Asset price and wealth dynamics with heterogeneous expectations [Elektronische Ressource] : a deterministic nonlinear structural model approach / Florian Heitger

Asset Price and Wealth Dynamics
with Heterogeneous Expectations
A Deterministic Nonlinear Structural Model Approach
Florian Heitger
Department of Economics
Faculty of Economics, Business, and Social Sciences
Christian-Albrechts-University of Kiel
September 2010Asset Price and Wealth Dynamics
with Heterogeneous Expectations
A Deterministic Nonlinear Structural Model Approach
Inaugural-Dissertation zur Erlangung des akademischen Grades
eines Doktors der Wirtschafts- und Sozialwissenschaften
der Wirtschafts- und Sozialwissenschaftlichen Fakult¨at
der Christian-Albrechts-Universit¨at zu Kiel
vorgelegt von
Diplom-Physiker Florian Heitger
aus Hamm/Westf.
Kiel, September 2010
Christian-Albrechts-Universita¨t zu Kiel
¨Institut fur Volkswirtschaftslehre
Lehrstuhl fu¨r Geld, Wa¨hrung und Internationale Finanzma¨rkte
Wilhelm-Seelig-Platz 1
24118 Kiel
GermanyGedruckt mit Genehmigung der
Wirtschafts- und Sozialwissenschaftlichen Fakulta¨t
der Christian-Albrechts-Universita¨t zu Kiel
Dekan:
Prof. Dr. Birgit Friedl
Erstberichterstattender:
Prof. Dr. Thomas Lux
Zweitberichterstattender:
Prof. Dr. Hans-Werner Wohltmann
Tag der Abgabe der Arbeit:
27. Mai 2010
Tag der mu¨ndlichen Pru¨fung:
1. September 2010Contents
List of Figures VI
List of Tables XI
Introduction 1
Part One: Basic Concepts and Literature Review 9
1 Prologue: Three Paradigms of Economic Modeling 10
1.1 Expectations in Economics and Physics . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Rational Expectation Hypothesis (REH) . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.1 Rational Expectations Equilibrium . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Efficient Market Hypothesis (EMH) . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.1 Fair Price and Martingale Property in Mathematical Finance Literature . 19
1.3.2 Classification of the Information Set . . . . . . . . . . . . . . . . . . . . . . 20
1.3.3 The Grossman-Stiglitz-Paradoxon . . . . . . . . . . . . . . . . . . . . . . . 21
1.4 Representative Agent Hypothesis (RAH) . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Appendix to Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
A 1.1 Derivation of Eq. (1.8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
A 1.2 Further Notes on the Martingale Property of Prices . . . . . . . . . . . . . 26
A 1.3 Excursion I: Basic Concepts of Utility Theory under Uncertainty . . . . . . 27
2 Behavioral Models of Financial Markets 32
2.1 The Brock-Hommes (BH) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.1.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1.2 Selected Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2 Selected Modified Versions of the BH Model . . . . . . . . . . . . . . . . . . . . . . 41
2.2.1 Gaunersdorfer’s Extension of the BH Model . . . . . . . . . . . . . . . . . 42
2.2.2 A More Realistic Model Setup of Gaunersdorfer’s Version of the BH Model 48CONTENTS II
2.2.3 Gaunersdorfer and Hommes’ Extension of the BH Model . . . . . . . . . . 55
2.3 The Levy-Levy-Solomon (LLS) Model . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.3.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.3.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Appendix to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
A 2.1 The Standard Mean-Variance Maximization Approach . . . . . . . . . . . 76
A 2.2 Further Notes on Eq. (2.8) . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
A 2.3 Derivation of Eq. (2.13) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
A 2.4 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
A 2.5 Gaunersdorfer’s Risk Adjusted Profits Term . . . . . . . . . . . . . . . . . 82
A 2.6 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
A 2.7 Proof of Theorem 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
A 2.8 Proof of Theorem 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
A 2.9 Notes on Gaunersdorfer and Hommes’ Risk Adjusted Profits Term . . . . . 87
A 2.10 Excursion II: Theoretical Analysis in Nonlinear Dynamics. . . . . . . . . . 88
A 2.11 Excursion III: Short Review of Bifurcation Theory for Discrete Maps . . . 92
A 2.12 Excursion IV: Numerical Analysis in Nonlinear Dynamics . . . . . . . . . . 94
Part Two: Two Nonlinear Stationary Models of Asset Return 102
3 The Chiarella-He (CH) Model with a Power Utility Function 103
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.2 Model Description of the Chiarella-He Model . . . . . . . . . . . . . . . . . . . . . 106
3.2.1 Development of the Growth Model. . . . . . . . . . . . . . . . . . . . . . . 107
3.2.2 Stationary Formulation of the Model . . . . . . . . . . . . . . . . . . . . . 109
3.3 Implementation of the Risk Aversion Coefficient . . . . . . . . . . . . . . . . . . . 113
3.4 The Model under Homogeneous Expectations: An Analytical Treatment . . . . . . 114
3.4.1 Specification of the Trading Strategies . . . . . . . . . . . . . . . . . . . . . 114
3.4.2 Determination of the Steady States . . . . . . . . . . . . . . . . . . . . . . 115
3.4.3 Local Stability of the Steady States . . . . . . . . . . . . . . . . . . . . . . 122
3.5 The Model under Heterogeneous Expectations: A Numerical Treatment . . . . . . 123
3.5.1 Fundamentalists with Different Risk Aversion Coefficients. . . . . . . . . . 127
3.5.2 Fundamentalists with Different Choices of the Parameter δ . . . . . . . . . 130
3.5.3 Fundamentalist versus Trend Trader . . . . . . . . . . . . . . . . . . . . . . 132
3.5.4 Two Trend-Traders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3.6 Numerical Simulations with a Noise Term . . . . . . . . . . . . . . . . . . . . . . . 139CONTENTS III
3.7 The CH Model under LLS-like Assumptions . . . . . . . . . . . . . . . . . . . . . . 145
3.7.1 Numerical Results for the Homogeneous Model Version . . . . . . . . . . . 146
3.7.2 Numerical Results for the Heterogeneous Model Version . . . . . . . . . . . 149
3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Appendix to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
A 3.1 Heuristic Explanation for Formula (3.26) . . . . . . . . . . . . . . . . . . . 154
A 3.2 Proof of Theorem 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
A 3.3 Proof of Theorem 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
A 3.4 Proof of Lemma 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
A 3.5 Proof of Corollary 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
A 3.6 Proof of Remark 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
A 3.7 Proof of Lemma 3.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
A 3.8 Proof of Theorem 3.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
A 3.9 Proof of Corollary 3.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
A 3.10 Proof of Corollary 3.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
4 Asset Price and Wealth Dynamics under a Market Maker Scenario 163
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
4.2 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4.2.1 Notations and Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . 165
4.2.2 Excess Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
4.2.3 Fundamental Price and Return . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.2.4 Risk Premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
4.2.5 Market Maker Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
4.2.6 Summary of the Stationary Model in its State Variables (Simplified Version) 178
4.3 Market Maker Model with Fundamentalists and Chartists . . . . . . . . . . . . . . 179
4.3.1 Homogeneous Beliefs – Fundamentalists . . . . . . . . . . . . . . . . . . . . 180
4.3.2 Homogeneous Beliefs – Chartists . . . . . . . . . . . . . . . . . . . . . . . . 182
4.3.3 Heterogeneous Beliefs – Fundamentalists and Chartists . . . . . . . . . . . 187
4.4 Adaptive Version of the Market Maker Model . . . . . . . . . . . . . . . . . . . . . 193
4.4.1 Specification of the Trading Strategies in the Adaptive Model Version . . . 193
4.4.2 Heterogeneous Beliefs – Fundamentalists and Chartists . . . . . . . . . . . 194
4.5 Results of the Fully Developed Market Maker Model . . . . . . . . . . . . . . . . . 198
4.5.1 Model Dynamics of the Revised Market Maker Model . . . . . . . . . . . . 200
4.5.2 Summary of the Revised Stationary Model in its State Variables . . . . . . 201
4.5.3 Homogeneous Beliefs – Fundamental