{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:10:46Z","timestamp":1760242246510,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2017,1,21]],"date-time":"2017-01-21T00:00:00Z","timestamp":1484956800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002631","name":"Gachon University","doi-asserted-by":"publisher","award":["GCU-2014-0130"],"award-info":[{"award-number":["GCU-2014-0130"]}],"id":[{"id":"10.13039\/501100002631","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The Crank\u2013Nicolson method can be used to solve the Black\u2013Scholes partial differential equation in one-dimension when both accuracy and stability is of concern. In multi-dimensions, however, discretizing the computational grid with a Crank\u2013Nicolson scheme requires significantly large storage compared to the widely adopted Operator Splitting Method (OSM). We found that symmetrizing the system of equations resulting from the Crank\u2013Nicolson discretization help us to use the standard pre-conditioner for the iterative matrix solver and reduces the number of iterations to get an accurate option values. In addition, the number of iterations that is required to solve the preconditioned system, resulting from the proposed iterative Crank\u2013Nicolson scheme, does not grow with the size of the system. Thus, we can effectively reduce the order of complexity in multidimensional option pricing. The numerical results are compared to the one with implicit Operator Splitting Method (OSM) to show the effectiveness.<\/jats:p>","DOI":"10.3390\/sym9010012","type":"journal-article","created":{"date-parts":[[2017,1,23]],"date-time":"2017-01-23T10:40:33Z","timestamp":1485168033000},"page":"12","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Iterative Speedup by Utilizing Symmetric Data in Pricing Options with Two Risky Assets"],"prefix":"10.3390","volume":"9","author":[{"given":"Dohyun","family":"Pak","sequence":"first","affiliation":[{"name":"Department of Mathematics &amp; Finance, Gachon University, 1342 Seongnamdaero, Sujeong-gu, Seongnam-si, Gyeonggi-do 13120, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Changkyu","family":"Han","sequence":"additional","affiliation":[{"name":"Department of Risk Management, Kiwoom Securities Co., Ltd., 18 Yeouinaru-ro 4(sa)-gil, Yeongdeungpo-gu, Seoul 07331, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Won-Tak","family":"Hong","sequence":"additional","affiliation":[{"name":"Department of Mathematics &amp; Finance, Gachon University, 1342 Seongnamdaero, Sujeong-gu, Seongnam-si, Gyeonggi-do 13120, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,1,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"771","DOI":"10.4236\/am.2011.26103","article-title":"The Operator Splitting Method for Black\u2013Scholes Equation","volume":"2","author":"Daoud","year":"2011","journal-title":"Appl. Math."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Duffy, D.J. (2006). Finite Difference Methods in Financial Engineering: A Partial Differential Approach, John Wiley and Sons.","DOI":"10.1002\/9781118673447"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1016\/j.cam.2013.01.008","article-title":"A comparison study of ADI and operator splitting methods on option pricing models","volume":"247","author":"Jeong","year":"2013","journal-title":"J. Comput. Appl. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/s40314-013-0104-9","article-title":"A numerical algorithm for multidimensional modeling of scattered data points","volume":"34","author":"Cavoretto","year":"2015","journal-title":"Comput. Appl. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/j.camwa.2015.11.007","article-title":"Radial basis function partition of unity methods for pricing vanilla basket options","volume":"71","author":"Shcherbakov","year":"2016","journal-title":"Comput. Math. Appl."},{"key":"ref_6","unstructured":"Oosterlee, C.W., Leentvaar, C.C., and Huang, X. (2005). Accurate American Option Pricing by Grid Stretching and High Order Finite Differences, Delft University of Technology. Technical Report."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"31","DOI":"10.21314\/JCF.2014.289","article-title":"TR-BDF2 for Stable American Option Pricing","volume":"17","year":"2014","journal-title":"J. Comput. Finance"},{"key":"ref_8","unstructured":"Hull, J.C. (2014). Options, Futures and Other Derivatives, Prentice Hall. [9 ed.]."},{"key":"ref_9","unstructured":"Wilmott, P. (2006). Paul Wilmott on Quantitative Finance, Wiley. [2 ed.]."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Zhu, Y.L., Wu, X., and Chern, I. (2004). Derivative Securities and Difference Methods, Springer.","DOI":"10.1007\/978-1-4757-3938-1"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Morton, K.W., and Mayers, D. (2005). Numerical Solution of Partial Differential Equations, Cambridge University Press. [2 ed.].","DOI":"10.1017\/CBO9780511812248"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Saad, Y. (2003). Iterative Methods for Sparse Linear Systems, SIAM. [2 ed.].","DOI":"10.1137\/1.9780898718003"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"856","DOI":"10.1137\/0907058","article-title":"GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems","volume":"7","author":"Saad","year":"1986","journal-title":"SIAM J. Sci. Stat. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1757","DOI":"10.1016\/S0898-1221(03)00154-8","article-title":"An efficient method for constructing an ILU preconditioner for solving large sparse nonsymmetric linear systems by the GMRES method","volume":"45","author":"Mittal","year":"2003","journal-title":"Comput. Math. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Trefethen, L.N., and Iii, D.B. (1997). Numerical Linear Algebra, SIAM.","DOI":"10.1137\/1.9780898719574"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Krekel, M., de Kock, J., Korn, R., and Man, T.K. (2004). An Analysis of Pricing Methods for Baskets Options, Wilmott.","DOI":"10.1002\/wilm.42820040318"},{"key":"ref_17","unstructured":"Haug, E.G. (2006). The Complete Guide to Option Pricing Formulas, McGraw-Hill Education."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"79","DOI":"10.21314\/JCF.2002.088","article-title":"Pricing Asian and Basket options via taylor expansion","volume":"5","author":"Ju","year":"2002","journal-title":"J. Comput. Financ."},{"key":"ref_19","unstructured":"Han, C. (2014). Finite Difference Method with GMRES Solver for the 2D Black\u2013Scholes Equation. [Master\u2019s Thesis, Korea University]."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/9\/1\/12\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:26:42Z","timestamp":1760207202000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/9\/1\/12"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1,21]]},"references-count":19,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,1]]}},"alternative-id":["sym9010012"],"URL":"https:\/\/doi.org\/10.3390\/sym9010012","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2017,1,21]]}}}