uu.seUppsala University Publications

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Radial Basis Function generated Finite Difference Methods for Pricing of Financial DerivativesPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2018 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### 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

URN: urn:nbn:se:uu:diva-357220ISBN: 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)
##### Opponent

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##### Supervisors

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#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt553",{id:"formSmash:j_idt553",widgetVar:"widget_formSmash_j_idt553",multiple:true}); Available from: 2018-09-06 Created: 2018-08-14 Last updated: 2018-10-02
##### List of papers

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.

1. Radial basis function generated finite differences for option pricing problems$(function(){PrimeFaces.cw("OverlayPanel","overlay1167155",{id:"formSmash:j_idt603:0:j_idt607",widgetVar:"overlay1167155",target:"formSmash:j_idt603:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Pricing financial derivatives using radial basis function generated finite differences with polyharmonic splines on smoothly varying node layouts$(function(){PrimeFaces.cw("OverlayPanel","overlay1238521",{id:"formSmash:j_idt603:1:j_idt607",widgetVar:"overlay1238521",target:"formSmash:j_idt603:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. A high order method for pricing of financial derivatives using radial basis function generated finite differences$(function(){PrimeFaces.cw("OverlayPanel","overlay1238522",{id:"formSmash:j_idt603:2:j_idt607",widgetVar:"overlay1238522",target:"formSmash:j_idt603:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Pricing derivatives under multiple stochastic factors by localized radial basis function methods$(function(){PrimeFaces.cw("OverlayPanel","overlay1156673",{id:"formSmash:j_idt603:3:j_idt607",widgetVar:"overlay1156673",target:"formSmash:j_idt603:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. BENCHOP—The BENCHmarking project in Option Pricing$(function(){PrimeFaces.cw("OverlayPanel","overlay848689",{id:"formSmash:j_idt603:4:j_idt607",widgetVar:"overlay848689",target:"formSmash:j_idt603:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

6. BENCHOP–SLV: The BENCHmarking project in Option Pricing – Stochastic and local volatility problems$(function(){PrimeFaces.cw("OverlayPanel","overlay1238523",{id:"formSmash:j_idt603:5:j_idt607",widgetVar:"overlay1238523",target:"formSmash:j_idt603:5:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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