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First, the numerical solution is obtained by developing a high-order scheme with order (<jats:inline-formula><jats:alternatives><jats:tex-math>$$3-\\alpha $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mn>3<\/mml:mn>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:mi>\u03b1<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>) for the time discretisation. Some theoretical analyses such as stability and convergence are presented in order to verify the efficiency and accuracy of the proposed scheme. Secondly, we employ a modified hybrid Nelder-Mead simplex search and particle swarm optimization (MH-NMSS-PSO) to identify the fractional order <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03b1<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and implied volatility <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\sigma $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03c3<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of the TFBS equation, and explore the financial meanings of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03b1<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> under extreme stock market conditions such as the Covid-19 and the 2008 global financial crisis. We analyse the values of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03b1<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and compare the mean squared errors of both the TFBS model and the BS model. Our empirical results show that <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03b1<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> may be regarded as a market fluctuation indicator for classifying financial environments, and the TFBS model is more capable of fitting real option data compared with the BS model, especially for put options during the economic downturn. In addition, we find and discuss an interesting relation between <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03b1<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\sigma $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03c3<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> from both the TFBS model and the BS model in three expressions, which could be an open problem for further research.<\/jats:p>","DOI":"10.1007\/s11075-023-01563-4","type":"journal-article","created":{"date-parts":[[2023,6,27]],"date-time":"2023-06-27T13:02:59Z","timestamp":1687870979000},"page":"1-30","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Parameter estimation for time-fractional Black-Scholes equation with S &amp;P 500 index option"],"prefix":"10.1007","volume":"95","author":[{"given":"Xingyu","family":"An","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qingxia","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1034-2349","authenticated-orcid":false,"given":"Fawang","family":"Liu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vo V.","family":"Anh","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ian W.","family":"Turner","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,6,27]]},"reference":[{"issue":"3","key":"1563_CR1","doi-asserted-by":"publisher","first-page":"637","DOI":"10.1142\/9789814759588_0001","volume":"81","author":"F Black","year":"1973","unstructured":"Black, F., Scholes, M.: The pricing of options and corporate liabilities. 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