{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T13:49:23Z","timestamp":1773236963496,"version":"3.50.1"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2013,2,22]],"date-time":"2013-02-22T00:00:00Z","timestamp":1361491200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2014,4]]},"DOI":"10.1007\/s10957-013-0284-x","type":"journal-article","created":{"date-parts":[[2013,2,21]],"date-time":"2013-02-21T14:41:00Z","timestamp":1361457660000},"page":"164-178","source":"Crossref","is-referenced-by-count":2,"title":["A Robust Spectral Method for Solving Heston\u2019s Model"],"prefix":"10.1007","volume":"161","author":[{"given":"E.","family":"Ngounda","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"K. C.","family":"Patidar","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"E.","family":"Pindza","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2013,2,22]]},"reference":[{"issue":"2","key":"284_CR1","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1111\/j.1540-6261.1987.tb02568.x","volume":"42","author":"J.C. Hull","year":"1987","unstructured":"Hull, J.C., White, A.: The pricing of options on assets with stochastic volatility. J. Finance 42(2), 281\u2013300 (1987)","journal-title":"J. Finance"},{"key":"284_CR2","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/0304-405X(76)90022-2","volume":"3","author":"R.C. Merton","year":"1975","unstructured":"Merton, R.C.: Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3, 125\u2013144 (1975)","journal-title":"J. Financ. Econ."},{"issue":"2","key":"284_CR3","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1093\/rfs\/6.2.327","volume":"6","author":"S.L. Heston","year":"1993","unstructured":"Heston, S.L.: A\u00a0closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6(2), 327\u2013343 (1993)","journal-title":"Rev. Financ. Stud."},{"issue":"2","key":"284_CR4","first-page":"303","volume":"7","author":"K.J. In\u2019t\u00a0Hout","year":"2010","unstructured":"In\u2019t\u00a0Hout, K.J., Foulon, S.: ADI finite difference schemes for option pricing in the Heston model with correlation. Int. J. Numer. Anal. Model. 7(2), 303\u2013320 (2010)","journal-title":"Int. J. Numer. Anal. Model."},{"key":"284_CR5","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1090\/S0002-9947-1956-0084194-4","volume":"82","author":"J. Douglas","year":"1956","unstructured":"Douglas, J., Rachford, H.H.: On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. 82, 421\u2013439 (1956)","journal-title":"Trans. Am. Math. Soc."},{"issue":"1","key":"284_CR6","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1006\/jcph.1996.0120","volume":"126","author":"S. McKee","year":"1996","unstructured":"McKee, S., Wall, D.P., Wilson, S.K.: An alternating direction implicit scheme for parabolic equations with mixed derivative and convective terms. J. Comp. Physiol. 126(1), 64\u201376 (1996)","journal-title":"J. Comp. Physiol."},{"key":"284_CR7","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/0898-1221(88)90150-2","volume":"16","author":"I.J.D. Craig","year":"1988","unstructured":"Craig, I.J.D., Sneyd, A.D.: An alternating-direction implicit scheme for parabolic equations with mixed derivatives. Comput. Math. Appl. 16, 341\u2013350 (1988)","journal-title":"Comput. Math. Appl."},{"issue":"3\u20134","key":"284_CR8","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1016\/j.apnum.2008.03.016","volume":"59","author":"K.J. In\u2019t\u00a0Hout","year":"2009","unstructured":"In\u2019t\u00a0Hout, K.J., Welfert, B.D.: Unconditional stability of second-order ADI schemes applied to multi-dimensional diffusion equations with mixed derivative. Appl. Numer. Math. 59(3\u20134), 677\u2013692 (2009)","journal-title":"Appl. Numer. Math."},{"key":"284_CR9","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1016\/S0168-9274(01)00152-0","volume":"42","author":"W. Hundsdorfer","year":"2002","unstructured":"Hundsdorfer, W.: Accuracy and stability of splitting with stabilizing corrections. Appl. Numer. Math. 42, 213\u2013233 (2002)","journal-title":"Appl. Numer. Math."},{"key":"284_CR10","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-09017-6","volume-title":"Numerical Solution of Time-Dependent Advection-Diffusion\u2013Reaction Equations","author":"W. Hundsdorfer","year":"2003","unstructured":"Hundsdorfer, W., Verwer, J.G.: Numerical Solution of Time-Dependent Advection-Diffusion\u2013Reaction Equations. Springer, Berlin (2003)"},{"key":"284_CR11","first-page":"1995","volume-title":"AIP Conference Proceedings","author":"T. Haentjens","year":"2010","unstructured":"Haentjens, T.: ADI finite difference discretization of the Heston\u2013Hull\u2013White PDE. In: AIP Conference Proceedings, ICNAAM 2010: International Conference of Numerical Analysis and Applied Mathematics 2010, vol.\u00a01281, pp. 1995\u20131999 (2010)"},{"issue":"2","key":"284_CR12","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1007\/s00211-009-0227-5","volume":"113","author":"S. Ikonen","year":"2009","unstructured":"Ikonen, S., Toivanen, J.: Operator splitting methods for pricing American options under stochastic volatility. Numer. Math. 113(2), 299\u2013324 (2009)","journal-title":"Numer. Math."},{"key":"284_CR13","first-page":"165","volume":"15","author":"C.W. Oosterlee","year":"2003","unstructured":"Oosterlee, C.W.: On multigrid for linear complementarity problems with application to American-style options. Electron. Trans. Numer. Anal. 15, 165\u2013185 (2003)","journal-title":"Electron. Trans. Numer. Anal."},{"issue":"2","key":"284_CR14","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/S0377-0427(98)00037-5","volume":"91","author":"R. Zvan","year":"1998","unstructured":"Zvan, R., Forsyth, P.A., Vetzal, K.R.: Penalty methods for American option with stochastic volatility. J.\u00a0Comput. Appl. Math. 91(2), 199\u2013218 (1998)","journal-title":"J.\u00a0Comput. Appl. Math."},{"key":"284_CR15","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1016\/j.aml.2010.02.009","volume":"23","author":"S.P. Zhu","year":"2010","unstructured":"Zhu, S.P., Chen, W.T.: A\u00a0new analytical approximation for European puts with stochastic volatility. Appl. Math. Lett. 23, 687\u2013692 (2010)","journal-title":"Appl. Math. Lett."},{"key":"284_CR16","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1137\/090776081","volume":"33","author":"K.J. In\u2019t\u00a0Hout","year":"2010","unstructured":"In\u2019t\u00a0Hout, K.J., Weideman, J.A.C.: Appraisal of a contour integral method for the Black-Scholes and Heston equations. SIAM J. Sci. Comput. 33, 763\u2013785 (2010)","journal-title":"SIAM J. Sci. Comput."},{"key":"284_CR17","first-page":"83","volume":"January","author":"H. Albrecher","year":"2007","unstructured":"Albrecher, H., Mayer, P., Schoutens, W., Tistaert, J.: The little Heston trap. Wilmott Mag. January, 83\u201392 (2007)","journal-title":"Wilmott Mag."},{"issue":"4","key":"284_CR18","doi-asserted-by":"crossref","first-page":"61","DOI":"10.21314\/JCF.1999.043","volume":"2","author":"P. Carr","year":"1999","unstructured":"Carr, P., Madan, D.: Option valuation using the Fast Fourier Transform. J. Comput. Finance 2(4), 61\u201373 (1999)","journal-title":"J. Comput. Finance"},{"key":"284_CR19","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1080\/135048699334528","volume":"6","author":"N. Clarke","year":"1999","unstructured":"Clarke, N., Parrott, K.: Multigrid for American option pricing with stochastic volatility. Appl. Math. Finance 6, 177\u2013195 (1999)","journal-title":"Appl. Math. Finance"},{"key":"284_CR20","volume-title":"The Volatility Surface: A\u00a0Practioner\u2019s Guide","author":"J. Gatheral","year":"2006","unstructured":"Gatheral, J.: The Volatility Surface: A\u00a0Practioner\u2019s Guide. Wiley Finance, New York (2006)"},{"key":"284_CR21","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1137\/090756119","volume":"2","author":"L.A. Grzelak","year":"2011","unstructured":"Grzelak, L.A., Oosterlee, C.W.: On the Heston model with stochastic interest rates. SIAM J. Financ. Math. 2, 255\u2013286 (2011)","journal-title":"SIAM J. Financ. Math."},{"key":"284_CR22","first-page":"94","volume":"September","author":"C. Kahl","year":"2005","unstructured":"Kahl, C., J\u00e4ckel, P.: Not-so-complex logarithms in the Heston model. Wilmott Mag. September, 94\u2013103 (2005)","journal-title":"Wilmott Mag."},{"issue":"11","key":"284_CR23","doi-asserted-by":"crossref","first-page":"1796","DOI":"10.1016\/j.jedc.2012.05.005","volume":"36","author":"J. Kim","year":"2012","unstructured":"Kim, J., Kim, B., Moon, K.-S., Wee, I.-S.: Valuation of power options under Heston\u2019s stochastic volatility model. J. Econ. Dyn. Control 36(11), 1796\u20131813 (2012)","journal-title":"J. Econ. Dyn. Control"},{"issue":"1","key":"284_CR24","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/j.jempfin.2011.09.003","volume":"19","author":"R. Langrock","year":"2012","unstructured":"Langrock, R., MacDonald, I.L., Zucchini, W.: Some nonstandard stochastic volatility models and their estimation using structured hidden Markov models. J. Empir. Finance 19(1), 147\u2013161 (2012)","journal-title":"J. Empir. Finance"},{"issue":"1","key":"284_CR25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jeconom.2012.08.004","volume":"173","author":"S. Laurent","year":"2013","unstructured":"Laurent, S., Rombouts, J.V.K., Violante, F.: On loss functions and ranking forecasting performances of multivariate volatility models. J.\u00a0Economet. 173(1), 1\u201310 (2013)","journal-title":"J.\u00a0Economet."},{"issue":"175","key":"284_CR26","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1090\/S0025-5718-1986-0842136-8","volume":"47","author":"C. Schneider","year":"1986","unstructured":"Schneider, C., Werner, W.: Some new aspect of rational interpolation. Math. Comput. 47(175), 285\u2013299 (1986)","journal-title":"Math. Comput."},{"issue":"5","key":"284_CR27","first-page":"8","volume":"28","author":"W. Tee","year":"2006","unstructured":"Tee, W.: A\u00a0rational collocation with adaptively transformed Chebyshev grid points. SIAM J. Sci. Comput. 28(5), 8\u20131811 (2006)","journal-title":"SIAM J. Sci. Comput."},{"issue":"187","key":"284_CR28","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1090\/S0025-5718-1989-0972370-0","volume":"53","author":"W. Heinrichs","year":"1989","unstructured":"Heinrichs, W.: Improved condition number of spectral methods. Math. Comput. 53(187), 103\u2013119 (1989)","journal-title":"Math. Comput."},{"issue":"5","key":"284_CR29","doi-asserted-by":"crossref","first-page":"1480","DOI":"10.1016\/j.jcp.2008.10.043","volume":"228","author":"K. Julien","year":"2009","unstructured":"Julien, K., Watson, M.: Efficient multi-dimensional solution of PDEs using Chebyshev spectral methods. J. Comp. Physiol. 228(5), 1480\u20131503 (2009)","journal-title":"J. Comp. Physiol."},{"key":"284_CR30","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898719598","volume-title":"Spectral Method in MATLAB","author":"L.N. Trefethen","year":"2000","unstructured":"Trefethen, L.N.: Spectral Method in MATLAB. SIAM, Philadelphia (2000)"},{"issue":"1","key":"284_CR31","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1093\/imamat\/23.1.97","volume":"23","author":"A. Talbot","year":"1979","unstructured":"Talbot, A.: The accurate numerical inversion of Laplace transforms. IMA J. Appl. Math. 23(1), 97\u2013120 (1979)","journal-title":"IMA J. Appl. Math."},{"key":"284_CR32","volume-title":"Theory and Problems of Laplace Transforms","author":"M.R. Spiegel","year":"1965","unstructured":"Spiegel, M.R.: Theory and Problems of Laplace Transforms. McGraw-Hill, New York (1965)"},{"issue":"259","key":"284_CR33","doi-asserted-by":"crossref","first-page":"1341","DOI":"10.1090\/S0025-5718-07-01945-X","volume":"76","author":"J.A.C. Weideman","year":"2007","unstructured":"Weideman, J.A.C., Trefethen, L.N.: Parabolic and hyperbolic contours for computing the Bromwich integral. Math. Comput. 76(259), 1341\u20131356 (2007)","journal-title":"Math. Comput."},{"issue":"2\u20133","key":"284_CR34","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/j.apnum.2004.06.015","volume":"51","author":"M. L\u00f3pez-Fern\u00e1ndez","year":"2004","unstructured":"L\u00f3pez-Fern\u00e1ndez, M., Palencia, C.: On the numerical inversion of the Laplace transform of certain holomorphic mappings. Appl. Numer. Math. 51(2\u20133), 289\u2013303 (2004)","journal-title":"Appl. Numer. Math."},{"key":"284_CR35","first-page":"T83","volume":"48","author":"V.E. Martensen","year":"1968","unstructured":"Martensen, V.E.: Zur numerischen Auswertung uneigentlicher Integrale. Z.\u00a0Angew. Math. Mech. 48, T83\u2013T85 (1968)","journal-title":"Z.\u00a0Angew. Math. Mech."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-013-0284-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10957-013-0284-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-013-0284-x","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,2]],"date-time":"2019-06-02T05:55:21Z","timestamp":1559454921000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10957-013-0284-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,2,22]]},"references-count":35,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2014,4]]}},"alternative-id":["284"],"URL":"https:\/\/doi.org\/10.1007\/s10957-013-0284-x","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,2,22]]}}}