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Quantum Economics and Finance

Information Entropy of the Financial Market: Modelling Random Processes Using Open Quantum Systems

Will Hicks

Published Sep 21, 2024• 2 citations
Publisher

Abstract

We discusss the role of information entropy on the behaviour of random processes, and how this might take effect in the dynamics of financial market prices. We then go on to show how the Open Quantum Systems approach can be used as a more flexible alternative to classical methods in terms of modelling the entropy gain of a random process. We start by describing an open quantum system that can be used to model the state of a financial market. We then go on to show how to represent an essentially classical diffusion in this framework. Finally, we show how by relaxing certain assumptions, one can generate interesting and essentially non-classical results, which are highlighted through numerical simulations.

DOI
10.1177/29767032241279990

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We discusss the role of information entropy on the behaviour of random processes, and how this might take effect in the dynamics of financial market prices. We then go on to show how the Open Quantum Systems approach can be used as a more flexible alternative to classical methods in terms of modelling the entropy gain of a random process. We start by describing an open quantum system that can be used to model the state of a financial market. We then go on to show how to represent an essentially classical diffusion in this framework. Finally, we show how by relaxing certain assumptions, one can generate interesting and essentially non-classical results, which are highlighted through numerical simulations.
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We start with the idea that open quantum systems can be used to represent financial markets by modelling events from the external environment and their impact on the market price. We show how to characterize distinct orbits of the time evolution, and look at the development of the reduced density matrix, that represents the state of the market, over long time frames. In particular we distinguish between classical and non-classical modes of time evolution. We show that whilst both tend to the same set of maximum entropy states, this occurs faster in classical systems, with a knock on effect on the resulting probability distributions. We demonstrate how non-classical modes of time-evolution can be used to incorporate factors such as illiquid trades and imperfect trading mechanisms, and distinguish between different mechanisms of non-classical time evolution.
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.
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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 the 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).