorem 188
5.2.3 Pathproperties 190
5.2.4 Brownianmotioninhigherdimensions 191
5.3 Continuityanddifferentiability 192
5.4 Self-similarityandfractionalBrownianmotion 193
5.5 Countingprocesses 195
5.5.1 Thepoissonprocess 195
5.5.2 Thecompoundpoissonprocess 196
5.6 Le´vyprocesses 199
5.7 Notesandfurtherreading 201
References 201
6 Itoˆ Calculus 203
6.1 Totalandquadraticvariation 204
6.2 StochasticStieltjesintegration 208
6.3 TheItoˆ integral 212
6.4 Quadraticcovariation 225
6.5 Itoˆ’sformula 226
6.6 Itoˆ processes 229
6.7 Diffusionprocessesandergodicity 236
6.8 Numericalapproximationsandstatisticalestimation 238
6.9 Notesandfurtherreading 239
References 240
7 TheBlack–Scholesmodel 241
7.1 Themodelandfirstproperties 241
7.2 Girsanov’stheorem 247
7.3 Equivalentmartingalemeasure 251
7.4 Arbitrage-freepricingandhedgingclaims 252
7.5 Thedeltahedge 256
7.6 Time-dependentvolatility 257
7.7 ThegeneralizedBlack–Scholesmodel 259
7.8 Notesandfurtherreading 261
References 262
8 Limittheoryfordiscrete-timeprocesses 263
8.1 Limittheoremsforcorrelatedtimeseries 264
8.2 Aregressionmodelforfinancialtimeseries 273
8.2.1 Leastsquaresestimation 276
8.3 Limittheoremsformartingaledifference 278
viii CONTENTS
8.4 Asymptotics 283
8.5 Densityestimationandnonparametricregression 287
8.5.1 Multivariatedensityestimation 288
8.5.2 Nonparametricregression 295
8.6 TheCLTforlinearprocesses 302
8.7 Mixingprocesses 306
8.7.1 Mixingcoefficients 306
8.7.2 Inequalities 308
8.8 Limittheoremsformixingprocesses 313
8.9 Notesandfurtherreading 323
References 323
9 Specialtopics 325
9.1 Copulas–andthe2008financialcrisis 325
9.1.1 Copulas 326
9.1.2 Thefinancialcrisis 332
9.1.3 ModelsforcreditdefaultsandCDOs 335
9.2 LocalLinearnonparametricregression 338
9.2.1 Applications in finance: estimation of martingale measures and Itoˆ
diffusions 339
9.2.2 Methodandasymptotics 340
9.3 Change-pointdetectionandmonitoring 350
9.3.1 Offlinedetection 351
9.3.2 Onlinedetection 359
9.4 Unitrootsandrandomwalk 363
9.4.1 TheOLSestimatorinthestationaryAR(1)model 364
9.4.2 Nonparametricdefinitionsforthedegreeofintegration 368
9.4.3 TheDickey–Fullertest 370
9.4.4 Detectingunitrootsandstationarity 373
9.5 Notesandfurtherreading 381
References 382
AppendixA 385
A.1 (Stochastic)Landausymbols 385
A.2 Bochner’slemma 387
A.3 Conditionalexpectation 387
A.4 Inequalities 388
A.5 Randomseries 389
A.6 Localmartingalesindiscretetime 389
AppendixB Weakconvergenceandcentrallimittheorems 391
B.1 Convergenceindistribution 391
B.2 Weakconvergence 392
CONTENTS ix
B.3 Prohorov’stheorem 398
B.4 Sufficientcriteria 399
B.5 MoreonSkorohodspaces 401
B.6 Centrallimittheoremsformartingaledifferences 402
B.7 Functionalcentrallimittheorems 403
B.8 Strongapproximations 405
References 407
Index 409
Preface
Thistextbookintendstoprovideacarefulandcomprehensiveintroductiontosomeofthemost
importantmathematicaltopicsrequiredforathoroughunderstandingoffinancialmarketsand
the quantitative methods used there. For this reason, the book covers mathematical finance
inthenarrowsense,thatis,arbitragetheoryforpricingcontingentclaimssuchasoptionsand
therelatedmathematicalmachinery,aswellasstatisticalmodelsandmethodstoanalyzedata
fromfinancialmarkets.Theseareasevolvedmoreorlessseparatefromeachotherandthelack
ofmaterialthatcoversbothwasamajormotivationformetoworkoutthepresenttextbook.
Thus,IwroteabookthatIwouldhavelikedwhentakingupthesubject.Itaddressesmaster
and Ph.D. students as well as researchers and practitioners interested in a comprehensive
presentationofbothareas,althoughmanychapterscanalsobestudiedbyBachelorstudents
who have passed introductory courses in probability calculus and statistics. Apart from a
coupleofexceptions,allresultsareprovedindetail,althoughusuallynotintheirmostgeneral
form.Giventheplethoraofnotions,concepts,modelsandmethodsandtheresultinginherent
complexity,particularlythosecomingtothesubjectforthefirsttimecanacquireathorough
understanding more quickly, if they can easily follow the derivations and calculations. For
thisreason,themathematicalformalismandnotationiskeptaselementaryaspossible.Each
chaptercloseswithnotesandcommentsonselectedreferences,whichmaycomplementthe
presentedmaterialoraregoodstartingpointsforfurtherstudies.
Chapter1startswithabasicintroductiontoimportantnotions:financialinstrumentssuch
asoptionsandderivativesandrelatedelementarymethods.However,derivationsareusually
not given in order to focus on ideas, principles and basic results. It sets the scene for the
followingchaptersandintroducestherequiredfinancialslang.Cashflows,discountingand
thetermstructureofinterestratesarestudiedatanelementarylevel.Thereturnoveragiven
period of time, for assets usually a day, represents the most important economic object of
interestinfinance,aspricescanbereconstructedfromreturnsandinvestmentsarejudgedby
comparingtheirreturn.Statisticalmeasuresfortheirlocation,dispersionandskewnesshave
importanteconomicinterpretations,andtherelevantstatisticalapproachestoestimatethem
are carefully introduced. Measuring the risk associated with an investment requires being
aware of the properties of related statistical estimates. For example, volatility is primarily
relatedtothestandarddeviationandvalue-at-risk,bydefinition,requiresthestudyofquantiles
andtheirstatisticalestimation.Thefirstchaptercloseswithaprimeronoptionpricing,which
introducesthemostimportantnotionsofthefieldofmathematicalfinanceinthenarrowsense,
namelytheprincipleofno-arbitrage,theprincipleofrisk-neutralpricingandtherelationof
thosenotionstoprobabilitycalculus,particularlytotheexistenceofanequivalentmartingale
measure.Indeed,thesebasicconceptsandacoupleoffundamentalinsightscanbeunderstood
bystudyingtheminthemostelementaryformorsimplybyexamples.
Chapter 2 then discusses arbitrage theory and the pricing of contingent claims within a
one-periodmodel.Attime0onesetsupaportfolioandattime1welookattheresult.Within
thissimpleframework,thebasicresultsdiscussedinChapter1aretreatedwithmathematical
rigor and extended from a finite probability space, where only a finite number of scenarios
xii PREFACE
can occur, to a general underlying probability space that models the real financial market.
Mathematical separation theorems, which tell us how one can separate a given point from
convex sets, are applied in order to establish the equivalence of the exclusion of arbitrage
opportunities and the existence of an equivalent martingale measure. For this reason, those
separationtheoremsareexplicitlyproved.Theconstructionofequivalentmartingalemeasures
basedontheEsschertransformisdiscussedaswell.
Chapter 3 provides a careful introduction to stochastic processes in discrete time (time
series),coveringmartingales,martingaledifferences,linearprocesses,ARMAandGARCH
processes as well as long-memory series. The notion of a martingale is fundamental for
mathematicalfinance,asoneofthekeyresultsassertsthatinanyfinancialmarketthatexcludes
arbitrage,thereexistsaprobabilitymeasuresuchthatthediscountedpriceseriesofarisky
asset forms a martingale and the pricing of contingent claims can be done by risk-neutral
pricingunderthatmeasure.Thesekeyinsightsallowustoapplytheelaboratedmathematical
theoryofmartingales.However,thetreatmentinChapter3isrestrictedtothemostimportant
findingsofthattheory,whicharereallyusedlater.Takingfirst-orderdifferencesofamartingale
leadsnaturallytomartingaledifferencesequences,whichformwhitenoiseprocessesandare
acommonreplacementfortheunrealistici.i.d.errortermsinstochasticmodelsforfinancial
data and, more generally, economic data. A key empirical insight of the statistical analysis
of financial return series is that they can often be assumed to be uncorrelated, but they are
usuallynotindependent.However,otherseriesmayexhibitsubstantialserialdependencethat
hastobetakenintoaccount.Appropriateparametricclassesoftime-seriesmodelsareARMA
processes,whichbelongtothemoregeneralandinfinite-dimensionalclassoflinearprocesses.
BasicapproachestoestimateautocovariancefunctionsandtheparametersofARMAmodels
arediscussed.Manyfinancialseriesexhibitthephenomenonofconditionalheteroscedasticity,
whichhasgivenrisetotheclassof(G)ARCHmodels.Lastly,fractionaldifferencesandlong-
memoryprocessesareintroduced.
Chapter4discussesindetailarbitragetheoryinadiscrete-timemultiperiodmodel.Here,
tradingisallowedatafinitenumberoftimepointsandateachtimepointthetradingstrategy
canbeupdatedusingallavailableinformationonmarketprices.Usingthemartingaletheory
indiscretetimestudiedinChapter3,itallowsustoinvestigatethepricingofoptionsandother
derivativesonarbitrage-freefinancialmarkets.TheCox–Ross–Rubinsteinbinomialmodelis
studiedingreaterdetail,sinceitisastandardtoolinpracticeandalsoprovidesthebasisto
derivethefamousBlack–ScholespricingformulaforaEuropeancall.Inaddition,thepricing
ofAmericanclaimsisstudied,whichrequiressomemoreadvancedresultsfromthetheory
ofoptimalstopping.
Chapter 5 introduces the reader to stochastic processes in continuous time. Brownian
motion will be the random source that governs the price processes of our financial market
model in continuous time. Nevertheless, to keep the chapter concise, the presentation of
Brownian motion is limited to its definition and the most important properties. Brownian
motionhaspuzzlingpropertiessuchascontinuouspathsthatarenowheredifferentiableorof
boundedvariation.AdvancedmodelsalsoincorporatefractionalBrownianmotionandLe´vy
processes, respectively. Le´vy processes inherit independent increments but allow for non-
normaldistributionsofthoseincrementsincludingheavytailsandjump.FractionalBrownian
motionisaGaussianprocessasisBrownianmotion,butitallowsforlong-rangedependent
incrementswheretemporalcorrelationsdieoutveryslowly.
Chapter6treatsthetheoryofstochasticintegration.Assumingthatthereaderisfamiliar
withintegrationinthesenseofRiemannorLebesgue,westartwithadiscussionofstochastic