Information and Arbitrage: Applications of Quantum Groups in Mathematical Finance

TL;DR

This paper explores the use of quantum groups in mathematical finance, proposing a quantum economic framework that extends classical models and offers new insights into pricing derivatives.

Contribution

It introduces a quantum group-based axiomatic framework for finance, connecting quantum logic with economic principles like no-arbitrage and equivalence.

Findings

Quantum groups naturally underpin stochastic and functional calculus in finance.

A holographic duality exchanges states and observables, creating dual economic models.

Noncommutativity provides a new modeling resource for derivative pricing.

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.