数理经济学和金融学Mathematical Economics and Finance
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Mathematical Economics and Finance
Michael Harrison Patrick Waldron
December 2, 1998
CONTENTS i
Contents
List of Tables iii
List of Figures v
PREFACE vii
What Is Economics? . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
What Is Mathematics? . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
NOTATION ix
I MATHEMATICS 1
1 LINEAR ALGEBRA 3
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Systems of Linear Equations and Matrices . . . . . . . . . . . . . 3
1.3 Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Matrix Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Vectors and Vector Spaces . . . . . . . . . . . . . . . . . . . . . 11
1.6 Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7 Bases and Dimension . . . . . . . . . . . . . . . . . . . . . . . . 12
1.8 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.9 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . 14
1.10 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.11 Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.12 Definite Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 VECTOR CALCULUS 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Basic Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Vector-valued Functions and Functions of Several Variables . . . 18
Revised: December 2, 1998
ii CONTENTS
2.4 Partial and Total Derivatives . . . . . . . . . . . . . . . . . . . . 20
2.5 The Chain Rule and Product Rule . . . . . . . . . . . . . . . . . 21
2.6 The Implicit Function Theorem . . . . . . . . . . . . . . . . . . . 23
2.7 Directional Derivatives . . . . . . . . . . . . . . . . . . . . . . . 24
2.8 Taylor’s Theorem: Deterministic Version . . . . . . . . . . . . . 25
2.9 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . 26
3 CONVEXITY AND OPTIMISATION 27
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Convexity and Concavity . . . . . . . . . . . . . . . . . . . . . . 27
3.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.2 Properties of concave functions . . . . . . . . . . . . . . 29
3.2.3 Convexity and differentiability . . . . . . . . . . . . . . . 30
3.2.4 Variations on the convexity theme . . . . . . . . . . . . . 34
3.3 Unconstrained Optimisation . . . . . . . . . . . . . . . . . . . . 39
3.4 Equality Constrained Optimisation:
The Lagrange Multiplier Theorems . . . . . . . . . . . . . . . . . 43
3.5 Inequality Constrained Optimisation:
The Kuhn-Tucker Theorems . . . . . . . . . . . . . . . . . . . . 50
3.6 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
II APPLICATIONS 61
4 CHOICE UNDER CERTAINTY 63
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4 Optimal Response Functions:
Marshallian and Hicksian Demand . . . . . . . . . . . . . . . . . 69
4.4.1 The consumer’s problem . . . . . . . . . . . . . . . . . . 69
4.4.2 The No Arbitrage Principle . . . . . . . . . . . . . . . . . 70
4.4.3 Other Properties of Marshallian demand . . . . . . . . . . 71
4.4.4 The dual problem . . . . . . . . . . . . . . . . . . . . . . 72
4.4.5 Properties of Hicksian demands . . . . . . . . . . . . . . 73
4.5 Envelope Functions:
Indirect Utility and Expenditure . . . . . . . . . . . . . . . . . . 73
4.6 Further Results in Demand Theory . . . . . . . . . . . . . . . . . 75
4.7 General Equilibrium Theory . . . . . . . . . . . . . . . . . . . . 78
4.7.1 Walras’ law . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.7.2 Brouwer’s fixed point theorem . . . . . . . . . . . . . . . 78
Revised: December 2, 1998
CONTENTS iii
4.7.3 Existence of equilibrium . . . . . . . . . . . . . . . . . . 78
4.8 The Welfare Theorems . . . . . . . . . . . . . . . . . . . . . . . 78
4.8.1 The Edgeworth box . . . . . . . . . . . . . . . . . . . . . 78
4.8.2 Pareto efficiency . . . . . . . . . . . . . . . . . . . . . . 78
4.8.3 The First Welfare Theorem . . . . . . . . . . . . . . . . . 79
4.8.4 The Separating Hyperplane Theorem . . . . . . . . . . . 80
4.8.5 The Second Welfare Theorem . . . . . . . . . . . . . . . 80
4.8.6 Complete markets . . . . . . . . . . . . . . . . . . . . . 82
4.8.7 Other characterizations of Pareto efficient allocations . . . 82
4.9 Multi-period General Equilibrium . . . . . . . . . . . . . . . . . 84
5 CHOICE UNDER UNCERTAINTY 85
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 Review of Basic Probability . . . . . . . . . . . . . . . . . . . . 85
5.3 Taylor’s Theorem: Stochastic Version . . . . . . . . . . . . . . . 88
5.4 Pricing State-Contingent Claims . . . . . . . . . . . . . . . . . . 88
5.4.1 Completion of markets using options . . . . . . . . . . . 90
5.4.2 Restrictions on security values implied by allocational efficiency
and covariance with aggregate consumption . . . 91
5.4.3 Completing markets with options on aggregate consumption 92
5.4.4 Replicating elementary claims with a butterfly spread . . . 93
5.5 The Expected Utility Paradigm . . . . . . . . . . . . . . . . . . . 93
5.5.1 Further axioms . . . . . . . . . . . . . . . . . . . . . . . 93
5.5.2 Existence of expected utility functions . . . . . . . . . . . 95
5.6 Jensen’s Inequality and Siegel’s Paradox . . . . . . . . . . . . . . 97
5.7 Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.8 The Mean-Variance Paradigm . . . . . . . . . . . . . . . . . . . 102
5.9 The Kelly Strategy . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.10 Alternative Non-Expected Utility Approaches . . . . . . . . . . . 104
6 PORTFOLIO THEORY 105
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2 Notation and preliminaries . . . . . . . . . . . . . . . . . . . . . 105
6.2.1 Measuring rates of return . . . . . . . . . . . . . . . . . . 105
6.2.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.3 The Single-period Portfolio Choice Problem . . . . . . . . . . . . 110
6.3.1 The canonical portfolio problem . . . . . . . . . . . . . . 110
6.3.2 Risk aversion and portfolio composition . . . . . . . . . . 112
6.3.3 Mutual fund separation . . . . . . . . . . . . . . . . . . . 114
6.4 Mathematics of the Portfolio Frontier . . . . . . . . . . . . . . . 116
Revised: December 2, 1998
iv CONTENTS
6.4.1 The portfolio frontier in <N:
risky assets only . . . . . . . . . . . . . . . . . . . . . . 116
6.4.2 The portfolio frontier in mean-variance space:
risky assets only . . . . . . . . . . . . . . . . . . . . . . 124
6.4.3 The portfolio frontier in <N:
riskfree and risky assets . . . . . . . . . . . . . . . . . . 129
6.4.4 The portfolio frontier in mean-variance space:
riskfree and risky assets . . . . . . . . . . . . . . . . . . 129
6.5 Market Equilibrium and the CAPM . . . . . . . . . . . . . . . . 130
6.5.1 Pricing assets and predicting security returns . . . . . . . 130
6.5.2 Properties of the market portfolio . . . . . . . . . . . . . 131
6.5.3 The zero-beta CAPM . . . . . . . . . . . . . . . . . . . . 131
6.5.4 The traditional CAPM . . . . . . . . . . . . . . . . . . . 132
7 INVESTMENT ANALYSIS 137
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.2 Arbitrage and Pricing Derivative Securities . . . . . . . . . . . . 137
7.2.1 The binomial option pricing model . . . . . . . . . . . . 137
7.2.2 The Black-Scholes option pricing model . . . . . . . . . . 137
7.3 Multi-period Investment Problems . . . . . . . . . . . . . . . . . 140
7.4 Continuous Time Investment Problems . . . . . . . . . . . . . . . 140
Revised: December 2, 1998
LIST OF TABLES v
List of Tables
3.1 Sign conditions for inequality constrained optimisation . . . . . . 51
5.1 Payoffs for Call Options on the Aggregate Consumption . . . . . 92
6.1 The effect of an interest rate of 10% per annum at different frequencies
of compounding. . . . . . . . . . . . . . . . . . . . . . 106
6.2 Notation for portfolio choice problem . . . . . . . . . . . . . . . 108
Revised: December 2, 1998
vi LIST OF TABLES
Revised: December 2, 1998
LIST OF FIGURES vii
List of Figures
Revised: December 2, 1998
viii LIST OF FIGURES
Revised: December 2, 1998
PREFACE ix
PREFACE
This book is based on courses MA381 and EC3080, taught at Trinity College
Dublin since 1992.
Comments on content and presentation in the present draft are welcome for the
benefit of future generations of students.
An electronic version of this book (in LATEX) is available on the World Wide
Web at http://pwaldron.bess.tcd.ie/teaching/ma381/notes/
although it may not always be the current version.
The book is not intended as a substitute for students’ own lecture notes. In particular,
many examples and diagrams are omitted and some material may be presented
in a different sequence from year to year.
In recent years, mathematics graduates have been increasingly expected to have
additional skills in practical subjects such as economics and finance, while economics
graduates have been expected to have an increasingly strong grounding in
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 economics
who wish to branch out into the other subject.
The present version lacks supporting materials in Mathematica or Maple, such as
are provided with competing works like ?.
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 questions
from the perspective of the outsider.
What Is Economics?
This section will consist of a brief verbal introduction to economics for mathematicians
and an outline of the course.
Revised: December 2, 1998
x PREFACE
What is economics?
1. Basic microeconomics is about the allocation of wealth or expenditure among
different physical goods. This gives us relative prices.
2. Basic finance is about the allocation of expenditure across two or more time
periods. This gives us the term structure of interest rates.
3. The next step is the allocation of expenditure across (a finite number or a
continuum of) states of nature. This gives us rates of return on risky assets,
which are random variables.
Then we can try to combine 2 and 3.
Finally we can try to combine 1 and 2 and 3.
Thus finance is just a subset of micoreconomics.
What do consumers do?
They maximise ‘utility’ given a budget constraint, based on prices and income.
What do firms do?
They maximise profits, given technological constraints (and input and output prices).
Microeconomics is ultimately the theory of the determination of prices by the interaction
of all these decisions: all agents simultaneously maximise their objective
functions subject to market clearing conditions.
What is Mathematics?
This section will have all the stuff about logic and proof and so on moved into it.
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