NULIR
The National University of Lesotho Institutional Repository (NULIR) is a digital archive that collects, preserves, and provides open access to the scholarly and intellectual output of the University.
Recent Submissions
Analytical solutions to the arithmetic asian options pricing model using lie symmetry methods
(National University of Lesotho, 2025-07) Kubeletsane, Monts’uoe Edward; Nchejane, Ngaka John
This dissertation explores the use of Lie symmetry methods to find analytical solutions for arithmetic Asian options; path-dependent financial derivatives widely used for risk management in commodity markets. The pricing problem is formulated as a partial differential equation (PDE) involving the asset price, time, and the running average of the asset price. Lie symmetry analysis is applied to the PDE to compute its infinitesimal generators, determine an optimal system of one-dimensional sub-algebras, and perform symmetry reductions. Each reduced PDE obtained through this process also admits further symmetries, allowing for successive reductions and the construction of exact invariant solutions. Techniques such as Riccati reductions and the Frobenius method are employed to solve the resulting ordinary differential equations (ODEs). The study further examines the influence of key financial parameters (volatility, interest rate, and time to maturity) on the structure and behavior of the pricing solutions. The findings contribute to the theoretical understanding of Asian option pricing and provide analytical benchmarks for validating numerical approaches.
Symmetry analysis of the parabolic system for european option pricing with liquidity shocks
(National University of Lesotho, 2025) Moreboli, Bonang; Nchejane, Ngaka John; Poka, Wetsi David
We address the knowledge gap between financial theory and practice by developing a framework to analyse European option pricing under liquidity shocks, which reduce market completeness and challenge traditional models like Black–Scholes. To tackle this, we undertake a Lie symmetry analysis of a coupled system designed to model European options subject to liquidity shocks. The system consists of a degenerate parabolic equation and another is a first-order nonlinear ordinary differential equation (ODE) in time with no spatial derivatives. Our analysis has uncovered the parabolic system’s Lie symmetry group and infinitesimal generators. We have then proceeded to investigate the system dynamics by constructing commutative tables that illustrate the relationships between the vector fields under study and the one-dimensional optimal system of symmetry subalgebras associated with the initial equation. Employing similarity reductions, we have applied the Lie symmetry methodology to decompose the system into several nonlinear ordinary differential equations (ODEs) corresponding
to each symmetry subalgebra. Finally, we present obtained invariant solutions through simulations as 2D and 3D graphs.
Solving the barrier options model with linear time-dependent volatility numerically
(National University of Lesotho, 2025-07) Phate, Seeiso; Nchejane, Ngaka John
The main purp ose of this work is to approximate the solution of the barrier option pricing model whose evolution is described in terms of partial differential equation called Black-Scholes model. In this model, we consider volatility as a linear function of time. This is done primarily using a numerical approximation technique by the name of finite difference method. We consider Crank Nicolson scheme and forward difference scheme in time derivative to discretize the model and represent it as a tridiagonal matrix. Furthermore, we analyse the stability of the discretized model using Von Neumann stability analysis. We finally find the numerical solution to find key insights and the implications.
Solving the barrier options model with linear time-dependent volatility by symmetry analysis
(National University of Lesotho, 2025-07) Kometsi, Relebohile Veronecca; Nchejane, Ngaka John
This project aims to solve the barrier options model with linear time-dependent volatility, described as a Partial Differential Equation (PDE) using symmetry analysis. The only PDE to be considered will be the Black-Scholes model. Firstly, we use Lie symmetry to find symmetries of our model from our determining equations. Once the symmetries are found, they will b e picked at random to form characteristic equations needed to find invariant solutions. Lastly, the solutions will b e interpreted and graphed.
Integrating arts into curriculum to enhance entrepreneurship skills: A case of three Botha-Bothe Schools
(National University of Lesotho, 2025-10) Makhoabenyane, Palesa; Motebang, B. M.
The introduction of the integrated curriculum in Lesotho led to a shift from individual subjects to a grouping of subjects under learning areas. Amongst the five learning areas in the new integrated curriculum is the Arts and Entrepreneurship learning area, which comprises content from subjects such as Business Education, ICT, Home Economics, Art, Crafts, Technical subjects, Music, Drama and Theatre. The Arts and Entrepreneurship learning area aims at developing creativity and entrepreneurial talents and abilities to curb societal problems such as unemployment. This research therefore studied the integration of arts into the curriculum and its significance in enhancing entrepreneurship skills. This investigation used a qualitative approach employing a case study research design which enabled an in-depth analysis of the phenomenon of arts in enhancing entrepreneurship skills. The participants in this research were Business Education teachers currently teaching Arts and Entrepreneurship in three Botha Bothe secondary schools. Two curriculum policy documents were analysed. Also, students’ artefacts were analysed as a form of triangulation. The findings in this study revealed that theintegration of arts into curriculum through its activities develops skills such as creativity, collaboration, problem-solving, and critical-thinking. Literature classifies the aforementioned skills as entrepreneurship skills, which implies that integrating arts into curriculum instills entrepreneurship skills. Therefore, it is crucial for arts to be integrated into curriculum as it enhances entrepreneurship skills.