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Radial basis function methods for pricing multi-asset options
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.
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: urn:nbn:se:uu:diva-284306OAI: oai:DiVA.org:uu-284306DiVA: diva2:920152
Supervisors
Projects
eSSENCE
Available from: 2016-01-08 Created: 2016-04-16 Last updated: 2017-08-31Bibliographically approved
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, 185-200 p.Article 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, 2361-2379 p.Article 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: 2017-12-04Bibliographically 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, 1401-1423 p.Article 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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Shcherbakov, Victor

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