{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,27]],"date-time":"2026-07-27T10:59:15Z","timestamp":1785149955200,"version":"3.55.0"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"7","license":[{"start":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T00:00:00Z","timestamp":1781568000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T00:00:00Z","timestamp":1781568000000},"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":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2026,7]]},"DOI":"10.1007\/s12190-026-02830-y","type":"journal-article","created":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T14:58:52Z","timestamp":1781621932000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A variable step-size implicit-explicit runge-kutta approach to option pricing for jump-diffusion models with nonsmooth payoff functions"],"prefix":"10.1007","volume":"72","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3556-0975","authenticated-orcid":false,"given":"Jincheng","family":"Ren","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-4722-7317","authenticated-orcid":false,"given":"Shiyu","family":"He","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,6,16]]},"reference":[{"issue":"3","key":"2830_CR1","doi-asserted-by":"publisher","first-page":"637","DOI":"10.1086\/260062","volume":"81","author":"F. Black","year":"1973","unstructured":"Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637\u2013654 (1973). https:\/\/doi.org\/10.1086\/260062.","journal-title":"J. Polit. Econ."},{"issue":"1","key":"2830_CR2","doi-asserted-by":"publisher","first-page":"141","DOI":"10.2307\/3003143","volume":"4","author":"R.C. Merton","year":"1973","unstructured":"Merton, R.C.: Theory of rational option pricing. Bell J. Econ. Manage. Sci. 4(1), 141 (1973). https:\/\/doi.org\/10.2307\/3003143.","journal-title":"Bell J. Econ. Manage. Sci."},{"issue":"4","key":"2830_CR3","doi-asserted-by":"publisher","first-page":"1596","DOI":"10.1137\/S0036142903436186","volume":"43","author":"R. Cont","year":"2005","unstructured":"Cont, R., Voltchkova, E.: A finite difference scheme for option pricing in jump diffusion and exponential l\u00e9vy models. SIAM J. Numer. Anal. 43(4), 1596\u20131626 (2005). https:\/\/doi.org\/10.1137\/S0036142903436186.","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"2830_CR4","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/s10092-007-0128-x","volume":"44","author":"M. Briani","year":"2007","unstructured":"Briani, M., Natalini, R., Russo, G.: Implicit\u2013explicit numerical schemes for jump\u2013diffusion processes. Calcolo 44(1), 33\u201357 (2007). https:\/\/doi.org\/10.1007\/s10092-007-0128-x.","journal-title":"Calcolo"},{"issue":"6","key":"2830_CR5","doi-asserted-by":"publisher","first-page":"2598","DOI":"10.1137\/090777529","volume":"49","author":"Y. Kwon","year":"2011","unstructured":"Kwon, Y., Lee, Y.: A second-order finite difference method for option pricing under jump-diffusion models. SIAM J. Numer. Anal. 49(6), 2598\u20132617 (2011). https:\/\/doi.org\/10.1137\/090777529.","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"2830_CR6","doi-asserted-by":"publisher","first-page":"979","DOI":"10.1007\/s10915-015-0001-z","volume":"65","author":"M.K. Kadalbajoo","year":"2015","unstructured":"Kadalbajoo, M.K., Tripathi, L.P., Kumar, A.: Second order accurate IMEX methods for option pricing under Merton and Kou jump-diffusion models. J Sci Comput. 65(3), 979\u20131024 (2015). https:\/\/doi.org\/10.1007\/s10915-015-0001-z.","journal-title":"J Sci Comput."},{"issue":"3","key":"2830_CR7","doi-asserted-by":"publisher","first-page":"1289","DOI":"10.1137\/18M1194328","volume":"57","author":"W. Wang","year":"2019","unstructured":"Wang, W., Chen, Y., Fang, H.: On the variable two-step IMEX BDF method for parabolic integro-differential equations with nonsmooth initial data arising in finance. SIAM J. Numer. Anal. 57(3), 1289\u20131317 (2019). https:\/\/doi.org\/10.1137\/18M1194328.","journal-title":"SIAM J. Numer. Anal."},{"issue":"1\u20132","key":"2830_CR8","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1016\/0304-405X(76)90022-2","volume":"3","author":"R.C. Merton","year":"1976","unstructured":"Merton, R.C.: Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3(1\u20132), 125\u2013144 (1976). https:\/\/doi.org\/10.1016\/0304-405X(76)90022-2.","journal-title":"J. Financ. Econ."},{"issue":"8","key":"2830_CR9","doi-asserted-by":"publisher","first-page":"1086","DOI":"10.1287\/mnsc.48.8.1086.166","volume":"48","author":"Kou","year":"2002","unstructured":"Kou, Kou, S.G.: A jump-diffusion model for option pricing. Manage. Sci. 48(8), 1086\u20131101 (2002). https:\/\/doi.org\/10.1287\/mnsc.48.8.1086.166.","journal-title":"Manage. Sci."},{"key":"2830_CR10","doi-asserted-by":"publisher","unstructured":"Salmi, S., Toivanen, J., Sydow, L.: An IMEX-scheme for pricing options under stochastic volatility models with jumps. SIAM J Sci Comput. 36(5), B817\u2013B834 (2014). https:\/\/doi.org\/10.1137\/130924905.","DOI":"10.1137\/130924905"},{"issue":"12","key":"2830_CR11","doi-asserted-by":"publisher","first-page":"2515","DOI":"10.1080\/00207160.2015.1072173","volume":"92","author":"L. Sydow","year":"2015","unstructured":"Sydow, L., Toivanen, J., Zhang, C.: Adaptive finite differences and IMEX time-stepping to price options under Bates model. Int J Comput Math. 92(12), 2515\u20132529 (2015). https:\/\/doi.org\/10.1080\/00207160.2015.1072173.","journal-title":"Int J Comput Math."},{"key":"2830_CR12","doi-asserted-by":"publisher","first-page":"114","DOI":"10.1016\/j.apnum.2020.02.004","volume":"153","author":"L. Boen","year":"2020","unstructured":"Boen, L., In\u2019t Hout, K.J.: Operator splitting schemes for American options under the two-asset Merton jump-diffusion model. Appl. Numer. Math. 153, 114\u2013131 (2020). https:\/\/doi.org\/10.1016\/j.apnum.2020.02.004.","journal-title":"Appl. Numer. Math."},{"issue":"3","key":"2830_CR13","doi-asserted-by":"publisher","first-page":"1079","DOI":"10.1002\/num.20677","volume":"28","author":"S.T. Lee","year":"2012","unstructured":"Lee, S.T., Sun, H.: Fourth-order compact scheme with local mesh refinement for option pricing in jump-diffusion model. Numer. Methods Partial 28(3), 1079\u20131098 (2012). https:\/\/doi.org\/10.1002\/num.20677.","journal-title":"Numer. Methods Partial"},{"issue":"3","key":"2830_CR14","doi-asserted-by":"publisher","first-page":"913","DOI":"10.1051\/m2an\/2021012","volume":"55","author":"W. Wang","year":"2021","unstructured":"Wang, W., Mao, M., Wang, Z.: An efficient variable step-size method for options pricing under jump-diffusion models with nonsmooth payoff function. Esaim: M2an 55(3), 913\u2013938 (2021). https:\/\/doi.org\/10.1051\/m2an\/2021012.","journal-title":"Esaim: M2an"},{"issue":"6","key":"2830_CR15","doi-asserted-by":"publisher","first-page":"1569","DOI":"10.4310\/CMS.2024.v22.n6.a6","volume":"22","author":"W. Wang","year":"2024","unstructured":"Wang, W., Mao, M., Li, Z.: IMEX variable step-size Runge-Kutta methods for parabolic integro-differential equations with nonsmooth initial data. Commun Math Sci. 22(6), 1569\u20131599 (2024). https:\/\/doi.org\/10.4310\/CMS.2024.v22.n6.a6.","journal-title":"Commun Math Sci."},{"key":"2830_CR16","doi-asserted-by":"crossref","unstructured":"Chen, Y., Wang, W.: Implicit\u2013explicit high-order methods for pricing options under merton\u2019s jump-diffusion models. J. Appl. Math. Comput. 1\u201330 (2025).","DOI":"10.22541\/au.172589766.61663669\/v1"},{"key":"2830_CR17","doi-asserted-by":"crossref","unstructured":"Liao, H., Wang, X., Wen, C.: Average energy dissipation rates of additive implicit-explicit Runge-Kutta methods for gradient flow problems. CSIAM Trans Appl Math,accepted (2026).","DOI":"10.4208\/csiam-am.SO-2025-0095"},{"issue":"4","key":"2830_CR18","doi-asserted-by":"publisher","first-page":"1808","DOI":"10.1137\/24M1701770","volume":"63","author":"X. Wang","year":"2025","unstructured":"Wang, X., Zhao, X., Liao, H.: A unified framework on the original energy laws of three effective classes of Runge\u2013Kutta methods for phase field crystal type models. SIAM J. Numer. Anal. 63(4), 1808\u20131832 (2025). https:\/\/doi.org\/10.1137\/24M1701770.","journal-title":"SIAM J. Numer. Anal."},{"issue":"354","key":"2830_CR19","doi-asserted-by":"publisher","first-page":"1721","DOI":"10.1090\/mcom\/4015","volume":"94","author":"H. Liao","year":"2024","unstructured":"Liao, H., Wang, X.: Average energy dissipation rates of explicit exponential Runge-Kutta methods for gradient flow problems. Math. Comp. 94(354), 1721\u20131759 (2024). https:\/\/doi.org\/10.1090\/mcom\/4015.","journal-title":"Math. Comp."},{"key":"2830_CR20","doi-asserted-by":"publisher","first-page":"519","DOI":"10.1016\/j.jcp.2024.113456","volume":"519","author":"H. Liao","year":"2024","unstructured":"Liao, H., Wang, X., Wen, C.: Original energy dissipation preserving corrections of integrating factor Runge-Kutta methods for gradient flow problems. J. Comput. Phys. 519, 519 (2024). https:\/\/doi.org\/10.1016\/j.jcp.2024.113456.","journal-title":"J. Comput. Phys."},{"issue":"2\u20133","key":"2830_CR21","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1016\/S0168-9274(97)00056-1","volume":"25","author":"U.M. Ascher","year":"1997","unstructured":"Ascher, U.M., Ruuth, S.J., Spiteri, R.J.: Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25(2\u20133), 151\u2013167 (1997). https:\/\/doi.org\/10.1016\/S0168-9274(97)00056-1.","journal-title":"Appl. Numer. Math."},{"issue":"3","key":"2830_CR22","doi-asserted-by":"publisher","first-page":"991","DOI":"10.1016\/j.ejor.2005.01.013","volume":"171","author":"J. Frutos","year":"2006","unstructured":"Frutos, J.: Implicit\u2013explicit Runge\u2013Kutta methods for financial derivatives pricing models. Eur. J. Oper. Res. 171(3), 991\u20131004 (2006). https:\/\/doi.org\/10.1016\/j.ejor.2005.01.013.","journal-title":"Eur. J. Oper. Res."},{"key":"2830_CR23","doi-asserted-by":"publisher","unstructured":"Boscarino, S., Pareschi, L., Russo, G.: Implicit-Explicit Runge--Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit. SIAM J Sci Comput. 35(1), A22\u2013A51 (2013). https:\/\/doi.org\/10.1137\/110842855.","DOI":"10.1137\/110842855"},{"key":"2830_CR24","doi-asserted-by":"publisher","unstructured":"Boscarino, S., B\u00fcrger, R., Mulet, P., Russo, G., Villada, L.M.: Linearly implicit IMEX Runge--Kutta methods for a class of degenerate convection-diffusion problems. SIAM J Sci Comput. 37(2), B305\u2013B331 (2015). https:\/\/doi.org\/10.1137\/140967544.","DOI":"10.1137\/140967544"},{"key":"2830_CR25","unstructured":"Kennedy, C.A., Carpenter, M.H.: Diagonally implicit Runge-Kutta methods for ordinary differential equations. A review. Technical Memorandum NASA\/TM\u20132016\u2013219173 (2016)."},{"issue":"2","key":"2830_CR26","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1023\/A:1022399225591","volume":"40","author":"H. Olsson","year":"2000","unstructured":"Olsson, H., Sderlind, G.: The approximate Runge-Kutta computational process. BIT Numer Math 40(2), 351\u2013373 (2000). https:\/\/doi.org\/10.1023\/A:1022399225591.","journal-title":"BIT Numer Math"},{"issue":"359","key":"2830_CR27","doi-asserted-by":"publisher","first-page":"1293","DOI":"10.1090\/mcom\/4090","volume":"95","author":"H. Liao","year":"2025","unstructured":"Liao, H., Tang, T., Wang, X., Zhou, Zhou, T.: A class of refined implicit-explicit Runge-Kutta methods with robust time adaptability and unconditional convergence for the Cahn-Hilliard model. Math Comp 95(359), 1293\u20131325 (2025). https:\/\/doi.org\/10.1090\/mcom\/4090.","journal-title":"Math Comp"},{"issue":"3","key":"2830_CR28","doi-asserted-by":"publisher","first-page":"1438","DOI":"10.1007\/s10915-018-0815-6","volume":"78","author":"Q. Du","year":"2019","unstructured":"Du, Q., Ju, L., Lu, J.: Analysis of fully discrete approximations for dissipative systems and application to time-dependent nonlocal diffusion problems. J Sci Comput. 78(3), 1438\u20131466 (2019). https:\/\/doi.org\/10.1007\/s10915-018-0815-6.","journal-title":"J Sci Comput."},{"issue":"161","key":"2830_CR29","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1090\/S0025-5718-1983-0679441-1","volume":"40","author":"G.J. Cooper","year":"1983","unstructured":"Cooper, G.J., Sayfy, A.: Additive Runge-Kutta methods for stiff ordinary differential equations. Math. Comp. 40(161), 207\u2013218 (1983). https:\/\/doi.org\/10.1090\/S0025-5718-1983-0679441-1.","journal-title":"Math. Comp."},{"key":"2830_CR30","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1016\/j.apnum.2016.10.018","volume":"113","author":"G. Izzo","year":"2017","unstructured":"Izzo, G., Jackiewicz, Z.: Highly stable implicit\u2013explicit Runge\u2013Kutta methods. Appl. Numer. Math. 113, 71\u201392 (2017). https:\/\/doi.org\/10.1016\/j.apnum.2016.10.018.","journal-title":"Appl. Numer. Math."},{"key":"2830_CR31","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1016\/j.jcp.2017.07.006","volume":"347","author":"J. Shin","year":"2017","unstructured":"Shin, J., Lee, H.G., Lee, J.Y.: Unconditionally stable methods for gradient flow using convex splitting Runge\u2013Kutta scheme. J. Comput. Phys. 347, 367\u2013381 (2017). https:\/\/doi.org\/10.1016\/j.jcp.2017.07.006.","journal-title":"J. Comput. Phys."},{"key":"2830_CR32","unstructured":"Tavella, D., Randall, C.: Pricing financial instruments: the finite difference method. In: Wiley Series in Financial Engineering. John Wiley & Sons, New York (2000)."},{"key":"2830_CR33","doi-asserted-by":"publisher","unstructured":"Huang, J., Yang, C., Wei, Y.: Parallel energy-stable solver for a coupled Allen--Cahn and Cahn--Hilliard System. SIAM J Sci Comput. 42(5), C294\u2013C312 (2020). https:\/\/doi.org\/10.1137\/20M1331160.","DOI":"10.1137\/20M1331160"},{"issue":"1","key":"2830_CR34","doi-asserted-by":"publisher","first-page":"649","DOI":"10.1093\/imanum\/draa075","volume":"42","author":"H. Liao","year":"2022","unstructured":"Liao, H., Ji, B., Zhang, L.: An adaptive BDF2 implicit time-stepping method for the phase field crystal model. IMA J. Numer. Anal. 42(1), 649\u2013679 (2022). https:\/\/doi.org\/10.1093\/imanum\/draa075.","journal-title":"IMA J. Numer. Anal."},{"key":"2830_CR35","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139020411","volume-title":"Matrix Analysis","author":"R.A. Horn","year":"2012","unstructured":"Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn edn. Cambridge university press, Cambridge, UK (2012).","edition":"2nd edn"}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-026-02830-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-026-02830-y","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-026-02830-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,27]],"date-time":"2026-07-27T10:02:07Z","timestamp":1785146527000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-026-02830-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,16]]},"references-count":35,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2026,7]]}},"alternative-id":["2830"],"URL":"https:\/\/doi.org\/10.1007\/s12190-026-02830-y","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,6,16]]},"assertion":[{"value":"10 February 2026","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 May 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 May 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 June 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare that they have no conflict of interest.","order":1,"name":"Ethics","label":"Conflict of interest","group":{"name":"EthicsHeading","label":"Declarations"}}],"article-number":"189"}}