{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T11:48:36Z","timestamp":1773661716933,"version":"3.50.1"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2024,8,20]],"date-time":"2024-08-20T00:00:00Z","timestamp":1724112000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,8,20]],"date-time":"2024-08-20T00:00:00Z","timestamp":1724112000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Soft Comput"],"published-print":{"date-parts":[[2026,3]]},"DOI":"10.1007\/s00500-024-09663-6","type":"journal-article","created":{"date-parts":[[2024,8,20]],"date-time":"2024-08-20T03:03:37Z","timestamp":1724123017000},"page":"2039-2050","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["The option pricing problem based on the uncertain fractional volatility stock model"],"prefix":"10.1007","volume":"30","author":[{"given":"Wenxiu","family":"Gong","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4938-9518","authenticated-orcid":false,"given":"Miao","family":"Tian","sequence":"additional","affiliation":[]},{"given":"Xiangfeng","family":"Yang","sequence":"additional","affiliation":[]},{"given":"Yesen","family":"Sun","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,8,20]]},"reference":[{"key":"9663_CR1","doi-asserted-by":"publisher","first-page":"637","DOI":"10.1086\/260062","volume":"81","author":"F Black","year":"1973","unstructured":"Black F, Scholes M (1973) The pricing of option and corporate liabilities. J Polit Econ 81:637\u2013654","journal-title":"J Polit Econ"},{"issue":"2","key":"9663_CR2","first-page":"32","volume":"8","author":"X Chen","year":"2011","unstructured":"Chen X (2011) American option pricing formula for uncertain financial market. Int J Oper Res 8(2):32\u201337","journal-title":"Int J Oper Res"},{"key":"9663_CR3","first-page":"5835","volume":"22","author":"X Chen","year":"2018","unstructured":"Chen X, Gao J (2018) Two-factor term structure model with uncertain volatility risk. Oft Comput 22:5835\u20135841","journal-title":"Oft Comput"},{"key":"9663_CR4","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1023\/A:1016592219341","volume":"29","author":"K Diethelm","year":"2002","unstructured":"Diethelm K, Ford N, Freed A (2002) A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn 29:3\u201322","journal-title":"Nonlinear Dyn"},{"key":"9663_CR5","unstructured":"Dunn R, Hauser P, Seibold T, Gong H. Estimating option prices with Heston\u2019s stochastic volatility model, https:\/\/www.valpo.edu\/mathematics-statistics\/files\/2015\/07\/Estimating-Option-Prices-with-Heston%E2%80%99s-Stochastic-Volatility-Model.pdf"},{"issue":"2","key":"9663_CR6","doi-asserted-by":"publisher","first-page":"2150011","DOI":"10.1142\/S1752890921500112","volume":"14","author":"R Gao","year":"2021","unstructured":"Gao R, Yin X (2021) American basket option pricing formulas in uncertain environment. J Uncertain Syst 14(2):2150011","journal-title":"J Uncertain Syst"},{"issue":"17","key":"9663_CR7","doi-asserted-by":"publisher","first-page":"5647","DOI":"10.1007\/s00500-017-2558-y","volume":"22","author":"Y Gao","year":"2018","unstructured":"Gao Y, Yang X, Fu Z (2018) Lookback option pricing problem of uncertain exponential Ornstein-Uhlenbeck model. Soft Comput 22(17):5647\u20135654","journal-title":"Soft Comput"},{"key":"9663_CR8","doi-asserted-by":"publisher","first-page":"97846","DOI":"10.1109\/ACCESS.2019.2928029","volume":"7","author":"R Gao","year":"2019","unstructured":"Gao R, Liu K, Li Z, Lv R (2019) American barrier option pricing formulas for stock model in uncertain environment. IEEE Access 7:97846\u201397856","journal-title":"IEEE Access"},{"key":"9663_CR9","unstructured":"Hagan P, Kumar D, Lesniewski A, Woodward E (2002) Managing smile risk, Wilmott Magazine, 84\u2013108"},{"key":"9663_CR10","doi-asserted-by":"publisher","first-page":"4153","DOI":"10.1007\/s00500-017-2633-4","volume":"22","author":"S Hassanzadeh","year":"2018","unstructured":"Hassanzadeh S, Mehrdoust F (2018) Valuation of European option under uncertain volatility model. Soft Comput 22:4153\u20134163","journal-title":"Soft Comput"},{"key":"9663_CR11","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1093\/rfs\/6.2.327","volume":"6","author":"S Heston","year":"1993","unstructured":"Heston S (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6:327\u2013343","journal-title":"Rev. Financ. Stud."},{"key":"9663_CR12","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1111\/j.1540-6261.1987.tb02568.x","volume":"42","author":"J Hull","year":"1987","unstructured":"Hull J, White A (1987) The pricing of options on assets with stochastic volatilities. J. Financ. 42:281\u2013300","journal-title":"J. Financ."},{"key":"9663_CR13","doi-asserted-by":"publisher","first-page":"347","DOI":"10.1016\/0304-405X(82)90007-1","volume":"10","author":"R Jarrow","year":"1982","unstructured":"Jarrow R, Rudd A (1982) Approximate option valuation for arbitrary stochastic processes. J Financ Econ 10:347\u2013369","journal-title":"J Financ Econ"},{"key":"9663_CR14","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1016\/j.matcom.2021.05.018","volume":"190","author":"T Jin","year":"2021","unstructured":"Jin T, Yang X (2021) Monotonicity theorem for the uncertain fractional differential equation and application to uncertain financial market. Math. Comput. Simulat. 190:203\u2013221","journal-title":"Math. Comput. Simulat."},{"key":"9663_CR15","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2020.109836","volume":"137","author":"T Jin","year":"2020","unstructured":"Jin T, Zhu Y (2020) First hitting time about solution for an uncertain fractional differential equation and application to an uncertain risk index model, Chaos. Soliton. Fract. 137:109836","journal-title":"Soliton. Fract."},{"key":"9663_CR16","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2019.122357","volume":"534","author":"T Jin","year":"2019","unstructured":"Jin T, Sun Y, Zhu Y (2019) Extreme values for solution to uncertain fractional differential equation and application to American option pricing model. Phys A 534:122357","journal-title":"Phys A"},{"key":"9663_CR17","volume":"372","author":"T Jin","year":"2020","unstructured":"Jin T, Sun Y, Zhu Y (2020) Time integral about solution of an uncertain fractional order differential equation and application to zero-coupon bond model. Appl Math Comput 372:124991","journal-title":"Appl Math Comput"},{"key":"9663_CR18","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2020.110409","volume":"142","author":"T Jin","year":"2021","unstructured":"Jin T, Ding H, Xia H, Bao J (2021) Reliability index and Asian barrier option pricing formulas of the uncertain fractional first-hitting time model with Caputo type, Chaos. Soliton. Fract. 142:110409","journal-title":"Soliton. Fract."},{"key":"9663_CR19","doi-asserted-by":"publisher","first-page":"11587","DOI":"10.1007\/s00500-023-08153-5","volume":"27","author":"T Jin","year":"2023","unstructured":"Jin T, Li F, Peng H, Li B, Jiang D (2023) Uncertain barrier swaption pricing problem based on the fractional differential equation in Caputo sense. Soft Comput 27:11587\u201311602","journal-title":"Soft Comput"},{"key":"9663_CR20","doi-asserted-by":"crossref","unstructured":"Jin T, Xia H (2021) Lookback option pricing models based on the uncertain fractional-order differential equation with Caputo type. J. Amb. Intel. Hum. Comput., 1-14","DOI":"10.1007\/s12652-021-03516-y"},{"key":"9663_CR21","doi-asserted-by":"publisher","first-page":"13369","DOI":"10.1007\/s00500-022-07538-2","volume":"26","author":"X Li","year":"2022","unstructured":"Li X, Sheng Y (2022) Uncertain exponential currency model and currency option pricing. Soft Comput 26:13369\u201313380","journal-title":"Soft Comput"},{"key":"9663_CR22","volume-title":"Uncertainty Theory","author":"B Liu","year":"2007","unstructured":"Liu B (2007) Uncertainty Theory, 2nd edn. Springer, Berlin","edition":"2"},{"key":"9663_CR23","first-page":"137","volume":"343","author":"Z Lu","year":"2019","unstructured":"Lu Z, Zhu Y (2019) Numerical approach for solution to an uncertain fractional differential equation. Appl Math Comput 343:137\u2013148","journal-title":"Appl Math Comput"},{"key":"9663_CR24","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1007\/s10700-018-9293-4","volume":"18","author":"Z Lu","year":"2019","unstructured":"Lu Z, Yan H, Zhu Y (2019) European option pricing model based on uncertain fractional differential equation. Fuzzy Optim. Decis. Mak. 18:199\u2013217","journal-title":"Fuzzy Optim. Decis. Mak."},{"issue":"2","key":"9663_CR25","first-page":"18","volume":"8","author":"J Peng","year":"2011","unstructured":"Peng J, Yao K (2011) A new option pricing model for stocks in uncertainty markets. Int. J. Operat. Res. 8(2):18\u201326","journal-title":"Int. J. Operat. Res."},{"key":"9663_CR26","doi-asserted-by":"publisher","first-page":"419","DOI":"10.2307\/2330793","volume":"22","author":"L Scott","year":"1987","unstructured":"Scott L (1987) Option pricing when the variance changes randomly: theory, estimation, and an application. J. Financ. Quant. Anal. 22:419\u2013438","journal-title":"J. Financ. Quant. Anal."},{"key":"9663_CR27","doi-asserted-by":"publisher","first-page":"727","DOI":"10.1093\/rfs\/4.4.727","volume":"4","author":"E Stein","year":"1991","unstructured":"Stein E, Stein J (1991) Stock price distributions with stochastic volatility: an analytic approach. Rev. Financ. Stud. 4:727\u2013752","journal-title":"Rev. Financ. Stud."},{"key":"9663_CR28","first-page":"1","volume":"1","author":"J Sun","year":"2015","unstructured":"Sun J, Chen X (2015) Asian option pricing formula for uncertain financial market. J. Uncert. Anal. Appl. 1:1\u201311","journal-title":"J. Uncert. Anal. Appl."},{"issue":"2","key":"9663_CR29","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1007\/s10700-016-9247-7","volume":"16","author":"Y Sun","year":"2017","unstructured":"Sun Y, Su T (2017) Mean-reverting stock model with floating interest rate in uncertain environment. Fuzzy Optim. Decis. Mak. 16(2):235\u2013255","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"9663_CR30","doi-asserted-by":"publisher","first-page":"5583","DOI":"10.1007\/s00500-017-2524-8","volume":"22","author":"Y Sun","year":"2018","unstructured":"Sun Y, Yao K, Dong J (2018) Asian option pricing of uncertain mean-reverting stock model. Soft Comput 22:5583\u20135592","journal-title":"Soft Comput"},{"key":"9663_CR31","doi-asserted-by":"publisher","first-page":"126","DOI":"10.1016\/j.matcom.2019.04.009","volume":"166","author":"M Tian","year":"2019","unstructured":"Tian M, Yang X, Zhang Y (2019) Barrier option pricing of mean-reverting stock model in uncertain environment. Math. Comput. Simulat. 166:126\u2013143","journal-title":"Math. Comput. Simulat."},{"issue":"5","key":"9663_CR32","doi-asserted-by":"publisher","first-page":"2703","DOI":"10.3934\/jimo.2020090","volume":"17","author":"M Tian","year":"2021","unstructured":"Tian M, Yang X, Zhang Y (2021) Lookback option pricing of mean-reverting stock model in uncertain environment. J. Ind. Manag. Optim. 17(5):2703\u20132714","journal-title":"J. Ind. Manag. Optim."},{"key":"9663_CR33","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2021.111566","volume":"153","author":"W Wang","year":"2021","unstructured":"Wang W, Ralescu D (2021) Valuation of lookback option under uncertain volatility model, Chaos. Soliton. Fract. 153:111566","journal-title":"Soliton. Fract."},{"key":"9663_CR34","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1016\/0304-405X(87)90009-2","volume":"19","author":"J Wiggins","year":"1987","unstructured":"Wiggins J (1987) Option values under stochastic volatilities. J Financ Econ 19:351\u2013372","journal-title":"J Financ Econ"},{"key":"9663_CR35","doi-asserted-by":"crossref","unstructured":"Yang X, Ralescu D (2015) Adams method for solving uncertain differential equations. Appl Math Comput 270:993\u20131003","DOI":"10.1016\/j.amc.2015.08.109"},{"key":"9663_CR36","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s40467-015-0038-4","volume":"3","author":"X Yang","year":"2015","unstructured":"Yang X, Shen Y (2015) Runge-Kutta method for solving uncertain differential equations. J. Uncert. Anal. Appl. 3:1\u201312","journal-title":"J. Uncert. Anal. Appl."},{"issue":"1","key":"9663_CR37","doi-asserted-by":"publisher","first-page":"2150002","DOI":"10.1142\/S1752890921500021","volume":"14","author":"G Yang","year":"2021","unstructured":"Yang G, Zhu Y (2021) Critical value-based power options pricing problems in uncertain financial markets. J. Uncert. Syst. 14(1):2150002","journal-title":"J. Uncert. Syst."},{"key":"9663_CR38","first-page":"1","volume":"1","author":"K Yao","year":"2013","unstructured":"Yao K (2013) Extreme values and integral of solution of uncertain differential equations. J. Uncert. Anal. Appl. 1:1\u201321","journal-title":"J. Uncert. Anal. Appl."},{"issue":"4","key":"9663_CR39","doi-asserted-by":"publisher","first-page":"399","DOI":"10.1007\/s10700-015-9211-y","volume":"14","author":"K Yao","year":"2015","unstructured":"Yao K (2015) Uncertain contour process and its application in stock model with floating interest rate. Fuzzy Optim. Decis. Ma. 14(4):399\u2013424","journal-title":"Fuzzy Optim. Decis. Ma."},{"issue":"3","key":"9663_CR40","first-page":"825","volume":"25","author":"K Yao","year":"2013","unstructured":"Yao K, Chen X (2013) A numerical method for solving uncertain differential equations. J. Intell. Fuzzy Syst. 25(3):825\u2013832","journal-title":"J. Intell. Fuzzy Syst."},{"issue":"2","key":"9663_CR41","doi-asserted-by":"publisher","first-page":"2250012","DOI":"10.1142\/S175289092250012X","volume":"16","author":"P Yu","year":"2023","unstructured":"Yu P, Zhu Y (2023) Pricing problem of geometric average Asian barrier option based on uncertain fractional differential equation. J. Uncert. syst. 16(2):2250012","journal-title":"J. Uncert. syst."},{"key":"9663_CR42","doi-asserted-by":"publisher","first-page":"5335","DOI":"10.1007\/s00500-018-3028-x","volume":"22","author":"Y Zhang","year":"2018","unstructured":"Zhang Y, Gao J, An Q (2018) International investing in uncertain financial market. Soft Comput 22:5335\u20135346","journal-title":"Soft Comput"},{"issue":"15","key":"9663_CR43","doi-asserted-by":"publisher","first-page":"3359","DOI":"10.1002\/mma.3335","volume":"38","author":"Y Zhu","year":"2015","unstructured":"Zhu Y (2015) Uncertain fractional differential equations and an interest rate model. Math Methods Appl Sci 38(15):3359\u20133368","journal-title":"Math Methods Appl Sci"},{"issue":"5","key":"9663_CR44","first-page":"1","volume":"3","author":"Y Zhu","year":"2015","unstructured":"Zhu Y (2015) Existence and uniqueness of the solution to uncertain fractional differential equation. J. Uncert. Anal. Appl. 3(5):1\u201311","journal-title":"J. Uncert. Anal. Appl."}],"container-title":["Soft Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-024-09663-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00500-024-09663-6","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-024-09663-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T10:48:37Z","timestamp":1773658117000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00500-024-09663-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,20]]},"references-count":44,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["9663"],"URL":"https:\/\/doi.org\/10.1007\/s00500-024-09663-6","relation":{},"ISSN":["1432-7643","1433-7479"],"issn-type":[{"value":"1432-7643","type":"print"},{"value":"1433-7479","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,20]]},"assertion":[{"value":"9 January 2024","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 August 2024","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest regarding the publication of this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"This article does not contain any studies with human participants performed by the author.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}}]}}