Mathematics conquers the stock exchanges

Banks use the latest methods of financial mathematics in order to reduce the risks of trading transactions. Insurance companies can also benefit and save millions of marks.

Lecture by Professor Walter Schachermayer on 25 August 1998

The financial wizards juggling with currencies, stocks and shares can make huge profits, but their speculations can also involve considerable risks. However, these can be minimised by means of modern mathematics. Sophisticated modern financial products can be used by business companies to guard themselves against fluctuations in rates of exchange or stock values. Insurers can also use these so-called derivatives to make it easier and cheaper to arrange reinsurance.

"No bank which is active in the growing derivatives markets can afford to neglect the new mathematical methods if they hope to remain competitive", says Walter Schachermayer, Professor of Applied Mathematics and Statistics at the University of Vienna. "In recent years there has been a boom on financial mathematics. Graduates in this field have no difficulty in finding interesting jobs with banks and insurance companies." Academic laurels have also been distributed. For instance, the Nobel Prize for Economics last year went to Myron. S. Scholes and Robert C. Merton, who together with their late colleague Fischer Black (who passed away in 1995) developed the mathematical foundations of the new financial techniques. Prof. Schachermayer himself received the highest scientific award in Austria, the Wittgenstein Prize, valued at 15 million Austrian schillings (or approximately DM 2.1 million). The internationally renowned expert in financial mathematics will be coming to Berlin for the International Congress of Mathematicians. On Tuesday, (25 August 1998, 7.30 p.m., Berliner Urania, Kleiststraße 13 - 14, Berlin-Schöneberg) he will be giving a talk on "The role of mathematics in financial markets", which is intended for the general public.

The principle of derivatives can be illustrated by means of a simple example. A company from Germany exports kitchen equipment to the United States, and has calculated its prices on the basis of the current exchange rate. If the dollar falls relative to the German mark, it will suffer from a drop in its profits on the deal. Therefore the company wishes to insure itself against exchange rate fluctuations. For a fee, the company's bank will agree to exchange dollars in six months time at today's exchange rate. If the dollar has remained stable or risen in value relative to the German mark then the option is not needed, since the company can get the same or better profit anyway. The bank is also satisfied - it gets to keep the fee it charged as its profit.

"The ideal situation would now be to bring together two people with opposing risks," explains Schachermayer. "For example, an American exporter might have the same worries about the exchange rate as the German entrepreneur. They could both arrange to cover their risk on the exchange." It is becoming increasingly common to obtain cover for risks on the financial markets. Options, futures and swaps, and other new financial products are not only used for currencies, but also for stocks and shares, etc.

Even natural catastrophes such as storm and flood damage are traded on the markets. "Catastrophe bonds" offer an excellent opportunity for speculation. Someone wants cover for possible damage, and someone else may be willing to take on part of the risk in the hope that there will not be a catastrophe. A Swiss insurer, for example, issues "Hail Bonds" which only pay a dividend if no hail falls in a defined area of Switzerland over a certain period. Similarly, in California it is possible to speculate on rain. Here derivatives only pay if there is between 43 and 69 centimetres of rainfall from Autumn to Spring at the airport.

"The fundamental idea of every insurance is to spread risk over a larger number of people. For large damage payments an insurance company establishes various layers of reinsurance, in order to have some back-up," says Schachermayer. "But that is relatively expensive and cumbersome compared with the financial markets." If it is possible to transfer reinsurance to the financial markets then millions can be saved in administrative costs.

The role of mathematics is to calculate the appropriate price for the derivatives. It is necessary to have information about expected changes in exchange rates or share values, or how often catastrophes are expected. Of course, mathematicians can no more see into the future than anyone else, but they can use the available data to produce probability distributions. It is then possible by means of complicated formulas to establish a fair price for the derivatives. In the Seventies, Black, Scholes and Merton made fundamental contributions, but there is still a need for research. "The mathematical models we are currently using could be improved considerably", says Walter Schachermayer, who has established his international reputation by providing mathematical proof of central principles in the world of finance. Numerous private and university research groups from all over the world are researching into financial mathematics. In Berlin these include the group around Prof. Hans Föllmer at the Humboldt University and Prof. Martin Schweizer from the Technical University Berlin.

"The fascinating thing about this branch of mathematics is that it has very direct practical implications," explains Schachermayer. "In physics, no matter what model you use to describe the motions of the planets they still keep on going around just the same. But when our models are put into practice they immediately change the financial world." Theory and application are in constant exchange. And Schachermayer, who left the academic world for two years to work for an insurance company, maintains contacts to mathematicians working for banks and insurance companies in Vienna, Frankfurt and London.

Vasco Alexander Schmidt

Walter Schachermayer (born 24 July 1950) studied economics and mathematics in Vienna. In 1980 he obtained his habilitation at Linz University, but then worked for an insurance company for two years. Since 1993 he has been Professor of Applied Mathematics and Statistics at the University of Vienna. In September he will move to the Technical University of Vienna, taking over the Chair for Financial and Insurance Mathematics.