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  • 1.
    Ahlkrona, Josefin
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Shcherbakov, Victor
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    A meshfree approach to non-Newtonian free surface ice flow: Application to the Haut Glacier d'Arolla2017In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 330, p. 633-649Article in journal (Refereed)
  • 2.
    Ahlkrona, Josefin
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Shcherbakov, Victor
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    A meshfree approach to non-Newtonian free surface ice flow: Application to the Haut Glacier d'Arolla2016Report (Other academic)
  • 3.
    Cheng, Gong
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Shcherbakov, Victor
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Anisotropic radial basis function methods for continental size ice sheet simulations2018In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 372, p. 161-177Article in journal (Refereed)
  • 4.
    Larsson, Elisabeth
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Shcherbakov, Victor
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Heryudono, Alfa
    A least squares radial basis function partition of unity method for solving PDEs2017In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 39, p. A2538-A2563Article in journal (Refereed)
  • 5.
    Milovanović, Slobodan
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Shcherbakov, Victor
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Pricing derivatives under multiple stochastic factors by localized radial basis function methods2017In: Computing Research Repository, no 1711.09852Article in journal (Other academic)
  • 6.
    Shcherbakov, Victor
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Localised Radial Basis Function Methods for Partial Differential Equations2018Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    Radial basis function methods exhibit several very attractive properties such as a high order convergence of the approximated solution and flexibility to the domain geometry. However the method in its classical formulation becomes impractical for problems with relatively large numbers of degrees of freedom due to the ill-conditioning and dense structure of coefficient matrix. To overcome the latter issue we employ a localisation technique, namely a partition of unity method, while the former issue was previously addressed by several authors and was of less concern in this thesis.

    In this thesis we develop radial basis function partition of unity methods for partial differential equations arising in financial mathematics and glaciology. In the applications of financial mathematics we focus on pricing multi-asset equity and credit derivatives whose models involve several stochastic factors. We demonstrate that localised radial basis function methods are very effective and well-suited for financial applications thanks to the high order approximation properties that allow for the reduction of storage and computational requirements, which is crucial in multi-dimensional problems to cope with the curse of dimensionality. In the glaciology application we in the first place make use of the meshfree nature of the methods and their flexibility with respect to the irregular geometries of ice sheets and glaciers. Also, we exploit the fact that radial basis function methods are stated in strong form, which is advantageous for approximating velocity fields of non-Newtonian viscous liquids such as ice, since it allows to avoid a full coefficient matrix reassembly within the nonlinear iteration.

    In addition to the applied problems we develop a least squares radial basis function partition of unity method that is robust with respect to the node layout. The method allows for scaling to problem sizes of a few hundred thousand nodes without encountering the issue of large condition numbers of the coefficient matrix. This property is enabled by the possibility to control the coefficient matrix condition number by the rate of oversampling and the mode of refinement.

    List of papers
    1. Radial basis function partition of unity methods for pricing vanilla basket options
    Open this publication in new window or tab >>Radial basis function partition of unity methods for pricing vanilla basket options
    2016 (English)In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 71, p. 185-200Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-272085 (URN)10.1016/j.camwa.2015.11.007 (DOI)000369455000012 ()
    Projects
    eSSENCE
    Available from: 2015-12-03 Created: 2016-01-11 Last updated: 2017-11-30Bibliographically approved
    2. BENCHOP—The BENCHmarking project in Option Pricing
    Open this publication in new window or tab >>BENCHOP—The BENCHmarking project in Option Pricing
    Show others...
    2015 (English)In: International Journal of Computer Mathematics, ISSN 0020-7160, E-ISSN 1029-0265, Vol. 92, p. 2361-2379Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-260897 (URN)10.1080/00207160.2015.1072172 (DOI)000363753800003 ()
    Projects
    eSSENCE
    Available from: 2015-09-21 Created: 2015-08-25 Last updated: 2018-08-21Bibliographically approved
    3. Radial basis function partition of unity operator splitting method for pricing multi-asset American options
    Open this publication in new window or tab >>Radial basis function partition of unity operator splitting method for pricing multi-asset American options
    2016 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 56, p. 1401-1423Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-284299 (URN)10.1007/s10543-016-0616-y (DOI)000388968500012 ()
    Projects
    eSSENCE
    Available from: 2016-04-08 Created: 2016-04-16 Last updated: 2017-11-30Bibliographically approved
    4. Pricing derivatives under multiple stochastic factors by localized radial basis function methods
    Open this publication in new window or tab >>Pricing derivatives under multiple stochastic factors by localized radial basis function methods
    2017 (English)In: Computing Research Repository, no 1711.09852Article in journal (Other academic) Submitted
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-333468 (URN)
    Projects
    eSSENCE
    Available from: 2017-11-27 Created: 2017-11-14 Last updated: 2018-08-22Bibliographically approved
    5. Inuence of jump-at-default in IR and FX on Quanto CDS prices
    Open this publication in new window or tab >>Inuence of jump-at-default in IR and FX on Quanto CDS prices
    (English)Manuscript (preprint) (Other academic)
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-333467 (URN)
    Available from: 2017-11-13 Created: 2017-11-13 Last updated: 2017-11-21
    6. A meshfree approach to non-Newtonian free surface ice flow: Application to the Haut Glacier d'Arolla
    Open this publication in new window or tab >>A meshfree approach to non-Newtonian free surface ice flow: Application to the Haut Glacier d'Arolla
    2017 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 330, p. 633-649Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-310706 (URN)10.1016/j.jcp.2016.10.045 (DOI)000394408900034 ()
    Projects
    eSSENCE
    Available from: 2016-10-24 Created: 2016-12-19 Last updated: 2017-11-21Bibliographically approved
    7. Anisotropic radial basis function methods for continental size ice sheet simulations
    Open this publication in new window or tab >>Anisotropic radial basis function methods for continental size ice sheet simulations
    2018 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 372, p. 161-177Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-333469 (URN)10.1016/j.jcp.2018.06.020 (DOI)000443284400008 ()
    Projects
    eSSENCE
    Available from: 2018-06-15 Created: 2017-11-14 Last updated: 2018-11-10Bibliographically approved
    8. A least squares radial basis function partition of unity method for solving PDEs
    Open this publication in new window or tab >>A least squares radial basis function partition of unity method for solving PDEs
    2017 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 39, p. A2538-A2563Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-316488 (URN)10.1137/17M1118087 (DOI)000418659900017 ()
    Projects
    eSSENCE
    Available from: 2017-11-09 Created: 2017-03-01 Last updated: 2018-06-16Bibliographically approved
  • 7.
    Shcherbakov, Victor
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Radial basis function methods for pricing multi-asset options2016Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    The price of an option can under some assumptions be determined by the solution of the Black–Scholes partial differential equation. Often options are issued on more than one asset. In this case it turns out that the option price is governed by the multi-dimensional version of the Black–Scholes equation. Options issued on a large number of underlying assets, such as index options, are of particular interest, but pricing such options is a challenge due to the "curse of dimensionality". The multi-dimensional PDE turn out to be computationally expensive to solve accurately even in quite a low number of dimensions.

    In this thesis we develop a radial basis function partition of unity method for pricing multi-asset options up to moderately high dimensions. Our approach requires the use of a lower number of node points per dimension than other standard PDE methods, such as finite differences or finite elements, thanks to a high order convergence rate. Our method shows good results for both European style options and American style options, which allow early exercise. For the options which do not allow early exercise, the method exhibits an exponential convergence rate under node refinement. For options that allow early exercise the option pricing problem becomes a free boundary problem. We incorporate two different approaches for handling the free boundary into the radial basis function partition of unity method: a penalty method, which leads to a nonlinear problem, and an operator splitting method, which leads to a splitting scheme. We show that both methods allow for locally high algebraic convergence rates, but it turns out that the operator splitting method is computationally more efficient than the penalty method. The main reason is that there is no need to solve a nonlinear problem, which is the case in the penalty formulation.

    List of papers
    1. Radial basis function partition of unity methods for pricing vanilla basket options
    Open this publication in new window or tab >>Radial basis function partition of unity methods for pricing vanilla basket options
    2016 (English)In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 71, p. 185-200Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-272085 (URN)10.1016/j.camwa.2015.11.007 (DOI)000369455000012 ()
    Projects
    eSSENCE
    Available from: 2015-12-03 Created: 2016-01-11 Last updated: 2017-11-30Bibliographically approved
    2. BENCHOP—The BENCHmarking project in Option Pricing
    Open this publication in new window or tab >>BENCHOP—The BENCHmarking project in Option Pricing
    Show others...
    2015 (English)In: International Journal of Computer Mathematics, ISSN 0020-7160, E-ISSN 1029-0265, Vol. 92, p. 2361-2379Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-260897 (URN)10.1080/00207160.2015.1072172 (DOI)000363753800003 ()
    Projects
    eSSENCE
    Available from: 2015-09-21 Created: 2015-08-25 Last updated: 2018-08-21Bibliographically approved
    3. Radial basis function partition of unity operator splitting method for pricing multi-asset American options
    Open this publication in new window or tab >>Radial basis function partition of unity operator splitting method for pricing multi-asset American options
    2016 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 56, p. 1401-1423Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-284299 (URN)10.1007/s10543-016-0616-y (DOI)000388968500012 ()
    Projects
    eSSENCE
    Available from: 2016-04-08 Created: 2016-04-16 Last updated: 2017-11-30Bibliographically approved
  • 8.
    Shcherbakov, Victor
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Radial basis function partition of unity operator splitting method for pricing multi-asset American options2016In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 56, p. 1401-1423Article in journal (Refereed)
  • 9.
    Shcherbakov, Victor
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Larsson, Elisabeth
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Radial basis function partition of unity methods for pricing vanilla basket options2015Report (Other academic)
  • 10.
    Shcherbakov, Victor
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Larsson, Elisabeth
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Radial basis function partition of unity methods for pricing vanilla basket options2016In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 71, p. 185-200Article in journal (Refereed)
  • 11.
    von Sydow, Lina
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Höök, Lars Josef
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Larsson, Elisabeth
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Lindström, Erik
    Milovanović, Slobodan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Persson, Jonas
    Shcherbakov, Victor
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Shpolyanskiy, Yuri
    Sirén, Samuel
    Toivanen, Jari
    Waldén, Johan
    Wiktorsson, Magnus
    Levesley, Jeremy
    Li, Juxi
    Oosterlee, Cornelis W.
    Ruijter, Maria J.
    Toropov, Alexander
    Zhao, Yangzhang
    BENCHOP—The BENCHmarking project in Option Pricing2015In: International Journal of Computer Mathematics, ISSN 0020-7160, E-ISSN 1029-0265, Vol. 92, p. 2361-2379Article in journal (Refereed)
  • 12.
    von Sydow, Lina
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Milovanović, Slobodan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Larsson, Elisabeth
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
    In't Hout, Karel
    Wiktorsson, Magnus
    Oosterlee, Cornelis W.
    Shcherbakov, Victor
    Wyns, Maarten
    Leitao, Alvaro
    Jain, Shashi
    Haentjens, Tinne
    Waldén, Johan
    BENCHOP–SLV: The BENCHmarking project in Option Pricing – Stochastic and local volatility problems2019In: International Journal of Computer Mathematics, ISSN 0020-7160, E-ISSN 1029-0265, Vol. 96Article in journal (Refereed)
1 - 12 of 12
CiteExportLink to result list
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Cite
Citation style
  • apa
  • ieee
  • modern-language-association
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf