Quantum Groups in Mathematical Finance

• URL: http://arxiv.org/abs/1711.07279v8
• Date: Mon, 27 Feb 2023 12:47:59 GMT
• Title: Quantum Groups in Mathematical Finance
• Abstract summary: The principles that govern pricing are reviewed in the context of the functional calculus of quantum processes. The essay concludes with the construction of interest rate models from the irreducible representations of Lie algebras.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Mathematical finance explores the consistency relationships between the prices of securities imposed by elementary economic principles. Commonplace among these are the absence of arbitrage and the equivalence of expectation and price, both essentially algebraic constraints on the valuation map. The principles that govern pricing are here reviewed in the context of the stochastic and functional calculus of quantum processes. Framed in terms of the duality between states (the arbitrage-free valuation maps) and observables (the contractual settlements of securities), quantum groups are central to the approach. Translating the economic principles into this framework, a link is made between option pricing and von Neumann algebras that is illuminating in both directions. The essay concludes with the construction of interest rate models from the irreducible representations of semisimple Lie algebras, demonstrating their application in the pricing of European and Bermudan swaptions.

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