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Quantum Valuation: A Wavefunction Formalism for Price, Uncertainty, Tension, and Noncommutative Arbitrage

Lukas Saul

Published Oct 6, 2026• 0 citations
Publisher

Abstract

We develop a quantum-style formalism for economic valuation in which the value of a good is represented by a wavefunction over price space. The observed price is associated with measurement of a price operator, while uncertainty in valuation is represented by the width of the valuation distribution. The state evolves according to a Schr\"odinger-type equation containing a valuation potential, a price-inertia term, and, for an open market, a non-Hermitian source/sink term. The purpose of the formalism is not to claim that financial markets are microscopic quantum systems. Rather, quantum mechanics provides a compact mathematical language for situations in which valuation is uncertain, sequential operations need not be interchangeable, and cyclic trading can reveal discrepancies that are invisible when each transaction is considered in isolation. We derive the canonical price--momentum commutator, the moment hierarchy, Ehrenfest equations, and a price--velocity uncertainty relation. We then introduce a lattice of mutually interacting assets through a neighbor-tension potential. The resulting model supports propagating valuation disturbances and provides a simple mathematical description of herding and synchronization. The principal new construction concerns arbitrage. Rather than identifying arbitrage directly with a geometric phase, we represent individual trading operations by operators. When two trading operations commute, their order is immaterial. When they do not commute, a round trip constructed from their group commutator fails to close in valuation space. This nonclosure is a natural mathematical signature of path dependence. An economically observable arbitrage opportunity arises when the resulting cycle transformation produces a positive net payoff after transaction costs. The formalism therefore connects four ideas: valuation uncertainty, neighbor interaction, noncommuting trading operations, and cyclic arbitrage. A harmonic valuation well provides a solvable benchmark, while a three-asset trading loop provides the simplest setting in which the distinction between ordinary price differences and genuinely noncommutative cycle effects becomes visible.

DOI
10.5281/zenodo.23176429

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•Jan 1, 2017•1 citation
We develop a quantum-style formalism for economic valuation in which the value of a good is represented by a wavefunction over price space. The observed price is associated with measurement of a price operator, while uncertainty in valuation is represented by the width of the valuation distribution. The state evolves according to a Schr"odinger-type equation containing a valuation potential, a price-inertia term, and, for an open market, a non-Hermitian source/sink term. The purpose of the formalism is not to claim that financial markets are microscopic quantum systems. Rather, quantum mechanics provides a compact mathematical language for situations in which valuation is uncertain, sequential operations need not be interchangeable, and cyclic trading can reveal discrepancies that are invisible when each transaction is considered in isolation. We derive the canonical price--momentum commutator, the moment hierarchy, Ehrenfest equations, and a price--velocity uncertainty relation. We then introduce a lattice of mutually interacting assets through a neighbor-tension potential. The resulting model supports propagating valuation disturbances and provides a simple mathematical description of herding and synchronization. The principal new construction concerns arbitrage. Rather than identifying arbitrage directly with a geometric phase, we represent individual trading operations by operators. When two trading operations commute, their order is immaterial. When they do not commute, a round trip constructed from their group commutator fails to close in valuation space. This nonclosure is a natural mathematical signature of path dependence. An economically observable arbitrage opportunity arises when the resulting cycle transformation produces a positive net payoff after transaction costs. The formalism therefore connects four ideas: valuation uncertainty, neighbor interaction, noncommuting trading operations, and cyclic arbitrage. A harmonic valuation well provides a solvable benchmark, while a three-asset trading loop provides the simplest setting in which the distinction between ordinary price differences and genuinely noncommutative cycle effects becomes visible.
A central topic in economics and finance is the valuation of strategic games and bets, of investments providing uncertain future cashflows, and of other asset-like structures and contracts. Often, such valuation involves a probabilistic model, random variables, and computing expectation values of complex payoffs. Extending beyond this classical framework, a scenario emerges where assets are non-commuting operators and the inherent uncertainty is described by quantum states. In this work, we combine such notions of quantum finance with a graph-theoretic framework and ideas from contextuality. We discuss exclusive quantum assets and introduce scenarios when holding such assets can provide advantages over holding classical assets. In particular, we consider a "quantum market" based on nonlocal games such as the CHSH game and demonstrate that this setting can function as a market with well-defined financial structures. We provide upper bounds on the fair prices of the quantum assets based on graph-theoretic quantities like the independence number and the Lovász theta number. Our insights into markets of non-commuting assets help to elucidate what a financial system in a world filled with quantum technology could look like.
Quantum computers have the potential to provide an advantage for financial pricing problems by the use of quantum estimation. In a broader context, it is reasonable to ask about situations where the market and the assets traded on the market themselves have quantum properties. In this work, we consider a financial setting where instead of by classical probabilities the market is described by a pure quantum state or, more generally, a quantum density operator. This setting naturally leads to a new asset class, which we call quantum assets. Under the assumption that such assets have a price and can be traded, we develop an extended definition of arbitrage to quantify gains without the corresponding risk. Our main result is a quantum version of the first fundamental theorem of asset pricing. If and only if there is no arbitrage, there exists a risk-free density operator under which all assets are martingales. This density operator is used for the pricing of quantum derivatives. To prove the theorem, we study the density operator version of the Radon-Nikodym measure change. We provide examples to illustrate the theory.
The relationship between expectation and price is commonly established with two principles: no-arbitrage, which asserts that both maps are positive; and equivalence, which asserts that the maps share the same null events. Constructed from the Arrow-Debreu securities, classical and quantum models of economics are then distinguished by their respective use of classical and quantum logic, following the program of von Neumann. In this essay, the operations and axioms of quantum groups are discovered in the minimal preconditions of stochastic and functional calculus, making this the natural domain for the axiomatic development of mathematical finance. Quantum economics emerges from the twin pillars of the Gelfand-Naimark-Segal construction, implementing the principle of no-arbitrage, and the Radon-Nikodym theorem, implementing the principle of equivalence. Exploiting quantum group duality, a holographic principle that exchanges the roles of state and observable creates two distinct economic models from the same set of elementary valuations. Advocating on the grounds that this contains and extends classical economics, noncommutativity is presented as a modelling resource, with novel applications in the pricing of options and other derivative securities.
Price, cost, and income (PCI) methods are traditionally used to approximate the value state of an economic commodity such as a property. Based on the estimates of these methods, we explore how quantum theory represents the fundamental process of value valuation in practice. We propose that the mathematical formalism of quantum theory is a promising view and measure of economic value. To ground our exploration, we first map traditional PCI estimates onto three-dimensional spherical coordinates, which were then transformed into two-dimensional quantum states using the Bloch sphere. This step enabled the computation of eigenvalues and eigenvectors of the Hamiltonian matrix, from which the value state measures were derived. The results exhibit practical applications as well as fundamental insights into potential connections between economic and quantum value states.
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.
Mathematical finance explores the consistency relationships between the prices of securities imposed by elementary economic principles. Commonplace among these are replicability and the absence of arbitrage, both essentially algebraic constraints on the valuation map from a security to its price. The discussion is framed in terms of observables, the securities, and states, the linear and positive maps from security to price. Founded on the principles of replicability and the absence of arbitrage, mathematical finance then equates to the theory of positive linear maps and their numeraire invariances. This acknowledges the algebraic nature of the defining principles which, crucially, may be applied in the context of quantum probability as well as the more familiar classical setting. Quantum groups are here defined to be dual pairs of *-Hopf algebras, and the central claim of this thesis is that the model for the dynamics of information relies solely on the quantum group properties of observables and states, as demonstrated by the application to finance. This naturally leads to the study of models based on restrictions of the *-Hopf algebras, such as the Quadratic Gauss model, that retain much of the phenomenology of their parent within a more tractable domain, and extensions of the *-Hopf algebras, such as the Linear Dirac model, with novel features unattainable in the classical case.