Wild Markets: The FractalMultifractal View of Risk, Ruin, and Reward Benoit Mandelbrot Sterling Prof

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Wild Markets The Fractal/Multifractal View of
Risk, Ruin, and RewardBenoit Mandelbrot
Sterling Professor of Mathematical
SciencesYale University, New Haven CT 06520-8283
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Rough surface with an indication of scale
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The indication of scale was deliberately
misleading!
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Cauliflower as example of self-similar fractal
surface
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Human lung cast (E. Weibel) an example of
self-similar fractal
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A fractal mountain that never was (Richard F.
Voss)
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On Rocks. A stone, when it is examined will be
found a mountain in miniature. The fineness of
Natures work is so great, that, into a single
block, a foot or two in diameter, she can
compress as many changes of form and structure,
on a small scale, as she needs for her mountains
on a large one and, taking moss for forests, and
grains of crystal for crags the surface of a
stone, in by far the plurality of instances, is
more interesting than the surface of an ordinary
hill more fantastic in form and incomparably
richer in colour the last quality being most
noble in stones of good birth (that is to say,
fallen from the crystalline mountain ranges).
J. Ruskin, Modern Painters 5, Ch. 18 (1860)
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PROGRAM FOR A SCIENCE OF ROUGHNESS

• Identify occurrences of pure (fractal) roughness
• Identify or invent tools to handle pure roughness
• 3.1 Explain
• 3.2 Learn to avoid or minimize roughness
• 3.3 Learn to seek roughness

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Fractality is not a panacea Fractality is never
the last word Allows for a great variety of
behaviors Often seems easy in its early stages,
very misleadingly so. Fontenelle Know the
facts before you see the causes. True, this
method is slow but avoids the ridicule of
finding the cause of what is not. (1687)
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A recursive fractal and a painting by S. Dali
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One of the Hundred Views of Mount Fuji by K.
Hokusai
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BROWNIAN CLUSTER
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D4/3 BROWNIAN ISLAND
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D4/3 BROWNIAN CLUSTER/ISLAND
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Definition Brownian cluster the set of
values of a complex Brownian bridge (a planar
Brownian motion that returned to the origin)

• A proper graphic representation was given in The
  Fractal Geometry of Nature
• Triggered much activity in pure
  mathematics
• Illustrates a distinctive feature of fractal
  geometry
• Token of the value of the eyes recent return
  into pure mathematics

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Assertion The Hausdorff-Besicovich dimension of
the boundary of the Brownian cluster is 4/3

• 1982 A conjecture on p. 243 of F.G.N.
• Provoked a rich literature, a near-proof
  by B. Duplantier
• 2000 Proved - to great acclaim -
  by G. Lawler, O.
  Schramm, and W. Werner
• The problems cry out for less bulky proofs

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A Distinctive Feature of Fractal Geometry

• First stages are simple and famously easy
• A few steps are known to yield
  extremely complex gorgeous
  pictures
• A few steps also yield new conjectures
  that everyone can understand and no
  mathematician can
  provefor a while
• At the same time, fractals are
  simple, complex and open-ended
  A good reason for fractals becoming
  an almost standard topic
  in secondary education

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The eye returns to the hard sciences,helped by
computer graphics

• The 4/3 concerns the boundary of
  any planar Brownian motion
• The conjecture did not arise from pure thought
• Raw pictures seldom talk. For the Brownian they
  do not showcase but hide the boundary
• A refined cluster picture looks like an island
• Value of D confirmed by a numerical test

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First picture of a multifractal measure
Mandelbrot 1972.
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Mandelbrot Set A pretty picture and a
conjecture open after 25 years
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1980
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1980
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1980
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1979
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1979
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1979
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1979
1980
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The Variation of Financial Prices
Stack of price increments actual data mixed
with simulations Brownian, unifractal,
mesofractal, and multifractal
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Ten sigma events probability, according to the
Gaussian distribution, is a few
millionths of a millionth
of a millionth of a
millionth (Inverse of the Avogadro
number!) Absurd. The Gaussian is not a
norm. It grossly fails to fit reality.
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Least peaked bell Gaussian Most peaked bell
Cauchy In between bell Lévy stable distribution.
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Cartoons of Price Variation
Fractal model founded on scaling or
self-affinity, a principle of invariance under
reduction or dilation.

• Generator is symmetric,
• hence defined by its first break point
• Recursive roughening
• implemented by a cascade

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Cartoons Output Simple to Complex

• A cascades outcome
• is varied and variable
• is tunable from overly simple to overly complex

• Guarantee these cartoons hide no additive
• beyond shuffling

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Recursive fractal cartoon of Brownian motion
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Recursive fractal cartoon of Lévy stable motion
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Recursive fractal cartoon of fractional Brownian
motions
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Recursive cartoons of multifractal functions
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Eight samples from one multifractal population
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The plots coordinatesdefine the first break of
the cartoon generator
Cartoons Phase Diagram
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States of Randomness/Variability The Mild
State

• The common apparatus of probability/statistics
  law of large numbers, central limit theorem,
  asymptotically negligible addends and correlation
• Constitutes a mild or passive state of
  randomness/variability, patterned on the Brownian
• Implemented by the isolated Fickian point
• This state cannot create structure,
  only blurs existing structure
• Mild randomness was the first stage of
  indeterminism but does not exhaust it
  indeterminism extends beyond this first stage.

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States Of Randomness/Variability The Wild
State

• Non-Fickian cartoons exhibit
  long tails and/or long dependence
• As a result, the common apparatus does not apply
• The wild, active or creative randomness
  does not average out
• It actually mimics structure- or creates its
  appearance
• Concentration absent, mesofractal or
  multifractal
• Cartoons, models, and three-state representations

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Emperical Test of the Prices Multifractality
determination of f(a) as an envelope
determination of t(q)
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• The step from mild to wild variability,from the
  first to the secondstage of indeterminism,
  marks a sharp increase in complexity a
  frontier for science
• For the reductionist the chastening examples of
  turbulence and 1/f noises

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Roughness is a frontier that science long
ignored now it must be faced

• The rms measures of volatility (in finance,
  metallurgy, etc.) assume mild variability
• Surprising riches fractals everywhere!
• Legitimate concern too good to be true
• Resolution roughness must be faced
  it clearly
  contradicts mild variability
  wildly variable fractals often face it

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W. H. Freeman Co., 1982.
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Springer, 1997
Springer, 1999
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Springer, 2002
Springer, 2004
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Mathematical Association of America, 2002
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Basic Books, 2004
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