This chapter attempts to delve deeper into the question on how quantum mechanical techniques can be brought closer into the realm of economics and finance. The (non-)Hermiticity of finance-based operators? Hermiticity of operators was discussed in Chapter 4 of the book. We again take up this very important concept in the context of financial asset pricing. It is a classical result from quantum mechanics that the existence of Hermiticity of the Hamiltonian operator is intimately linked with the notion of conservation of probability. The existence of Hermiticity is also known to be closely linked to the concept of spatial localization. Please see below. Baaquie [1] makes the important argument that the Black-Scholes Hamiltonian is non-Hermitian and this condition provides for the need to satisfy the martingale condition. Please recall that the martingale property was covered in Chapter 2, Section 2.9. It is also important to mention that Luigi Accardi has indicated that it is white noise which may be the cause of non-Hermiticity in a finance context. One can argue that within an economics/finance context, the equivalent of the state function, using Baaquie [1] [2], can be the option price function. A similar interpretation can also be found in the paper by Li and Zhang [3] (see also Haven [4]). Please note that the work of Li and Zhang is covered in Section 13.12 of this chapter.
This chapter attempts to delve deeper into the question on how quantum mechanical techniques can be brought closer into the realm of economics and finance. The (non-)Hermiticity of finance-based operators? Hermiticity of operators was discussed in Chapter 4 of the book. We again take up this very important concept in the context of financial asset pricing. It is a classical result from quantum mechanics that the existence of Hermiticity of the Hamiltonian operator is intimately linked with the notion of conservation of probability. The existence of Hermiticity is also known to be closely linked to the concept of spatial localization. Please see below. Baaquie [1] makes the important argument that the Black-Scholes Hamiltonian is non-Hermitian and this condition provides for the need to satisfy the martingale condition. Please recall that the martingale property was covered in Chapter 2, Section 2.9. It is also important to mention that Luigi Accardi has indicated that it is white noise which may be the cause of non-Hermiticity in a finance context. One can argue that within an economics/finance context, the equivalent of the state function, using Baaquie [1] [2], can be the option price function. A similar interpretation can also be found in the paper by Li and Zhang [3] (see also Haven [4]). Please note that the work of Li and Zhang is covered in Section 13.12 of this chapter.
This chapter starts with the idea of a quantized version of supply and demand. We also provide for a discussion why quantum probability is useful in both option pricing and investment choice decisions. We also discuss briefly quantum-like Bayesian networks within the setting of missing financial data. We also hint toward the usefulness of open systems in social science and then dive into the very important observation that Black–Scholes Hamiltonians are non-Hermitian. Toward the end of the chapter, we give an overview of models (with and without memory and (a)symmetry between past and future times). We give some examples and then conclude the chapter with thoughts on the limitations of mathematical modelling.
In this chapter and Chapters 11, 12, and 13, we will discuss quantum-like models with financial applications in mind. The very last chapter of this book will analyze the sources of quantum-like processing in the brain. The concept of non-arbitrage, as we have indicated before (see Section 4.18.3) is of key importance in financial models. We reiterate its importance in this chapter. We also briefly mention a popular finance model. In the coming chapters, the wave function will sometimes have an information interpretation. Therefore we also discuss the informational aspects of the classical financial theory (see Section 10.1). This chapter is meant to introduce the reader to some of the key achievements in financial theory. By doing so, we hope to convince the reader, in later chapters, that some of those very achievements can possibly be reformulated with the help of the wave function approach. Section 10.3 in this chapter will give a flavor of how Fourier integration can possibly be used to model George Soros' concept of reflexivity. Relevance of the concepts of efficiency and non-arbitrage: a brief discussion With the 2008 financial crisis, the concept of efficiency may have been seriously questioned. Indeed the financial market in September 2008 did take a tremendous hit and the ensuing price movements would not be consistent with a so-called efficient market.
In this talk we intend to provide for an overview of the uses we can make in financial economics of several quantum physical concepts. We introduce and briefly discuss the so called information wave function. We discuss how the information wave function can be of use in financial option pricing. We then briefly allude on how the information wave function can be used in arbitrage. We briefly discuss how the information wave concept can be connected to measures of information. We round off the paper on recent work we are doing with Andrei Khrennikov on i) testing probability interference in psychology (Khrennikov and Haven 2006 (I)) and ii) finding uses of ‘financial ’ nonlocality and entanglement (Khrennikov and Haven 2006 (II)). In summary this talk addresses interference (probability interference), financial entanglement and looks at using the wave function in a variety of ways within an economics/finance – political science – social interaction environment.
In this chapter the concept of the Hamiltonian is introduced in the pricing of options. Hamiltonians occur naturally in finance; to demonstrate this the analysis of the Black–Scholes equation is recast in the formalism of quantum mechanics. It is then shown how the Hamiltonian plays a central role in the general theory of option pricing. The Hamiltonian formulation provides new tools for obtaining solutions for option pricing; two key concepts related to the Hamiltonian are (a) eigenfunctions and (b) potentials. Knowledge of all the eigenfunctions of a Hamiltonian yields an exact solution for a large class of path-dependent and path-independent options. For example, barrier options can be modelled by placing constraints on the eigenfunctions of the Hamiltonian. The potentials are a means for defining new financial instruments, and for modelling path-dependent options. Essentials of quantum mechanics It is shown in this chapter that option pricing in finance has a mathematical description that is identical to a quantum system; hence the key features of quantum theory are briefly reviewed. Quantum theory is a vast subject that forms the bedrock of contemporary physics, chemistry and biology. Only those aspects of quantum mechanics are reviewed that are relevant for the analysis of option pricing. In classical mechanics the position of a particle at time t , denoted by x t , is a deterministic function of t , and is given by Newton's law of motion. Classical mechanics is analogous to the case of the evolution of a stock price with zero volatility (σ = 0) that yields a deterministic evolution of the stock price.
The pilot wave function and its uses outside of quantum mechanics Let us recall our discussion on Bohmian mechanics in Chapter 6. The idea of using Bohmian mechanics outside of quantum mechanics is not new anymore. Khrennikov [1] provides for an overview on how this model can contribute in precise terms to areas such as economics and finance.1 Historically, the work by Bohm and Hiley [2] and also Hiley and Pylkkanen [3] brought forward the idea that the pilot wave function could be seen as a wave function containing information. In Khrennikov [4], Choustova [5], and Haven [6], the idea of using pilot wave theory to finance was investigated. Khrennikov [1] (p. 160) remarks that “the force induced by the pilot wave field does not depend on the amplitude of the wave.” He cites the argument made by Bohm and Hiley [2] that because of this property the pilot wave is an information wave. The various interpretations (with applications) of the wave function will be covered in detail in Chapter 13. Here are some of the key features which can be of use in an financial/economics setting. When the wave function is not factorized, then a change in the price of a stock i will affect the prices of stocks, j , with j ≠ i . See Khrennikov [1] (p. 161). The Bohmian theory, as we remarked already in Chapters 1 and 6, is non-local.
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.