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Radial Basis Function generated Finite Difference Methods for Pricing of Financial Derivatives
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.ORCID iD: 0000-0003-3164-5242
2018 (English)Doctoral 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.

##### Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2018. , p. 63
##### Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1702
##### Keywords [en]
Radial basis function, Finite difference, Computational finance, Pricing of financial derivatives, Option pricing, Partial differential equation
##### National Category
Computational Mathematics
##### Research subject
Scientific Computing with specialization in Numerical Analysis
##### Identifiers
ISBN: 978-91-513-0403-8 (print)OAI: oai:DiVA.org:uu-357220DiVA, id: diva2:1238525
##### Public defence
2018-09-28, ITC 2446, Polacksbacken, Lägerhyddsvägen 2, Uppsala, 10:15 (English)
##### Supervisors
Available from: 2018-09-06 Created: 2018-08-14 Last updated: 2018-10-02
##### 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
##### 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
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
2019 (English)In: International Journal of Computer Mathematics, ISSN 0020-7160, E-ISSN 1029-0265, Vol. 96, no 10, p. 1910-1923Article in journal (Refereed) Published
##### Abstract [en]

In the recent project BENCHOP – the BENCHmarking project in Option Pricing we found that Stochastic and Local Volatility problems were particularly challenging. Here we continue the effort by introducing a set of benchmark problems for this type of problems. Eight different methods targeted for the Stochastic Differential Equation (SDE) formulation and the Partial Differential Equation (PDE) formulation of the problem, as well as Fourier methods making use of the characteristic function, were implemented to solve these problems. Comparisons are made with respect to time to reach a certain error level in the computed solution for the different methods. The implemented Fourier method was superior to all others for the two problems where it was implemented. Generally, methods targeting the PDE formulation of the problem outperformed the methods for the SDE formulation. Among the methods for the PDE formulation the ADI method stood out as the best performing one.

##### Keywords
Option pricing, stochastic and local volatility, numerical methods, benchmark problem, stochastic differential equation, partial differential equation, characteristic function
##### National Category
Computational Mathematics
##### Identifiers
urn:nbn:se:uu:diva-357218 (URN)10.1080/00207160.2018.1544368 (DOI)000475440700002 ()
eSSENCE
##### Funder
Swedish National Infrastructure for Computing (SNIC), SNIC 2017/1-236 Available from: 2018-11-07 Created: 2018-08-14 Last updated: 2019-08-22Bibliographically approved

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Cite
Citation style
• apa
• ieee
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