{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T11:36:14Z","timestamp":1774956974242,"version":"3.50.1"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2020,8,27]],"date-time":"2020-08-27T00:00:00Z","timestamp":1598486400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,8,27]],"date-time":"2020-08-27T00:00:00Z","timestamp":1598486400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100004733","name":"Universidade de Macau","doi-asserted-by":"publisher","award":["MYRG2018-00025-FST"],"award-info":[{"award-number":["MYRG2018-00025-FST"]}],"id":[{"id":"10.13039\/501100004733","id-type":"DOI","asserted-by":"publisher"}]},{"name":"The Science and Technology Development Fundx0FF0C;Macau SAR","award":["081\/2016\/A2"],"award-info":[{"award-number":["081\/2016\/A2"]}]},{"DOI":"10.13039\/501100003453","name":"Natural Science Foundation of Guangdong Province","doi-asserted-by":"publisher","award":["2017A030313400"],"award-info":[{"award-number":["2017A030313400"]}],"id":[{"id":"10.13039\/501100003453","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2021,7]]},"DOI":"10.1007\/s11075-020-00994-7","type":"journal-article","created":{"date-parts":[[2020,8,27]],"date-time":"2020-08-27T02:02:15Z","timestamp":1598493735000},"page":"939-965","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["An implicit-explicit preconditioned direct method for pricing options under regime-switching tempered fractional partial differential models"],"prefix":"10.1007","volume":"87","author":[{"given":"Xu","family":"Chen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Deng","family":"Ding","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Siu-Long","family":"Lei","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wenfei","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,8,27]]},"reference":[{"key":"994_CR1","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1007\/s10690-016-9219-5","volume":"23","author":"P Asiimwe","year":"2016","unstructured":"Asiimwe, P., Mahera, C.W., Menoukeu-Pamen, O.: On the price of risk under a regime switching CGMY process. Asia-pacific Finan. Markets 23, 305 (2016)","journal-title":"Asia-pacific Finan. Markets"},{"key":"994_CR2","doi-asserted-by":"publisher","first-page":"146","DOI":"10.1016\/j.cam.2019.01.044","volume":"357","author":"J Bao","year":"2019","unstructured":"Bao, J., Zhao, Y.: Option pricing in Markov-modulated exponential Levy\u0301 models with stochastic interest rates. J. Comput. Appl. Math. 357, 146\u2013160 (2019)","journal-title":"J. Comput. Appl. Math."},{"issue":"3","key":"994_CR3","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)","journal-title":"J. Polit. Econ."},{"key":"994_CR4","doi-asserted-by":"publisher","DOI":"10.1142\/4955","volume-title":"Non-Gaussian Merton-Black-Scholes Theory","author":"S Boyarchenko","year":"2002","unstructured":"Boyarchenko, S., Levendorskii, S.: Non-Gaussian Merton-Black-Scholes Theory. World Scientific, Singapore (2002)"},{"issue":"2","key":"994_CR5","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1086\/338705","volume":"75","author":"P Carr","year":"2002","unstructured":"Carr, P., Geman, H., Madan, D.B., Yor, M.: The fine structure of asset returns: an empirical investigation. J. Bus. 75(2), 305\u2013333 (2002)","journal-title":"J. Bus."},{"issue":"2","key":"994_CR6","doi-asserted-by":"publisher","first-page":"753","DOI":"10.1111\/1540-6261.00544","volume":"58","author":"P Carr","year":"2003","unstructured":"Carr, P., Wu, L.: The finite moment log stable process and option pricing. J. Financ. 58(2), 753\u2013777 (2003)","journal-title":"J. Financ."},{"issue":"2","key":"994_CR7","doi-asserted-by":"publisher","first-page":"749","DOI":"10.1016\/j.physa.2006.08.071","volume":"374","author":"A Cartea","year":"2007","unstructured":"Cartea, A., del Castillo-Negrete, D.: Fractional diffusion models of option prices in markets with jumps. Physica A 374(2), 749\u2013763 (2007)","journal-title":"Physica A"},{"key":"994_CR8","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718850","volume-title":"An Introduction to Iterative Toeplitz Solvers","author":"R Chan","year":"2007","unstructured":"Chan, R., Jin, X.: An Introduction to Iterative Toeplitz Solvers. SIAM, Philadelphia (2007)"},{"issue":"2","key":"994_CR9","doi-asserted-by":"publisher","first-page":"A890","DOI":"10.1137\/14097207X","volume":"37","author":"M Chen","year":"2015","unstructured":"Chen, M., Deng, W.: High order algorithms for the fractional substantial diffusion equation with truncated L\u00e9vy flights. SIAM J. Sci. Comput. 37(2), A890\u2013A917 (2015)","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"994_CR10","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1016\/j.camwa.2013.10.007","volume":"67","author":"W Chen","year":"2014","unstructured":"Chen, W., Wang, S.: A penalty method for a fractional order parabolic variational inequality governing American put option valuation. Comput. Math. Appl. 67(1), 77\u201390 (2014)","journal-title":"Comput. Math. Appl."},{"key":"994_CR11","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1016\/j.apnum.2015.06.004","volume":"97","author":"W Chen","year":"2015","unstructured":"Chen, W., Xu, X., Zhu, S.: A predictor-corrector approach for pricing American options under the finite moment log-stable model. Appl. Numer. Math. 97, 15\u201329 (2015)","journal-title":"Appl. Numer. Math."},{"key":"994_CR12","doi-asserted-by":"publisher","first-page":"1932","DOI":"10.1016\/j.camwa.2017.02.040","volume":"73","author":"X Chen","year":"2017","unstructured":"Chen, X., Wang, W., Ding, D., Lei, S.: A fast preconditioned policy iteration method for solving the tempered fractional HJB equation governing American options valuation. Comput. Math. Appl. 73, 1932\u20131944 (2017)","journal-title":"Comput. Math. Appl."},{"key":"994_CR13","volume-title":"Financial Modelling with Jump Processes","author":"R Cont","year":"2003","unstructured":"Cont, R., Tankov, P.: Financial Modelling with Jump Processes. CRC press, London (2003)"},{"key":"994_CR14","doi-asserted-by":"publisher","first-page":"250","DOI":"10.1007\/s00780-006-0004-6","volume":"10","author":"RJ Elliott","year":"2006","unstructured":"Elliott, R.J., Osakwe, C.J.U.: Option pricing for pure jump processes with Markov switching compensators. Financ. Stoch. 10, 250 (2006)","journal-title":"Financ. Stoch."},{"key":"994_CR15","doi-asserted-by":"publisher","first-page":"730","DOI":"10.1007\/BF01200697","volume":"15","author":"I Gohberg","year":"1992","unstructured":"Gohberg, I., Olshevsky, V.: Circulants, displacements and decompositions of matrices. Integr. Equat. Oper. Th. 15, 730\u2013743 (1992)","journal-title":"Integr. Equat. Oper. Th."},{"key":"994_CR16","doi-asserted-by":"publisher","first-page":"1529","DOI":"10.1080\/14697688.2016.1158854","volume":"16","author":"X Guo","year":"2016","unstructured":"Guo, X., Li, Y.: Valuation of American options under the CGMY model. Quant. Financ. 16, 1529\u20131539 (2016)","journal-title":"Quant. Financ."},{"key":"994_CR17","first-page":"A3322","volume":"40","author":"X Guo","year":"2018","unstructured":"Guo, X., Li, Y., Wang, H.: A high order finite difference method for tempered fractional diffusion equations with application to the CGMY model. SIAM. J. Sci. Comput 40, A3322\u2013A3343 (2018)","journal-title":"J. Sci. Comput"},{"key":"994_CR18","doi-asserted-by":"publisher","first-page":"1500","DOI":"10.1016\/j.camwa.2018.07.002","volume":"76","author":"X Guo","year":"2018","unstructured":"Guo, X., Li, Y., Wang, H.: Tempered fractional diffusion equations for pricing multi-asset options under CGMYe process. Comput. Math. Appl. 76, 1500\u20131514 (2018)","journal-title":"Comput. Math. Appl."},{"key":"994_CR19","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1007\/s10614-011-9269-8","volume":"40","author":"A Itkin","year":"2012","unstructured":"Itkin, A., Carr, P.: Using pseudo-parabolic and fractional equations for option pricing in jump diffusion models. Comput. Econ. 40, 63\u2013104 (2012)","journal-title":"Comput. Econ."},{"key":"994_CR20","doi-asserted-by":"crossref","unstructured":"Kazmi, K., An, IMEX: Predictor\u2013corrector method for pricing options under regime-switching jump-diffusion models. Int. J. Comput. Math., pp. 1\u201321 (2018)","DOI":"10.1080\/00207160.2018.1446526"},{"key":"994_CR21","doi-asserted-by":"publisher","first-page":"1062","DOI":"10.1080\/00207160.2016.1149579","volume":"94","author":"S Lei","year":"2016","unstructured":"Lei, S., Huang, Y.: Fast algorithms for high-order numerical methods for space-fractional diffusion equations. Int. J. Comput. Math. 94, 1062\u20131078 (2016)","journal-title":"Int. J. Comput. Math."},{"key":"994_CR22","doi-asserted-by":"publisher","first-page":"715","DOI":"10.1016\/j.jcp.2013.02.025","volume":"242","author":"S Lei","year":"2013","unstructured":"Lei, S., Sun, H.: A circulant preconditioner for fractional diffusion equations. J. Comput. Phys. 242, 715\u2013725 (2013)","journal-title":"J. Comput. Phys."},{"key":"994_CR23","doi-asserted-by":"publisher","first-page":"1633","DOI":"10.1007\/s10915-017-0602-9","volume":"75","author":"S Lei","year":"2018","unstructured":"Lei, S., Wang, W., Chen, X., Ding, D.: A fast preconditioned penalty method for American options pricing under regime-switching tempered fractional diffusion models. J. Sci. Comput. 75, 1633\u20131655 (2018)","journal-title":"J. Sci. Comput."},{"issue":"5","key":"994_CR24","doi-asserted-by":"publisher","first-page":"1382","DOI":"10.1002\/num.21948","volume":"31","author":"Q Meng","year":"2014","unstructured":"Meng, Q., Ding, D., Sheng, Q.: Preconditioned iterative methods for fractional diffusion models in finance. Numer. Meth. Part Differ. Equ. 31(5), 1382\u20131395 (2014)","journal-title":"Numer. Meth. Part Differ. Equ."},{"key":"994_CR25","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1007\/s11009-014-9399-2","volume":"18","author":"RH Momeya","year":"2016","unstructured":"Momeya, R.H., Morales, M.: On the price of risk of the underlying Markov chain in a regime-switching exponential Levy\u0301 model. Methodol. Comput. Appl. Probab. 18, 107 (2016)","journal-title":"Methodol. Comput. Appl. Probab."},{"issue":"2","key":"994_CR26","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1016\/0024-3795(77)90073-8","volume":"18","author":"RJ Plemmons","year":"1977","unstructured":"Plemmons, R.J.: M-matrix characterizations.I\u2013nonsingular M-matrices. Linear Algebra Appl. 18(2), 175\u2013188 (1977)","journal-title":"Linear Algebra Appl."},{"key":"994_CR27","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1016\/j.cam.2015.12.019","volume":"300","author":"NA Rambeerich","year":"2016","unstructured":"Rambeerich, N.A., Pantelous, A.: A high order finite element scheme for pricing options under regime switching jump diffusion processes. J. Comput. Appl. Math. 300, 83\u201396 (2016)","journal-title":"J. Comput. Appl. Math."},{"issue":"3","key":"994_CR28","doi-asserted-by":"publisher","first-page":"856","DOI":"10.1137\/0907058","volume":"7","author":"Y Saad","year":"1986","unstructured":"Saad, Y., Schultz, M.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7 (3), 856\u2013869 (1986)","journal-title":"SIAM J. Sci. Statist. Comput."},{"key":"994_CR29","doi-asserted-by":"publisher","first-page":"673","DOI":"10.1080\/14697688.2017.1412494","volume":"18","author":"G Tour","year":"2018","unstructured":"Tour, G., Thakoor, N., Khaliq, A.Q.M., Tangman, D.Y.: COS Method for option pricing under a regime-switching model with time-changed Levy\u0301 processes. Quant. Financ. 18, 673\u2013692 (2018)","journal-title":"Quant. Financ."},{"issue":"12","key":"994_CR30","doi-asserted-by":"publisher","first-page":"2596","DOI":"10.1080\/00207160.2015.1077948","volume":"92","author":"W Wang","year":"2015","unstructured":"Wang, W., Chen, X., Ding, D., Lei, S.: Circulant preconditioning technique for barrier options pricing under fractional diffusion models. Int. J. Comput. Math. 92(12), 2596\u20132614 (2015)","journal-title":"Int. J. Comput. Math."},{"key":"994_CR31","doi-asserted-by":"publisher","first-page":"2989","DOI":"10.1016\/j.camwa.2018.01.026","volume":"75","author":"M Yousuf","year":"2018","unstructured":"Yousuf, M., Khaliq, A.Q.M., Alrabeei, S.: Solving complex PIDE systems for pricing American option under multi-state regime switching jump\u2013diffusion model. Comput. Math. Appl. 75, 2989\u20133001 (2018)","journal-title":"Comput. Math. Appl."},{"key":"994_CR32","doi-asserted-by":"publisher","first-page":"186","DOI":"10.1016\/j.cam.2017.09.045","volume":"339","author":"H Zhang","year":"2018","unstructured":"Zhang, H., Liu, F., Chen, S., Anh, V., Chen, J.: Fast numerical simulation of a new time-space fractional option pricing model governing European call option. Appl. Math. Comput. 339, 186\u2013198 (2018)","journal-title":"Appl. Math. Comput."},{"issue":"11","key":"994_CR33","doi-asserted-by":"publisher","first-page":"5819","DOI":"10.1016\/j.apm.2016.01.027","volume":"40","author":"H Zhang","year":"2016","unstructured":"Zhang, H., Liu, F., Turner, I., Chen, S.: The numerical simulation of the tempered fractional Black-Scholes equation for European double barrier option. Appl. Math. Model. 40(11), 5819\u20135834 (2016)","journal-title":"Appl. Math. Model."},{"key":"994_CR34","doi-asserted-by":"publisher","first-page":"569","DOI":"10.1007\/s11075-016-0212-x","volume":"75","author":"H Zhang","year":"2017","unstructured":"Zhang, H., Liu, F., Turner, I., Chen, S., Yang, Q.: Numerical simulation of a Finite Moment Log Stable model for a European call option. Numer. Algor. 75, 569\u2013585 (2017)","journal-title":"Numer. Algor."},{"key":"994_CR35","doi-asserted-by":"publisher","first-page":"782","DOI":"10.4208\/eajam.130218.290618","volume":"8","author":"Z Zhou","year":"2018","unstructured":"Zhou, Z., Ma, J., Gao, X.: Convergence of iterative Laplace transform methods for a system of fractional PDEs and PIDEs arising in option pricing. East Asian J. Appl. Math. 8, 782\u2013808 (2018)","journal-title":"East Asian J. Appl. Math."},{"key":"994_CR36","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1007\/s10915-017-0423-x","volume":"74","author":"Z Zhou","year":"2018","unstructured":"Zhou, Z., Ma, J., Sun, H.: Fast Laplace transform methods for free-boundary problems of fractional diffusion equations. J. Sci. Comput. 74, 49 (2018)","journal-title":"J. Sci. Comput."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-020-00994-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-020-00994-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-020-00994-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,8,26]],"date-time":"2021-08-26T23:53:07Z","timestamp":1630021987000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-020-00994-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,27]]},"references-count":36,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,7]]}},"alternative-id":["994"],"URL":"https:\/\/doi.org\/10.1007\/s11075-020-00994-7","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,8,27]]},"assertion":[{"value":"6 November 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 July 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 August 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}