### Mathematics Awareness Week 1996

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## Decisive Mathematics for Risk Management

### Decisive Mathematics for Risk Management

Thousands of times every day of national and international companies use mathematics to measure and control financial risks. And not just the typical school or college mathematics (although that is there as well), but advanced mathematics that has only been developed in recent years.

The benefits for a company of insuring against the risk of fire are well known. These same companies are now recognizing that they need to insure against other risks that are potentially even more damaging, and certainly more frequent.

Although these risks have always been there, it's only recently that they have reached such significant levels. Now mathematical techniques are available that allow treasurers and risk managers to measure them precisely and to make rational decisions to reduce and control them.

Consider the case of a company exporting computers to Germany. Suppose that in a year it will receive one million marks which it will need to convert immediately to U.S. dollars to pay for manufacturing expenses. This means, in effect, that the company will spend one million marks buying dollars at whatever the rate is in a year's time. If the dollar appreciates against the mark, the company will get fewer dollars. Conversely, if the dollar depreciates, the dollar profit will be greater.

In the past, profit margins were generally large enough to cover the possibility of adverse movements of exchange rates. Now, with national and international competition at record highs, rarely does a company have this luxury. Companies that don't cut their prices to the minimum find their sales dropping away.

A similar situation holds for mining companies. The viability of a copper mine, for example, depends not just on current copper prices, but also on what prices (and production costs) will be over the next years.

Interest rates are another example. Suppose you are in charge of funding the construction of a shopping complex. Every month for the next three years millions of dollars will need to be borrowed to complete the project. If this can be done for, say, eight percent interest then the center will be within its budget. If not, the whole project may be a financial failure.

The solution in all these cases is the careful and systematic use of options. An option gives the right to a specific financial transaction in the future without any obligation to carry it out.

Consider the exchange rate case and suppose that the current exchange rate is 0.68 dollars per mark. The exporter could purchase an option that gives the right to buy one million marks worth of dollars in a years time for the rate of, say, 0.70.

The exporter now knows that the company will never have to pay more then 0.70 marks per dollar -- if the exchange rate is less than this, the marks can be purchased on the open market, and if it is more, then the option is 'exercised'and the marks are purchased at the agreed strike price.

Similar scenarios are possible for mine and construction companies. Options can be bought that guarantee that their final costs will never be above a certain limit. Also, the option still leaves open the possibility of profiting from favorable movements in copper prices and interest rates.

It seems that options were first traded in the seventeenth century in Holland during a period of extreme speculation in the prices of tulip bulbs. In the U.S.A. options in agricultural commodities were available from the eighteenth century. But the option market was fragmented and irregular until 1973 when the Chicago Board Options Exchange began trading standardized option contracts on stocks.

In the same year two American mathematics professors by the names of Fischer Black and Myron Scholes published a paper that revolutionized option markets around the world.

Before their work it was thought option prices would depend on the opinions of the buyers and sellers as to whether prices would increase or decrease.

It came as a surprise to everyone when Black and Scholes proved mathematically that there was a rational price for options independent of any views of market direction. Further, if the option was traded at a price different from this, then a certain profit could be made.

There is now a bewildering array of different types of options available to treasurers and risk managers through exchanges and financial institutions: averaging, barrier, quanto, digital, and so on. Even though they all trace trace their origins back to the ideas of Black and Scholes, the theories and techniques that are now used go far beyond this pioneering work.

So today's decisions in risk management involving millions, sometimes billions of dollars, depend crucially on mathematics developed over the past few years. And, more importantly, on mathematicians continuing to develop powerful theories and techniques which can be implemented throughout the industry.

John Price
Department of Mathematics
Maharishi University of Management
Fairfield, IA 52557-1127
email: jprice@mum.edu