Quantum Economics and Finance: An Applied Mathematics Introduction

248 Pages·6.917 MB·English

by  David Orrell

QUANTUM ECONOMICS
AND FINANCE
An Applied Mathematics
Introduction
This edition first published 2022 by Panda Ohana Publishing
First edition published 2020 by Panda Ohana Publishing
www.pandaohana.com
[email protected]
© 2020, 2021, 2022 David Orrell
All rights reserved. This book or any portion thereof may not be reprinted or
reproduced or utilized in any form or by any electronic, mechanical or other
means, now known or hereafter invented, including photocopying, scanning or
recording, or in any information and retrieval system, without the express
written permission of the publisher.
The publisher and the author of this book shall not be liable in the event of
incidental or consequential damages in connection with, or arising out of
reference to or reliance on any information in this book or use of the
suggestions contained in any part of this book. The publisher and the author
make no representation or warranties with respect to the accuracy or
completeness of the contents of this book and specifically disclaim all
warranties, including without limitation warranties of fitness for a particular
purpose. The methods contained herein may not be suitable for every
situation. The information in this book does not constitute advice.
ISBN 978-1-9160816-2-8
Library of Congress Control Number: 2021937950
Published in New York City, NY
QUANTUM ECONOMICS
AND FINANCE
An Applied Mathematics
Introduction
Third Edition
David Orrell
Panda Ohana Publishing
Other books by David Orrell
Money, Magic, and How to Dismantle a Financial Bomb: Quantum Economics
for the Real World
Quantum Economics: The New Science of Money
The Money Formula: Dodgy Finance, Pseudo Science, and How
Mathematicians Took Over the Markets (with Paul Wilmott)
Economyths: 11 Ways Economics Gets it Wrong
The Evolution of Money (with Roman Chlupatý)
Truth or Beauty: Science and the Quest for Order
Soumrak Homo Economicus (with Tomas Sedlacek and Roman Chlupatý)
The Other Side of the Coin: The Emerging Vision of Economics and Our Place
in The World
Apollo’s Arrow: The Science of Prediction and the Future of Everything
Introducing Economics: A Graphic Guide (with Borin Van Loon)
Behavioural Economics: Psychology, Neuroscience, and the Human Side of
Economics
Instant Economics: Key Thinkers, Theories, Discoveries and Concepts
A Brief History of Money: 4000 Years of Markets, Currencies, Debt and Crisis
For Beatriz, Isabel and Emma

Contents
Introduction 1
1 Why quantum? 9
1.1 Not so strange . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 A quantum of value . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Transactions as a measurement process . . . . . . . . . . . . . 12
1.4 Entanglements . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 Money creation . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Quantum economic person . . . . . . . . . . . . . . . . . . . . 17
1.7 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.8 How much . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 Some basics 25
2.1 The Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Quantum measurements. . . . . . . . . . . . . . . . . . . . . . 27
2.3 Example: a binary decision . . . . . . . . . . . . . . . . . . . . 28
2.4 Entropic force . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5 Density matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.6 Quantum coin . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.7 Two-state systems . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.8 Complications . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3 Quantum propensity 41
3.1 Expected utility theory . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Behavioural economics . . . . . . . . . . . . . . . . . . . . . . 43
3.3 The sure thing effect . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 The unsure thing effect . . . . . . . . . . . . . . . . . . . . . . 46
3.5 Loss aversion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
vii
viii CONTENTS
3.6 Ellsberg paradox . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.7 Allais effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 Mental interference 53
4.1 Wave splitter. . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Projection sequences . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 The disjunction effect . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 The attraction function . . . . . . . . . . . . . . . . . . . . . . 58
4.5 The conjunction effect. . . . . . . . . . . . . . . . . . . . . . . 59
4.6 The order effect . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.7 The quarter law . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5 Computing in colour 65
5.1 Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2 Basic logic gates . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.3 Gates on multiple qubits . . . . . . . . . . . . . . . . . . . . . 68
5.4 Circuit representation . . . . . . . . . . . . . . . . . . . . . . . 70
5.5 Example: An entanglement circuit . . . . . . . . . . . . . . . . 71
5.6 Example: The quantum walk . . . . . . . . . . . . . . . . . . . 74
5.7 Example: Quantum learning . . . . . . . . . . . . . . . . . . . 75
5.8 Quantum programming . . . . . . . . . . . . . . . . . . . . . . 75
5.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6 The quantum walk 79
6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2 The classical random walk . . . . . . . . . . . . . . . . . . . . 80
6.3 The quantum version . . . . . . . . . . . . . . . . . . . . . . . 81
6.4 Quantum diffusion . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.5 Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7 Option pricing 91
7.1 Walking the line . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.2 Quantum fluctuations . . . . . . . . . . . . . . . . . . . . . . . 93
7.3 The binomial model . . . . . . . . . . . . . . . . . . . . . . . . 94
7.4 Pricing options. . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.5 Quantum anomalies . . . . . . . . . . . . . . . . . . . . . . . . 96
7.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
CONTENTS ix
8 Quantum games 105
8.1 The penny flip . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8.2 The prisoner’s dilemma . . . . . . . . . . . . . . . . . . . . . . 108
8.3 Quantum rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.4 Entangled prisoners . . . . . . . . . . . . . . . . . . . . . . . . 110
8.5 The projection game. . . . . . . . . . . . . . . . . . . . . . . . 114
8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
9 Quantum oscillations 117
9.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
9.2 Quantize this . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
9.3 The harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . 121
9.4 Creation and annihilation . . . . . . . . . . . . . . . . . . . . . 124
9.5 Coherent state . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
9.6 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
9.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
9.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
9.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 130
10 Probabilistic supply and demand 131
10.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
10.2 Case with single buyer and seller . . . . . . . . . . . . . . . . . 132
10.3 Market makers . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
10.4 Multiple agents . . . . . . . . . . . . . . . . . . . . . . . . . . 136
10.5 Effects on price flexibility . . . . . . . . . . . . . . . . . . . . . 138
10.6 Entropic force interpretation . . . . . . . . . . . . . . . . . . . 139
10.7 The entropic oscillator. . . . . . . . . . . . . . . . . . . . . . . 140
10.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
10.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
10.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 143
11 Quantized supply and demand 145
11.1 The quantum financial oscillator . . . . . . . . . . . . . . . . . 146
11.2 Quantum mass. . . . . . . . . . . . . . . . . . . . . . . . . . . 147
11.3 Quantum versus classical . . . . . . . . . . . . . . . . . . . . . 149
11.4 A model of stock markets . . . . . . . . . . . . . . . . . . . . . 151
11.5 Price impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
11.6 The quantum supply-demand curve . . . . . . . . . . . . . . . . 156
11.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
11.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
11.9 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
x CONTENTS
12 The quantum market 165
12.1 Financial bosons . . . . . . . . . . . . . . . . . . . . . . . . . . 165
12.2 Monetary exchange . . . . . . . . . . . . . . . . . . . . . . . . 166
12.3 Cash flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
12.4 Quantum (in)security . . . . . . . . . . . . . . . . . . . . . . . 169
12.5 The colour of money . . . . . . . . . . . . . . . . . . . . . . . 170
12.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
12.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
12.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 173
13 Entanglements 175
13.1 Preference reversal. . . . . . . . . . . . . . . . . . . . . . . . . 176
13.2 Mortgage default . . . . . . . . . . . . . . . . . . . . . . . . . 178
13.3 A quantum of debt . . . . . . . . . . . . . . . . . . . . . . . . 180
13.4 Social entanglement . . . . . . . . . . . . . . . . . . . . . . . . 181
13.5 Financial entanglement . . . . . . . . . . . . . . . . . . . . . . 182
13.6 Entanglement versus correlation . . . . . . . . . . . . . . . . . 184
13.7 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
13.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
13.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
13.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 188
14 Building quantum models 189
14.1 Principles for modelling . . . . . . . . . . . . . . . . . . . . . . 190
14.2 Probabilistic transactions . . . . . . . . . . . . . . . . . . . . . 191
14.3 Option pricing with supply and demand . . . . . . . . . . . . . 193
14.4 Quantum agent-based models . . . . . . . . . . . . . . . . . . . 195
14.5 The market maker model . . . . . . . . . . . . . . . . . . . . . 199
14.6 New foundations . . . . . . . . . . . . . . . . . . . . . . . . . . 201
14.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
15 Quantum gateway 205
15.1 The will is our pendulum . . . . . . . . . . . . . . . . . . . . . 205
15.2 A new operating system . . . . . . . . . . . . . . . . . . . . . . 207
15.3 Shut up and calculate . . . . . . . . . . . . . . . . . . . . . . . 208
15.4 Reasons to be quantum . . . . . . . . . . . . . . . . . . . . . . 210
15.5 The prediction test . . . . . . . . . . . . . . . . . . . . . . . . 211
15.6 Quantum by another name . . . . . . . . . . . . . . . . . . . . 214
15.7 Quantum calculation . . . . . . . . . . . . . . . . . . . . . . . 216
Bibliography 219
Index 233