School of Mathematics (ETDs)

Permanent URI for this communityhttps://hdl.handle.net/10539/38013

Browse

Filters

2023 - 20242

Settings

Has files: YesSubject: search.filters.subject.SDG-9: Industry, innovation and infrastructure

Search Results

Now showing 1 - 2 of 2
  • No Thumbnail Available
    Item
    A Study of Financial Models and their Symmetry Driven Analytical Solutions
    (University of the Witwatersrand, Johannesburg, 2024-07) Maphanga, Rivoningo; Jamal, Sameerah
    The theory of financial models play a crucial role in understanding and predicting the behaviour of various financial instruments. In this thesis, we explore the application of Lie symmetries and boundary conditions in four prominent financial models: the Black-Scholes, a generalized bond-pricing, a CEV type, and an option-pricing model. These models revolutionized the field of mathematical finance by introducing a framework for valuing options or bonds. We investigate the Lie symmetries underlying these equations and explore their implications in financial mathematics. By employing Lie symmetries, we are able to identify invariant solutions, leading to a deeper understanding of the dynamics and behaviour of the equations. Furthermore, the thesis delves into the role of boundary conditions in financial models. Boundary conditions play a vital role in defining the behaviour of financial instruments, and their accurate specification is essential for obtaining meaningful results. We analyze the impact of different boundary or terminal conditions on option and bond pricing models. By examining the effects of boundary conditions, we enhance our understanding of the limitations and nuances of these models in different financial scenarios. Bond pricing models are vital in the valuation and risk management of fixed-income securities and their investigation provides insights into the behaviour of bond prices and yields. By uncovering the underlying symmetries and understanding the implications of boundary conditions, we aim to enhance the accuracy and predictive power of bond and option pricing models.
  • Thumbnail Image
    Item
    Distance measures, independence number and chromatic number
    (University of the Witwatersrand, Johannesburg, 2023-03) Moholane, Letlhogonolo; Jonck, Betsie; Mukwembi, Simon
    There are numerous parameters in graph theory. In this dissertation, we pay a special attention to average distance, independence number, average eccentricity, order and the chromatic number of a graph. In 1975, Doyle and Graver proved an upper bound on the average distance with respect to the order of the graph. This gave rise to studies that focus on upper and lower bounds on average distance in terms of other graph parameters. Approximately, three decades after Doyle and Graver proved their result, Dankelmann, Goddard, and Swart in 2004 produced a study that gave an upper bound on average eccentricity in terms of minimum degree and order of the graph, initiating studies that focus on giving bounds on average eccentricity with respect to other known graph parameters. In this dissertation, we investigate bounds on average eccentricity and on average distance. We give upper bounds on average eccentricity in terms of independence number of the graph and order of the graph. Then, we present bounds on average eccentricity when order and chromatic number of the graph are prescribed. The second part of the dissertation is dedicated to presenting upper bounds on average distance with respect to independence number and order of the graph, and again, in terms of chromatic number and order of the graph.