The Mathematics of Financial Modelingand Investment Management

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258 The Mathematics of Financial Modeling and Investment Management

yield very different results. He argued that the numerical solutions of

extremely sensitive differential equations such as those he was using pro-

duced diverging results due to rounding-off errors made by the computer

system. His discovery was published in a meteorological journal where it

remained unnoticed for many years.

Fractals

While in principle deterministic chaotic systems are unpredictable

because of their sensitivity to initial conditions, the statistics of their

behavior can be studied. Consider, for example, the chaos laws that

describe the evolution of weather: while the weather is basically unpre-

dictable over long periods of time, long-run simulations are used to pre-

dict the statistics of weather.

It was discovered that probability distributions originating from cha-

otic systems exhibit fat tails in the sense that very large, extreme events

have nonnegligible probabilities.^5 It was also discovered that chaotic sys-

tems exhibit complex unexpected behavior. The motion of chaotic sys-

tems is often associated with self-similarity and fractal shapes.

Fractals were introduced in the 1960s by Benoit Mandelbrot, a

mathematician working at the IBM research center in Yorktown Heights,

New York. Starting from the empirical observation that cotton price

time-series are similar at different time scales, Mandelbrot developed a

powerful theory of fractal geometrical objects. Fractals are geometrical

objects that are geometrically similar to part of themselves. Stock prices

exhibit this property insofar as price time-series look the same at differ-

ent time scales.

Chaotic systems are also sensitive to changes in their parameters. In

a chaotic system, only some regions of the parameter space exhibit cha-

otic behavior. The change in behavior is abrupt and, in general, it can-

not be predicted analytically. In addition, chaotic behavior appears in

systems that are apparently very simple.

While the intuition that chaotic systems might exist is not new, the

systematic exploration of chaotic systems started only in the 1970s. The

discovery of the existence of nonlinear chaotic systems marked a con-

ceptual crisis in the physical sciences: it challenges the very notion of the

applicability of mathematics to the description of reality. Chaos laws

are not testable on a large scale; their applicability cannot be predicted

(^5) See W. Brock, D. Hsieh, and B. LeBaron, Nonlinear Dynamics, Chaos, and Insta-
bility (Cambridge, MA: MIT Press, 1991) and D. Hsieh, “Chaos and Nonlinear Dy-
namics: Application to Financial Markets,” Journal of Finance 46 (1991), pp. 1839–
1877.