1

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

2

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)

3

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)

4

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)

5

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

6

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

7

Stylized facts for financial data

• Distributions with heavy tails

• Clusters of volatility,• Multifractality,• Leverage effect,• Super-exponential

growth of speculative bubbles.

8

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

9

Implications of the two models• Practical consequences :

•Extreme risk assessment,•Multi-moment asset pricing methods.

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.

11

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

12

13

Forecast of Financial Volatility

14

scale

time

Arneodo, Muzy and Sornette (1998)

Causal cascade of volatility from large to small time scales

15

16

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

17

D. Sornette, Y. Malevergne and J.F. MuzyVolatility fingerprints of large shocks: Endogeneous versusexogeneous,Risk Magazine (2003)(http://arXiv.org/abs/cond-mat/0204626)

18

Real Data and Multifractal Random Walk model

19

Forecasting historical and implied volatility with the MRW

Comparison with RiskMetrics and GARCH(1,1)

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

21

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

22

D. Sornette and A. HelmstetterEndogeneous Versus Exogeneous Shocks in Systems with Memory, Physica A 318, 577 (2003)

Epidemic branching process of word-of-mouth

23

endogenous

Exogenousrelaxation

Exogenous precursor

θ=0.3±0.1

Real data averaged over +100 books

24

Predicting Financial Crashes

Each bubble has been rescaled vertically and translatedto end at the time of the crash

time (~2 years)

price

25

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)

26

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)

27

Disorder : K small

OrderK large

Critical:K=criticalvalue

Renormalization group:Organization of thedescription scale by scale

Scale invariance

28

The bubble and Crash of Oct. 1997Continuous line: first-order LPPLDashed line: second-order LPPL

29

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

30

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

31

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