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Shcherbakov, Victor
Publications (10 of 12) Show all publications
von Sydow, L., Milovanović, S., Larsson, E., In't Hout, K., Wiktorsson, M., Oosterlee, C. W., . . . Waldén, J. (2019). BENCHOP–SLV: The BENCHmarking project in Option Pricing – Stochastic and local volatility problems. International Journal of Computer Mathematics, 96
Open this publication in new window or tab >>BENCHOP–SLV: The BENCHmarking project in Option Pricing – Stochastic and local volatility problems
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2019 (English)In: International Journal of Computer Mathematics, ISSN 0020-7160, E-ISSN 1029-0265, Vol. 96Article in journal (Refereed) Epub ahead of print
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-357218 (URN)10.1080/00207160.2018.1544368 (DOI)
Projects
eSSENCE
Available from: 2018-11-07 Created: 2018-08-14 Last updated: 2018-11-10Bibliographically approved
Cheng, G. & Shcherbakov, V. (2018). Anisotropic radial basis function methods for continental size ice sheet simulations. Journal of Computational Physics, 372, 161-177
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
Shcherbakov, V. (2018). Localised Radial Basis Function Methods for Partial Differential Equations. (Doctoral dissertation). Uppsala: Acta Universitatis Upsaliensis
Open this publication in new window or tab >>Localised Radial Basis Function Methods for Partial Differential Equations
2018 (English)Doctoral 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.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2018. p. 54
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1600
Keywords
Radial basis function, Partition of unity, Computational finance, Option pricing, Credit default swap, Glaciology, Fluid dynamics, Non-Newtonian flow, Anisotropic RBF
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-332715 (URN)978-91-513-0157-0 (ISBN)
Public defence
2018-01-19, ITC 2446, Polacksbacken, Lägerhyddsvägen 2, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2017-12-14 Created: 2017-11-21 Last updated: 2018-03-08
Larsson, E., Shcherbakov, V. & Heryudono, A. (2017). A least squares radial basis function partition of unity method for solving PDEs. SIAM Journal on Scientific Computing, 39, A2538-A2563
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
Ahlkrona, J. & Shcherbakov, V. (2017). A meshfree approach to non-Newtonian free surface ice flow: Application to the Haut Glacier d'Arolla. Journal of Computational Physics, 330, 633-649
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
Milovanović, S. & Shcherbakov, V. (2017). Pricing derivatives under multiple stochastic factors by localized radial basis function methods. Computing Research Repository (1711.09852)
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
Ahlkrona, J. & Shcherbakov, V. (2016). 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
2016 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2016-005
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-283437 (URN)
Projects
eSSENCE
Available from: 2016-04-19 Created: 2016-04-13 Last updated: 2016-05-16Bibliographically approved
Shcherbakov, V. (2016). Radial basis function methods for pricing multi-asset options. (Licentiate dissertation). Uppsala University
Open this publication in new window or tab >>Radial basis function methods for pricing multi-asset options
2016 (English)Licentiate 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.

Place, publisher, year, edition, pages
Uppsala University, 2016
Series
Information technology licentiate theses: Licentiate theses from the Department of Information Technology, ISSN 1404-5117 ; 2016-001
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-284306 (URN)
Supervisors
Projects
eSSENCE
Available from: 2016-01-08 Created: 2016-04-16 Last updated: 2017-08-31Bibliographically approved
Shcherbakov, V. & Larsson, E. (2016). Radial basis function partition of unity methods for pricing vanilla basket options. Computers and Mathematics with Applications, 71, 185-200
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
Shcherbakov, V. (2016). Radial basis function partition of unity operator splitting method for pricing multi-asset American options. BIT Numerical Mathematics, 56, 1401-1423
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
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