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arXiv:1207.3412 · AIP conference proceedings

A quantum mechanical model for the relationship between stock price and stock ownership

Liviu‐Adrian Cotfas

Published Jan 1, 2012• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/1.4766764

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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.
The price of a given stock is exactly known only at the time of sale when the stock is between the traders. If we know the price (owner) then we have no information on the owner (price). A more general description including cases when we have partial information on both price and ownership is obtained by using the quantum mechanics methods. The relation price-ownership is similar to the relation position-momentum. Our approach is based on the mathematical formalism used in the case of quantum systems with finite-dimensional Hilbert space. The linear operator corresponding to the ownership is obtained from the linear operator corresponding to the price by using the finite Fourier transform. In our idealized model, the Schrodinger type equation describing the time evolution of the stock price is solved numerically.
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.
Writing the article "Time independent pricing of options in range bound markets" [1], the question in the title came naturally to my mind.It is stated, in the above article, that in certain market conditions the stock price is subjected to an equation that exactly matches a time independent Schrodinger equation.The time independent equation for options valuation is used further to explain a stock market phenomenon that resembles an α particle decay tunneling effect.The transmission coefficient for the stock price tunneling effect it is also deduced.Although, it may not have important impact in quantum physics, the philosophical aspects residing in the use of quantum mechanics for stock market specific are very important.
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.
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.