Robust time spectral methods for solving fractional differential equations in finance

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dc.contributor.advisor Mare, Eben
dc.contributor.coadvisor Pindza, Edson
dc.contributor.postgraduate Bambe Moutsinga, Claude Rodrigue
dc.date.accessioned 2021-02-26T12:06:56Z
dc.date.available 2021-02-26T12:06:56Z
dc.date.created 2021-05-05
dc.date.issued 2021
dc.description Thesis (PhD)--University of Pretoria, 2021. en_ZA
dc.description.abstract In this work, we construct numerical methods to solve a wide range of problems in finance. This includes the valuation under affine jump diffusion processes, chaotic and hyperchaotic systems, and pricing fractional cryptocurrency models. These problems are of extreme importance in the area of finance. With today’s rapid economic growth one has to get a reliable method to solve chaotic problems which are found in economic systems while allowing synchronization. Moreover, the internet of things is changing the appearance of money. In the last decade, a new form of financial assets known as cryptocurrencies or cryptoassets have emerged. These assets rely on a decentralized distributed ledger called the blockchain where transactions are settled in real time. Their transparency and simplicity have attracted the main stream economy players, i.e, banks, financial institutions and governments to name these only. Therefore it is very important to propose new mathematical models that help to understand their dynamics. In this thesis we propose a model based on fractional differential equations. Modeling these problems in most cases leads to solving systems of nonlinear ordinary or fractional differential equations. These equations are known for their stiffness, i.e., very sensitive to initial conditions generating chaos and of multiple fractional order. For these reason we design numerical methods involving Chebyshev polynomials. The work is done from the frequency space rather than the physical space as most spectral methods do. The method is tested for valuing assets under jump diffusion processes, chaotic and hyperchaotic finance systems, and also adapted for asset price valuation under fraction Cryptocurrency. In all cases the methods prove to be very accurate, reliable and practically easy for the financial manager. en_ZA
dc.description.availability Unrestricted en_ZA
dc.description.degree PhD en_ZA
dc.description.department Mathematics and Applied Mathematics en_ZA
dc.identifier.citation In this thesis entitled, Robust time spectral methods for solving fractional differential equations in finance, the promovendus studied numerical methods that accommodate a wide range of problems encountered in the area of finance. With the digitalization of finance, more complicated problems have emerged. This creates a need of fast and robust methods for solving differential equations. To this extend, he proposes spectral methods based on Chebyshev polynomials where operations are executed from the spectral space of coefficients rather than the physical space. A tremendous gain in computation is therefore achieved. He also coupled the method with a splitting technique to solve large time scale problems. The designed numerical methods are tested and proved to be reliable and fast convergent when compared to other numerical methods in the field. The methods will have its applications in pricing financial derivatives in the JSE. en_ZA
dc.identifier.other A2021 en_ZA
dc.identifier.uri http://hdl.handle.net/2263/78864
dc.publisher University of Pretoria
dc.rights © 2019 University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria.
dc.subject spectral methods en_ZA
dc.subject Chebyshev polynomials
dc.subject Fractional derivatives
dc.subject Chaotic finance systems
dc.subject UCTD
dc.title Robust time spectral methods for solving fractional differential equations in finance en_ZA
dc.type Thesis en_ZA


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