The recent crash demonstrated (once again) that the description of the financial market by present financial mathematics cannot be considered as totally satisfactory. We remind that nowadays financial mathematics is heavily based on the use of random variables and stochastic processes which are described by Kolmogorov’s measuretheoretic model for probability (“classical probabilistic model”). I speculate that the present financial crises is a sign (a kind of experiment to test validity of classical probability theory at the financial market) that the use of this model in finances should be either totally rejected or at least completed. One of the best candidates for a new probabilistic financial model is quantum probability or its generalizations, so to say quantum-like (QL) models. Speculations that the financial market may be nonclassical have been present in scientific literature for many years. The aim of this note is to move from the domain of speculation to rigorous statistical arguments in favor of probabilistic nonclassicality of the financial market. I design a corresponding statistical test which is based on violation of the formula of total probability (FTP). The latter is the basic in classical probability and its violation would be a strong sign in favor of QL behavior at the market. 1 1
Recently the mathematical formalism of quantum mechanics, especially methods of quantum probability theory, started to be widely used in a variety of applications outside of physics, e.g., cognition and psychology as well as economy and finances. To distinguish such models from genuine quantum physical models, they often called quantum-like (although often people simply speak about, e.g., "quantum cognition"). These novel applications generate a number of foundational questions. Nowadays we can speak about a new science - foundations of quantum-like modeling. At the first stage this science was mainly about comparison of classical and quantum models, mainly in the probabilistic setting. It was found that statistical data from cognitive psychology violate some basic constraints posed on data by classical probability theory (Kolmogorov, 1933); in particular, the constraints given by the formula of total probability and Bell's type inequalities. Recently another question attracted some attention. In spite of real success in applications, there are no reason to believe that the quantum probability would cover completely all problems of, e.g., cognition. May be more general probability models have to be explored. A similar problem attracted a lot of attention in foundations of quantum physics culminating in a series of experiments to check Sorkin's equality for the triple-slit experiment by Weihs' group. In this note we present a similar test in the cognitive experimental setting. Performance of this test would either give further confirmation of the adequacy of the quantum probability model to cognitive applications or rejection of the conventional quantum model. Thus this note opens the door for a series of exciting experimental tests for the quantum-like model of cognition.
We analyze complexity of financial (and general economic) processes by comparing classical and quantum-like models for randomness. Our analysis implies that it might be that a quantum-like probabilistic description is more natural for financial market than the classical one. A part of our analysis is devoted to study the possibility of application of the quantum probabilistic model to agents of financial market. We show that, although the direct quantum (physical) reduction (based on using the scales of quantum mechanics) is meaningless, one may apply so called quantum-like models. In our approach quantum-like probabilistic behaviour is a consequence of contextualy of statistical data in finances (and economics in general). However, our hypothesis on "quantumness" of financial data should be tested experimentally (as opposed to the conventional description based on the noncontextual classical probabilistic approach). We present a new statistical test based on a generalization of the well known in quantum physics Bell's inequality.
We apply methods of quantum mechanics for mathematical modeling of price dynamics at the financial market. We propose to describe behavioral financial factors (e.g., expectations of traders) by using the pilot wave (Bohmian) model of quantum mechanics. Trajectories of prices are determined by two financial potentials: classical-like
V(q) ("hard" market conditions, e.g., natural resources) and quantum-like
U(q) (behavioral market conditions). On one hand, our Bohmian model is a quantum-like model for the financial market, cf. with works of W. Segal, I. E. Segal, E. Haven, E. W.Piotrowski, J. Sladkowski. On the other hand, (since Bohmian mechanics provides the possibility to describe individual price trajectories) it belongs to the domain of extended research on deterministic dynamics for financial assets (C.W. J. Granger, W.A. Barnett, A. J. Benhabib, W.A. Brock, C. Sayers, J. Y. Campbell, A. W. Lo, A. C. MacKinlay, A. Serletis, S. Kuchta, M. Frank, R. Gencay, T. Stengos, M. J. Hinich, D. Patterson, D. A. Hsieh, D. T. Caplan, J.A. Scheinkman, B. LeBaron and many others).
We apply methods of quantum mechanics for mathe-matical modeling of price dynamics at financial mar-ket. We propose to describe behavioral financial factors (e.g., expectations of traders) by using the pilot wave (Bohmian) model of quantum mechanics. Our Bohmian model is a quantum-like model for the financial market, cf. with works of W. Segal, I. E. Segal, E. Haven. Our model expresses complexity of financial market: tra-ditional description of price dynamics is completed by Schrödinger’s dynamics for the pilot wave of expecta-tions of traders. This is a kind of socio-economic model for financial market
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.