Browsing by Author "Jamal, Sameerah"

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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.
A symmetry perspective of third-order polynomial evolution equations
(University of the Witwatersrand, Johannesburg, 2024) Gwaxa, Bongumusa; Jamal, Sameerah
In this thesis, we analyse the full class of ten Fujimoto-Watanabe equations. In particular, these are highly nonlinear third-order and two fifth-order equations. With the aid of computer algebra software such as Mathematica, we calculate symmetries for these equations and we construct their commutator tables. The one dimensional system of optimal subalgebras is obtained via adjoint operators. Finally, we reduce these higher-order partial differential equations into ordinary differential equations, derive their solutions via a power series solution method and show how convergence may be tested. Lastly, we determine some conservation laws
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.
An analysis of the invariance and conservation laws of some classes of nonlinear wave equations
(2011-07-20) Jamal, Sameerah
We analyse nonlinear partial di erential equations arising from the modelling of wave phenomena. A large class of wave equations with dissipation and source terms are studied using a symmetry approach and the construction of conservation laws. Some previously unknown conservation laws and symmetries are obtained. We then proceed to use the multiplier (and homotopy) approach to construct conservation laws from which we obtain some surprisingly interesting higher-order variational symmetries. We also nd the corresponding conserved quantities for a large class of Gordon-type equations similar to those of the sine-Gordon equation and the relativistic Klein-Gordon equation. In particular, we direct our research and analysis towards a wave equation with non-constant coe cient terms, that is, coe cients dependent on time and space. Finally, we study a class of multi-dimensional wave equations.