Gauge symmetries and the Higgs mechanism in Quantum Finance

TL;DR

This paper applies concepts from gauge symmetry and the Higgs mechanism to quantum finance, showing how stochastic volatility arises naturally from symmetry principles in the Hamiltonian formulation of financial equations.

Contribution

It introduces a novel framework linking gauge symmetries and the Higgs mechanism to stochastic volatility in financial models, providing new insights into their underlying structure.

Findings

Merton-Garman equation emerges from Black-Scholes via gauge invariance

Stochastic volatility is interpreted as a gauge field acquiring mass

Constraints on model parameters are derived from symmetry considerations

Abstract

By using the Hamiltonian formulation, we demonstrate that the Merton-Garman equation emerges naturally from the Black-Scholes equation after imposing invariance (symmetry) under local (gauge) transformations over changes in the stock price. This is the case because imposing gauge symmetry implies the appearance of an additional field, which corresponds to the stochastic volatility. The gauge symmetry then imposes some constraints over the free-parameters of the Merton-Garman Hamiltonian. Finally, we analyze how the stochastic volatility gets massive dynamically via Higgs mechanism.