Last twenty years have seen a tremendous growth of the financial mar- kets both in trading volumes and in sophistication of instruments. This ever-increasing complexity of the market structure necessitates use of math- ematically advanced models from the side of market participants. So far, the prevalent paradigm for these models has been the stochastic analysis as a branch of applied mathematics. In the last few years however, there has been an influx of purely physical concepts and methodology, constituting nascent field of econophysics. To what extent this new approach is useful remains, however, an open question. In my bachelor thesis I will focus on one subfield of econophysics, namely quantum finance. First, I will give an overview of both stochastic analysis and the new quantum finance paradigm. Then using the framework of quantum theory and quantum field theory I will construct a model of European stock options. 1
Last twenty years have seen a tremendous growth of the financial mar- kets both in trading volumes and in sophistication of instruments. This ever-increasing complexity of the market structure necessitates use of math- ematically advanced models from the side of market participants. So far, the prevalent paradigm for these models has been the stochastic analysis as a branch of applied mathematics. In the last few years however, there has been an influx of purely physical concepts and methodology, constituting nascent field of econophysics. To what extent this new approach is useful remains, however, an open question. In my bachelor thesis I will focus on one subfield of econophysics, namely quantum finance. First, I will give an overview of both stochastic analysis and the new quantum finance paradigm. Then using the framework of quantum theory and quantum field theory I will construct a model of European stock options. 1
Two underlying themes run through this book: first, defining and analyzing the subject of quantitative finance in the conceptual and mathematical framework of quantum theory, with special emphasis on its path-integral formulation, and, second, the introduction of the techniques and methodology of quantum field theory in the study of interest rates. No attempt is made to apply quantum theory in re-working the fundamental principles of finance. Instead, the term ‘quantum’ refers to the abstract mathematical constructs of quantum theory that include probability theory, state space, operators, Hamiltonians, commutation equations, Lagrangians, path integrals, quantized fields, bosons, fermions and so on. All these theoretical structures find natural and useful applications in finance. The path integral and Hamiltonian formulations of (random) quantum processes have been given special emphasis since they are equivalent to, as well as independent of, the formalism of stochastic calculus – which currently is one of the cornerstones of mathematical finance. The starting point for the application of path integrals and Hamiltonians in finance is in stock option pricing. Path integrals are subsequently applied to the modelling of linear and nonlinear theories of interest rates as a two-dimensional quantum field, something that is beyond the scope of stochastic calculus. Path integrals have the additional advantage of providing a framework for efficiently implementing the mathematical procedure of renormalization which is necessary in the study of nonlinear quantum field theories. The term ‘Quantum Finance’ represents the synthesis of the concepts, methods and mathematics of quantum theory, with the field of theoretical and applied finance.
At present, mainstream theoretical finance is almost completely dominated by stochastic calculus. Quantum finance , in contrast, presents a formulation of finance that is completely independent of stochastic calculus. Quantum finance addressed only a few problems of finance. The onus thus fell on me to demonstrate that a wide class of problems of finance, which at present are understood in terms of stochastic calculus, have a natural quantum finance formulation and generalization. It needed to be established that the quantum formulation of finance is equivalent to the one based on stochastic calculus and – if it is to be useful for the practitioners – is conceptually transparent and computationally tractable. Furthermore, it needed to be shown that quantum finance is not merely equivalent to stochastic calculus but goes beyond it; namely, that models based on quantum finance are more general than those based on stochastic calculus. Interest rates and coupon bonds and their derivatives form a major component of the debt market. They are, by far, the most complex and intricate types of financial instruments that also have a rich mathematical structure. Debt instruments provided an ideal testing ground for quantum finance . I gravitated to the study of interest rates and coupon bonds, since these financial instruments provide an excellent arena for exhibiting the main features of quantum finance – as well as for illustrating its point of departure from stochastic calculus. No attempt has been made to survey the subject of debt instruments, for which there exists a vast corpus of literature. Instead, this book has focused on only new results that have been obtained by applying quantum finance to debt instruments.
Abstract Finance stands as the most quantitatively advanced branch of the social sciences, yet it is also the most humbling. The mean-variance efficient frontier, the CAPM, arbitrage pricing relations, the Black–Scholes–Merton option formula and the efficient market hypothesis possess unparalleled formal elegance, yet they suffered a broad empirical refutation in the global financial crisis of 2008. Standard models understate tail risk, misinterpret the comovement of assets that are supposed to diversify risk, and presuppose liquidity and rationality — two features that vanish precisely when they are most needed. Behavioural finance has documented an extensive catalogue of cognitive biases, yet it remains a patchwork of isolated effects without a unified calculus of probability. This monograph is the first systematic work to establish quantum finance. It argues that the mathematical formalism of quantum mechanics supplies this missing unified structure: financial states are state vectors in Hilbert space; beliefs and valuations coexist as superposed amplitudes; trading and information disclosure act as measurements that collapse states in accordance with the Born rule; assets and markets are entangled with one another; and path interference generates fat tails and herding, which are inadmissible under classical additive probability. The book comprises six parts and thirty chapters in total. Part I lays the philosophical and formal foundations. Starting from the paradigm crisis, it proceeds through quantum probability and quantum stochastic processes to construct the analytical framework built on three core pillars of quantum finance. Part II develops quantum asset pricing, covering the quantum CAPM, quantum APT, quantum option pricing and the term structure of interest rates. Part III advances quantum risk management, including quantum risk measures, Value-at-Risk (VaR) and expected shortfall, stress testing, credit risk and phase transitions in systemic risk. Part IV addresses quantum portfolios and asset allocation, ranging from the quantum meanvariance frontier, strategic and tactical asset allocation, to factor investing and hedge fund strategies. Part V turns to market microstructure and behavioural finance, examining quantum order flows, marketmaker liquidity, quantum prospect theory and market sentiment. Part VI looks ahead to frontier avenues: quantum computational optimization, quantum machine learning, blockchains, postquantum cryptography and quantuminformed financial regulation. This book rigorously recovers the classical CAPM, efficient market hypothesis and Gaussian risk models as special cases under decoherence limits. It is intended for graduate students and faculty in finance and financial engineering, as well as researchers and practitioners working on asset pricing, risk management, portfolio allocation and financial regulation.
This book applies the mathematics and concepts of quantum mechanics and quantum field theory to the modelling of interest rates and the theory of options. Particular emphasis is placed on path integrals and Hamiltonians. Financial mathematics is dominated by stochastic calculus. The present book offers a formulation that is completely independent of that approach. As such many results emerge from the ideas developed by the author. This work will be of interest to physicists and mathematicians working in the field of finance, to quantitative analysts in banks and finance firms and to practitioners in the field of fixed income securities and foreign exchange. The book can also be used as a graduate text for courses in financial physics and financial mathematics.
A new paradigm based on quantum probability that can model stock markets and price options—and also upends classical quantitative finance. In The Quantum Stock Market, David Orrell combines theory and new empirical findings to argue for a new approach to finance based on quantum probability. The main finding, a property called q-variance, may be unique in finance in that it was discovered as the result of a prediction, affects financial markets in general, has no tunable parameter, and has important consequences for a range of topics including the pricing of financial options. The property, which somehow eluded detection for over a century, has been verified by experts in the field. And the fact that it is incompatible with classical finance—which is perhaps why it went undetected—suggests that the field must be reinvented. Orrell argues for nothing less than a paradigm shift in finance. As with any new work that challenges existing methods the ideas have already received pushback, from leaders in the field of finance to physicists skeptical of applying quantum probability outside its traditional domain. Written in an interdisciplinary style that mixes mathematical equations, data analysis, and informed comments on the field of finance, The Quantum Stock Market should be read by anyone who wants to be on the right side of a paradigm shift.
Foreword Preface Acknowledgements 1. Synopsis Part I. Fundamental Concepts of Finance: 2. Introduction to finance 3. Derivative securities Part II. Systems with Finite Number of Degrees of Freedom: 4. Hamiltonians and stock options 5. Path integrals and stock options 6. Stochastic interest rates' Hamiltonians and path integrals Part III. Quantum Field Theory of Interest Rates Models: 7. Quantum field theory of forward interest rates 8. Empirical forward interest rates and field theory models 9. Field theory of Treasury Bonds' derivatives and hedging 10. Field theory Hamiltonian of forward interest rates 11. Conclusions Appendix A: mathematical background Brief glossary of financial terms Brief glossary of physics terms List of main symbols References Index.