Scaling and Memoryin Stock Market and Currency Variations:

Similarities to Earthquakes

Shlomo HavlinBar-Ilan, Israel

in collaboration with 

Kazuko Yamasaki Tokyo, Japan

Valerie Livina, Sergey Tuzov, Lev MuchnikBar-Ilan, Israel

 Armin Bunde

Giessen, Germany 

H. Eugene StanleyBoston, USA

Challenges:(a) Are there scaling laws in return intervals?(b) Is there memory in the records of return intervals?(c) Are there similarities between economy and earthquakes?(d) How can we improve forecast of extreme events?

)1(

)1()(

tR

tRtR

Return intervalsStock market data Currency series

EarthquakesNormalized absolute return

Scaling in Zipf plots

Stock market Currency Earthquakes

Length return interval for a given

threshold q

Ranking in decreasing length

Scaling function

rankx

RxfRx qq

)()(

Scaling in distributions

probability distribution to have

a return interval for a given q

)(qP

Stock market Currency Earthquakes

Scaling function

qq Rf

RP

1)(

IBMYen-Dollar Japan

Memory in the records

Conditional probability

for having a return interval after

for

)|( 0qP

0

4/30

4/10

Memory in the distributions

Stock market Currency

Earthquakes

Clustering of extreme events

Scaling function

qq Rf

RP

1)|( 0

Memory in the averagesStock market Currency

Earthquakes

mean conditionalreturn interval

)(ˆ 0

Summary

Scaling of return intervals Well approximated by single scaled function. Strong effect of memory

Origin: long-term correlations in the volatilities. Strong similarity in both scaling (for different q) and memory to earthquakes. Application: improving risk assessment.

qqq RRfP /)/()(

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q RforPP

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qqq

qqq

q RforPP

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V. Livina, S. Tuzov, S. Havlin, A.Bunde, Recurrence intervals between earthquakes strongly depend on history, preprint physics/0410274 (Physica A, in press).

A. Bunde, J. Eichner, J. Kantelhardt, S. Havlin, Long-term memory: natural mechanism for the clustering ofextreme events and anomalous residual times in climateRecords (PRL, to appear).

K. Yamasaki, S. Havlin, A. Bunde, H. E. Stanley, Scaling and memory in volatility return intervals in stock markets(to appear)

Bibliography