Group Quantization of Quadratic Hamiltonians in Finance

TL;DR

This paper applies the group quantization formalism to construct functional spaces and operators for quadratic Hamiltonians in finance, including models like Black-Scholes and Ho-Lee, revealing underlying symmetries.

Contribution

It introduces a novel application of group quantization to financial models with quadratic potentials, identifying the symmetry group structure involved.

Findings

Constructed functional spaces for Gaussian pricing kernels

Identified the symmetry group as a semi-direct extension of Heisenberg-Weyl by SL(2,R)

Applied formalism to models like Black-Scholes and Ho-Lee

Abstract

The Group Quantization formalism is a scheme for constructing a functional space that is an irreducible infinite dimensional representation of the Lie algebra belonging to a dynamical symmetry group. We apply this formalism to the construction of functional space and operators for quadratic potentials -- gaussian pricing kernels in finance. We describe the Black-Scholes theory, the Ho-Lee interest rate model and the Euclidean repulsive and attractive oscillators. The symmetry group used in this work has the structure of a principal bundle with base (dynamical) group a semi-direct extension of the Heisenberg-Weyl group by SL(2,R), and structure group (fiber) the positive real line.