Posthumous tributes to Benoit Mandelbrot (1924-2010) have highlighted his remarkable influence on the natural sciences, from geometry to meteorology, to theories with non-Euclidean spaces and geospatial models approach. Mandelbrot culminates a series of major thinkers going back to classical Greece: a Greek among RomansMandelbrot, Fractals; Finance, Financial Markets, Epistemology, Thomas Kuhn

Estrada, Fernando

English

Research Papers in Economics

«History of Economic Ideas», xix/2011/1
BENOIT MANDELBROT (1924-1910):
A GREEK AMONG ROMANS
Fernando Estrada*
Universidad Externado de Colombia
Centro de Investigaciones y Proyectos Especiales cipe
osthumous tributes to Benoit Mandelbrot (1924-2010) have high-
lightedhis remarkable influence on the natural sciences, from geom-
etry to meteorology, to theories with non-Euclidean spaces and geospa-
tial models approach. Mandelbrot culminates a series of major thinkers
going back to classical Greece: a Greek among Romans (Estrada 2010a).
The trajectory of his considerable influence is also related to theoreti-
cal and experimental economics, specifically the theory of financial
markets and the psychology of economic behavior, the theory of frac-
tals: powerful enough explanations to understand a day of panic in the
international stock exchanges.
Indeed, besides is famous work: The fractal geometry of nature (1982),
his contributions to economic theory are widely recognized (Mandel-
brot 1997, 2004, 2005). And his extraordinary work: The (mis) Behavior of
Markets: A Fractal View of Risk, Ruin and Reward (2004). Mandelbrot de-
veloped for financial theory an analogy of the physical world in the
three states of matter: solid, liquid and gas. Three stages of random-
ness: ‘docile’, ‘quiet’, ‘wild’. Three times in the markets that can be con-
trasted with the mathematics of fractal geometry.
Conventional economic theory assumes that price variation can be
modeled by random processes that are consistent with the pattern  more
simple, like every rise and fall determined by the toss of a coin. Howev-
er, according to Mandelbrot, the fractal shown to us is that the actual
price behavior deviates substantially from this standard. A more accu-
rate model, the variation multifractal ‘wild’ price, provides the founda-
tion for a new economic theory of greater scope and more reliability.
Mandelbrot offers an analytical framework for understanding the
fractal randomness of runaway rough forms in different manifestations:
from the turbulence caused by a hurricane, electrical noise, or the path
of the stock quotes. This is not a theory for getting rich, but its fractal
theory helps to see the emergence, development and evolution of dis-
* Address for correspondence: persuacion@gmail.com.
P
HEI 1 2011_Impaginato  28/03/11  07.26  Pagina 9asters. Mandelbrot knocked prevailing myths in the modern economy
that lead to underestimate the real risk in financial markets.
Taleb yields the best posthumous tribute to Benoit Mandelbrot to call
him: “A Greek among Romans”. In the second half of the twentieth
century few mathematicians achieved universal influence as remark-
able as Mandelbrot’s. The international scientific community will not
cease to mourn the death of one of its finest sons.
A short commentary describes the trajectory of the fractal mod-
els/multifractal F/M by Benoit Mandelbrot (Estrada 2010b). The geo-
metrical discoveries of Mandelbrot cover a broad area of research fields,
from meteorology and mathematical physics to individual and collec-
tive behavior in society, besides his contributions to the analysis of the
financial crisis in his wonderful essay on The (mis) Behavior of Markets. A
fractal view of Risk, Ruin and Reward (2004). Mandelbrot’s arguments
have revealed significant anomalies in the prevailing paradigms. Is this
a new paradigm in Kuhn’s sense as stated by the same Mandelbrot?
Mandelbrot’s hypothesis has been extended creatively to the field of
financial modeling, the mathematical approach to the markets since the
fractals geometric structures of various levels of size, each of which re-
peats a small scale the overall structure. The hypothesis suggests that
Mandelbrot fractal rough measures reflect the nature and allow a non-
Euclidean approach: geometric structures such as clouds, coasts, wind
gusts and conferences in the bags. To Mandelbrot the main feature in a
market price, as in various phenomena of culture and nature, is fractal
geometry.
Summarizing his argument is as follows:
The fractal geometry of roughness measured intrinsically. Thus marking the begin-
ning of a specific quantity theory: the roughness in all its manifestations
(Mand05)
Roughness of his behavior ubiquitous in nature and in culture (in-
cluding financial markets). Fractality is scattered everywhere as fractal
geometry objects are multiform. An overview of fractals and multi-
fractal is described by Mandelbrot in a long chapter (Mand02). Despite
its unquestionable antiquity, the study of roughness has been quite be-
hind when compared with older concepts of physics such as slope (of
a road or trend), weight, tone, warmth, color and similar categories.
The problem, as emphasized by Mandelbrot is that the density of
roughness was not measured. Work in this direction had to wait until
the postulates of this thinker.
A brief explanation of the author can see the contrast between the
implications of fractal geometry in two seemingly divergent worlds:
the physical and financial. The inclination of a slope is defined by the
10 Fernando Estrada
HEI 1 2011_Impaginato  28/03/11  07.26  Pagina 10derivative of the height h (x) along the slope. In theory, this definition
implies that increasing the radius dh / dx tends to the limit dx ¬ 0. Cus-
tom has made this explanation to be ‘normal’. In practice, just dh/dx
is almost constant. However, by definition, rugged profiles and sur-
faces are such that dh/dx vary in an unlimited extension. By contrast,
a basic factor in many models of price variation – in Bachelier model
and the derivative models fractal/multifractal – is not a suitable in-
strument. However, there is an ‘anomalous (d/x) height (d/h) radius
lenght. Its existence is a manifestation of ‘escalation’. According to the
model of Bachelier this radio has a limit · = ½.The same in all in-
stances for all financial data as a fundamental property, but an advan-
tage and a disadvantage because a maximum of the ½ is not a valid
parameter for the data set. On the contrary, fractal models  /  multi-
fractal allow
·_ = ½.
First, the language of nature does not always correspond with the state-
ment of theoretical schemes. Normal science is established on ideal
conditions. But the opposite is true that both physical and cultural
forms of reality are often a result of increasing complexity. At a dis-
tance, without limit given values need to be constant. The case applies
for extension inertia in the financial sector with the Bachelier model.
Secondly, the measurement criterion is inefficient, according to Man-
delbrot. The derivatives are not suitable when you have an irregular be-
havior of prices. The explanation in this case demands a Popperian
measure. The M/F contribute to measures that may be out of a domi-
nant factor for the standard explanation: the prediction.
To explain problems of forecasting, Mandelbrot estimated that val-
ues can be constant but different from ½. In general, decreasing situa-
tions occur in both nature and culture, the author refers to similar cas-
es (Mand02). The value of · may vary in a specific way in a matter of
moments, that characterizes the multifractal. The value of ·may be the
same at all points in time but different from ½. This characterizes the
fractal or hhm model (Hurst-Hölder-Mandelbrot). There is also an im-
portant intermediate event called “mesofractalidad” or plm model
(Pareto-Lévy-Mandelbrot).
In the case of judges Mandelbrot derived the mathematical intuitive
concept predates the slope. That is, a quantitative measure of the intu-
itive notion of ‘slope’ would come before his mathematical concept, the
concept of ·takes the opposite direction. Mandelbrot is originally who
modified a concept introduced in 1870 by Holderlin, a purely mathe-
matical concept totally separated from intuition. However, Mandel-
brot’s work is identified with the Hölder exponent, a central aspect of
Benoit Mandelbrot (1924-2010): a Greek among Romans 11
HEI 1 2011_Impaginato  28/03/11  07.26  Pagina 11the roughness. This proximity can be seen in the work of Mandelbrot
fractal geometry applied to the cartoons (Mand02). Both Mandelbrot
and Hölder point out how intuition perceived roughness levels differ.
In summary, fractal geometry begins by measuring the roughness by
means of · and / or concepts related fractals. The value and/or distri-
bution of ·is directly observable. This is not an elusive concept that has
to be discovered indirectly from many observations. The predominant
role that has this exponent is a Mandelbrot aspect related to the princi-
ple of parsimony.
For reasons of space numerous details related to financial theory
(Mand97) cannot be explained. Mandelbrot developed the arguments
against the Brownian promotion in the following manner: first, the
value · = ½ roughness characterized a form of ‘mild’. Second, the
roughness found in financial data takes an expression ‘wild’ that ex-
cludes · = ½.
While substitutes for Brownian theses were rejected, Mandelbrot’s
objections were heard and many alternative models reacted to save the
apparent ‘anomalies’ discovered. The pioneering observations of the
discontinuity of the dominant paradigm were made by the our author.
In light of the fractal approach / multifractal (F/M) reviews had some-
thing in common: local roughness reestablished the ½.
In the field of financial theory, an often required adjustment is vari-
able – stochastic – volatility. The postulate states that short records fol-
low a Brownian motion, but the variance of the movement changes in-
termittently. Mandelbrot’s critique is that the model boundaries are not
only not eliminated but that pushes to the rules vary the volatility
process. If these variations are fast, the Brownian effect is dissolved,
“and in the homeopathic medicine” (Mandelbrot).
If the variance changes slowly, a second criticism activated by Man-
delbrot is that only if the volatility measures the level of Brownian mo-
tion this aspect changes the variance variable in the model. Whether to
proceed with the generally accepted premise that · = 1/2. Mandelbrot
argues that this conclusion has not been verified. The evidence suggests
the opposite (Mand05).
Mandelbrot points out that in any case their work responds to a dif-
ferent epistemological framework. Their models are definitely not of
volatility variables.
In the short term, according to Mandelbrot the patches can be de-
fended as an immediate response to urgent needs. However, the expla-
nation in science can not do likewise. When the number of arrange-
ments in a formula exceeds its limits, it falls under its own weight and
must begin again. The affirmation of scoliosis in (Mand05) is an impor-
tant statement for those working in the philosophy of science:
12 Fernando Estrada
HEI 1 2011_Impaginato  28/03/11  07.26  Pagina 12Thomas Kuhn brilliantly described this process as a “paradigm shift”.
What I introduced myself as a good alternative paradigm fractal/mul-
tifractal (F/M).
The Scholium Mandelbrot (Mand05) helps to highlight a phenome-
non until now invisible to a majority of financial experts, namely that
the explanatory models in analyzing investments, making portfolios,
option pricing or risk management does not contribute to solving fun-
damental problems, because the models themselves are part of the
problem. Financial theory continues to work within traditional models
of classical physics. Anomalies within the explanatory schemes are due
to transfer of linear responses to problems whose budgets are not linear.
The evolution and dynamics of the concept of paradigm in Kuhn sug-
gest that we can tell with relative caution when theoretical functions are
extended within a particular field of scientific knowledge (Estrada
2010c), when the application set is irrelevant for the precariousness of
their paradigmatic exemplars. Fits within a paradigm may not remain
forever, for there are conditions that require fundamental changes. Clas-
sical Newtonian Particle Mechanics (mcp) were displaced in some areas
of theoretical physics as their explanations were limited and presented
disadvantages (Moul02). Key components of the concept of paradigm
in Kuhn are discussed in greater scope in the reconstruction by
Stegmüller (Steg83). Structuralism version allows us to extend the scope
of the paradigm in several aspects: the role of theoretical concepts, the
paradigmatic exemplars, methodological values and rules within the sci-
entific community. The hypothesis with which Stegmüller operates re-
processing can be linked to the need to explore the paradigm proposed
by Mandelbrot, from which there are, hitherto, unpublished aspects:
what does it finance the core K of a theory?, What paradigmatic exem-
plars offer a feature extensive description in the same jobs   Mandelbrot?
(revisar gramaticalmente estas dos preguntas: no sé que quieren decir)
Financial theory pragmatism is necessary to estimate the nature of
Kuhnian paradigms, conventional financial models get caught in the
standard descriptions of scientific knowledge. They become danger-
ously wrong in conditions that can affect people’s basic income. An ob-
servation of the relationship between the scientific community and the
financial discipline paradigm suggests a separate section of epistemol-
ogy of science.
References
Kuhn T. S. 1996, The structure of scientific revolutions, 3rd edn., Chicago, University of
Chicago Press.
— 2000, The road since structure: Philosophical essays, 1970–1993, with an autobiographical
interview, ed. by J. Conant and J. Haugeland, Chicago, University of Chicago Press.
Benoit Mandelbrot (1924-2010): a Greek among Romans 13
HEI 1 2011_Impaginato  28/03/11  07.26  Pagina 13Mandelbrot B. B. 1982, The fractal geometry of nature, NewYork, Freeman.
— 2005, «Parallel cartoons of fractal models of finance», Annals of Finance, 1, 179-192.
Mandelbrot B. B. 2002, Gaussian self-affinity and fractals, Berlin-Heidelberg-New
York, Springer.
— 1997, Fractals and scaling in finance, Berlin-Heidelberg-NewYork, Springer.
Estrada F. 2010a, «Theory of Argumentation in Financial Markets», available at:
http://ideas.repec.org/a/srs/jasf12/2v1y2010i1p18-22.html
— 2010b, «Brief Commentary on Benoit Mandelbrot Models», available at: http://
papers.ssrn.com/sol3/papers.cfm?abstract_id=1558156
— 2010c, «Financial Theory has a Paradigm a la Kuhn? A Preliminary Exploration of
Financial Epistemology», available at: http://papers.ssrn.com/sol3/papers.cfm?
abstract_id=1556530
Popper K. 1959, The Logic of Scientific Discovery, London, Hutchinson.
Stegmüller W .  1983,  Estructura y Dinámica de Teorias, transl. by U. Moulines,
Barcelona, Ariel.
14 Fernando Estrada
HEI 1 2011_Impaginato  28/03/11  07.26  Pagina 14

(1924-2010): a Greek among

Estructura y Dinámica de Teorias, transl. by U. Moulines,

Fractals and scaling in finance,

Gaussian self-affinity and fractals,

Parallel cartoons of fractal models of finance»,

The road since structure: Philosophical essays, 1970–1993, with an autobiographical interview,

The structure of scientific revolutions, 3rd edn.,

Benoit Mandelbrot (1924 - 2011 ) : A Greek among Romans

Benoit Mandelbrot (1924 - 2011 ) : A Greek among Romans

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