{"id":"https://openalex.org/W2140935522","doi":"https://doi.org/10.1111/j.0960-1627.2005.00224.x","title":"THE BLACK\u2010SCHOLES EQUATION REVISITED: ASYMPTOTIC EXPANSIONS AND SINGULAR PERTURBATIONS","display_name":"THE BLACK\u2010SCHOLES EQUATION REVISITED: ASYMPTOTIC EXPANSIONS AND SINGULAR PERTURBATIONS","publication_year":2005,"publication_date":"2005-03-18","ids":{"openalex":"https://openalex.org/W2140935522","doi":"https://doi.org/10.1111/j.0960-1627.2005.00224.x","mag":"2140935522"},"language":"en","primary_location":{"id":"doi:10.1111/j.0960-1627.2005.00224.x","doi":"https://doi.org/10.1111/j.0960-1627.2005.00224.x","is_oa":false,"landing_page_url":"https://doi.org/10.1111/j.0960-1627.2005.00224.x","pdf_url":null,"source":{"id":"https://openalex.org/S133335107","display_name":"Mathematical Finance","issn_l":"0960-1627","issn":["0960-1627","1467-9965"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["abdc-a","anvur-class-a","anvur-scientific","cwts-core","dongbi-b","fnege-2","jufo-2","ki-jl-2","norway-2","poland-140","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Finance","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://papers.ssrn.com/sol3/Delivery.cfm/mafi_224.pdf?abstractid=687247&mirid=1","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5011501150","display_name":"Martin Widdicks","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I28407311","display_name":"University of Manchester","ror":"https://ror.org/027m9bs27","country_code":"GB","type":"education","lineage":["https://openalex.org/I28407311"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Martin Widdicks","raw_affiliation_strings":["University of Manchester","University of Manchester Manchester Business School"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Manchester","institution_ids":["https://openalex.org/I28407311"]},{"raw_affiliation_string":"University of Manchester Manchester Business School","institution_ids":["https://openalex.org/I28407311"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5069206635","display_name":"Peter W. Duck","orcid":"https://orcid.org/0000-0002-7915-5982","observed_orcids":["https://orcid.org/0000-0002-7915-5982"]},"institutions":[{"id":"https://openalex.org/I28407311","display_name":"University of Manchester","ror":"https://ror.org/027m9bs27","country_code":"GB","type":"education","lineage":["https://openalex.org/I28407311"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Peter W. Duck","raw_affiliation_strings":["University of Manchester","University of Manchester - Department of Mathematics"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Manchester","institution_ids":["https://openalex.org/I28407311"]},{"raw_affiliation_string":"University of Manchester - Department of Mathematics","institution_ids":["https://openalex.org/I28407311"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5053170118","display_name":"Ari D. Andricopoulos","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I28407311","display_name":"University of Manchester","ror":"https://ror.org/027m9bs27","country_code":"GB","type":"education","lineage":["https://openalex.org/I28407311"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Ari D. Andricopoulos","raw_affiliation_strings":["University of Manchester","University of Manchester - Department of Mathematics"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Manchester","institution_ids":["https://openalex.org/I28407311"]},{"raw_affiliation_string":"University of Manchester - Department of Mathematics","institution_ids":["https://openalex.org/I28407311"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5108038392","display_name":"David Newton","orcid":"https://orcid.org/0000-0003-0470-8113","observed_orcids":["https://orcid.org/0000-0003-0470-8113","https://orcid.org/0000-0001-6891-5686"]},"institutions":[{"id":"https://openalex.org/I142263535","display_name":"University of Nottingham","ror":"https://ror.org/01ee9ar58","country_code":"GB","type":"education","lineage":["https://openalex.org/I142263535"]},{"id":"https://openalex.org/I28407311","display_name":"University of Manchester","ror":"https://ror.org/027m9bs27","country_code":"GB","type":"education","lineage":["https://openalex.org/I28407311"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"David P. Newton","raw_affiliation_strings":["University of Manchester","Nottingham University Business School (NUBS),"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Manchester","institution_ids":["https://openalex.org/I28407311"]},{"raw_affiliation_string":"Nottingham University Business School (NUBS),","institution_ids":["https://openalex.org/I142263535"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5069206635"],"corresponding_institution_ids":["https://openalex.org/I28407311"],"apc_list":{"value":3840,"currency":"USD","value_usd":3840},"apc_paid":null,"fwci":3.985,"has_fulltext":false,"cited_by_count":49,"citation_normalized_percentile":{"value":0.93975956,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"15","issue":"2","first_page":"373","last_page":"391"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.9990000128746033,"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"}},"topics":[{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.9990000128746033,"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"}},{"id":"https://openalex.org/T12727","display_name":"Differential Equations and Numerical Methods","score":0.0005000000237487257,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T10047","display_name":"Financial Markets and Investment Strategies","score":0.0,"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/black-scholes-equation","display_name":"Black-Scholes equation","score":0.9909999966621399},{"id":"https://openalex.org/keywords/option-pricing","display_name":"option pricing","score":0.734000027179718},{"id":"https://openalex.org/keywords/american-options","display_name":"American options","score":0.8510000109672546},{"id":"https://openalex.org/keywords/barrier-options","display_name":"barrier options","score":0.9700000286102295},{"id":"https://openalex.org/keywords/european-options","display_name":"European options","score":0.9950000047683716},{"id":"https://openalex.org/keywords/singular-perturbation","display_name":"singular perturbation","score":0.7599999904632568},{"id":"https://openalex.org/keywords/asymptotic-expansion","display_name":"asymptotic expansion","score":0.9940000176429749}],"concepts":[{"id":"https://openalex.org/C165160513","wikidata":"https://www.wikidata.org/wiki/Q7524249","display_name":"Singular perturbation","level":2,"score":0.7642815709114075},{"id":"https://openalex.org/C2776836400","wikidata":"https://www.wikidata.org/wiki/Q31087","display_name":"Calculator","level":2,"score":0.7066779136657715},{"id":"https://openalex.org/C186027771","wikidata":"https://www.wikidata.org/wiki/Q4008379","display_name":"Valuation (finance)","level":2,"score":0.6780095100402832},{"id":"https://openalex.org/C163128081","wikidata":"https://www.wikidata.org/wiki/Q1338307","display_name":"Black\u2013Scholes model","level":3,"score":0.5912273526191711},{"id":"https://openalex.org/C194483076","wikidata":"https://www.wikidata.org/wiki/Q380382","display_name":"Valuation of options","level":2,"score":0.5455869436264038},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.43202853202819824},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.40859314799308777},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.4009239077568054},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.3505690097808838},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.33108216524124146},{"id":"https://openalex.org/C149782125","wikidata":"https://www.wikidata.org/wiki/Q160039","display_name":"Econometrics","level":1,"score":0.32114264369010925},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.26802533864974976},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.25687769055366516},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.1322309374809265},{"id":"https://openalex.org/C91602232","wikidata":"https://www.wikidata.org/wiki/Q756115","display_name":"Volatility (finance)","level":2,"score":0.06480109691619873},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1111/j.0960-1627.2005.00224.x","doi":"https://doi.org/10.1111/j.0960-1627.2005.00224.x","is_oa":false,"landing_page_url":"https://doi.org/10.1111/j.0960-1627.2005.00224.x","pdf_url":null,"source":{"id":"https://openalex.org/S133335107","display_name":"Mathematical Finance","issn_l":"0960-1627","issn":["0960-1627","1467-9965"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["abdc-a","anvur-class-a","anvur-scientific","cwts-core","dongbi-b","fnege-2","jufo-2","ki-jl-2","norway-2","poland-140","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Finance","raw_type":"journal-article"},{"id":"mag:2140935522","doi":null,"is_oa":true,"landing_page_url":"https://papers.ssrn.com/sol3/Delivery.cfm/mafi_224.pdf?abstractid=687247&mirid=1","pdf_url":null,"source":{"id":"https://openalex.org/S4210172589","display_name":"SSRN Electronic Journal","issn_l":"1556-5068","issn":["1556-5068"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I1318003438","host_organization_name":"RELX Group (Netherlands)","host_organization_lineage":["https://openalex.org/I1318003438"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SSRN Electronic Journal","raw_type":null},{"id":"pmh:oai:eprints.lancs.ac.uk:44928","doi":null,"is_oa":false,"landing_page_url":"https://eprints.lancs.ac.uk/id/eprint/44928/","pdf_url":null,"source":{"id":"https://openalex.org/S4306401916","display_name":"Lancaster EPrints (Lancaster University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67415387","host_organization_name":"Lancaster University","host_organization_lineage":["https://openalex.org/I67415387"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Journal Article"}],"best_oa_location":{"id":"mag:2140935522","doi":null,"is_oa":true,"landing_page_url":"https://papers.ssrn.com/sol3/Delivery.cfm/mafi_224.pdf?abstractid=687247&mirid=1","pdf_url":null,"source":{"id":"https://openalex.org/S4210172589","display_name":"SSRN Electronic Journal","issn_l":"1556-5068","issn":["1556-5068"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I1318003438","host_organization_name":"RELX Group (Netherlands)","host_organization_lineage":["https://openalex.org/I1318003438"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SSRN Electronic Journal","raw_type":null},"sustainable_development_goals":[],"sustainable_development_goals_aurora":[{"id":"https://metadata.un.org/sdg/8","score":0.6600000262260437,"display_name":"Decent work and economic growth"}],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":14,"referenced_works":["https://openalex.org/W51500993","https://openalex.org/W69750804","https://openalex.org/W1528840936","https://openalex.org/W1608234917","https://openalex.org/W1988982671","https://openalex.org/W1995086075","https://openalex.org/W2026385856","https://openalex.org/W2069625813","https://openalex.org/W2077791698","https://openalex.org/W2078644288","https://openalex.org/W2083719702","https://openalex.org/W2114971059","https://openalex.org/W2169032495","https://openalex.org/W3121904677"],"related_works":["https://openalex.org/W2612205261","https://openalex.org/W2383399106","https://openalex.org/W225085924","https://openalex.org/W122990458","https://openalex.org/W2009254302","https://openalex.org/W2514337737","https://openalex.org/W4246504678","https://openalex.org/W1568823047","https://openalex.org/W7097864483","https://openalex.org/W3123209934"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"novel":[3],"singular":[4],"perturbation":[5],"techniques":[6],"are":[7],"applied":[8],"to":[9,21,45],"price":[10],"European,":[11],"American,":[12],"and":[13,81],"barrier":[14,82],"options.":[15,83],"Employment":[16],"of":[17,25,35,65,70],"these":[18],"methods":[19],"leads":[20],"a":[22,46,53],"significant":[23],"simplification":[24],"the":[26,33,40,63,71,74],"problem":[27,42],"in":[28],"all":[29,69],"cases,":[30],"by":[31],"reducing":[32],"number":[34],"parameters.":[36],"For":[37],"American":[38,80],"options,":[39],"valuation":[41],"is":[43],"reduced":[44],"procedure":[47],"that":[48],"may":[49],"be":[50],"performed":[51],"on":[52,62],"rudimentary":[54],"handheld":[55],"calculator.":[56],"The":[57],"method":[58],"also":[59],"sheds":[60],"light":[61],"evolution":[64],"option":[66],"prices":[67],"for":[68,79],"cases":[72],"considered,":[73],"results":[75],"being":[76],"particularly":[77],"illuminating":[78]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":6},{"year":2012,"cited_by_count":2}],"updated_date":"2026-10-16T09:59:22.235173","created_date":"2016-06-24T00:00:00"}