The Black–Scholes–Merton Model as an Idealization of Discrete-Time Economies

218 Pages·2019·1.333 MB·English

by  David M. Kreps

TheBlack–Scholes–MertonModelasanIdealization
ofDiscrete-TimeEconomies
This book examines whether continuous-time models in frictionless financial
economiescanbewellapproximatedbydiscrete-timemodels.Itspecificallylooks
toanswerthequestion:InwhatsenseandtowhatextentdoesthefamousBlack–
Scholes–Merton(BSM)continuous-timemodeloffinancialmarketsidealizemore
realistic discrete-time models of those markets? While it is well known that the
BSM model is an idealization of discrete-time economies where the stock price
process is driven by a binomial random walk, it is less known that the BSM
model idealizes discrete-time economies whose stock price process is driven by
more general random walks. After recounting the foundations of discrete- and
continuous-time models of financial markets, David M. Kreps develops the gen-
eraltheoryofconvergenceofdiscrete-timemodelstotheircontinuous-timelimits,
bothinmathematicalandeconomicterms.Theexpositionlowerstheentrybarriers
totheliteratureoffinancialmathematicsforless-technicalreaders,whileprovid-
ing a fuller understanding of the connections between BSM and nearby discrete
economies.
David M. Kreps is the Adams Distinguished Professor of Management, Emeri-
tus,attheStanfordGraduateSchoolofBusiness.Hehasbeenhonoredwithmany
awards,includingtheJohnBatesClarkMedalbytheAmericanEconomicAssoci-
ationin1989andtheJohnJ.CartyAwardfortheAdvancementofSciencebythe
NationalAcademyofSciencesin2018.
EconometricSocietyMonographSeries
Editors:
AndreaPrat,ColumbiaUniversity
StéphaneBonhomme,UniversityofChicago
TheEconometricSocietyisaninternationalsocietyfortheadvancementofeconomictheoryin
relation to statistics andmathematics. The EconometricSociety Monographseries is designed
to promote the publication of original research contributions of high quality in mathematical
economicsandtheoreticalandappliedeconometrics.
BooksintheSeries
W.Thomson,HowtoDivideWhenThereIsn’tEnough:FromAristotle,theTalmud,and
MaimonidestotheAxiomaticsofResourceAllocation,2019
O.Compte&A.Postlewaite,IgnoranceandUncertainty,2018
I.Molchanov&F.Molinari,RandomSetsinEconometrics,2018
B.Honoré,A.Pakes,M.Piazzesi,&L.Samuelson(eds.),AdvancesinEconomicsand
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R.Blundell,W.Newey,&T.Persson,(eds.),AdvancesinEconomicsandEconometrics:Theory
andApplications,NinthWorldCongress,Vols.I,II,&III,2006
J.Roemer,Democracy,Education,andEquality,2006
C.Blackorby,W.Bossert,&D.Donaldson,PopulationIssuesinSocialChoiceTheory,Welfare
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C.Hsiao,AnalysisofPanelData,2nded.,2003
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Econometrics:TheoryandApplications,EighthWorldCongress,Vols.I,II,&III,2003
E.Ghysels,N.Swanson,&M.Watson(eds.),EssaysinEconometrics:CollectedPapersofClive
W.J.Granger,Vols.I&II,2001
S.Strøm(ed.),EconometricsandEconomicTheoryinthe20thCentury:TheRagnarFrisch
CentennialSymposium,1999
A.C.Cameron&P.K.Trivedi,RegressionAnalysisofCount-Data,1998
D.Jacobs,E.Kalai,&M.Kamien(eds.),FrontiersofResearchinEconomicTheory:TheNancy
L. SchwartzMemorialLectures,1998
Continuedonpagefollowingtheindex

The Black–Scholes–Merton Model
as an Idealization of Discrete-Time
Economies
David M. Kreps
StanfordUniversity,California
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781108486361
DOI:10.1017/9781108626903
(cid:2)c DavidM.Kreps2019
Thispublicationisincopyright.Subjecttostatutoryexception
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noreproductionofanypartmaytakeplacewithoutthewritten
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Firstpublished2019
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accurateorappropriate.
Contents
Preface ix
1. Introduction 1
Proofs 5
Website 5
2. FinitelyManyStatesandTimes 6
TheExplicitModel 6
TheQuestionsAddressed 8
WhatSortofConsumers? 9
⇐⇒ ⇐⇒ ⇐⇒
Viability NoFreeLunches PricingKernel
EquivalentMartingaleMeasure 13
TwoGeneralizations 18
TwoExamples 20
WhichContingentClaimsCanBeSynthesized? 23
ArbitrageBoundsonthePriceofClaims 24
BibliographicNotes 28
3. CountinuousTimeandtheBlack–Scholes–Merton(BSM)Model 29
The(Limited)ScopeofThisChapter 29
TheBSMModel 30
DoublingStrategies 31
SimpleTradingStrategies 33
Fo¨llmer’sExample 37
GeneralizingProposition2.1: TheFundamentalTheorem
ofAssetPricing(FTAP) 38
“ArbitragePricing”byViabilityArguments 42
“ArbitragePricing”bySynthesis 44
TheFinancial-GainsIntegral 48
IntheBSMModel,withContinuousTrading,
MarketsAreComplete 51
Editorial: Continuous-TimeTradingasanIdealization 52
BibliographicNotes 54
4. BSMasanIdealizationofBinomial-Random-WalkEconomies 56
OneStateSpace 57
C[0,1] versus D[0,1] 58
vi Contents
OneStateSpace,Continued 60
GeneralizingCox,Ross,andRubinstein 62
WeakConvergenceinGeneral 64
Donsker’sTheorem 68
ProofofProposition4.1 69
UnboundedandDiscontinuousContingentClaims 74
ConvergenceofSolutionstotheConsumer’sProblem: Formulation 77
ExistenceandCharacterizationofSolutions
forDiscrete-TimeEconomies 79
ExistenceandCharacterizationofSolutionsfortheBSMEconomy 82
EmployingProposition4.4 87
ConvergenceofSolutionstotheConsumer’sProblem,Completed 88
ConcerningtheAssumptioninProposition4.4thatthe
Bounds(4.2)Hold 96
OtherBinomialRandomWalks 96
SoIsBSMaGoodIdealizationofthe
Binomial-Random-WalkEconomies? 98
BibliographicRemarks 99
5. GeneralRandom-WalkModels 100
TheEsscherTransformandemm 102
TheTrinomialRandomWalkandNearSynthesisoftheCallOption 107
AsymptoticSynthesisofContingentClaimsforGeneral
Random-WalkEconomies 112
ProofStageI:AVerySpecialCase 114
ProofStageII:ALessSpecialCase,Begun 118
TheSkorohodRepresentationTheorem 119
StageIICompleted 120
ProofStageIII:GeneralContingentClaimsinThreeSteps 124
ConcerningtheProofofProposition5.1band
theBoundednessoftheSupportof ζ 126
UnboundedClaims 127
What’sGoingonHere? “Arbitrage,”asDefined
intheLiterature,AsksTooMuch 129
ArbitrageursversusConsumers 130
ConvergenceofOptimalConsumption,1: BoundedSupport 132
ConvergenceofOptimalConsumption,2: DiscreteTimeIs
AsymptoticallyatLeastasGoodasContinuousTime 133
Contents vii
ConvergenceofOptimalConsumption,3: IsDiscreteTime
AsymptoticallyNoBetterthanContinuousTime? 137
ConvergenceofOptimalConsumption,4: CRRA(andCARA)Utility 139
Chapter4versusChapter5 146
TheBottomLine? 146
BibliographicNotes 147
6. Barlow’sExample 148
Background: TheCox–Ross–RubinsteinModels 149
Barlow’sExample 152
VariationsonBarlow’sExample 155
BarlowwithConsecutiveStepsCombined 155
ATwo-by-TwoSummaryTable 156
7. ThePo¨tzelberger–SchlumprechtExampleand
AsymptoticArbitrage 158
TheExample 158
AsymptoticArbitrage–thePhilosophy 163
AsymptoticArbitrage–theTheory 167
NoAsymptoticArbitrage–theConsequences 169
8. ConcludingRemarks,Part1: HowRobustanIdealizationisBSM?171
9. ConcludingRemarks,Part2: Continuous-TimeModels
asIdealizationsofDiscreteTime 174
Appendix 177
TwoEquivalentWaystoRepresentaFieldofEvents
foraFiniteStateSpace 177
ProofofProposition4.2a 178
ProofofProposition4.4 183
SomeCalculationsfortheGeneralBinomialModel 186
(Asymptotic)SolutionsoftheConsumer’sExpected-Utility
MaximizationProblemwithCARAUtility 188
References 194
AuthorIndex 199
SubjectIndex 200