{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:00:32Z","timestamp":1760058032011,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,3,15]],"date-time":"2025-03-15T00:00:00Z","timestamp":1741996800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>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\u2013Scholes (BS) equation. When this occurs, a Merton\u2013Garman-like equation emerges. From the perspective of manifolds, this means that the Black\u2013Scholes and Merton\u2013Garman (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.<\/jats:p>","DOI":"10.3390\/axioms14030215","type":"journal-article","created":{"date-parts":[[2025,3,17]],"date-time":"2025-03-17T06:36:23Z","timestamp":1742193383000},"page":"215","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Local Equivalence of the Black\u2013Scholes and Merton\u2013Garman Equations"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2509-5048","authenticated-orcid":false,"given":"Ivan","family":"Arraut","sequence":"first","affiliation":[{"name":"FBL, University of Saint Joseph, Estrada Marginal da Ilha Verde, 14-17, Macao SAR, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1086\/260062","article-title":"The pricing of Options and Corporate Liabilities","volume":"81","author":"Black","year":"1973","journal-title":"J. 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