{"id":434074,"date":"2026-08-24T17:23:58","date_gmt":"2026-08-24T09:23:58","guid":{"rendered":"https:\/\/library.usj.edu.mo\/?post_type=tnc_col_19837_item&#038;p=434074"},"modified":"2026-08-24T18:33:05","modified_gmt":"2026-08-24T10:33:05","slug":"local-equivalence-of-the-black-scholes-and-merton-garman-equations","status":"publish","type":"tnc_col_19837_item","link":"https:\/\/library.usj.edu.mo\/zh\/scholarly-work-usj\/local-equivalence-of-the-black-scholes-and-merton-garman-equations\/","title":{"rendered":"LOCAL EQUIVALENCE OF THE BLACK\u2013SCHOLES AND MERTON\u2013GARMAN EQUATIONS"},"content":{"rendered":"<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.<\/p>","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","format":"standard","class_list":["post-434074","tnc_col_19837_item","type-tnc_col_19837_item","status-publish","format-standard","hentry","tnc_tax_19818-english","tnc_tax_19716-faculty-of-business-and-law","tnc_tax_19659-journal-article"],"taxonomy_info":[],"featured_image_src_large":false,"author_info":[],"comment_info":0,"featured_img":false,"coauthors":[],"author_meta":{"author_link":"https:\/\/library.usj.edu.mo\/zh\/author\/emilychan\/","display_name":"emily chan"},"relative_dates":{"created":"Posted 4 \u5929 ago","modified":"Updated 4 \u5929 ago"},"absolute_dates":{"created":"Posted on 24 8 \u6708, 2026","modified":"Updated on 24 8 \u6708, 2026"},"absolute_dates_time":{"created":"Posted on 24 8 \u6708, 2026 5:23 \u4e0b\u5348","modified":"Updated on 24 8 \u6708, 2026 6:33 \u4e0b\u5348"},"featured_img_caption":"","tax_additional":{"tnc_tax_19993":{"linked":[],"unlinked":[],"slug":"tnc_tax_19993","name":"SDGs"},"tnc_tax_19818":{"linked":["<a href=\"https:\/\/library.usj.edu.mo\/zh\/language\/english\/\" class=\"advgb-post-tax-term\">ENGLISH<\/a>"],"unlinked":["<span class=\"advgb-post-tax-term\">ENGLISH<\/span>"],"slug":"tnc_tax_19818","name":"LANGUAGE"},"tnc_tax_19716":{"linked":["<a href=\"https:\/\/library.usj.edu.mo\/zh\/faculty-research-unit\/faculty-of-business-and-law\/\" class=\"advgb-post-tax-term\">FACULTY OF BUSINESS AND LAW<\/a>"],"unlinked":["<span class=\"advgb-post-tax-term\">FACULTY OF BUSINESS AND LAW<\/span>"],"slug":"tnc_tax_19716","name":"FACULTY \/ RESEARCH UNIT"},"tnc_tax_19664":{"linked":[],"unlinked":[],"slug":"tnc_tax_19664","name":"SUBJECT"},"tnc_tax_19659":{"linked":["<a href=\"https:\/\/library.usj.edu.mo\/zh\/resource-type\/journal-article\/\" class=\"advgb-post-tax-term\">Journal Article<\/a>"],"unlinked":["<span class=\"advgb-post-tax-term\">Journal Article<\/span>"],"slug":"tnc_tax_19659","name":"Document Type"}},"series_order":"","_links":{"self":[{"href":"https:\/\/library.usj.edu.mo\/zh\/wp-json\/wp\/v2\/tnc_col_19837_item\/434074","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/library.usj.edu.mo\/zh\/wp-json\/wp\/v2\/tnc_col_19837_item"}],"about":[{"href":"https:\/\/library.usj.edu.mo\/zh\/wp-json\/wp\/v2\/types\/tnc_col_19837_item"}],"replies":[{"embeddable":true,"href":"https:\/\/library.usj.edu.mo\/zh\/wp-json\/wp\/v2\/comments?post=434074"}],"version-history":[{"count":1,"href":"https:\/\/library.usj.edu.mo\/zh\/wp-json\/wp\/v2\/tnc_col_19837_item\/434074\/revisions"}],"predecessor-version":[{"id":436314,"href":"https:\/\/library.usj.edu.mo\/zh\/wp-json\/wp\/v2\/tnc_col_19837_item\/434074\/revisions\/436314"}],"wp:attachment":[{"href":"https:\/\/library.usj.edu.mo\/zh\/wp-json\/wp\/v2\/media?parent=434074"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}