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arXiv:1110.5350 · AIP conference proceedings

A quantum-like approach to the stock market

Diederik Aerts, Bart D’Hooghe, Sandro Sozzo

Published Jan 1, 2012• 4 citations
arXivHTMLPDF

Abstract

Modern approaches to stock pricing in quantitative finance are typically founded on the 'Black-Scholes model' and the underlying 'random walk hypothesis'. Empirical data indicate that this hypothesis works well in stable situations but, in abrupt transitions such as during an economical crisis, the random walk model fails and alternative descriptions are needed. For this reason, several proposals have been recently forwarded which are based on the formalism of quantum mechanics. In this paper we apply the 'SCoP formalism', elaborated to provide an operational foundation of quantum mechanics, to the stock market. We argue that a stock market is an intrinsically contextual system where agents' decisions globally influence the market system and stocks prices, determining a nonclassical behavior. More specifically, we maintain that a given stock does not generally have a definite value, e.g., a price, but its value is actualized as a consequence of the contextual interactions in the trading process. This contextual influence is responsible of the non-Kolmogorovian quantum-like behavior of the market at a statistical level. Then, we propose a 'sphere model' within our 'hidden measurement formalism' that describes a buying/selling process of a stock and shows that it is intuitively reasonable to assume that the stock has not a definite price until it is traded. This result is relevant in our opinion since it provides a theoretical support to the use of quantum models in finance.

DOI
10.1063/1.3689002

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Modern approaches to stock pricing in quantitative finance are typically founded on the 'Black-Scholes model' and the underlying 'random walk hypothesis'. Empirical data indicate that this hypothesis works well in stable situations but, in abrupt transitions such as during an economical crisis, the random walk model fails and alternative descriptions are needed. For this reason, several proposals have been recently forwarded which are based on the formalism of quantum mechanics. In this paper we apply the 'SCoP formalism', elaborated to provide an operational foundation of quantum mechanics, to the stock market. We argue that a stock market is an intrinsically contextual system where agents' decisions globally influence the market system and stocks prices, determining a nonclassical behavior. More specifically, we maintain that a given stock does not generally have a definite value, e.g., a price, but its value is actualized as a consequence of the contextual interactions in the trading process. This contextual influence is responsible of the non-Kolmogorovian quantum-like behavior of the market at a statistical level. Then, we propose a 'sphere model' within our 'hidden measurement formalism' that describes a buying/selling process of a stock and shows that it is intuitively reasonable to assume that the stock has not a definite price until it is traded. This result is relevant in our opinion since it provides a theoretical support to the use of quantum models in finance.
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.
The problem of measurement in economic models and the possibility of their quantum-mechanical description are considered. It is revealed that the apparent paradox of such a description is associated with a priori requirement of conformity of the model to all the alternatives of free choice of the observer. The measurement of the state of a trader on a stock exchange is formally defined as his responses to the proposals of sale at a fixed price. It is shown that an analogue of Bell's inequalities for this measurement model is violated at the most general assumptions related to the strategy of the trader and requires a quantum-mechanical description of the dynamics of his condition. In the framework of the theory of weak continuous quantum measurements, the equation of stock price dynamics and the quantum-mechanical generalization of the F. Black and M. Scholes model for pricing options are obtained. The fundamental distinctions between the obtained model and the classical one are discussed.
We pursue the quantum-mechanical challenge to the efficient market hypothesis for the stock market by employing the quantum Brownian motion model. We utilize the quantum Caldeira-Leggett master equation as a possible phenomenological model for the stock-market-prices fluctuations while introducing the external harmonic field for the Brownian particle. Two quantum regimes are of particular interest: the exact regime as well as the approximate regime of the pure decoherence ("recoilless") limit of the Caldeira-Leggett equation. By calculating the standard deviation and the kurtosis for the particle's position observable, we can detect deviations of the quantum-mechanical behavior from the classical counterpart, which bases the efficient market hypothesis. By varying the damping factor, temperature as well as the oscillator's frequency, we are able to provide interpretation of different economic scenarios and possible situations that are not normally recognized by the efficient market hypothesis. Hence we recognize the quantum Brownian oscillator as a possibly useful model for the realistic behavior of stock prices.
Uncertainty in economics still poses some fundamental problems illustrated, e.g., by the Allais and Ellsberg paradoxes.To overcome these difficulties, economists have introduced an interesting distinction between risk and ambiguity depending on the existence of a (classical Kolmogorovian) probabilistic structure modeling these uncertainty situations.On the other hand, evidence of everyday life suggests that context plays a fundamental role in human decisions under uncertainty.Moreover, it is well known from physics that any probabilistic structure modeling contextual interactions between entities structurally needs a non-Kolmogorovian quantum-like framework.In this paper we introduce the notion of contextual risk with the aim of modeling a substantial part of the situations in which usually only ambiguity is present.More precisely, we firstly introduce the essentials of an operational formalism called the hidden measurement approach in which probability is introduced as a consequence of fluctuations in the interaction between entities and contexts.Within the hidden measurement approach we propose a sphere model as a mathematical tool for situations in which contextual risk occurs.We show that a probabilistic model of this kind is necessarily non-Kolmogorovian, hence it requires either the formalism of quantum mechanics or a generalization of it.This insight is relevant, for it explains the presence of quantum or, better, quantum-like, structures in economics, as suggested by some authors, and can serve to solve the aforementioned paradoxes.
The trade of a fixed stock can be regarded as the basic process that measures its momentary price. The stock price is exactly known only at the time of sale when the stock is between traders, that is, only in the case when the owner is unknown. We show that the stock price can be better described by a function indicating at any moment of time the probabilities for the possible values of price if a transaction takes place. This more general description contains partial information on the stock price, but it also contains partial information on the stock owner. By following the analogy with quantum mechanics, we assume that the time evolution of the function describing the stock price can be described by a Schrodinger type equation.
Measurement is a key element of quantum theory where only one possible outcome comes to pass, while other ones disappear. Similar phenomena have been observed in psychology, language processing, and economics. Mathematical descriptions that employ elements of quantum theory have successfully modeled effects observable outside of subatomic physics. Price and value are important concepts in finance. We distinguish between price as something fixed in a transaction and publicly available, while value is an agent’s private assessment. Measurement and updates of both may happen in asynchronous timelines and the frequency of measurements might impact change. When considering derivative markets, usually the focus is on price dynamics given by a stochastic process – market liquidity is often assumed to be infinite. A more detailed consideration of liquidity in relation to price dynamics is desirable and this work outlines possible angles from quantum-inspired models. We discuss how market liquidity and volatility relate to each other in the residential real estate and stock market. We investigate quantum walks, quantum models of psychological decision making by individuals and groups, and the role that the framing of a question plays, all with the goal of an improved understanding of the link between observed price volatility and liquidity.