Scaling and sizedependence in stock market trading activity

1
Scaling and size-dependence in stock market
trading activity

• Zoltán Eisler, János Kertész
• Budapest University of Technology and Economics

2
Outline

• Correlations of traded value
• company size ? strength of correlations in the
  trading of the stock
• nonuniversality
• 2647 stocks at NYSE in 2000-2002
• Fluctuation scaling
• a phenomenological scaling framework

3
Traded value

• traded value during t, t?t

P. Gopikrishnan et al., Phys. Rev. E 62, 4493
(2000)
4
Non-universal correlations of traded value

• The distribution of f is NOT Levy stable
• standard deviation exists on all time scales

Z. Eisler, J. Kertész, Eur. Phys. J. B 51, 145
(2006) S.M.D. Queirós, Europhys. Lett. 71, 339
(2005)
5
Non-universal correlations of traded value

• The distribution of f is NOT Levy stable
• standard deviation exists on all time scales
• for a stock i, the Hurst exponent H(i) can be
  defined as (DFA)
• persistent H gt 0.5
• uncorrelated H 0.5

Z. Eisler, J. Kertész, Eur. Phys. J. B 51, 145
(2006)
6
Non-universal correlations of traded value

• Crossover
• from weak correlation to strong correlation
• around ?t 1 day

Example GE
0.93
log s(?t)
0.53
log ?t
Z. Eisler, J. Kertész, Phys. Rev. E 73, 046109
(2006)
7
Non-universal correlations of traded value

• Stocks display a crossover
• at 20 min lt ?t lt 300 min
• from no correlation to strong correlation

Example GE
0.93
log s(?t)
0.53
log ?t
Z. Eisler, J. Kertész, Phys. Rev. E 73, 046109
(2006)
8
Non-universal correlations of traded value

• the crossover appears with growing company size
• intraday weak correlations
• day-to-day strength of correlations depends on
  company size

log s(?t)-0.5log ?t
log ?t
Z. Eisler, J. Kertész, physics/0606161, 145
(2006)
9
Non-universal correlations of traded value

• H non-universal!
• ?(intraday) 0.00 0.01
• ?(day-to-day) 0.06 0.01
• company size parameter

Z. Eisler, J. Kertész, Phys. Rev. E 73, 046109
(2006)
10
Fluctuation scaling

• take all stocks
• stocks i1...2647
• the trading activity of stock i fi(t) 0
• calculate the mean and the standard deviation

M.A. Menezes, A.-L. Barabási, Phys. Rev. Lett.
92, 28701 (2004) Z. Eisler et al., Europhys.
Lett. 69, 664 (2005)
11
Fluctuation scaling

• scaling over 5 orders of magnitude

M.A. Menezes, A.-L. Barabási, Phys. Rev. Lett.
92, 28701 (2004) Z. Eisler et al., Europhys.
Lett. 69, 664 (2005)
12
Fluctuation scaling

• scaling over 5 orders of magnitude
• a depends strongly on the window size ?t

M.A. Menezes, A.-L. Barabási, Phys. Rev. Lett.
92, 28701 (2004) Z. Eisler et al., Europhys.
Lett. 69, 664 (2005)
13
Fluctuation scaling

• the ?t?0 value can be explained (some other time)

M.A. Menezes, A.-L. Barabási, Phys. Rev. Lett.
92, 28701 (2004) Z. Eisler et al., Europhys.
Lett. 69, 664 (2005)
14
Fluctuation scaling

• the ?t?0 value can be explained (some other time)
• two linear regimes

M.A. Menezes, A.-L. Barabási, Phys. Rev. Lett.
92, 28701 (2004) Z. Eisler et al., Europhys.
Lett. 69, 664 (2005)
15
The common point
Fluctuation scaling
Size dependent correl.
Z. Eisler, J. Kertész, Phys. Rev. E 73, 046109
(2006)
16
The common point

• if both scaling laws hold simultaneously

Z. Eisler, J. Kertész, Phys. Rev. E 73, 046109
(2006)
17
The common point

• if both scaling laws hold simultaneously
• one finds there is no other way, but

Z. Eisler, J. Kertész, Phys. Rev. E 73, 046109
(2006)
18
The common point
19
The common point
20
The common point
21
Why do we care?

• Not because the two slopes are equal
• But because the two laws persist
• This is the ONLY way

22
Summary

• Scaling law between companies

Z. Eisler, J. Kertész, Phys. Rev. E 73, 046109
(2006)
23
Summary

• Scaling law between companies
• One can calculate a when ?t?0 (not
  discussed today)
• Correlations account
  for a(?t) dependence

Z. Eisler, J. Kertész, Phys. Rev. E 73, 046109
(2006)
24
Summary

• Scaling law between companies
• One can calculate a when ?t?0 (not
  discussed today)
• Correlations account
  for a(?t) dependence
• The coexistence of H and a is only possible, when

Z. Eisler, J. Kertész, Phys. Rev. E 73, 046109
(2006)
25
1 Z. Eisler, J. Kertész, Phys. Rev. E 73,
046109 (2006) 2 Z. Eisler, J. Kertész, Eur.
Phys. J. B 51, 145 (2006) 3 Z. Eisler et at.,
Europhys. Lett. 69, 664 (2005) 4 Z. Eisler, J.
Kertész, arXivphysics/0606161 (2006) 5 A.
Yuen, P.Ch. Ivanov, arXivphysics/0508203
(2005) 6 M.A. Menezes, A.-L. Barabási, PRL 92,
28701 (2004) 7 P. Gopikrishnan et al., Phys.
Rev. E 62, 4493 (2000) http//maxwell.phy.bme.hu/
eisler

• Thank you!

26
Universality

• different systems behave the same
• curves collapse
• power law behavior

PHYSICS

• no fundamental reason
• What are stylized facts?
• sort of universality ? universality

ECONOPHYSICS
27
Fluctuation scaling
NYSE (2000-2002)
NASDAQ (2000-2002)
0.66
0.64
?0.06
?0.05
Z. Eisler, J. Kertész, Eur. Phys. J. B 51, 145
(2006) Z. Eisler, J. Kertész, Phys. Rev. E 73,
046109 (2006)
28
Universality

• different systems behave the same
• curves collapse
• power law behavior, e.g.,

PHYSICS

• inverse cube law
• inverse cube half law
• we do see power laws

ECONOPHYSICS
29
World Wide Web

• fi(t) visitations of web page i on day t

• 3000 web pages
• daily resolution, 30 days

M.A. Menezes, A.-L. Barabási, Phys. Rev. Lett.
92, 28701 (2004)
30
World Wide Web

• fi(t) visitations of web page i on day t

1

• 3000 web pages
• daily resolution, 30 days

M.A. Menezes, A.-L. Barabási, Phys. Rev. Lett.
92, 28701 (2004)
31
Universality

• curves collapse
• different systems behave the same
• power law behavior, e.g.,

PHYSICS

• inverse cube law
• inverse cube half law
• delicate questions related to extrapolation and
  robustness

ECONOPHYSICS
inverse cube half law
inverse cube law
P. Gopikrishnan et al., Physica A 287, 362-373
(2000) P. Gopikrishnan et al., Phys. Rev. E 62,
4493 (2000)
32
This is exactly what happens
NASDAQ (2000-2002)
0.06
0.00
33
SummarySize dependent correlations

• traded values (and intertrade intervals) display
  size dependent correlations

Z. Eisler, J. Kertész, Eur. Phys. J. B 51, 145
(2006) A. Yuen, P. Ch. Ivanov, arXivphysics/05082
03 (2005) Z. Eisler, J. Kertész, Phys. Rev. E 73,
046109 (2006)
34
SummarySize dependent correlations

• traded values (and intertrade intervals) display
  size dependent correlations
• related to fluctuation scaling

Z. Eisler, J. Kertész, Eur. Phys. J. B 51, 145
(2006) A. Yuen, P. Ch. Ivanov, arXivphysics/05082
03 (2005) Z. Eisler, J. Kertész, Phys. Rev. E 73,
046109 (2006)
35
SummaryFluctuation scaling

• Scaling law between companies
• Trades stick together
• ß accounts for ?t lt 1 day
• correlations account
  for ?t gt 1 day
• The coexistence of H and a is only possible, when

• The concept of universality has to be used with
  care in finance (averaging, distributions, etc.)
• Company size/capitalization is a relevant
  parameter that influences many observables in a
  non-trivial way (non-universality, multiscaling)

Z. Eisler, J. Kertész, arXivphysics/0510058
(2005)
36
Non-universal correlations of intertrade times
NYSE
Z. Eisler, J. Kertész, arXivphysics/0508156
(2005) A. Yuen, P. Ch. Ivanov, arXivphysics/05082
03 (2005)
37

• Ivanov et al. (2004) find universally for N
  larger than the usual daily nr. of trades
• for 30 large companies HTa2 0.94 0.05

P. Ch. Ivanov et al., Phys Rev. E 69, 56107 (2004)
38
Basic measures of trading activity (2)

• company capitalization the total value of all
  stocks, proxy of company size

39
Basic measures of trading activity (2)

• company capitalization the total value of all
  stocks, proxy of company size

• trivially there must be an influence
• but what is it like?

40
Average activity

• G. Zumbach (2004)
• FTSE-100 (large)
• power laws

G. Zumbach, Quant. Fin. 4, 441-456 (2004)
41
Average activity

• NYSE, 3347 stocks (2000)
• fitted by power law
• breakdown for high capitalization

Z. Eisler, J. Kertész, arXivphysics/0508156
42
Average activity

• G. Zumbach (2004)
• FTSE-100 (large)
• power laws
• We drop the restriction to high capitalization
• all 3347 stocks traded at NYSE during 2000

43
Average activity

• NYSE, 3347 stocks (2000)
• fitted by power law
• breakdown for high capitalization

Z. Eisler, J. Kertész, arXivphysics/0508156
44
Multiscaling distribution of intertrade times (1)

• Ivanov et al. (2004) find universally

• Normalized distribution is independent of stock
  (for 30 large companies)

P. Ch. Ivanov et al., Phys Rev. E 69, 56107 (2004)
45
Multiscaling distribution of intertrade times (2)

• This implies that the distribution of T should
  show gap scaling
• with -t(q)q.

Z. Eisler, J. Kertész, arXivphysics/0508156
46

• q1,2,4,6,8,12,16
• bottom to top

Z. Eisler, J. Kertész, arXivphysics/0508156
47

• q1,2,4,6,8,12,16
• bottom to top

Z. Eisler, J. Kertész, arXivphysics/0508156
48
Summary

• Trades tend to stick together Value/trade
  increases with increasing company size

49
Summary

• Trades tend to stick together Value/trade
  increases with increasing company size

50
Summary

• Trades tend to stick together Value/trade
  increases with increasing company size
• Persistence of traded value and intertrade time
  increases for times longer than 60-300 mins

51
Summary

• Trades tend to stick together Value/trade
  increases with increasing company size
• Persistence of traded value and intertrade time
  increases for times longer than 60-300 mins

52
Summary

• Trades tend to stick together Value/trade
  increases with increasing company size
• Persistence of traded value and intertrade time
  increases for times longer than 60-300 mins
• The Hurst exponent of the f-process (T-process)
  increases (decreases) logarithmically with ltfgt
  (ltTgt)

53
Summary

• Trades tend to stick together Value/trade
  increases with increasing company size
• Persistence of traded value and intertrade time
  increases for times longer than 60-300 mins
• The Hurst exponent of the f-process (T-process)
  increases (decreases) logarithmically with ltfgt
  (ltTgt)

54
Summary

• Trades tend to stick together Value/trade
  increases with increasing company size
• Persistence of traded value and intertrade time
  increases for times longer than 60-300 mins
• The Hurst exponent of the f-process (T-process)
  increases (decreases) logarithmically with ltfgt
  (ltTgt)
• Distribution of intertrade times shows
  multiscaling

55
Summary

• Trades tend to stick together Value/trade
  increases with increasing company size
• Persistence of traded value and intertrade time
  increases for times longer than 60-300 mins
• The Hurst exponent of the f-process (T-process)
  increases (decreases) logarithmically with ltfgt
  (ltTgt)
• Distribution of intertrade times shows
  multiscaling

56
Summary

• Trades tend to stick together Value/trade
  increases with increasing company size
• Persistence of traded value and intertrade time
  increases for times longer than 60-300 mins
• The Hurst exponent of the f-process (T-process)
  increases (decreases) logarithmically with ltfgt
  (ltTgt)
• Distribution of intertrade times shows
  multiscaling

• The concept of universality has to be used with
  care in finance
• Company size/capitalization is a relevant
  parameter that influences many observables in a
  non-trivial way (non-universality, multiscaling)