Mathematical economics and finance

153 Pages·1998·1.327 MB·English

by  Harrison M., Waldron P.

Mathematical Economics and Finance
Michael Harrison Patrick Waldron
December 2, 1998
CONTENTS i
Contents
ListofTables iii
ListofFigures v
PREFACE vii
WhatIsEconomics? . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
WhatIsMathematics?. . . . . . . . . . . . . . . . . . . . . . . . . . . viii
NOTATION ix
I MATHEMATICS 1
1 LINEARALGEBRA 3
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 SystemsofLinearEquationsandMatrices . . . . . . . . . . . . . 3
1.3 MatrixOperations . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 MatrixArithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 VectorsandVectorSpaces . . . . . . . . . . . . . . . . . . . . . 11
1.6 LinearIndependence . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7 BasesandDimension . . . . . . . . . . . . . . . . . . . . . . . . 12
1.8 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.9 EigenvaluesandEigenvectors . . . . . . . . . . . . . . . . . . . . 14
1.10 QuadraticForms . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.11 SymmetricMatrices . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.12 DefiniteMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 VECTORCALCULUS 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 BasicTopology . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Vector-valuedFunctionsandFunctionsofSeveralVariables . . . 18
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ii CONTENTS
2.4 PartialandTotalDerivatives . . . . . . . . . . . . . . . . . . . . 20
2.5 TheChainRuleandProductRule . . . . . . . . . . . . . . . . . 21
2.6 TheImplicitFunctionTheorem . . . . . . . . . . . . . . . . . . . 23
2.7 DirectionalDerivatives . . . . . . . . . . . . . . . . . . . . . . . 24
2.8 Taylor’sTheorem: DeterministicVersion . . . . . . . . . . . . . 25
2.9 TheFundamentalTheoremofCalculus . . . . . . . . . . . . . . 26
3 CONVEXITYANDOPTIMISATION 27
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 ConvexityandConcavity . . . . . . . . . . . . . . . . . . . . . . 27
3.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.2 Propertiesofconcavefunctions . . . . . . . . . . . . . . 29
3.2.3 Convexityanddifferentiability . . . . . . . . . . . . . . . 30
3.2.4 Variationsontheconvexitytheme . . . . . . . . . . . . . 34
3.3 UnconstrainedOptimisation . . . . . . . . . . . . . . . . . . . . 39
3.4 EqualityConstrainedOptimisation:
TheLagrangeMultiplierTheorems . . . . . . . . . . . . . . . . . 43
3.5 InequalityConstrainedOptimisation:
TheKuhn-TuckerTheorems . . . . . . . . . . . . . . . . . . . . 50
3.6 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
II APPLICATIONS 61
4 CHOICEUNDERCERTAINTY 63
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4 OptimalResponseFunctions:
MarshallianandHicksianDemand . . . . . . . . . . . . . . . . . 69
4.4.1 Theconsumer’sproblem . . . . . . . . . . . . . . . . . . 69
4.4.2 TheNoArbitragePrinciple . . . . . . . . . . . . . . . . . 70
4.4.3 OtherPropertiesofMarshalliandemand . . . . . . . . . . 71
4.4.4 Thedualproblem . . . . . . . . . . . . . . . . . . . . . . 72
4.4.5 PropertiesofHicksiandemands . . . . . . . . . . . . . . 73
4.5 EnvelopeFunctions:
IndirectUtilityandExpenditure . . . . . . . . . . . . . . . . . . 73
4.6 FurtherResultsinDemandTheory . . . . . . . . . . . . . . . . . 75
4.7 GeneralEquilibriumTheory . . . . . . . . . . . . . . . . . . . . 78
4.7.1 Walras’law . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.7.2 Brouwer’sfixedpointtheorem . . . . . . . . . . . . . . . 78
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CONTENTS iii
4.7.3 Existenceofequilibrium . . . . . . . . . . . . . . . . . . 78
4.8 TheWelfareTheorems . . . . . . . . . . . . . . . . . . . . . . . 78
4.8.1 TheEdgeworthbox . . . . . . . . . . . . . . . . . . . . . 78
4.8.2 Paretoefficiency . . . . . . . . . . . . . . . . . . . . . . 78
4.8.3 TheFirstWelfareTheorem . . . . . . . . . . . . . . . . . 79
4.8.4 TheSeparatingHyperplaneTheorem . . . . . . . . . . . 80
4.8.5 TheSecondWelfareTheorem . . . . . . . . . . . . . . . 80
4.8.6 Completemarkets . . . . . . . . . . . . . . . . . . . . . 82
4.8.7 OthercharacterizationsofParetoefficientallocations . . . 82
4.9 Multi-periodGeneralEquilibrium . . . . . . . . . . . . . . . . . 84
5 CHOICEUNDERUNCERTAINTY 85
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 ReviewofBasicProbability . . . . . . . . . . . . . . . . . . . . 85
5.3 Taylor’sTheorem: StochasticVersion . . . . . . . . . . . . . . . 88
5.4 PricingState-ContingentClaims . . . . . . . . . . . . . . . . . . 88
5.4.1 Completionofmarketsusingoptions . . . . . . . . . . . 90
5.4.2 Restrictionsonsecurityvaluesimpliedbyallocationalef-
ficiencyandcovariancewithaggregateconsumption . . . 91
5.4.3 Completingmarketswithoptionsonaggregateconsumption 92
5.4.4 Replicatingelementaryclaimswithabutterflyspread . . . 93
5.5 TheExpectedUtilityParadigm . . . . . . . . . . . . . . . . . . . 93
5.5.1 Furtheraxioms . . . . . . . . . . . . . . . . . . . . . . . 93
5.5.2 Existenceofexpectedutilityfunctions . . . . . . . . . . . 95
5.6 Jensen’sInequalityandSiegel’sParadox . . . . . . . . . . . . . . 97
5.7 RiskAversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.8 TheMean-VarianceParadigm . . . . . . . . . . . . . . . . . . . 102
5.9 TheKellyStrategy . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.10 AlternativeNon-ExpectedUtilityApproaches . . . . . . . . . . . 104
6 PORTFOLIOTHEORY 105
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2 Notationandpreliminaries . . . . . . . . . . . . . . . . . . . . . 105
6.2.1 Measuringratesofreturn . . . . . . . . . . . . . . . . . . 105
6.2.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.3 TheSingle-periodPortfolioChoiceProblem . . . . . . . . . . . . 110
6.3.1 Thecanonicalportfolioproblem . . . . . . . . . . . . . . 110
6.3.2 Riskaversionandportfoliocomposition . . . . . . . . . . 112
6.3.3 Mutualfundseparation . . . . . . . . . . . . . . . . . . . 114
6.4 MathematicsofthePortfolioFrontier . . . . . . . . . . . . . . . 116
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iv CONTENTS
6.4.1 Theportfoliofrontierin(cid:60)N:
riskyassetsonly . . . . . . . . . . . . . . . . . . . . . . 116
6.4.2 Theportfoliofrontierinmean-variancespace:
riskyassetsonly . . . . . . . . . . . . . . . . . . . . . . 124
6.4.3 Theportfoliofrontierin(cid:60)N:
riskfreeandriskyassets . . . . . . . . . . . . . . . . . . 129
6.4.4 Theportfoliofrontierinmean-variancespace:
riskfreeandriskyassets . . . . . . . . . . . . . . . . . . 129
6.5 MarketEquilibriumandtheCAPM . . . . . . . . . . . . . . . . 130
6.5.1 Pricingassetsandpredictingsecurityreturns . . . . . . . 130
6.5.2 Propertiesofthemarketportfolio . . . . . . . . . . . . . 131
6.5.3 Thezero-betaCAPM . . . . . . . . . . . . . . . . . . . . 131
6.5.4 ThetraditionalCAPM . . . . . . . . . . . . . . . . . . . 132
7 INVESTMENTANALYSIS 137
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.2 ArbitrageandPricingDerivativeSecurities . . . . . . . . . . . . 137
7.2.1 Thebinomialoptionpricingmodel . . . . . . . . . . . . 137
7.2.2 TheBlack-Scholesoptionpricingmodel . . . . . . . . . . 137
7.3 Multi-periodInvestmentProblems . . . . . . . . . . . . . . . . . 140
7.4 ContinuousTimeInvestmentProblems . . . . . . . . . . . . . . . 140
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LISTOFTABLES v
List of Tables
3.1 Signconditionsforinequalityconstrainedoptimisation . . . . . . 51
5.1 PayoffsforCallOptionsontheAggregateConsumption . . . . . 92
6.1 The effect of an interest rate of 10% per annum at different fre-
quenciesofcompounding. . . . . . . . . . . . . . . . . . . . . . 106
6.2 Notationforportfoliochoiceproblem . . . . . . . . . . . . . . . 108
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vi LISTOFTABLES
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LISTOFFIGURES vii
List of Figures
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viii LISTOFFIGURES
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PREFACE ix
PREFACE
This book is based on courses MA381 and EC3080, taught at Trinity College
Dublinsince1992.
Comments on content and presentation in the present draft are welcome for the
benefitoffuturegenerationsofstudents.
An electronic version of this book (in LATEX) is available on the World Wide
Web at http://pwaldron.bess.tcd.ie/teaching/ma381/notes/
althoughitmaynotalwaysbethecurrentversion.
Thebookisnotintendedasasubstituteforstudents’ownlecturenotes. Inparticu-
lar,manyexamplesanddiagramsareomittedandsomematerialmaybepresented
inadifferentsequencefromyeartoyear.
In recent years, mathematics graduates have been increasingly expected to have
additional skills in practical subjects such as economics and finance, while eco-
nomicsgraduateshavebeenexpectedtohaveanincreasinglystronggroundingin
mathematics. The increasing need for those working in economics and finance to
have a strong grounding in mathematics has been highlighted by such layman’s
guides as ?, ?, ? (adapted from ?) and ?. In the light of these trends, the present
book is aimed at advanced undergraduate students of either mathematics or eco-
nomicswhowishtobranchoutintotheothersubject.
The present versionlacks supportingmaterialsin Mathematicaor Maple, such as
areprovidedwithcompetingworkslike?.
Before starting to work through this book, mathematics students should think
about the nature, subject matter and scientific methodology of economics while
economics students should think about the nature, subject matter and scientific
methodology of mathematics. The following sections briefly address these ques-
tionsfromtheperspectiveoftheoutsider.
What Is Economics?
This section will consist of a brief verbal introduction to economics for mathe-
maticiansandanoutlineofthecourse.
Revised: December2,1998