Upload
vuduong
View
215
Download
0
Embed Size (px)
Citation preview
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Entrepreneurial RisksLecture 2:
Power Laws and Dragon-kingsHeavy Tails and Long Tails
-normal versus power laws-calculation tools-new business model: the long tail-many examples of power laws in nature and society-scale invariance, fractal and multifractals-a few (among many) mechanisms for power laws-beyond power laws: black swans vs dragon-kings
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Entrepreneurial RisksLecture 2:
Power Laws vs Long Tails
- Chapters 5, 14 and 15 ofD. Sornette, Critical Phenomena in Natural Sciences, Chaos, Fractals, Self-organization and Disorder: Concepts and Tools, 2nd ed. (Springer Series in Synergetics, Heidelberg, 2004)
-Wolfgang Amann, Cuno Pümpin and Didier Sornette, How power laws may shape the face of corporate strategy, Critical Eye 17, 21-26 (2008)
-D. Sornette, Dragon-Kings, Black Swans and the Prediction of Crises, International Journal of Terraspace Science and Engineering 2(1), 1-18 (2009) (http://arXiv.org/abs/0907.4290)
-D. Sornette and G. Ouillon, Dragon-kings: mechanisms, statistical methods and empirical evidence, to appear in European Physical Journal Special Topics (2012) (special issue on power laws and dragon-kings)
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Readings
Risk
• Loss x Chance
• Not a number, but a curve
• Not just a single curve
• Not necessarily quantitative
Chance of a severe hurricane in 2010 is 20%
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Development of Normal Distribution
• Jacob Bernoulli (ca. 1700) First experiments with distributions
• De Moivre (1733): Tossing coins• Laplace (1749 – 1828) / Poisson (ca. 1800): Law
of large numbers• Gauss (1807)• Quetelet (1847): considers normal distribution as
a fundamental law of humanity
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Normal distribution and CLT
(CLT: Central Limit Theorem)
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Key Concepts in Asset Management
• MPT: Markowitz Portfolio Theory (1952)– risk = std dev of return
• CAPM: Capital Asset Pricing Model (Sharpe 1964)– std risk remunerated only when not diversifiable (market risk)
• Efficient Market Hypothesis (Fama 1966...)• Black-Scholes-Merton: option theory (1973)• Value at Risk (first applications before WW II)
All concepts developed between 1950 - 1975
All concepts use standard deviation.
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Capital Allocation Line (1945 – 2001)
Quelle: Morgan Stanley, Ibbotson Associates
Annual Returns
Volatility: std
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Normal laws have “typical scales”
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
“What is the probability that someone has twice your height ?Essentially zero! The height, weight and many other variables are distributed with ʻmildʼ probability distribution functions with a well defined typical value and relatively small variations around it.
Mild forms and wild forms of societal self-organization
Didier Sornette (2004), 2nd ed., Critical Phenomena in Natural Sciences. Chaos, Fractals, Self-organization and Disorder: Concepts and Tools, Springer, Heidelberg
Crow J F Journal of Heredity 2004;95:365-374 © 2004 The American Genetic Association
(Top) A living histogram from the Connecticut State Agricultural College (J. Heredity 5:511–518, 1914). (Bottom) A modern version from the same university, arranged by Linda Strausbaugh (Genetics 147:5, 1997). The mean height of males in 1914 was 67.3 inches and 70.1 in 1997. In 1997 the height of females, shown in white, was 64.8 inches
Source: http://jhered.oxfordjournals.org/content/95/5/365.full
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
“What is the probability that someone has twice your height ?Essentially zero! The height, weight and many other variables are distributed with ʻmildʼ probability distribution functions with a well defined typical value and relatively small variations around it.
What is the probability that someone has twice your wealth? The answer of course depends somewhat on your wealth but in general there is a non-vanishing fraction of the population twice, ten times, or even one hundred times wealthier as you are.”
Mild forms and wild forms of societal self-organization
Didier Sornette (2004), 2nd ed., Critical Phenomena in Natural Sciences. Chaos, Fractals, Self-organization and Disorder: Concepts and Tools, Springer, Heidelberg
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
October 1987 NYSE: 21.6 standard deviations
Return time=
44‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘00‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘
000‘000‘000 years (= 44 * 1099)
(Age of Universe = 10 * 109 years)
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Shocks are quite often 1994 interest rates schock
1996 Asia boom
1997 Asia crisis
1998 Russia crisis / LTCM
2000 Internet rally
2001 Bursting of internet bubble, 11.9.01
2002 Nemax - 68,8 %
2003 Irak crisis
2003 Stock exchange rally
2005 Japan, Arabian markets (Boom)
2005 / 2006 Februar: Crash of Arabian markets
You can often hear or read of reports of a 5, …, 9, … sigma event!
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Few large earthquakes, many small ones
Few large cities, many small ones
Few frequently occurring words (“and”, “the”, “of”), manywhich rarely occur
Few extremely wealthy individuals, many poor
Few popular websites (Google, Yahoo, YouTube),many millions of others that are rarely visited.
Examples where “Normal” distribution does not apply
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
How do we represent this graphically?
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
“Long Tail” vs “Heavy Tail”
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Jean Le Rond d'Alembert (November 16, 1717 – October 29, 1783)
Life duration, years
Paradox: Average life duration is 26 years, while the chances to die before 8 and live more than 8 years are the same. 26 yrs
Average lifeduration
8 yrs
Die before 8
Die after 8
Basic Paradox of Heavy Tails
mean median median
mean
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
The Power Law: a “Heavy Tailed” distribution
f(x) =C
x1+β
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Probability density function f(x)
The distribution is called heavy tailed if it has infinite second moment:
For power-law distributions (Pareto distributions):
Note also that if(Infinite expectation)
Heavy Tails
1 (0,1)
n
iiX nm
Nnσ
=
−→
∑1
1/ ( )
n
i niX b
SC n β β=
−→
∑
Central Limit Theorem Generalized Central Limit Theorem
1
1 ( , ) ,n
ii
N nm nX N m mn n n
σ σ=
⎛ ⎞→ → →⎜ ⎟⎝ ⎠∑
1
1
( ) ,1 21 ( ) /, 1
n
i ni
bX CS n b n
n
ββ β β
ββ
−
=
< ∞ < <⎧→ + → ⎨ ∞ ≤⎩∑
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Limit Theorems
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Heavy Tails:how to determine the optimal price
�
S N p(x)dxαS
+∞
∫⎡
⎣ ⎢
⎤
⎦ ⎥ ∝ SN 1
x1+βdx ∝
αS
+∞
∫ NS1−β
Product price Number of customers who can afford the price
•β < 1: S⇑ (luxury goods)
•β > 1: S⇓ (mass product)
Total sales:
p(x): PDF of customers’ wealth (buying power)
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Normal Distribution vs. Power Law
• Normal distributions:– uniform distribution, vertex, no fat tails– mostly in static systems with weak interactions
• Power Laws:– weak vertex, continuous decent from highest
point, strong fat tails– System elements are in long-range interaction – Systems grow in a dynamic / evolutionary way
→ Out-of-Equilibrium Systems
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
• Earthquakes• Avalanches• Wealth, rich get richer• Extinction of Dinosaurs• Forest Fires• Epidemics• Pulsars• Scientific theories• Size of towns• Revolutions• Wars• Board membership• Movie actors• Proteins• Financial markets
Examples of Heavy Tails
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Cities
Left: histogram of the populations of all US cities with population of 10 000 or more. Right: another histogram of the same data, but plotted on logarithmic scales. The approximate straight-line form of the histogram in the right panel implies that the distribution follows a power law. Data from the 2000 US Census.
31
Heavy tails in pdf of earthquakes
Heavy tails in ruptures
Heavy tails in pdf of seismic rates
Harvard catalog
(CNES, France)
Turcotte (1999)
Heavy tails in pdf of rock falls, Landslides, mountain collapses
SCEC, 1985-2003, m≥2, grid of 5x5 km, time step=1 day
(Saichev and Sornette, 2005)
32
Heavy tails in cdf of Solar flares
Heavy tails in cdf of Hurricane losses
1000
104
105
1 10
Damage values for top 30 damaging hurricanes normalized to 1995 dollars by inflation, personal
property increases and coastal county population change
Normalized1925Normalized1900N
Dam
age
(mill
ion
1995
dol
lars
)
RANK
Y = M0*XM1
57911M0-0.80871M10.97899R
(Newman, 2005)
Heavy tails in pdf of rain events
Peters et al. (2002)
Heavy tails in pdf of forest fires
Malamud et al., Science 281 (1998)
33
OUTLIERS OUTLIERS
Heavy-tail of movie sales
Heavy-tail of price financialreturns
Firm sizes (Zipf’s law)
City sizes (Zipf’s law)
34
Heavy-tail of pdf of war sizes
Levy (1983); Turcotte (1999)
Heavy-tail of pdf of health care costs
Rupper et al. (2002)
Heavy-tail of cdf of book sales
Heavy-tail of cdf of terrorist intensityJohnson et al. (2006)
Survivor Cdf
Sales per day
Heavy-tail of cdf of cyber risks
b=0.7
ID Thefts
Heavy-tail of YouTube view counts
0
200
400
600
800
1000
1200
1 2 3 4 5 6 7 8 9 10
After-tax present value in millions of 1990 dollars
DBC
1980-84 pharmaceuticals in groups of deciles
Exponential model 1dataExponential model 2
Heavy-tail of Pharmaceutical sales
Number
waiting time
Software vulnerabilities
views
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
(Axtell, Science, 2001)
Heavy tails in pdf of firm sizes
37
Numbers of occurrences of words in the novel Moby Dick by Hermann Melville.
Numbers of citations to scientific papers published in 1981, from time of publication until June 1997.
Numbers of hits on web sites by 60 000 users of the America Online Internet service for the day of 1 December 1997.
Numbers of copies of bestselling books sold in the US between 1895 and 1965.
Number of calls received by AT&Ttelephone customers in the US for a single day.
M. E. J. Newman, Power laws, Pareto distributions and Zipf’s law (2005)
38
Peak gamma-ray intensity of solar flares in counts per second, measured from Earth orbit between February1980 and November 1989.
Intensity of wars from 1816 to 1980, measured as battle deaths per 10 000 of the population of theparticipating countries.
Diameter of craters on the moon. Vertical axis is measured per squarekilometre.
Frequencyof occurrence of family names in the US in the year 1990.
M. E. J. Newman, Power laws, Pareto distributions and Zipf’s law (2005)
39
Not heavy tailed ⇒ most probablyWILDLY UNDERESTIMATED
Frequency of fatalities due to man-caused events
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
What do these heavy tails mean?
Why do we see power laws everywhere?
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Coherent-noise mechanism
List of mechanisms for power laws I
(Self-organized criticality)
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
List of mechanisms for power laws II
• Sweeping of an instability
•Avalanches in hysteretic loops
(Coupling of sub-critical bifurcations)
43
Mitzenmacher M (2004) A brief history of generative models for power law and lognormal distributions, Internet Mathematics 1, 226-251.
Newman MEJ (2005) Power laws, Pareto distributions and Zipf’s law, Contemporary Physics 46, 323-351.
D. Sornette (2004) Probability Distributions in Complex Systems, Encyclopedia of Complexity and System Science (Springer Science), 2004
D. Sornette (2006) Critical Phenomena in Natural Sciences,Chaos, Fractals, Self-organization and Disorder: Concepts and Tools,2nd ed., 2nd print, pp.528, 102 figs. , 4 tabs (Springer Series in Synergetics, Heidelberg)
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Power laws, Scale Invariance and Fractals
Chapter 5Didier Sornette (2004), 2nd ed., Critical Phenomena in Natural Sciences. Chaos, Fractals, Self-organization and Disorder: Concepts and Tools, Springer, Heidelberg
Under a change of scale of the control parameter
Functional equation whose solution is
Power laws are the hallmark of scale invariance:
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Magnification symmetry => fractals
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Illustration of scale invariance
Allometric scaling for cities
here, the exponent ~1.1 > 1 ! => positive feedbacks / synergies
Bettencourt, l. M. a., lobo, J., Helbing, d., Kuhnert, C. & West, G. B. Proc. Natl Acad. Sci. USA 104, 7301–7306 (2007).
Allometric scaling for open source softwares
Number of lines changed as a function of the number of developers active per time bins for the Apache HTTP server (left) and the GNU C compiler (right)
Maillart et al. (2012)
Exponent ~2expresses superlinear positive feedbacks
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Size Number1 11/3 21/9 41/27 8
1/3^n 2^n
Self-similarity and power law
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Size Number1 11/3 21/9 41/27 8
1/3^n 2^n
Fractal dimension:
N(r) =1
rD
2n =1
[(1/3)n]D
Self-similarity and power law
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Self-similarity and power law
6
B.B. Mandelbrot, The Fractal Geometry Of Nature, 1983, W.H.Freeman
G.A. Edgar, Measure, Topology and Fractal Geometry,1990, Springer-Verlag
M. Barnsley, Fractals Everywhere, 1988, Academic Press
J. Feder, Fractals, 1988, Plenum
Books About Fractals
Movie: courtesy of Nicola Pestalozzi (ETH Zurich)
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Coherent-noise mechanism
List of mechanisms for power laws I
(Self-organized criticality)
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
List of mechanisms for power laws II
• Sweeping of an instability
•Avalanches in hysteretic loops
(Coupling of sub-critical bifurcations)
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Preferential attachment
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Source: Lada A. Adamic
Growth with Preferential Attachment
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
MECHANISMS FOR POWER LAWSComplex system approach
• One general idea: System elements are interconnected • Network with hubs and knots• Hubs with many connections have a information
advantage.
Chapters 14 and 15Didier Sornette (2004), 2nd ed., Critical Phenomena in Natural Sciences. Chaos, Fractals, Self-organization and Disorder: Concepts and Tools, Springer, Heidelberg
Saint Matthew effect
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
: number of pages with in-degree j and total number t of pages
Probability that increases is
Growth with Preferential Attachment
this page.
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Recurrence equation
For large j:
Yule (1925); Simon (1955); Lokta, Zipf, …., Barabasi
Steady state
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Stochastic Recurrence Equations
with ‘a’ stochastic
Law of proportional growth:
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
whose solution is
with
Note relationship with mechanism of “competition between exponentials”
Xt+1 = atXt + bt
Stochastic Recurrence Equations
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Stochastic Recurrence Equations
Model of financial bubbles (Rational Expectation bubbles)
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Why do we care?
Very general and ubiquitous mechanism: multiplicative process with a source or minimum size
Insight into the dynamics of firms and other economic entities
For instance, for firms, recent research show that the Zipf law of firms’sizes is mainly due to risks while returns are subdominant: risk/luck dominates! (Malevergne, Saichev, Sornette, 2007)
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Source: Lada A. Adamic
Growth with Preferential Attachment
Cumulative distributions of market share of digital cameras measured for 4 snapshots in Japan
(Ryohei Hisano, Didier Sornette, Takayuki Mizuno, 2011)
Test of Gibrat’s law of proportional growth for market share of products until January 31 2008
(Ryohei Hisano, Didier Sornette, Takayuki Mizuno, 2011)
The panel depicts the distribution of lifetimes of digital cameras using the definition mentioned in the text. Green circles denote the distribution of lifetimes of products which died during 2005, blue squares correspond to 2006, red triangles to 2007, and black crosses mixing all of these years together.
(Ryohei Hisano, Didier Sornette, Takayuki Mizuno, 2011)
(Ryohei Hisano, Didier Sornette, Takayuki Mizuno, 2011)
Comparison between the theoretical predicted power law exponents μ(TH) and the empirical exponents μ(MLE).
95
Self-organized criticality
Earthquakes Cannot Be PredictedRobert J. Geller, David D. Jackson, Yan Y. Kagan, Francesco MulargiaScience 275, 1616-1617 (1997)Turcotte (1999)
(Bak, Tang, Wiesenfeld, 1987)
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Sandpile model as paradigm of Self-Organization
Bak, P., Tang, C. and Wiesenfeld, K. (1987). "Self-organized criticality: an explanation of 1 / f noise". Physical Review Letters 59 (4): 381–384.
Artwork by Elaine Wiesenfeld (from Bak, How Nature Works, 1996)
1996
Aggregate Fluctuations from Independent Sectoral Shocks: Self-Organized Criticality in a Model of Production and Inventory Dynamics
Per Bak, Kan Chen (1992)
J.A. Scheinkman and M. Woodford, Self-organized criticality and economic fluctuations,American Economic Review 84 (2), 417-421 (1994)
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
PDF of avalanche sizesPDF of avalanche durations
Self-organization + power laws = Self-organized Criticality
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Stress field immediately before (a) and after (b) a mainshock. The stress change due to the mainshock in shownin (c). The elements that broke during the avalanche are shown in dark in (c) (stress decrease) and were mostly close to the rupture threshold before the mainshock [light gray in (a)]. The upper panels show the whole grid of size L=1024 and the lower plots represent a subset of thegrid delineated by the square in the upper plot.
(Helmsetter et al., Phys. Rev. E 70, 046120, 2004)
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Forest Fire Model (FFM)
During each time step the system is updated according to the rules:
The Forest Fire Model is a stochastic 3-state cellular automaton defined on a d-dimensional lattice with Ld sites.
Each site is occupied by a tree, a burning tree, or is empty.
• empty site → tree with the growth rate probability p • tree → burning tree with the lightning rate probability f, if
no nearest neighbour is burning • tree → burning tree with the probability 1-g, if at least
one nearest neighbour is burning, where g defines immunity.
• burning tree → empty site SOC in the limit f << p <<1
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Michael Biggs, Strikes as Forest Fires: Chicago and Paris in the Late Nineteenth CenturyAmerican Journal of Sociology, volume 110 (2005), pages 1684–1714
•Strikes
•Power grid dynamics
•Network of firms and credit risk dependences
•Economic policy implications
•Spatial SIR epidemic with recurrence
Forest Fire Model (FFM)
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
A typical simulation
http://www.shodor.org/interactivate/activities/ABetterFire/?version=1.5.0_13&browser=Mozilla&vendor=Apple_Inc.
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Power laws and large risks
• Power laws are ubiquitous• They express scale invariance• Large and extreme events
-example of height vs wealth
• Gaussian approach inappropriate: underestimation of the real risks
– Markowitz mean-variance portfolio– Black-Scholes option pricing and hedging– Asset valuation (CAPM, APT, factor models)– Financial crashes
TWO PROBLEMS What tail? What risks?
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
What model(s) for the Distributions (of Returns)?
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Implications of the two models
Practical consequences :•Extreme risk assessment,•Multi-moment asset pricing methods.
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
The Long Tail defined
In Mathematics The name for a statistical
distribution curve based on a high amplitude population followed by a low frequency population which gradually tails off (Wikipedia)
Any Industry where there is demand / availability for specialized products can and will be influenced by the long tail as accessible selection increases
and transaction friction decreases
In Business Products that are in low demand
that can collectively make up a market share that rivals the relatively few current bestsellers when distributed over such a big channel as the Internet (Chris Anderson)
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Power law dynamics in the online worldCompanies and consumers discover the long-tail
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
The diversity of the ecosystem makes it a fertile environment for small players. You don’t have to dominate the food chain to get by in the Web world; you can find a productive niche and thrive…
-Steven Johnson, “Web 2.0 Arrives”
Power law dynamics in the online worldCompanies and consumers discover the long-tail
Analogy with the power of bacteria on Earth...
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Power law dynamics in the online world
Different products and different strategies emerge
Bestsellers (Head):• Highly visible• Marketed to us• Long sales cycle
Long Tail Products: Rarely visible Word-of-mouth Sales, not cycles
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
Amazon Barnes & Noble
130000
2300000
Total Inventory (books)
Rhapsody Walmart
39000
735000
Total Inventory (songs)
Netflix Blockbuster
3000
25000
Total Inventory (movies)
Power law dynamics in the online worldSome examples
22.00
78.00
Rhapsody % Sales from Long Tail
57.00
43.00
Amazon % Sales from Long Tail
20.00
80.00
Netflix % Sales from Long Tail
Source: jotspot
Movie sales example
Didier Sornette and Daniel Zajdenweber, The economic return of research\,: the Pareto law and its implications, European Physical Journal B, 8 (4), 653-664 (1999).
µ � 1.5
131
Beyond power laws: 7 examples of “Dragons”
Material science: failure and rupture processes.
Geophysics: Gutenberg-Richter law and characteristic earthquakes.
Hydrodynamics: Extreme dragon events in the pdf of turbulent velocity fluctuations.
Financial economics: Outliers and dragons in the distribution of financial drawdowns.
Population geography: Paris (London) as the dragon-king in the Zipf distribution of French (English) city sizes.
Brain medicine: Epileptic seizures
Metastable states in complex optimization problems: Self-organized critical random directed “polymers”
132
Traditional emphasis onDaily returns do not revealany anomalous events
Crashes as “Black swans”?
“Black swans”
A. Johansen and D. Sornette, Stock market crashes are outliers,European Physical Journal B 1, 141-143 (1998)
A. Johansen and D. Sornette, Large Stock Market Price Drawdowns Are Outliers, Journal of Risk 4(2), 69-110, Winter 2001/02
“Dragons” of financial risks
10% daily drop on Nasdaq : 1/1000 probability
1 in 1000 days => 1 day in 4 years
30% drop in three consecutive days?
(1/1000)*(1/1000)*(1/1000) = (1/1000’000’000)
=> one event in 4 millions years!
137
Beyond power laws: 7 examples of “Dragons”
Material science: failure and rupture processes.
Geophysics: Gutenberg-Richter law and characteristic earthquakes.
Hydrodynamics: Extreme dragon events in the pdf of turbulent velocity fluctuations.
Financial economics: Outliers and dragons in the distribution of financial drawdowns.
Population geography: Paris as the dragon-king in the Zipf distribution of French city sizes.
Brain medicine: Epileptic seizures
Metastable states in complex optimization problems: Self-organized critical random directed “polymers”
138
Paris as a dragon-king
Jean Laherrere and Didier Sornette, Stretched exponential distributions in Nature and Economy: ``Fat tails''with characteristic scales, European Physical Journal B 2, 525-539 (1998)
2009
(Size)c
139
Beyond power laws: 7 examples of “Dragons”
Material science: failure and rupture processes.
Geophysics: Gutenberg-Richter law and characteristic earthquakes.
Hydrodynamics: Extreme dragon events in the pdf of turbulent velocity fluctuations.
Financial economics: Outliers and dragons in the distribution of financial drawdowns.
Population geography: Paris as the dragon-king in the Zipf distribution of French city sizes.
Brain medicine: Epileptic seizures
Metastable states in complex optimization problems: Self-organized critical random directed “polymers”
Energy distribution for the [+-62] specimen #4 at different times, for 5 time windows with 3400events each. The average time (in seconds) of events in each window is given in the caption.
H. Nechad, A. Helmstetter, R. El Guerjouma and D. Sornette, Andrade and Critical Time-to-Failure Laws in Fiber-Matrix Composites: Experiments and Model, Journal of Mechanics and Physics of Solids (JMPS) 53, 1099-1127 (2005)
...
time-to-failure analysisS.G. Sammis and D. Sornette, Positive Feedback, Memory and the Predictability of Earthquakes, Proceedings of the National Academy of Sciences USA, V99 SUPP1:2501-2508 (2002 FEB 19)
142
Beyond power laws: 7 examples of “Dragons”
Material science: failure and rupture processes.
Geophysics: Gutenberg-Richter law and characteristic earthquakes.
Hydrodynamics: Extreme dragon events in the pdf of turbulent velocity fluctuations.
Financial economics: Outliers and dragons in the distribution of financial drawdowns.
Population geography: Paris as the dragon-king in the Zipf distribution of French city sizes.
Brain medicine: Epileptic seizures
Metastable states in complex optimization problems: Self-organized critical random directed “polymers”
L'vov, V.S., Pomyalov, A. and Procaccia, I. (2001) Outliers, Extreme Events and Multiscaling,Physical Review E 6305 (5), 6118, U158-U166.
Pdf of the square of theVelocity as in the previous figure but for a much longertime series, so that the tailof the distributions for large Fluctuations is much betterconstrained. The hypothesisthat there are no outliers is tested here by collapsing the distributions for the three shown layers. While this is a success for small fluctuations, the tails of the distributions for large events are very different, indicating that extreme fluctuations belong to a different class of their own and hence are outliers.
L'vov, V.S., Pomyalov, A. and Procaccia, I. (2001) Outliers, Extreme Events and Multiscaling,Physical Review E 6305 (5), 6118, U158-U166.
Collapse ~of positions and amplitudes! for five intensivepeaks belonging to the 20th shell.
146
Beyond power laws: 7 examples of “Dragons”
Material science: failure and rupture processes.
Geophysics: Gutenberg-Richter law and characteristic earthquakes.
Hydrodynamics: Extreme dragon events in the pdf of turbulent velocity fluctuations.
Financial economics: Outliers and dragons in the distribution of financial drawdowns.
Population geography: Paris as the dragon-king in the Zipf distribution of French city sizes.
Brain medicine: Epileptic seizures
Metastable states in complex optimization problems: Self-organized critical random directed “polymers”
LTAD 1-6(1-6)
LTMD 1-6(17-22)
LTPD 1-6(33-38)
RTAD 1-6(41-46)
RTMD 1-6(25-30)
RTAD 1-6(9-14)
RFD 1-8(57-64)
LFD 1-8(49-56)
Depth Needle Electrodes Contact Numbering: N … 3 2 1
Key: L=Left R=Right A=Anterior M=Mesial P=Posterior D=Depth T=Temporal F=Frontal
Focus
Epileptic Seizures – Quakes of the Brain?with Ivan Osorio – KUMC & FHS
Mark G. Frei - FHSJohn Milton -The Claremont Colleges
(arxiv.org/abs/0712.3929)
149
Gutenberg-Richter distribution of sizes Omori law: Direct and Inverse
pdf of inter-event waiting times The longer it has been since the last event, the longer it will be since the next one!
SYNCHRONISATION AND COLLECTIVE EFFECTSIN EXTENDED STOCHASTIC SYSTEMS
Fireflies
Miltenberger et al. (1993)
Earthquake-fault model
(Prof. R.E. Amritkar)
Interaction (coupling) strength
Heterogeneity; level of compartmentalization
10
1
0.1
0.01
0.0010.001 0.01 0.1 1 10
SYNCHRONIZATIONEXTREME RISKS
SELF-ORGANIZED CRITICALITY
+
+
+
++
+
+
*
*
*
*
*
*
*
Coexistence of SOCand Synchronized behavior
INCOHERENT
Generic diagram for coupled threshold oscillators of relaxation
153
The pdf’s of the seizure energies and of the inter-seizure waiting times for subject 21.
Note the shoulder in each distribution, demonstrating the presence of a characteristic size and time scale, qualifying the periodic regime.
Some humans are like rats with large doses of
convulsant
154
Beyond power laws: 7 examples of “Dragons”
Material science: failure and rupture processes.
Geophysics: Gutenberg-Richter law and characteristic earthquakes.
Hydrodynamics: Extreme dragon events in the pdf of turbulent velocity fluctuations.
Financial economics: Outliers and dragons in the distribution of financial drawdowns.
Population geography: Paris as the dragon-king in the Zipf distribution of French city sizes.
Brain medicine: Epileptic seizures
Metastable states in random media: Self-organized critical random directed polymers
Singh, et. al., 1983, BSSA 73,
1779-1796
Southern California
Knopoff, 2000, PNAS 97, 11880-11884
Main, 1995, BSSA 85, 1299-1308
Complex magnitude distributions
Characteristic earthquakes?
Wesnousky, 1996, BSSA 86, 286-291
Probability distribution of size and power output of individual aurora region. (a) size distribution during quiet times. (b) size distribution during substorms. (c) power distribution during quiet times.(d) power distribution during substorms.
Lui et al, GEOPHYSICAL RESEARCH LETTERS, VOL. 27, NO. 7, PAGES 911-914, APRIL 1,2000
Chapman et al.,, GEOPHYSICAL RESEARCH LETTERS, VOL. 25, NO. 13, PAGES 2397-2400, JULY 1, 1998
A simple avalanche model as an analogue formagnetospheric activity
Global auroral energy deposition
Dynamic clustering in N balls in a billiardTwo balls are D-neighbors at epoch t if they collided during the time interval [t-D, t].Any connected component of this neighbor relation is called a D-cluster at epoch t .
Power-law cluster size distribution at critical instant tc for seven models with fixed N=5000 and rho=0.1, 0.01..., 0.00000001.
Critical mass of the largest cluster as a function of the billiard density rho at fixed N=5000.
Gabrielov, A., V. Keilis-Borok, Y. Sinai, and I. Zaliapin (2008) Statistical properties of the cluster dynamics of the systems of statistical mechanics. ESI Lecture Notes in Mathematics and Physics:Boltzmann's Legacy, European Mathematical Society, G. Gallavotti, W. Reiter and J. Yngvason (Eds.), 203-216.
Mechanisms for Dragon-kings
•Generalized correlated percolation
•Partial global synchronization
•A kind of condensation (a la Bose-Einstein)
D-MTEC Chair of Entrepreneurial Risks
Prof. Dr. Didier Sornette www.er.ethz.ch
•The ubiquity of “Heavy tails” (power laws)
•What are large and extremes events? (power law, stretched exponentials, dragon-king regime...)
•Promote outliers, dragon-kings … both at the individual and collective levels => “Social Bubble” hypothesis
•Entrepreneurs’ role to explore new scenarios, new horizons for wealth creation
•“Dragon-Kings vs black swans” view of the world
Highlights