Abstract

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 scale invariances. This recognises that the defining principles are in essence algebraic, and may be satisfied in circumstances more general than the classical probabilistic setting. These slides consider the dual Hopf algebraic structures of observables and states, and how they may be utilised to create efficient models for pricing. This naturally leads to the study of models based on restrictions of the dual Hopf algebras, such as the Quadratic Gauss model, that retain much of the phenomenology of their parents within a more tractable domain, and extensions of the dual Hopf algebras, such as the Linear Dirac model, with novel features unattainable in the classical case.

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