Robust time spectral methods for solving fractional differential equations in finance

Description

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. / Thesis (PhD)--University of Pretoria, 2021. / Mathematics and Applied Mathematics / PhD / Unrestricted

Links & Downloads

1. http://hdl.handle.net/2263/78864
2. 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.
3. A2021

Tags

spectral methods

Chebyshev polynomials

Fractional derivatives

Chaotic finance systems

UCTD

Additional Fields

Identifer	oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:up/oai:repository.up.ac.za:2263/78864
Date	January 2021
Creators	Bambe Moutsinga, Claude Rodrigue
Contributors	Mare, Eben, [e-mail], Pindza, Edson
Publisher	University of Pretoria
Source Sets	South African National ETD Portal
Detected Language	English
Type	Thesis
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.