but still not take us there rapidly. See Bouchaud and Potters (2003).
Lognormal: There is an intermediate variety that is called the lognormal, emphasized by
one Gibrat (see Sutton [1997]) early in the twentieth century as an attempt to explain
the distribution of wealth. In this framework, it is not quite that the wealthy get
wealthier, in a pure preferential attachment situation, but that if your wealth is at 100
you will vary by 1, but when your wealth is at 1,000, you will vary by 10. The relative
changes in your wealth are Gaussian. So the lognormal superficially resembles
the fractal, in the sense that it may tolerate some large deviations, but it is dangerous
because these rapidly taper off at the end. The introduction of the lognormal was a
very bad compromise, but a way to conceal the flaws of the Gaussian.
Extinctions: Sterelny (2001). For extinctions from abrupt fractures, see Courtillot (1995)
and Courtillot and Gaudemer (1996). Jumps: Eldredge and Gould.
FRACTALS, POWER LAWS, and SCALE-FREE DISTRIBUTIONS
Definition: Technically, P>x= K x~a where a is supposed to be the power-law exponent.
It is said to be scale free, in the sense that it does not have a characteristic scale: relative
deviation of does not depend on x, but on n—for x "large enough." Now,
in the other class of distribution, the one that I can intuitively describe as nonscalable,
with the typical shape p(x) = Exp [-a x], the scale will be a.
Problem of "how large": Now the problem that is usually misunderstood. This scalability
might stop somewhere, but I do not know where, so I might consider it infinite.
The statements very large and I don't know how large and infinitely large are epistemologically
substitutable. There might be a point at which the distributions flip. This
will show once we look at them more graphically.
Log P>x = -a Log X +C1- for a scalable. When we do a log-log plot (i.e., plot P>x
and x on a logarithmic scale), as in Figures 15 and 16, we should see a straight line.
Fractals and power laws: Mandelbrot (1975,1982). Schroeder (1991) is imperative. John
Chipman's unpublished manuscript The Paretian Heritage (Chipman [2006]) is the
best review piece I've seen. See also Mitzenmacher (2003).
"To come very near true theory and to grasp its precise application are two very
different things as the history of science teaches us. Everything of importance has
been said before by somebody who did not discover it." Whitehead (1925).
Fractals in poetry: For the quote on Dickinson, see Fulton (1998).
Lacunarity: Brockman (2005). In the arts, Mandelbrot (1982).
Fractals in medicine: "New Tool to Diagnose and Treat Breast Cancer," Newswise,
July 18, 2006.
General reference books in statistical physics: The most complete (in relation to fat tails)
is Sornette (2004). See also Voit (2001) or the far deeper Bouchaud and Potters
(2002) for financial prices and econophysics. For "complexity" theory, technical
books: Bocarra (2004), Strogatz (1994), the popular Ruelle (1991), and also Prigogine
(1996).
Fitting processes: For the philosophy of the problem, Taleb and Pilpel (2004). See also
Pisarenko and Sornette (2004), Sornette et al. (2004), and Sornette and Ide (2001).
Poisson jump: Sometimes people propose a Gaussian distribution with a small probability
of a "Poisson" jump. This may be fine, but how do you know how large the jump
is going to be? Past data might not tell you how large the jump is.
NOTES 3 27
FIGURE 15: TYPICAL DISTRIBUTION WITH POWER-LAW TAILS (HERE A STUDENT T)
Nonscalable segment:
inconsequential in its
cumulative impact
Start of scalability:
can be progressive
1 f
0.001 0.01 0.1 1 10 100
L O G (X)
FIGURE 16
Scalable: straight line
(slope close to 1.5) to 'infinity* it might become vertical somewhere
(i.e., a - > -Infinity) but
1 2 5 10 20 50 100
L O G (X)
The two exhaustive domains of attraction: vertical or straight line with slopes either
negative infinity or constant negative a. Note that since probabilities need to add
up to 1 (even in France) there cannot be other alternatives to the two basins, which
is why I narrow it down to these two exclusively.
My ideas are made very simple with this clean cut polarization—added to the
problem of not knowing which basin we are in owing to the scarcity of data on the
far right.
3 2 8 NOTES
Small sample effect: Weron (2001). Officer (1972) is quite ignorant of the point.
Recursivity of statistics: Taleb and Pilpel (2004), Blyth et al. (2005).
Biology: Modern molecular biology pioneers Salvador Luria and Max Delbruck witnessed
a clustering phenomenon with the occasional occurrence of extremely large
mutants in a bacterial colony, larger than all other bacteria.
Thermodynamics: Entropy maximization without the constraints of a second moment
leads to a Levy-stable distribution—Mandelbrot's thesis of 1952 (see Mandelbrot
[1997a]). Tsallis's more sophisticated view of entropy leads to a Student T
Imitation chains and pathologies: An informational cascade is a process where a purely
rational agent elects a particular choice ignoring his own private information (or
judgment) to follow that of others. You run, I follow you, because you may be aware
of a danger I may be missing. It is efficient to do what others do instead of having to
reinvent the wheel every time. But this copying the behavior of others can lead to imitation
chains. Soon everyone is running in the same direction, and it can be for spurious
reasons. This behavior causes stock market bubbles and the formation of
massive cultural fads. Bikhchandani et al. (1992). In psychology, see Hansen and
Donoghue (1977). In biology/selection, Dugatkin (2001), Kirpatrick and Dugatkin
(1994).
Self-organized criticality: Bak and Chen (1991), Bak (1996).
Economic variables: Bundt and Murphy (2006). Most economic variables seem to follow
a "stable" distribution. They include foreign exchange, the GDP, the money supply,
interest rates (long and short term), and industrial production.
Statisticians not accepting scalability: Flawed reasoning mistaking for sampling error in
the tails for a boundedness: Perline (2005), for instance, does not understand the difference
between absence of evidence and evidence of absence.
Time series and memory: You can have "fractal memory," i.e., the effect of past events on
the present has an impact that has a "tail." It decays as power-law, not exponentially.
Marmott's work: Marmott (2004).
CHAPTER 18
Economists: Weintraub (2002), Szenberg (1992).
Portfolio theory and modern finance: Markowitz (1952, 1959), Huang and Litzenberger
(1988) and Sharpe (1994, 1996). What is called the Sharpe ratio is meaningless outside
of Mediocristan. The contents of Steve Ross's book (Ross [2004]) on "neoclassical
finance" are completely canceled if you consider Extremistan in spite of the
"elegant" mathematics and the beautiful top-down theories. "Anecdote" of Merton
minor in Merton (1992).
Obsession with measurement: Crosby (1997) is often shown to me as convincing evidence
that measuring was a great accomplishment not knowing that it applied to Mediocristan
and Mediocristan only. Bernstein (1996) makes the same error.
Power laws in finance: Mandelbrot (1963), Gabaix et al. (2003), and Stanley et al.
(2000). Kaizoji and Kaizoji (2004), Véhel and Walter (2002). Land prices: Kaizoji
(2003). Magisterial: Bouchaud and Potters (2003).
Equity premium puzzle: If you accept fat tails, there is no equity premium puzzle. Benartzi
and Thaler (1995) offer a psychological explanation, not realizing that variance is not
the measure. So do many others.
Covered writes: a sucker's game as you cut your upside—conditional on the upside being
breached, the stock should rally a lot more than intuitively accepted. For a representative
mistake, see Board et al. (2000).
Nobel family: "Nobel Descendant Slams Economics Prize," The Local, September 28,
2005, Stockholm.
Double bubble: The problem of derivatives is that if the underlying security has mild fat
tails and follows a mild power law (i.e., a tail exponent of three or higher), the derivative
will produce far fatter tails (if the payoff is in squares, then the tail exponent of
NOTES 3 2 9
the derivatives portfolio will be half that of the primitive). This makes the Black-
Scholes-Merton equation twice as unfit!
Poisson busting: The best way to figure out the problems of the Poisson as a substitute for
a scalable is to calibrate a Poisson and compute the errors out of sample. The same
applies to methods such as GARCH—they fare well in sample, but horribly, horribly
outside (even a trailing three-month past historical volatility or mean deviation will
outperform a GARCH of higher orders).
Why the Nobel: Derman and Taleb (2005), Haug (2007).
Claude Bernard and experimental medicine: "Empiricism pour le présent, avec direction
a aspiration scientifique pour l'avenir. " From Claude Bernard, Principe de la médecine
expérimentale. See also Fagot-Largeault (2002) and Ruffie (1977). Modern evidencebased
medicine: Ierodiakonou and Vandenbroucke (1993) and Vandenbroucke
(1996) discuss a stochastic approach to medicine.
CHAPTER 19
Popper quote; From Conjectures and Refutations, pages 95-97.
The lottery paradox: This is one example of scholars not understanding the high-impact
rare event. There is a well-known philosophical conundrum called the "lottery paradox,"
originally posed by the logician Henry Kyburg (see Rescher [2001] and Clark
[2002]), which goes as follows: "I do not believe that any ticket will win the lottery,
but I do believe that all tickets will win the lottery." To me (and a regular person) this
statement does not seem to have anything strange in it. Yet for an academic philosopher
trained in classical logic, this is a paradox. But it is only so if one tries to squeeze
probability statements into commonly used logic that dates from Aristotle and is all
or nothing. An all or nothing acceptance and rejection ("I believe" or "I do not believe")
is inadequate with the highly improbable. We need shades of belief, degrees of
faith you have in a statement other than 100% or 0%.
One final philosophical consideration. For my friend the options trader and
Talmudic scholar Rabbi Tony Glickman: life is convex and to be seen as a series of
derivatives. Simply put, when you cut the negative exposure, you limit your vulnerability
to unknowledge, Taleb (2005).
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