Quantum Duality in Mathematical Finance

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 numeraire invariances. This acknowledges the algebraic nature of the defining principles which, crucially, may be applied in the context of quantum probability as well as the more familiar classical setting.Quantum groups are here defined to be dual pairs of ∗-Hopf algebras, and the central claim of this thesis is that the model for the dynamics of information relies solely on the quantum group properties of observables and states, as demonstrated by the application to finance. This naturally leads to the study of models based on restrictions of the ∗-Hopf algebras, such as the Quadratic Gauss model, that retain much of the phenomenology of their parent within a more tractable domain, and extensions of the ∗-Hopf algebras, such as the Linear Dirac model, with novel features unattainable in the classical case