A Study of Financial Models and their Symmetry Driven Analytical Solutions - Maphanga, Rivoningo (University of the Witwatersrand, Johannesburg, 2024)
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.
A thesis submitted in requirement for the degree Doctor of Philosophy, to the Faculty of Science, School of Mathematics, University of the Witwatersrand, Johannesburg, 2024.
Keywords: PDEs; Lie Symmetries; Option Pricing Models; Heat Equation; Boundary Conditions; Convolution; UCTD; SDG-9: Industry, innovation and infrastructure; SDG-4: Quality education