Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management by Jean-Philippe Bouchaud

1

 Probability theory: basic notions
 All epistemological value of the theory of probability is based on this: that large scale
 random phenomena in their collective action create strict, non random regularity.
 (Gnedenko and Kolmogorov, Limit Distributions for Sums of Independent
 Random Variables.)
 1.1 Introduction
 Randomness stems from our incomplete knowledge of reality, from the lack of information
 which forbids a perfect prediction of the future. Randomness arises from complexity, from
 the fact that causes are diverse, that tiny perturbations may result in large effects. For over a
 century now, Science has abandoned Laplace’s deterministic vision, and has fully accepted
 the task of deciphering randomness and inventing adequate tools for its description. The
 surprise is that, after all, randomness has many facets and that there are many levels to
 uncertainty, but, above all, that a new form of predictability appears, which is no longer
 deterministic but statistical.
 Financial markets offer an ideal testing ground for these statistical ideas. The fact that
 a large number of participants, with divergent anticipations and conflicting interests, are
 simultaneously present in these markets, leads to unpredictable behaviour. Moreover, finan-
cial markets are (sometimes strongly) affected by external news–which are, both in date
 and in nature, to a large degree unexpected. The statistical approach consists in drawing
 from past observations some information on the frequency of possible price changes. If one
 then assumes that these frequencies reflect some intimate mechanism of the markets them-
selves, then one may hope that these frequencies will remain stable in the course of time.
 Forexample,themechanismunderlyingtherouletteorthegameofdiceisobviouslyalways
 the same, and one expects that the frequency of all possible outcomes will be invariant in
 time–although of course each individual outcome is random.
 This ‘bet’ that probabilities are stable (or better, stationary) is very reasonable in the
 case of roulette or dice;† it is nevertheless much less justified in the case of financial
 markets–despite the large number of participants which confer to the system a certain

 regularity, at least in the sense of Gnedenko and Kolmogorov. It is clear, for example, that
 financial markets do not behave now as they did 30 years ago: many factors contribute to
 the evolution of the way markets behave (development of derivative markets, world-wide
 and computer-aided trading, etc.). As will be mentioned below, ‘young’ markets (such as
 emergent countries markets) and more mature markets (exchange rate markets, interest rate
 markets, etc.) behave quite differently. The statistical approach to financial markets is based
 ontheideathatwhateverevolutiontakes place, this happens sufficiently slowly (on the scale
 of several years) so that the observation of the recent past is useful to describe a not too
 distant future. However, even this ‘weak stability’ hypothesis is sometimes badly in error,
 in particular in the case of a crisis, which marks a sudden change of market behaviour. The
 recent example of some Asian currencies indexed to the dollar (such as the Korean won or
 the Thai baht) is interesting, since the observation of past fluctuations is clearly of no help
 to predict the amplitude of the sudden turmoil of 1997, see Figure 1.1.

Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management by Jean-Philippe Bouchaud