Scaling and Memory

1
Scaling and Memory in Stock Market and Currency
Variations Similarities to Earthquakes
Shlomo Havlin Bar-Ilan, Israel in collaboration
with   Kazuko Yamasaki Tokyo, Japan Valerie
Livina, Sergey Tuzov, Lev Muchnik Bar-Ilan,
Israel   Armin Bunde Giessen, Germany   H.
Eugene Stanley Boston, USA
2
Return intervals
Stock market data
Currency series
Earthquakes
Normalized absolute return
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?
3
Scaling in Zipf plots
Stock market Currency
Earthquakes
Length return interval for a given threshold
q Ranking in decreasing length
Scaling function
4
Scaling in distributions
Stock market Currency
Earthquakes
probability distribution to have a
return interval for a given q
Yen-Dollar
Japan
IBM
Scaling function
5
Memory in the records
Conditional probability
for having a return interval
after for
6
Memory in the distributions
Clustering of extreme events
Stock market
Currency
Earthquakes
Scaling function
7
Memory in the averages
Stock market
Currency
mean conditional return interval
Earthquakes
8
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.

9
Bibliography

• V. Livina, S. Tuzov, S. Havlin, A.Bunde,
  Recurrence
• intervals between earthquakes strongly
  depend on
• history, preprint physics/0410274 (Physica A, in
  press).
• Bunde, J. Eichner, J. Kantelhardt, S. Havlin,
  Long-
• term memory natural mechanism for the clustering
  of
• extreme events and anomalous residual times in
  climate
• Records (PRL, to appear).
• K. Yamasaki, S. Havlin, A. Bunde, H. E. Stanley,
  Scaling
• and memory in volatility return intervals in
  stock markets
• (to appear)