Yearbooks

Programme: BScHons Mathematics of Finance

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Code Faculty Department
02240276 Faculty of Natural and Agricultural Sciences Department: Mathematics and Applied Mathematics
Credits Duration NQF level
Minimum duration of study: 1 year Total credits: 135 NQF level:  08

Admission requirements

  1. Relevant bachelor’s degree
  2. At least 60% for all mathematics and applied mathematics modules at final-year level
  3. A minimum of 60% for each of the following subjects/modules (or equivalent):
  • Real analysis at final-year level
  • Linear algebra at second-year level

Promotion to next study year

The progress of all honours candidates is monitored biannually by the postgraduate coordinator/head of department. A candidate’s study may be terminated if the progress is unsatisfactory or if the candidate is unable to finish his/her studies during the prescribed period.

Minimum credits: 135

Core credits:         120
Elective credits:      15

Additional information: 

  • WTW 732 and WTW 762 are presented as weekly lectures together with some extra block lectures.
  • Students must select either WTW 792 or WTW 795

Core modules

  • Module content:

    An introduction to the basic mathematical objects of linear functional analysis will be presented. These include metric spaces, Hilbert spaces and Banach spaces. Subspaces, linear operators and functionals will be discussed in detail. The fundamental theorems for normed spaces: The Hahn-Banach theorem, Banach-Steinhaus theorem, open mapping theorem and closed graph theorem. Hilbert space theory: Riesz' theorem, the basics of projections and orthonormal sets.

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  • Module content:

    Introduction to markets and instruments. Futures and options trading strategies, exotic options, arbitrage relationships, binomial option pricing method, mean variance hedging, volatility and the Greeks, volatility smiles, Black-Scholes PDE and solutions, derivative disasters.

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  • Module content:

    An analysis as well as an implementation (including computer programs) of methods are covered. Numerical linear algebra: Direct and iterative methods for linear systems and matrix eigenvalue problems: Iterative methods for nonlinear systems of equations. Finite difference method for partial differential equations: Linear elliptic, parabolic, hyperbolic and eigenvalue problems. Introduction to nonlinear problems. Numerical stability, error estimates and convergence are dealt with.

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  • Module content:

    Measure and integration theory: The Caratheodory extension procedure for measures defined on a ring, measurable functions, integration with respect to a measure on a σ-ring, in particular the Lebesgue integral, convergence theorems and Fubini's theorem.
    Probability theory: Measure theoretic modelling, random variables, expectation values and independence, the Borel-Cantelli lemmas, the law of large numbers. L¹-theory, L²-theory and the geometry of Hilbert space, Fourier series and the Fourier transform as an operator on L², applications of Fourier analysis to random walks, the central limit theorem.

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  • Module content:

    Exotic options, arbitrage relationships, Black-Scholes PDE and solutions, hedging and the Miller-Modigliani theory, static hedging, numerical methods, interest rate derivatives, BDT model, Vasicek and Hull-White models, complete markets, stochastic differential equations, equivalent Martingale measures.

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  • Module content:

    Mathematical modelling of Random walk. Conditional expectation and Martingales. Brownian motion and other Lévy processes. Stochastic integration. Ito's Lemma. Stochastic differential equations. Application to finance.

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  • Module content:

    Consult Department.

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  • Module content:

    Consult Department.

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Elective modules

  • Module content:

    Projection matrices and sums of squares of linear sets. Estimation and the Gauss-Markov theorem. Generalised t- and F- tests.

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  • Module content:

    The singular normal distribution. Distributions of quadratic forms. The general linear model. Multiple comparisons. Analysis of covariance. Generalised linear models. Analysis of categorical data.

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  • Module content:

    Matrix algebra. Some multivariate measures. Visualising multivariate data.  Multivariate distributions. Samples from multivariate normal populations. The Wishart distribution. Hotelling’s T ² statistic. Inferences about mean vectors.

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  • Module content:

    Discriminant analysis and classification. Principal component analysis. The biplot. Multidimensional scaling. Factor analysis. Probabilistic clustering.

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  • Module content:

    A selection of special topics will be presented that reflects the expertise of researchers in the Department. The presentation of a specific topic is contingent on student numbers. Consult the website of the Department of Mathematics and Applied Mathematics for more details.

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  • Module content:

    Classical optimisation:  Necessary and sufficient conditions for local minima.  Equality constraints and Lagrange multipliers.  Inequality constraints and the Kuhn-Tucker conditions.  Application of saddle point theorems to the solutions of the dual problem.  One-dimensional search techniques.  Gradient methods for unconstrained optimisation.  Quadratically terminating search algorithms.  The conjugate gradient method.  Fletcher-Reeves.  Second order variable metric methods:  DFP and BFCS.  Boundary following and penalty function methods for constrained problems.   Modern multiplier methods and sequential quadratic programming methods.  Practical design optimisation project.

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  • Module content:

    An analysis as well as an implementation (including computer programs) of methods is covered. Introduction to the theory of Sobolev spaces. Variational and weak formulation of elliptic, parabolic, hyperbolic and eigenvalue problems. Finite element approximation of problems in variational form, interpolation theory in Sobolev spaces, convergence and error estimates.

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  • Module content:

    This module aims at using advanced undergraduate mathematics and rigorously applying mathematical methods to concrete problems in various areas of natural science and engineering.
    The module will be taught by several lecturers from UP, industry and public sector. The content of the module may vary from year to year and is determined by relevant focus areas within the Department. The list of areas from which topics to be covered will be selected, includes: Systems of differential equations; dynamical systems; discrete structures; Fourier analysis; methods of optimisation; numerical methods; mathematical models in biology, finance, physics, etc.

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  • Module content:

    Field-theoretic and material models of mathematical physics. The Friedrichs-Sobolev spaces. Energy methods and Hilbert spaces, weak solutions – existence and uniqueness. Separation of variables, Laplace transform, eigenvalue problems and eigenfunction expansions. The regularity theorems for elliptic forms (without proofs) and their applications. Weak solutions for the heat/diffusion and related equations.

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The regulations and rules for the degrees published here are subject to change and may be amended after the publication of this information.

The General Academic Regulations (G Regulations) and General Student Rules apply to all faculties and registered students of the University, as well as all prospective students who have accepted an offer of a place at the University of Pretoria. On registering for a programme, the student bears the responsibility of ensuring that they familiarise themselves with the General Academic Regulations applicable to their registration, as well as the relevant faculty-specific and programme-specific regulations and information as stipulated in the relevant yearbook. Ignorance concerning these regulations will not be accepted as an excuse for any transgression, or basis for an exception to any of the aforementioned regulations.

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