Browsing by Author "Maphanga, Rivoningo"

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    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.
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    A Technique to Solve a Parabolic Equation by Point Symmetries that Incorporate Initial Data
    (Springer, 2025-03) Jamal, Sameerah; Maphanga, Rivoningo
    In this paper, we show how transformation techniques coupled with a convolution integral can be used to solve a generalised option-pricing model, including the Black–Scholes model. Such equations are parabolic and the special convolutions are extremely involved as they arise from an initial value problem. New symmetries are derived to obtain solutions through an application of the invariant surface condition. The main outcome is that the point symmetries are effective in producing exact solutions that satisfy a given initial condition, such as those represented by a call-option.
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    The analysis of PDEs arising from the Korteweg-de Vries hierarchies
    (2021) Maphanga, Rivoningo
    In this dissertation, we study the hierarchy commonly defined as an infinite sequence of partial differential equations which begins with the Korteweg-deVries equation and its modified version. We look at how Lie point symmetries and conservation laws of each of these hierarchies aid the solving of higher order partial differential equations through associations and transformations. An important feature of these hierarchies is their highly nonlinear property. In this regard, obtaining solutions for the members of these hierarchies poses a great problem, where in the past, it was impossible to calculate solutions. In this study, we establish a method to allow for the construction of new solutions to the full hierarchy. We begin by determining point symmetries and conserved vectors of each hierarchy, then proceed to testing association between the obtained symmetries and conserved vectors, and finally using transformations to construct solutions