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122 Risk management technology in financial services
rules presumed to underlie and reflect a given process. This could serve as a model
if the clauses are of a kind that permits to predict further occurrences from those
already experienced – albeit at a confidence level which may not be perfect.
Within the domain defined by the preceding paragraphs, the goal of this chapter
is to present to the reader a comprehensive approach to modelling, with particular
emphasis on models for stress testing (which characterizes also Chapter 11). This
population of models is developed for the long leg of the risk distribution–anew
culture in modelling, and most particularly in financial analysis and experimentation.
Because nearly all current models focus on the high frequency area which is the
bubble part of a normal distribution, the contribution to stress testing of classical
type models is limited. As sections 7.6 and 7.7 will bring to the reader’s attention,
the usual way to stress test is to assume 5, 10, 15 standard deviations from the risk
distribution’s mean. This permits to continue using rich statistical tables developed
for normal distributions, while examining extreme events.
The downside of the solution outlined in the preceding paragraph is that it does
not account for spikes. Yet, spikes which show up at the long leg of a risk distri-
bution are important and costly events. Therefore, we are interested in their origins,
nature, frequency and underlying reasons – which, quite often, tend to change over
time as market conditions evolve. In terms of their analysis, knowledge engineering
(Chapter 9) and genetic algorithms (Chapter 10) might be of help.
7.2 The development of mathematical science
To appreciate the way of mathematical thought accumulated over the ages, we should
turn to the ninth book of Euclid’s Elements. There we find a sequence of propositions
(precisely, 21 to 32) which have no connection whatever with what precedes or
follows them. Such propositions constitute a piece of Pythagorean mathematics, which
predated by centuries Euclid’s work, but still influences present-day science.
Pythagoras, the great philosopher and mathematician of ancient Greece, trans-
formed the science of mathematics, essentially geometry and analytics brought to
Greece from Egypt by Thales, one of the seven sages of antiquity. By all evidence,
the ancient Greeks had learned from the Egyptians the rules for determination of
areas and volumes – just like the Egyptians learned arithmetic from the Babylonians,
who knew how to solve systems of linear and quadratic equations with two or more
In terms of science and mathematics, the Babylonians, too, had learned a great
deal from other civilizations which preceded their own – particularly from Sumer
and India. This is not surprising because progress in science and technology is based
not only on the research spirit and innovative ideas (see Chapter 1) but also, if not
primarily, on the analytical thinking accumulating during the ages which permits
subsequent generations to leap forward.
An example of knowledge coming from ancient India and benefiting subsequent
generations till the present day is the invention of zero. By all accounts, this has
been the greatest contribution ever to mathematics. It took a real genius to identify
‘nothing’, call it ‘something’ and integrate it into a system of signs and rules.