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
Exploring the integration of Information and Communication Technology for Education (ICTE) to improve learners’ listening skills in the teaching and learning of French in one secondary school in Lesotho
(2025) Rapuleng, Moeketsi; Makumane, M
Information and Communication Technology for Education (ICTE) seems to revolutionise the teaching and learning of foreign languages such as French in the current century by improving quality teaching and learning. The purpose of this study was to explore the integration of ICTE to improve learners’ listening skills in the teaching and learning of French as a Foreign Language (FFL) in one secondary school in Lesotho. The study was underpinned by two theories of technology: Technological Pedagogical Content Knowledge (TPACK) and Technology Acceptance Model (TAM). The study was guided by the qualitative research approach employing the critical paradigm. The research design was action research – under participatory action research – conducted in three phases: diagnostic phase, transformative phase and reflective phase. Two methods – focus group discussions and observations – were used to generate data. One school in Maseru, Lesotho that offered FFL was selected and 15 Grade 11 learners of FFL who were going to write DELF A1 (Diplôme d’Études en Langue Française) – translated as (diploma in studying French language) French examination – participated in the study. Data was analysed using thematic data analysis. The findings of this study revealed that consistent integration of ICTE in the FFL classroom improves learners’ listening skills. Also, teaching methods that incorporate ICTE improve learners’ language skills. The study recommends that the government of Lesotho, through the Ministry of Education and Training, should provide schools with language laboratories and other necessary infrastructure to support teaching and learning through ICTE. In addition, the Basic Education ICT Policy 2024 should be fully implemented and supported at school level with clear implementation strategies and a practical roadmap. The study concludes that the FFL secondary curriculum should be localised in order to address the societal and professional needs of learners.
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.