Information and Arbitrage: Applications of Quantum Groups in Mathematical Finance

Abstract

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.

Figures (21)

• Figure 1: The operations of $\ast$-algebra are defined on the point measures and digital functions of a classical group, linearly and topologically completed as the convolution algebra of measures and the pointwise algebra of functions. The definition of the quantum group abstracts these operations, enabling noncommutativity of both products simultaneously. This extension from classical to quantum economics introduces Heisenberg uncertainty as a novel source of volatility, with applications in the pricing of options.
• Figure 2: The products of states and observables are combined to create reversible operations that perform stochastic integration and differentiation on discrete schedules. Implemented on quantum hardware, these are entanglement and disentanglement operations.
• Figure 3: Quantum relu activation with two internal dimensions matches classical squareplus activation, which is smooth and strictly greater than classical relu activation. Used as a model for options, this additional value generates a fat-tailed volatility smile as a consequence of the Heisenberg uncertainty of noncommuting matrices.
• Figure 4: The implied cumulative density and (unannualised) volatility smile in the quantum perceptron with five internal dimensions calibrated to four examples of the SABR model with parameters $(t,\sigma,\alpha,\beta,\rho)$ given by $(4,0.25,0.4,1,0)$, $(4,0.25,0.6,1,0)$, $(4,0.25,0.4,0.5,0)$, $(4,0.25,0.4,1,0.5)$ respectively.
• Figure 5: The empirical system is created from the theoretical system by expanding the state and observable spaces via topological completion and reducing them by eliminating indistinguishable valuations, leveraging the locally convex topology defined by the pairing.

Theorems & Definitions (5)

• Theorem 1: Cauchy-Schwarz inequality
• Theorem 2: Heisenberg uncertainty
• Theorem 3: Gelfand-Naimark-Segal
• Theorem 4: Radon-Nikodym
• Definition 1: Derivative security