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  • 1.
    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.
    Pricing financial derivatives using radial basis function generated finite differences with polyharmonic splines on smoothly varying node layouts2018In: Computing Research Repository, no 1808.02365Article in journal (Other academic)
  • 2.
    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.
    Radial Basis Function generated Finite Difference Methods for Pricing of Financial Derivatives2018Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    The purpose of this thesis is to present state of the art in radial basis function generated finite difference (RBF-FD) methods for pricing of financial derivatives. This work provides a detailed overview of RBF-FD properties and challenges that arise when the RBF-FD methods are used in financial applications.

    Across the financial markets of the world, financial derivatives such as futures, options, and others, are traded in substantial volumes. Knowing the prices of those financial instruments at any given time is of utmost importance. Many of the theoretical pricing models for financial derivatives can be represented using multidimensional PDEs, which are in most cases analytically unsolvable.

    We present RBF-FD as a recent numerical method with the potential to efficiently approximate solutions of PDEs in finance. As its name suggests, the RBF-FD method is of a finite difference (FD) type, from the radial basis function (RBF) group of methods. When used to approximate differential operators, the method is featured with a sparse differentiation matrix, and it is relatively simple to implement — like the standard FD methods. Moreover, the method is mesh-free, meaning that it does not require a structured discretization of the computational domain, and it is of a customizable order of accuracy — which are the features it inherits from the global RBF approximations.

    The results in this thesis demonstrate how to successfully apply RBF-FD to different pricing problems by studying the effects of RBF shape parameters for Gaussian RBF-FD approximations, improving the approximation of differential operators in multiple dimensions by using polyharmonic splines augmented with polynomials, constructing suitable node layouts, and smoothing of the initial data to enable high order convergence of the method. Finally, we compare RBF-FD with other available methods on a plethora of pricing problems to form an objective image of the method’s performance.

    Future development of RBF-FD is expected to result in a solid mesh-free high order method for multi-dimensional PDEs, that can be used together with dimension reduction techniques to efficiently solve problems of high dimensionality that we often encounter in finance.

    List of papers
    1. Radial basis function generated finite differences for option pricing problems
    Open this publication in new window or tab >>Radial basis function generated finite differences for option pricing problems
    2018 (English)In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 75, p. 1462-1481Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-336813 (URN)10.1016/j.camwa.2017.11.015 (DOI)000428100600024 ()
    Projects
    eSSENCE
    Available from: 2017-12-01 Created: 2017-12-18 Last updated: 2018-08-21Bibliographically approved
    2. Pricing financial derivatives using radial basis function generated finite differences with polyharmonic splines on smoothly varying node layouts
    Open this publication in new window or tab >>Pricing financial derivatives using radial basis function generated finite differences with polyharmonic splines on smoothly varying node layouts
    2018 (English)In: Computing Research Repository, no 1808.02365Article in journal (Other academic) Submitted
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-356834 (URN)
    Projects
    eSSENCE
    Available from: 2018-08-07 Created: 2018-08-14 Last updated: 2018-08-21Bibliographically approved
    3. A high order method for pricing of financial derivatives using radial basis function generated finite differences
    Open this publication in new window or tab >>A high order method for pricing of financial derivatives using radial basis function generated finite differences
    (English)Manuscript (preprint) (Other academic)
    Abstract [en]

    In this paper, we consider the numerical pricing of financial derivatives using Radial Basis Function generated Finite Differences in space. Such discretization methods have the advantage of not requiring Cartesian grids. Instead, the nodes can be placed with higher density in areas where there is a need for higher accuracy. Still, the discretization matrix is fairly sparse. As a model problem, we consider the pricing of European options in 2D. Since such options have a discontinuity in the first derivative of the payoff function which prohibits high order convergence, we smooth this function using an established technique for Cartesian grids. Numerical experiments show that we acquire a fourth order scheme in space, both for the uniform and the nonuniform node layouts that we use. The high order method with the nonuniform node layout achieves very high accuracy with relatively few nodes. This renders the potential for solving pricing problems in higher spatial dimensions since the computational memory and time demand become much smaller with this method compared to standard techniques.

    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-357217 (URN)
    Projects
    eSSENCE
    Available from: 2018-08-14 Created: 2018-08-14 Last updated: 2018-08-21
    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. 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
    6. BENCHOP–SLV: The BENCHmarking project in Option Pricing – Stochastic and local volatility problems
    Open this publication in new window or tab >>BENCHOP–SLV: The BENCHmarking project in Option Pricing – Stochastic and local volatility problems
    Show others...
    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
  • 3.
    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)
  • 4.
    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.
    von Sydow, Lina
    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 high order method for pricing of financial derivatives using radial basis function generated finite differencesManuscript (preprint) (Other academic)
    Abstract [en]

    In this paper, we consider the numerical pricing of financial derivatives using Radial Basis Function generated Finite Differences in space. Such discretization methods have the advantage of not requiring Cartesian grids. Instead, the nodes can be placed with higher density in areas where there is a need for higher accuracy. Still, the discretization matrix is fairly sparse. As a model problem, we consider the pricing of European options in 2D. Since such options have a discontinuity in the first derivative of the payoff function which prohibits high order convergence, we smooth this function using an established technique for Cartesian grids. Numerical experiments show that we acquire a fourth order scheme in space, both for the uniform and the nonuniform node layouts that we use. The high order method with the nonuniform node layout achieves very high accuracy with relatively few nodes. This renders the potential for solving pricing problems in higher spatial dimensions since the computational memory and time demand become much smaller with this method compared to standard techniques.

  • 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.
    von Sydow, Lina
    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 generated finite differences for option pricing problems2018In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 75, p. 1462-1481Article in journal (Refereed)
  • 6.
    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)
  • 7.
    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 - 7 of 7
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  • en-GB
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  • fi-FI
  • nn-NO
  • nn-NB
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