Size matters

1
Size matters

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

2
Outline

• Universality in physics
• Universality in finance?
• In-depth analysis of
• traded value distributions
• correlations
• intertrade times
• Summary

3
Universality in physics

• 5 different magnetic materials (CrBr3, EuO, Ni,
  YIG, Pd3Fe)
• the curves collapse

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

H.E. Stanley, Rev. Mod. Phys. 71, 358 (1999)
4
Universality in finance?

• the inverse cube law
• price changes (returns)
• 1000 NYSE companies
• outside Levy regime

a 3
P. Gopikrishnan et al., Physica A 287, 362-373
(2000)
5
Universality in finance?

• the inverse cube law
• the inverse half cube law
• trading volume (Q) or value (f)
• Levy stable

Delicate questions related to extrapolation
P. Gopikrishnan et al., Phys. Rev. E 62, 4493
(2000)
6
Basic measures of trading activity (1)

• number of trades during ?t
• exchanged value in trade n
• traded value (capital flow) during ?t

7
Basic measures of trading activity (1)

• number of trades during ?t
• exchanged value in trade n
• traded value (capital flow) during ?t

8
Basic measures of trading activity (2)

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

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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?

10
Average activity

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

G. Zumbach, Quant. Fin. 4, 441-456 (2004)
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An extended analysis

• We drop the restriction to high capitalization
• all 3347 stocks traded at NYSE during 2000
• Trading activity
• Traded value
• Intertrade times

12
Average activity

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

Z. Eisler, J. Kertész, arXivphysics/0508156
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Trades stick together

• At about 1 trade / 20 minutes, transaction size
  starts to grow
• 3108 USD in capitalization

• Different mechanism for different company sizes?

Z. Eisler, J. Kertész, arXivphysics/0508156
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Beyond averages The distribution of traded value

• the inverse cube half law
• trading volume (Q) or value (f)
• Levy stable

P. Gopikrishnan et al., Phys. Rev. E 62, 4493
(2000)
15
Beyond averages The distribution of traded value

• analysis of 1000 top companies using extensions
  of Hills method
• tail exponent greater than 2
• Gaussian stable

Z. Eisler, J. Kertész, arXivphysics/0508156
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Beyond averages The distribution of traded value

• fat tails
• tail exponent greater than 2
• analysis of 1000 top companies
• increasing effective exponents
• Gaussian stable

Z. Eisler, J. Kertész, arXivphysics/0508156
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Non-universal correlations of traded value (1)

• The measured tail exponents are greater than 2
• standard deviation exists on all time scales
• for a stock i, the Hurst exponent H(i) can be
  defined as
• persistent H gt 0.5
• uncorrelated H 0.5

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Non-universal correlations of traded value (2)

• Stocks display a crossover
• at ?t 390 min 1 day
• from weaker to stronger correlation

Z. Eisler, J. Kertész, arXivphysics/0508156
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Capitalization dependence

• H non-universal!
• depends on which is closely related
  to company size
• ?(?t lt 250 min) 0.016 0.001
• ?(?t gt 630 min) 0.063 0.002

Z. Eisler, J. Kertész, arXivphysics/0508156
20
Non-universal correlations of traded value (3)

• strongly depends on capitalization
• capitalization acts as a parameter that
  determines the strength of correlations present
  in trading activity
• the effect is weak on an intraday scale
• it is much stronger for day-to-day fluctuations
• a clear logarithmic law only the order of
  magnitude matters!

Z. Eisler, J. Kertész, arXivphysics/0508156
21
Non-universal correlations of intertrade times
(1)

• intertrade interval time between two consecutive
  trades, T(n1nmax)
• Hurst exponents in virtual time (n)
• Ivanov et al. (2004) find universally for N
  larger than the usual daily nr. of trades
• for 30 large companies HT 0.94 0.05

P. Ch. Ivanov et al., Phys Rev. E 69, 56107 (2004)
22

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

• on a narrow range of company sizes
  non-universality cannot be detected

Z. Eisler, J. Kertész, arXivphysics/0508156
23
Non-universal correlations of intertrade times
(2)

• an extended range of companies gives an
  approximate logarithmic law, ?T -0.10
  0.02
• ?T lt 0 larger companies show, stronger
  correlations

GE
Z. Eisler, J. Kertész, arXivphysics/0508156
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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)
25
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
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• q1,2,4,6,8,12,16
• bottom to top

Z. Eisler, J. Kertész, arXivphysics/0508156
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• q1,2,4,6,8,12,16
• bottom to top

Z. Eisler, J. Kertész, arXivphysics/0508156
28
Parametric multiscaling of intertrade times

• The Legendre transform of t(q)
• multifractal spectrum f(a)
• can be connected to a universal distribution of
  intertrade times
• Probability that
• is proportional to

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Parametric multiscaling of intertrade times (1)

• We found
• Let us introduce a family of measures
• a company i with N trades Ti(n1N-1)
• where L is the length of the dataset, so that for
  all companies

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Parametric multiscaling of intertrade times (2)

• it is easy to show, that
• where t(q)-?(q)
• one can map µ on the new variable a defined as

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• one can map µ on the new variable a defined as
• and assume separability so that
• A saddle point approximation yields
• Legendre transf. ? multifractal spectrum

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• Legendre transf. ? multifractal spectrum
• Or equivalently

q-th moment
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Summary

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

34
Summary

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

35
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 1 day

36
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 1 day

37
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 1 day
• The Hurst exponent of the f-process (T-process)
  increases (decreases) logarithmically with ltfgt
  (ltTgt)

38
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 1 day
• The Hurst exponent of the f-process (T-process)
  increases (decreases) logarithmically with ltfgt
  (ltTgt)

39
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 1 day
• The Hurst exponent of the f-process (T-process)
  increases (decreases) logarithmically with ltfgt
  (ltTgt)
• Distribution of intertrade times shows
  multiscaling

40
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 1 day
• The Hurst exponent of the f-process (T-process)
  increases (decreases) logarithmically with ltfgt
  (ltTgt)
• Distribution of intertrade times shows
  multiscaling

41
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 1 day
• 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)

42
Dont forget Size matters!
1 Z. Eisler, J. Kertész, arXivphysics/0508156
2 P. Ch. Ivanov et al., Phys Rev. E 69, 56107
(2004) 3 G. Zumbach, Quant. Fin. 4, 441-456
(2004) 4 T.C. Halsey et al., Phys. Rev. A 33,
1141-1151 (1986) 5 P. Gopikrishnan et al.,
Physica A 287, 362-373 (2000) 6 P. Gopikrishnan
et al., Phys. Rev. E 62, 4493-4496 (2000) 7
H.E. Stanley, Rev. Mod. Phys. 71, 358 (1999)

• Thank you for your attention!