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.
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.
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.
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.
Two underlying themes run through this book: first, defining and analyzing the subject of quantitative finance in the conceptual and mathematical framework of quantum theory, with special emphasis on its path-integral formulation, and, second, the introduction of the techniques and methodology of quantum field theory in the study of interest rates. No attempt is made to apply quantum theory in re-working the fundamental principles of finance. Instead, the term ‘quantum’ refers to the abstract mathematical constructs of quantum theory that include probability theory, state space, operators, Hamiltonians, commutation equations, Lagrangians, path integrals, quantized fields, bosons, fermions and so on. All these theoretical structures find natural and useful applications in finance. The path integral and Hamiltonian formulations of (random) quantum processes have been given special emphasis since they are equivalent to, as well as independent of, the formalism of stochastic calculus – which currently is one of the cornerstones of mathematical finance. The starting point for the application of path integrals and Hamiltonians in finance is in stock option pricing. Path integrals are subsequently applied to the modelling of linear and nonlinear theories of interest rates as a two-dimensional quantum field, something that is beyond the scope of stochastic calculus. Path integrals have the additional advantage of providing a framework for efficiently implementing the mathematical procedure of renormalization which is necessary in the study of nonlinear quantum field theories. The term ‘Quantum Finance’ represents the synthesis of the concepts, methods and mathematics of quantum theory, with the field of theoretical and applied finance.
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.
In this talk we intend to provide for an overview of the uses we can make in financial economics of several quantum physical concepts. We introduce and briefly discuss the so called information wave function. We discuss how the information wave function can be of use in financial option pricing. We then briefly allude on how the information wave function can be used in arbitrage. We briefly discuss how the information wave concept can be connected to measures of information. We round off the paper on recent work we are doing with Andrei Khrennikov on i) testing probability interference in psychology (Khrennikov and Haven 2006 (I)) and ii) finding uses of ‘financial ’ nonlocality and entanglement (Khrennikov and Haven 2006 (II)). In summary this talk addresses interference (probability interference), financial entanglement and looks at using the wave function in a variety of ways within an economics/finance – political science – social interaction environment.
This chapter attempts to delve deeper into the question on how quantum mechanical techniques can be brought closer into the realm of economics and finance. The (non-)Hermiticity of finance-based operators? Hermiticity of operators was discussed in Chapter 4 of the book. We again take up this very important concept in the context of financial asset pricing. It is a classical result from quantum mechanics that the existence of Hermiticity of the Hamiltonian operator is intimately linked with the notion of conservation of probability. The existence of Hermiticity is also known to be closely linked to the concept of spatial localization. Please see below. Baaquie [1] makes the important argument that the Black-Scholes Hamiltonian is non-Hermitian and this condition provides for the need to satisfy the martingale condition. Please recall that the martingale property was covered in Chapter 2, Section 2.9. It is also important to mention that Luigi Accardi has indicated that it is white noise which may be the cause of non-Hermiticity in a finance context. One can argue that within an economics/finance context, the equivalent of the state function, using Baaquie [1] [2], can be the option price function. A similar interpretation can also be found in the paper by Li and Zhang [3] (see also Haven [4]). Please note that the work of Li and Zhang is covered in Section 13.12 of this chapter.