{"id":"https://openalex.org/W2201300930","doi":"https://doi.org/10.2139/ssrn.1691384","title":"Notes on Solving the Black-Scholes Equation","display_name":"Notes on Solving the Black-Scholes Equation","publication_year":2009,"publication_date":"2009-01-01","ids":{"openalex":"https://openalex.org/W2201300930","doi":"https://doi.org/10.2139/ssrn.1691384","mag":"2201300930"},"language":"en","primary_location":{"id":"doi:10.2139/ssrn.1691384","doi":"https://doi.org/10.2139/ssrn.1691384","is_oa":true,"landing_page_url":"https://doi.org/10.2139/ssrn.1691384","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":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SSRN Electronic Journal","raw_type":"journal-article"},"volume":null,"type":"preprint","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.2139/ssrn.1691384","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5054123780","display_name":"Zhan Chen","orcid":"https://orcid.org/0000-0001-8687-8348","observed_orcids":["https://orcid.org/0000-0001-8687-8348"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Zhan Chen","raw_affiliation_strings":["Eola Investments, LLC, Aurora, IL 60504, United States","Eola Investments, LLC"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Eola Investments, LLC, Aurora, IL 60504, United States","institution_ids":[]},{"raw_affiliation_string":"Eola Investments, LLC","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5054123780"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"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/T12589","display_name":"Differential Equations and Boundary Problems","score":0.0005000000237487257,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T13806","display_name":"Mathematical and Computational Methods","score":9.999999747378752e-05,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/black-scholes-equation","display_name":"Black-Scholes equation","score":0.9990000128746033},{"id":"https://openalex.org/keywords/european-call-options","display_name":"European call options","score":0.9110000133514404},{"id":"https://openalex.org/keywords/analytical-solution","display_name":"analytical solution","score":0.984000027179718},{"id":"https://openalex.org/keywords/green-s-function","display_name":"Green's function","score":0.8199999928474426},{"id":"https://openalex.org/keywords/option-pricing","display_name":"option pricing","score":0.7120000123977661},{"id":"https://openalex.org/keywords/asymptotic-analysis","display_name":"asymptotic analysis","score":0.8360000252723694},{"id":"https://openalex.org/keywords/variable-transformation","display_name":"change of variables","score":0.8550000190734863},{"id":"https://openalex.org/keywords/fourier-transform","display_name":"Fourier transform","score":0.9620000123977661},{"id":"https://openalex.org/keywords/complex-integration","display_name":"complex integration","score":0.9919999837875366}],"concepts":[{"id":"https://openalex.org/C163128081","wikidata":"https://www.wikidata.org/wiki/Q1338307","display_name":"Black\u2013Scholes model","level":3,"score":0.6856802701950073},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.4445459246635437},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.40294331312179565},{"id":"https://openalex.org/C162118730","wikidata":"https://www.wikidata.org/wiki/Q1128453","display_name":"Actuarial science","level":1,"score":0.39720475673675537},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.3940085470676422},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.37860897183418274},{"id":"https://openalex.org/C149782125","wikidata":"https://www.wikidata.org/wiki/Q160039","display_name":"Econometrics","level":1,"score":0.32509565353393555},{"id":"https://openalex.org/C91602232","wikidata":"https://www.wikidata.org/wiki/Q756115","display_name":"Volatility (finance)","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.2139/ssrn.1691384","doi":"https://doi.org/10.2139/ssrn.1691384","is_oa":true,"landing_page_url":"https://doi.org/10.2139/ssrn.1691384","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":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SSRN Electronic Journal","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.2139/ssrn.1691384","doi":"https://doi.org/10.2139/ssrn.1691384","is_oa":true,"landing_page_url":"https://doi.org/10.2139/ssrn.1691384","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":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SSRN Electronic Journal","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":1,"referenced_works":["https://openalex.org/W2182987541"],"related_works":["https://openalex.org/W4403058284","https://openalex.org/W2623371705","https://openalex.org/W4387732180","https://openalex.org/W3005688696","https://openalex.org/W3035662362","https://openalex.org/W7098626052","https://openalex.org/W2375376466","https://openalex.org/W3125625051","https://openalex.org/W4247292125","https://openalex.org/W2389059585"],"abstract_inverted_index":null,"counts_by_year":[],"updated_date":"2026-08-04T08:18:43.703281","created_date":"2016-06-24T00:00:00"}
