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.
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.
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.
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.
In this paper, we present a quantum version of some portions of Mathematical Finance, including theory of arbitrage, asset pricing, and optional decomposition in financial markets based on finite dimensional quantum probability spaces. As examples, the quantum model of binomial markets is studied. We show that this quantum model ceases to pose the paradox which appears in the classical model of the binomial market. Furthermore, we re-deduce the Cox-Ross-Rubinstein binomial option pricing formula by considering multi-period quantum binomial markets.
The Group Quantization formalism is a scheme for constructing a functional space that is an irreducible infinite dimensional representation of the Lie algebra belonging to a dynamical symmetry group. We apply this formalism to the construction of functional space and operators for quadratic potentials -- gaussian pricing kernels in finance. We describe the Black-Scholes theory, the Ho-Lee interest rate model and the Euclidean repulsive and attractive oscillators. The symmetry group used in this work has the structure of a principal bundle with base (dynamical) group a semi-direct extension of the Heisenberg-Weyl group by SL(2,R), and structure group (fiber) the positive real line.
In this article, the authors discuss the use of quantum probability, that is, the probability theory of quantum mechanics, for option pricing and for finance in general. The authors discuss the motivations for applying quantum probability to finance. The critical issues are replacing random variables with operators, self-reflexivity of markets, and the existence of incompatible observations. The authors outline quantum probability theory, quantum stochastic processes, and the pricing of options in a quantum context. TOPICS:Options, portfolio theory, portfolio construction Key Findings • Quantum probability theory is a probabilistic theory of observations. Observations can change the system and be incompatible. • Quantum probability offers a more empirically faithful handling of large events and of uncertainty. • A better theory of valuation is offered by quantum probability theory than classical probability theory.
Abstract Finance stands as the most quantitatively advanced branch of the social sciences, yet it is also the most humbling. The mean-variance efficient frontier, the CAPM, arbitrage pricing relations, the Black–Scholes–Merton option formula and the efficient market hypothesis possess unparalleled formal elegance, yet they suffered a broad empirical refutation in the global financial crisis of 2008. Standard models understate tail risk, misinterpret the comovement of assets that are supposed to diversify risk, and presuppose liquidity and rationality — two features that vanish precisely when they are most needed. Behavioural finance has documented an extensive catalogue of cognitive biases, yet it remains a patchwork of isolated effects without a unified calculus of probability. This monograph is the first systematic work to establish quantum finance. It argues that the mathematical formalism of quantum mechanics supplies this missing unified structure: financial states are state vectors in Hilbert space; beliefs and valuations coexist as superposed amplitudes; trading and information disclosure act as measurements that collapse states in accordance with the Born rule; assets and markets are entangled with one another; and path interference generates fat tails and herding, which are inadmissible under classical additive probability. The book comprises six parts and thirty chapters in total. Part I lays the philosophical and formal foundations. Starting from the paradigm crisis, it proceeds through quantum probability and quantum stochastic processes to construct the analytical framework built on three core pillars of quantum finance. Part II develops quantum asset pricing, covering the quantum CAPM, quantum APT, quantum option pricing and the term structure of interest rates. Part III advances quantum risk management, including quantum risk measures, Value-at-Risk (VaR) and expected shortfall, stress testing, credit risk and phase transitions in systemic risk. Part IV addresses quantum portfolios and asset allocation, ranging from the quantum meanvariance frontier, strategic and tactical asset allocation, to factor investing and hedge fund strategies. Part V turns to market microstructure and behavioural finance, examining quantum order flows, marketmaker liquidity, quantum prospect theory and market sentiment. Part VI looks ahead to frontier avenues: quantum computational optimization, quantum machine learning, blockchains, postquantum cryptography and quantuminformed financial regulation. This book rigorously recovers the classical CAPM, efficient market hypothesis and Gaussian risk models as special cases under decoherence limits. It is intended for graduate students and faculty in finance and financial engineering, as well as researchers and practitioners working on asset pricing, risk management, portfolio allocation and financial regulation.