power laws and scaling in finance - eth z · power laws and large risks • power laws are...
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Power laws and scaling in financePractical applications for risk control
and management
ETH-ZurichChair of Entrepreneurial RisksDepartment of Management, Technology and Economics (D-MTEC)
http://www.mtec.ethz.ch/
D. SORNETTE
D-MTEC Chair of Entrepreneurial Risks
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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)
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Heavy tails in pdf of Solar flares
Heavy tails in pdf of Hurricane losses
1000
10 4
10 5
1 10
Damage values for top 30 damaging hurricanes normalized to 1995 dollars by inflation, personal
property increases and coastal county population change
Dam
age
(mill
ion
1995
dol
lars
)
RANK
Y = M0*X M1
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)
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Heavy-tail of price changes
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
OUTLIERS OUTLIERS
Heavy-tail of Pharmaceutical sales
Heavy-tail of movie sales
Heavy-tail of crash losses(drawdowns)
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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 pdf of book sales
Heavy-tail of pdf of terrorist intensityJohnson et al. (2006)
Survivor Cdf
Sales per day
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Power laws and large risks
• Power laws are ubiquitous• They express scale invariance• Probability of large excursion:
-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
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Stylized facts for financial data
• Distributions with heavy tails
• Clusters of volatility,• Multifractality,• Leverage effect,• Super-exponential
growth of speculative bubbles.
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Empirical Results about the Distributions of Returns
• Models in terms of Regularly varying distributions:
Longin (1996) , Lux (1996-2000), Pagan (1996), Gopikrishnan et al. (1998)…
• Models in terms of Weibull-like distributions:
Mantegna and Stanley (1994), Ebernlein et al.(1998), Gouriéroux and Jasiak (1998), Laherrère and Sornette (1999)…
[ ] μ−⋅=≥ xxxrt )(Pr L )43( −≈μ
[ ] [ ]ct xxxr ⋅−=≥ )(expPr L )1( <c
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Implications of the two models• Practical consequences :
•Extreme risk assessment,•Multi-moment asset pricing methods.
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10Complementary sample distribution function for the Standard & Poor’s 500 30-minute returns over the two decades 1980–1999. The plain (resp. dotted) line depicts the complementary distribution for the positive (the absolute value of negative) returns.
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Main Results• For sufficiently high thresholds, both the Power laws and
Weibull distributions comply with the data.• For both models, the evolution of the parameters is not
exhausted at the end of the range of available data.
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.2
0
0.2
0.4
0.6
0.8
1
1.2
MSE
MP
Quantile U
Dow Jones, Daily returns, 1896-2000 – positive tail
Weibull
PowerLaw
Power law
Tai
l ind
ex
bc
b value
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Forecast of Financial Volatility
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scale
time
Arneodo, Muzy and Sornette (1998)
Causal cascade of volatility from large to small time scales
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10 min40 min160 min1 day1 week1 month
The Multifractal Random Walk Model
Prediction Data
•Heavy tail is consequence of long-range time dependence
•Self-consistent coherent description of PDF and dependences at all scales simultaneously
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D. Sornette, Y. Malevergne and J.F. MuzyVolatility fingerprints of large shocks: Endogeneous versusexogeneous,Risk Magazine (2003)(http://arXiv.org/abs/cond-mat/0204626)
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Real Data and Multifractal Random Walk model
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Forecasting historical and implied volatility with the MRW
Comparison with RiskMetrics and GARCH(1,1)
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20D. Sornette et al., Phys. Rev. Letts. 93 (22), 228701 (2004); F. Deschatres and D. Sornette, The Dynamics of Book Sales: Endogenous versus Exogenous Shocks in Complex Networks, Phys. Rev. E 72, 016112 (2005)
PREDICTING COMMERCIAL SALES
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The Original “Crisis”
• On Friday January 17, 2003, WSMC jumped to rank 5 on Amazon.com’s sales ranking (with Harry Potter as #1!!!)
• Two days before: release of an interview on MSNBC’s MoneyCentral website
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D. Sornette and A. HelmstetterEndogeneous Versus Exogeneous Shocks in Systems with Memory, Physica A 318, 577 (2003)
Epidemic branching process of word-of-mouth
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endogenous
Exogenousrelaxation
Exogenous precursor
θ=0.3±0.1
Real data averaged over +100 books
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Predicting Financial Crashes
Each bubble has been rescaled vertically and translatedto end at the time of the crash
time (~2 years)
price
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950C
1Kg
1cm
97
1cm
1Kg
99
1Kg
101
The breaking of macroscopic linear extrapolation
?Extrapolation?
BOILING PHASE TRANSITIONMore is different: a single molecule does not boil at 100C0
Simplest Example of a “More is Different” TransitionWater level vs. temperature
(S. Solomon)
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95 97 99 101
Example of “MORE IS DIFFERENT” transition in Finance:
Instead of Water Level: -economic index(Dow-Jones etc…)
Crash = result of collective behavior of individual traders(S. Solomon)
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Disorder : K small
OrderK large
Critical:K=criticalvalue
Renormalization group:Organization of thedescription scale by scale
Scale invariance
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The bubble and Crash of Oct. 1997Continuous line: first-order LPPLDashed line: second-order LPPL
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Towards a methodology to identify crash risks
• Development of methods to diagnose bubbles• Crashes are not predictable• Only the end of bubbles can be forecasted• 2/3 of bubbles end in a crash• Multi-time-scales• Probability of crashes; alarm index
– Successful forward predictions: Oct. 1997; Aug. 1998, April 2000
– False alarms: Oct. 1997
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Summary
• Power laws are ubiquitous: large risks are common• Robust reliable prediction of VaR with sparse data
(hedge-funds)• Forecasts of financial volatility (option market maker)• Predicting commercial sales (books, CDs, movies…)• Predicting financial instabilities
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References
Y. MALEVERGNE, V. PISARENKO and D. SORNETTE(2005) “On the power of generalized extreme value (GEV) and generalized Pareto distribution (GPD) estimators for empirical distributions of log-returns.”Applied Financial Economics 16, 271-289 (2006)
Y. MALEVERGNE, V. PISARENKO and D. SORNETTE(2005) “Empirical distributions of stock returns: Exponential or power-like?” Quantitative Finance 5, 379-401.
Y. MALEVERGNE and D. SORNETTE (2005) Extreme Financial Risks. Springer. Chapter 2.
Nov 2005