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
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
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.