Quantum Groups in Mathematical Finance

35 Pages Posted: 2 May 2017 Last revised: 26 Jan 2024

See all articles by Paul McCloud

Paul McCloud

University College London - Department of Mathematics

Date Written: January 24, 2024

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.

Keywords: Derivative Pricing, Arbitrage, Noncommutativity, Duality, Quantum Groups, Hopf Algebras

JEL Classification: C02, G12, G13

Suggested Citation

McCloud, Paul, Quantum Groups in Mathematical Finance (January 24, 2024). Available at SSRN: https://ssrn.com/abstract=2960917 or http://dx.doi.org/10.2139/ssrn.2960917

Paul McCloud (Contact Author)

University College London - Department of Mathematics ( email )

Gower Street
London, WC1E 6BT
United Kingdom

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