Gauge symmetries and the Higgs mechanism in Quantum Finance

Paper: arXiv 2306.03237

Authors: Ivan Arraut

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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents an highly advanced theoretical application of quantum field theory concepts, such as gauge symmetries and the Higgs mechanism, to derive stochastic volatility models in finance. However, it is entirely theoretical with no empirical backtesting or real-world market data validation.

Research Flowchart

  flowchart TD
    A["Research Goal:<br>Derive Merton-Garman Model & Stochastic Volatility via Gauge Symmetry"] --> B["Inputs:<br>Black-Scholes Equation & Financial Gauge Transformations"]
    B --> C["Methodology:<br>Apply Hamiltonian Formulation"]
    C --> D["Impose Local Gauge Invariance<br>over Stock Price Changes"]
    D --> E["Emergence of Stochastic Volatility Field<br>& Constraint Derivation"]
    E --> F["Computational Process:<br>Apply Higgs Mechanism"]
    F --> G["Key Finding:<br>Dynamic Mass Generation for Volatility & Merton-Garman Derivation"]