Mathematical Finance in South Africa
12David Taylor & Glenn Brickhill
Abstract: Mathematical Finance in South African universities has a surprisingly long history.
This paper will give a short overview of the history of the subject in South Africa. It will also
cover the development of the various degrees and courses offered around the country and
report on the substantial involvement of the finance industry in this development. Finally, we
will discuss which techniques are most apposite in South African financial markets and what
research, both academic and commercial, is being pursued.
What is Mathematical Finance?
It is tempting to say that Mathematical Finance should consist merely of a study of the Black-Scholes equation (and its variants) and, for the purposes of many programmes in “Financial Engineering”, this may well be true.
However, the study of derivative securities and their pricing models, which have been developed over the past 30 years, is now much more complicated than merely finding solutions of a second-order, linear partial differential equation with exotic boundary conditions. Any complete degree course on martingale or risk-neutral pricing methodology cannot ignore the contributions from stochastic calculus and analysis, numerical approximation theory and, most recently, optimal stopping theory, to the study. Consequently, the mathematical tools that are needed by a Quantitative Analyst in an investment bank have become wide-ranging and complicated. With the increasing sophistication of our understanding of market behaviour comes a concurrent increase in sophistication of our mathematical description of it. Mathematical Finance offers a classically applied mathematical opportunity to invent new mathematics to deal with new problems.
Mathematical Finance is not Actuarial Science. This is not to say that Actuarial Science does not contribute to the research and debate in Mathematical Finance. However, it is clear from our experience, and from analyzing many degree programmes outside of South Africa, that there is a clear academic distinction between the disciplines. Mathematical Finance is also not finance in the traditional sense. Although some of the models thdo draw on the mathematical foundations of 20 century finance, it is the insights of Merton and Black &
Scholes that drive and motivate Mathematical Finance. In particular the ideas of completeness of the market and derivative replication through portfolio design lead to the theory of no-arbitrage and ultimately to the Black-Scholes analysis. As a consequence, much of the advancement in the field has been driven by work done by pure and applied mathematicians. In this sense then, Mathematical Finance should be regarded as a truly modern branch of applied science.
Why is it taught?
The trivial answer to this question is that there is a demand in the job market for graduates of Mathematical Finance programmes. In reality the demand is driven by a number of factors which are interlinked and interdependent. Globalisation has led to country specific markets being incapable of insulating themselves from international effects. Although South Africa does retain some arcane version of exchange control, recent years have seen a gradual relaxation of these controls and a(n externally and internally driven) move towards a freely floating currency. Under circumstances of an open economy and sporadic but aggressive interest in
1 Programme in Advanced Mathematics of Finance, School of Computational & Applied Mathematics, University of the Witwatersrand-Johannesburg, South Africa (email@example.com) 2 Rand Merchant Bank, Johannesburg, South Africa (Glenn.Brickhill@rmb.co.za)
emerging markets as a source of investment, risk and return, it is imperative that the participants in these markets are as sophisticated in training as their international counterparts.
All financial markets seem to lead eventually to derivative markets. In most cases the initial impulse for a derivative market seems to be the need for an “alternative” form of insurance. In some instances, the depth of
reserves in financial markets makes them preferable to traditional insurance options. However, it is usually the presence of hedgers, speculators and arbitrageurs that enhances and deepens the derivative markets. In contrast to ordinary financial markets, the pricing of derivatives is invariably dependent on an arbitrage-type argument. It is clear, then, that money can be made or lost through ignorance and mis-pricing. As a consequence it becomes increasingly important that financial institutions in South Africa, and elsewhere in emerging economies, employ financial engineers with highly developed quantitative skills.
In the brief, thirty-odd years since the birth of derivative pricing theory, the number of derivative products, as well as their complexity, has increased dramatically. The size of the market, in both value and notional amounts, is staggering. (OTC market: Notional = US$220 trillion (thousand billion), Value = US$6 to 7 3trillion. Exchange Traded Options: Notional = US$312 trillion.) In many cases, the notional amounts far
exceed the value of the underlying market. Pricing and hedging the risk inherent in these sometimes highly non-linear financial products is rapidly becoming an extremely non-trivial mathematical exercise. Consequently, it will be those institutions that are best prepared for this that will prosper.
At what level should it be taught?
In almost all programmes outside of South Africa, it seems that the core of Mathematical Finance is taught at the post-graduate level. In the USA, Europe and Britain, the emphasis is generally on a professional Masters level degree (akin to an MBA) although this may sometimes stifle progression to the PhD, so the Masters degree is often offered through, or in conjunction with, the Mathematics department (or some version of it.)
South Africa has a peculiarity at university level in that the first-year students are in general a year younger than elsewhere in the world. This has resulted in a three-year undergraduate degree being followed by an Honours year before Masters or PhD studies are undertaken. In order to be compatible with the rest of the world, professional degrees are routinely four years in length. In South Africa there are a number of examples of the Mathematics of Finance being taught from first-year undergraduate level. At the University of Johannesburg (formerly Rand Afrikaans University) (http://www.uj.ac.za), the North-West University
(formerly Potchefstroom University) (http://www.puk.ac.za), the University of KwaZulu-Natal (formerly
University of Natal – Durban) and at Pretoria University (http://www.up.ac.za) these programmes are often
coupled with actuarial training. In many instances, though, the students have to make a decision about their future path before reaching the fourth-year of study.
At two universities, the study of Mathematics of Finance takes place at the post-graduate level only. These are the University of the Witwatersrand and the University of Cape Town. The reason for this is both historical and market related. At the University of the Witwatersrand the lecture courses that formed the core of Mathematical Finance were originally taught within the Honours degree of the School of Computational & Applied Mathematics as an “area of interest”. It subsequently became a separate, professional degree for
branding reasons. Discussion with the employers of graduates, at the time, suggested that an undergraduate curriculum was unnecessary and even unsuitable. At the University of Cape Town the degree is at the Masters level. This was partly motivated by a desire to avoid competition with the established Honours degree at the University of the Witwatersrand and partly because the creation of professional Masters degrees was in vogue at the time of its creation.
3 Source: BIS (Bank of International Settlements) Quarterly Review, March 2005.
Where should it be taught?
From an analysis of European and American programmes, it would appear that geography plays little role in determining the existence or success of Financial Engineering programmes. The reasons for this may be manifold but obviously lie partly in the size of the graduate student community in developed economies. The minimum requirement for entry into the quantitative financial community in London or New York is now a PhD. This is not the case in South Africa.
It is, however, no coincidence that the more successful Financial Mathematics programmes throughout the world are those at universities near their country’s financial centre(s). The success of Columbia and NYU in New York, Chicago University in Chicago, ETH in Zurich, and Imperial and Kings’ Colleges in London are
testament to this. The situation in South Africa is comparable. The degree programmes offered at the University of the Witwatersrand, Pretoria University, the University of Johannesburg and the University of Cape Town produce the most successful graduates. Johannesburg is the financial capital of South Africa and Cape Town is the centre of the pension and fund management industries. Stellenbosch University has an (undergraduate) Financial Mathematics programme offered through its Mathematics & Statistics Departments. Graduates of this programme are, in general, absorbed by the Cape Town financial community. North-West University’s Centre for Business Mathematics and Informatics benefits from being relatively close to
Johannesburg and from a demand for its graduates from one of South Africa’s largest retail and investment banks, ABSA (Amalgamated Banks of South Africa). It is clear, however, that there are two considerations which influence the success of a programme. One is access to the industry itself, the other, the resources that are available to the university. The programme at the University of KwaZulu-Natal suffers from lack both of these. Durban is relatively removed from the financial centres of South Africa and the graduates of their Financial Mathematics degree have to find employment either in Johannesburg or Cape Town. However, there is a large pool of talented students in KwaZulu-Natal. Geographically, it is vital that these students have access to some sort of Financial Engineering programme. It would be economically damaging and punitive to deny them this opportunity. The resources available to the universities are often augmented by the financial community itself. The Centre for Business Mathematics & Informatics at the North-West University and the Programme in Advanced Mathematics of Finance at the University of the Witwatersrand have both received support from the banking industry which runs into millions of South African rands. It is unsurprising then that these programmes are amongst the most successful.
South African Academic Programmes
The University of the Witwatersrand - Johannesburg has the oldest programme in Financial Mathematics
in South Africa. It owes this to the vision of Dr Dawie de Jongh, formerly of ABSA Investment Bank and now at the North-West University. Dr de Jongh approached the university in the late 1980’s with the suggestion
that they form an area of study in financial engineering. Fortunately he was in the position to assist with developing the curriculum and teaching the material. Dr de Jongh is certainly the “father” of Financial Mathematics in South Africa. The programme commenced in 1990 (which makes it one of the oldest in the world!) In 1997 the programme was registered with the Department of Education as a distinct Honours degree in Advanced Mathematics of Finance and the first intake of five students was in 1998. The increase in interest in the degree was exponential. There were 19 registrations in 1999 and 31 in 2000. Since then, the number of enrolling students in the full-time Honours degree has remained fairly uniform at about 25 (although the class sizes were smaller in 2003 and 2004 because of the effect of 9/11). The reasons for controlling the class size lie in the staffing resources available and the level of quality that the programme has tried to maintain. The number of applicants to the programme has continued to increase, however, and each available place in the Honours degree is now five times over-subscribed. The staff complement has increased with the development of the area, and the programme now employees four teaching staff and one administrative assistant. Once the
teaching programme at Honours level was established, the emphasis moved to a research Masters degree. Success in a research initiative is much more difficult to establish and maintain, but is probably more vital to the success and survival of a programme than the teaching component. The key factors here are funding and stability of supervising staff. The latter has been the major inhibiting issue for most programmes in South Africa. Staff and students of Mathematical Finance programmes are employment targets for the commercial industry. Coupled to this is an alarming level of emigration of skilled labour from post-apartheid South Africa. This is an issue that affects all emerging markets and seems to have no immediate solution. The programme
has been extensively funded by the banking community throughout its existence.
Year Number of Full-time Number of Part-time Number of Masters
Honours Graduates Honours Graduates Graduates
1998 2 3
1999 15 4
2000 27 4 1
2001 29 7 1
2002 29 3 2
2003 15 2 1
2004 15 4 1 42005 24 3
Table 1. Graduates of the Programme in Advanced Mathematics of Finance at the University of the
Witwatersrand – Johannesburg, 1998 – 2004.
The University of Cape Town created their MSc degree in Mathematics of Finance in 1999, and it was first offered in 2000. The curriculum was developed by Mr. Hardy Hulley, who was a PhD student in the Department of Mathematics & Applied Mathematics at the time. Mr. Hulley subsequently taught and researched at the University of the Witwatersrand. The MSc is a collaboration between the Departments of Mathematics & Applied Mathematics and Statistical Sciences, although it was initially situated in the Department of Mathematics & Applied Mathematics. The degree runs over two years. An intensive and extensive series of lectured courses in the first year is followed by a research report component in the second year. Many students elect to complete the mini-dissertation component while employed in the finance industry. Coursework and dissertation are equally weighted, and a pass on both components must be obtained for the degree to be awarded. The annual intake of students varies, ranging from 12 to 25, but the drop-out rate is high, and approximately 50% of students fail to complete the coursework component. Approximately 15 students have obtained the degree during the first 5 years of the programme’s existence. The programme struggles to
produce research because of the twin restrictions of limited resources and over-stretched staff.
The University of KwaZulu-Natal currently offers an Honours degree in Mathematics of Finance. This