Local Equivalence of the Black–Scholes and Merton–Garman Equations
Year: 2025
Type: article
Abstract: It has been previously demonstrated that stochastic volatility emerges as the gauge field necessary to restore local symmetry under changes in stock prices in the Black–Scholes (BS) equation. When this occurs, a Merton–Garman-like equation emerges. From the perspective of manifolds, this means that the Black–Scholes and Merton–Garman (MG) equations can be considered locally equivalent. In this scenario, the MG Hamiltonian is a special case of a more general Hamiltonian, here referred to as the gauge Hamiltonian. We then show that the gauge character of volatility implies a specific functional relationship between stock prices and volatility. The connection between stock prices and volatility is a powerful tool for improving volatility estimations in the stock market, which is a key ingredient for investors to make good decisions. Finally, we define an extended version of the martingale condition, defined for the gauge Hamiltonian.
Source: Axioms
Author: Ivan Arraut
Institution: University of Saint Joseph
Language: English
Subfield: Finance
Domain: Social Sciences
Open Access status: gold
Estimated APC paid: $2,896
OpenAlex ID: W4408529460