{"id":"https://openalex.org/W4415813570","doi":"https://doi.org/10.3390/sym17111830","title":"Option Pricing Formulas of Uncertain Mean-Reverting Stock Model with Symmetry Analysis","display_name":"Option Pricing Formulas of Uncertain Mean-Reverting Stock Model with Symmetry Analysis","publication_year":2025,"publication_date":"2025-11-01","ids":{"openalex":"https://openalex.org/W4415813570","doi":"https://doi.org/10.3390/sym17111830"},"language":"en","primary_location":{"id":"doi:10.3390/sym17111830","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17111830","pdf_url":"https://www.mdpi.com/2073-8994/17/11/1830/pdf?version=1761970361","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/17/11/1830/pdf?version=1761970361","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5015041779","display_name":"Yuxing Jia","orcid":"https://orcid.org/0000-0002-0298-5996"},"institutions":[{"id":"https://openalex.org/I200845125","display_name":"Nanjing University of Information Science and Technology","ror":"https://ror.org/02y0rxk19","country_code":"CN","type":"education","lineage":["https://openalex.org/I200845125"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yuxing Jia","raw_affiliation_strings":["School of Management Science and Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China"],"affiliations":[{"raw_affiliation_string":"School of Management Science and Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China","institution_ids":["https://openalex.org/I200845125"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5066362361","display_name":"Kaixi Zhang","orcid":"https://orcid.org/0009-0003-0317-7629"},"institutions":[{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Kaixi Zhang","raw_affiliation_strings":["Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China","institution_ids":["https://openalex.org/I99065089"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5030735045","display_name":"Jinsheng Xie","orcid":"https://orcid.org/0000-0003-1869-2247"},"institutions":[{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jinsheng Xie","raw_affiliation_strings":["Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China"],"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China","institution_ids":["https://openalex.org/I99065089"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100411567","display_name":"Yuhan Sun","orcid":"https://orcid.org/0000-0002-2940-213X"},"institutions":[{"id":"https://openalex.org/I116953780","display_name":"Tongji University","ror":"https://ror.org/03rc6as71","country_code":"CN","type":"education","lineage":["https://openalex.org/I116953780"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yuhan Sun","raw_affiliation_strings":["School of Economics and Management, Tongji University, Shanghai 200092, China"],"affiliations":[{"raw_affiliation_string":"School of Economics and Management, Tongji University, Shanghai 200092, China","institution_ids":["https://openalex.org/I116953780"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5069061259","display_name":"L. Jeff Hong","orcid":"https://orcid.org/0000-0001-7011-4001"},"institutions":[{"id":"https://openalex.org/I20942203","display_name":"Hainan University","ror":"https://ror.org/03q648j11","country_code":"CN","type":"education","lineage":["https://openalex.org/I20942203"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Lifang Hong","raw_affiliation_strings":["Key Laboratory of Engineering Modeling and Statistical Computation of Hainan Province, Hainan University, Haikou 570228, China","School of Mathematics and Statistics, Hainan University, Haikou 570228, China"],"affiliations":[{"raw_affiliation_string":"Key Laboratory of Engineering Modeling and Statistical Computation of Hainan Province, Hainan University, Haikou 570228, China","institution_ids":["https://openalex.org/I20942203"]},{"raw_affiliation_string":"School of Mathematics and Statistics, Hainan University, Haikou 570228, China","institution_ids":["https://openalex.org/I20942203"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100414820","display_name":"Zhigang Wang","orcid":"https://orcid.org/0000-0001-7312-300X"},"institutions":[{"id":"https://openalex.org/I20942203","display_name":"Hainan University","ror":"https://ror.org/03q648j11","country_code":"CN","type":"education","lineage":["https://openalex.org/I20942203"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Zhigang Wang","raw_affiliation_strings":["Key Laboratory of Engineering Modeling and Statistical Computation of Hainan Province, Hainan University, Haikou 570228, China","School of Mathematics and Statistics, Hainan University, Haikou 570228, China"],"affiliations":[{"raw_affiliation_string":"Key Laboratory of Engineering Modeling and Statistical Computation of Hainan Province, Hainan University, Haikou 570228, China","institution_ids":["https://openalex.org/I20942203"]},{"raw_affiliation_string":"School of Mathematics and Statistics, Hainan University, Haikou 570228, China","institution_ids":["https://openalex.org/I20942203"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5100414820"],"corresponding_institution_ids":["https://openalex.org/I20942203"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.3020749,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"17","issue":"11","first_page":"1830","last_page":"1830"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12135","display_name":"Fuzzy Systems and Optimization","score":0.9327999949455261,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12135","display_name":"Fuzzy Systems and Optimization","score":0.9327999949455261,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11413","display_name":"Risk and Portfolio Optimization","score":0.03819999843835831,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.006000000052154064,"subfield":{"id":"https://openalex.org/subfields/2003","display_name":"Finance"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/stock","display_name":"Stock (firearms)","score":0.5831999778747559},{"id":"https://openalex.org/keywords/valuation-of-options","display_name":"Valuation of options","score":0.5271000266075134},{"id":"https://openalex.org/keywords/duality","display_name":"Duality (order theory)","score":0.42570000886917114},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.37709999084472656},{"id":"https://openalex.org/keywords/inverse","display_name":"Inverse","score":0.35929998755455017},{"id":"https://openalex.org/keywords/stochastic-differential-equation","display_name":"Stochastic differential equation","score":0.34279999136924744},{"id":"https://openalex.org/keywords/black\u2013scholes-model","display_name":"Black\u2013Scholes model","score":0.3425999879837036},{"id":"https://openalex.org/keywords/symmetry","display_name":"Symmetry (geometry)","score":0.33730000257492065}],"concepts":[{"id":"https://openalex.org/C204036174","wikidata":"https://www.wikidata.org/wiki/Q909380","display_name":"Stock (firearms)","level":2,"score":0.5831999778747559},{"id":"https://openalex.org/C194483076","wikidata":"https://www.wikidata.org/wiki/Q380382","display_name":"Valuation of options","level":2,"score":0.5271000266075134},{"id":"https://openalex.org/C149782125","wikidata":"https://www.wikidata.org/wiki/Q160039","display_name":"Econometrics","level":1,"score":0.4805000126361847},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4672999978065491},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.45089998841285706},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.4406000077724457},{"id":"https://openalex.org/C2778023678","wikidata":"https://www.wikidata.org/wiki/Q554403","display_name":"Duality (order theory)","level":2,"score":0.42570000886917114},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.37709999084472656},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3695000112056732},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.35929998755455017},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.35440000891685486},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3531999886035919},{"id":"https://openalex.org/C51955184","wikidata":"https://www.wikidata.org/wiki/Q1545585","display_name":"Stochastic differential equation","level":2,"score":0.34279999136924744},{"id":"https://openalex.org/C163128081","wikidata":"https://www.wikidata.org/wiki/Q1338307","display_name":"Black\u2013Scholes model","level":3,"score":0.3425999879837036},{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.33730000257492065},{"id":"https://openalex.org/C23925645","wikidata":"https://www.wikidata.org/wiki/Q5449731","display_name":"Financial modeling","level":2,"score":0.32440000772476196},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.3215999901294708},{"id":"https://openalex.org/C2988984586","wikidata":"https://www.wikidata.org/wiki/Q1020013","display_name":"Stock price","level":3,"score":0.296999990940094},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.28630000352859497},{"id":"https://openalex.org/C104859751","wikidata":"https://www.wikidata.org/wiki/Q5450390","display_name":"Finite difference methods for option pricing","level":4,"score":0.28529998660087585},{"id":"https://openalex.org/C93226319","wikidata":"https://www.wikidata.org/wiki/Q193137","display_name":"Differential (mechanical device)","level":2,"score":0.2736999988555908},{"id":"https://openalex.org/C51152595","wikidata":"https://www.wikidata.org/wiki/Q7882501","display_name":"Uncertainty theory","level":2,"score":0.26190000772476196},{"id":"https://openalex.org/C166827900","wikidata":"https://www.wikidata.org/wiki/Q7843049","display_name":"Trinomial tree","level":4,"score":0.2556000053882599},{"id":"https://openalex.org/C105235620","wikidata":"https://www.wikidata.org/wiki/Q1138253","display_name":"Binomial options pricing model","level":3,"score":0.250900000333786}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.3390/sym17111830","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17111830","pdf_url":"https://www.mdpi.com/2073-8994/17/11/1830/pdf?version=1761970361","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.3390/sym17111830","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17111830","pdf_url":"https://www.mdpi.com/2073-8994/17/11/1830/pdf?version=1761970361","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320324849","display_name":"Nanjing University of Information Science and Technology","ror":"https://ror.org/02y0rxk19"},{"id":"https://openalex.org/F4320335990","display_name":"Startup Foundation for Introducing Talent of Nanjing University of Information Science and Technology","ror":null}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4415813570.pdf","grobid_xml":"https://content.openalex.org/works/W4415813570.grobid-xml"},"referenced_works_count":24,"referenced_works":["https://openalex.org/W2057112219","https://openalex.org/W2063532481","https://openalex.org/W2139905755","https://openalex.org/W2163991121","https://openalex.org/W2401930306","https://openalex.org/W2590563850","https://openalex.org/W2606376523","https://openalex.org/W2616402233","https://openalex.org/W2802573242","https://openalex.org/W2927877475","https://openalex.org/W2946244885","https://openalex.org/W2960652284","https://openalex.org/W3026192632","https://openalex.org/W3043604544","https://openalex.org/W3047479302","https://openalex.org/W3160147592","https://openalex.org/W3195513686","https://openalex.org/W4220822722","https://openalex.org/W4281775731","https://openalex.org/W4293708200","https://openalex.org/W4385246374","https://openalex.org/W4387779022","https://openalex.org/W4389278990","https://openalex.org/W4412399347"],"related_works":[],"abstract_inverted_index":{"With":[0],"the":[1,20,25,32,39,47,67,75,80,90,105,108,112],"development":[2],"of":[3,34,49,111],"uncertain":[4,6,26,54,68],"finance,":[5],"stock":[7,15,28,70],"models":[8],"have":[9],"become":[10],"increasingly":[11],"popular":[12],"for":[13,57],"modeling":[14],"prices.":[16],"This":[17],"paper":[18],"explores":[19],"symmetric":[21,94],"properties":[22],"inherent":[23],"in":[24,31,74],"mean-reverting":[27,69],"model,":[29],"particularly":[30],"structure":[33],"its":[35],"differential":[36,55],"equations":[37],"and":[38,61,79,84,107],"resulting":[40],"pricing":[41,51],"formulas.":[42,114],"The":[43,72],"primary":[44],"findings":[45],"comprise":[46],"derivation":[48],"explicit":[50],"formulas,":[52],"via":[53],"equations,":[56],"European,":[58],"American,":[59],"Asian,":[60],"geometric":[62],"average":[63],"Asian":[64],"options":[65,86],"under":[66],"model.":[71],"symmetry":[73],"inverse":[76],"uncertainty":[77],"distributions":[78],"duality":[81],"between":[82],"call":[83],"put":[85],"are":[87,101],"highlighted,":[88],"demonstrating":[89],"model\u2019s":[91],"alignment":[92],"with":[93],"financial":[95],"principles.":[96],"Furthermore,":[97],"several":[98],"numerical":[99],"examples":[100],"provided":[102],"to":[103],"illustrate":[104],"applicability":[106],"symmetry-related":[109],"characteristics":[110],"derived":[113]},"counts_by_year":[],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-11-03T00:00:00"}