Part 2: Correlations and structure: some recent results

1
Part 2 Correlations and structure some recent
results

• Zoltán Eisler
• Dept. of Theor. Phys., Budapest Univ. of
  Technology and Economics

2
Risk and return

• How risky is it to invest into GE?

3
Risk and return

• How risky is it to invest into GE?
• one measure of the risk is volatility
• the standard deviation of returns

4
Risk and return

• How risky is it to invest into GE?
• one measure of the risk is volatility
• the standard deviation of returns
• Two factors in GE returns (and so volatility)
• rGE(t) ßGErMARKET(t) rGE(t)

5
Risk and return

• How risky is it to invest into GE?
• one measure of the risk is volatility
• the standard deviation of returns
• Two factors in GE returns (and so volatility)
• rGE(t) ßGErMARKET(t) rGE(t)
• To decrease risk portfolio

6
Risk and return

• Two factors in GE returns (and so volatility)
• rGE(t) ßGErMARKET(t) rGE(t)
• To decrease risk portfolio
• rPORTFOLIO(t) ßrMARKET(t) Siwiri(t)

7
Risk and return

• Two factors in GE returns (and so volatility)
• rGE(t) ßGErMARKET(t) rGE(t)
• To decrease risk portfolio
• rPORTFOLIO(t) ßrMARKET(t) Siwiri(t)
• How to select stocks that their specific returns
  ri are uncorrelated?

8
The portfolio selection problem

• How to select stocks that their specific returns
  ri are uncorrelated?

V. Plerou et al. Physica A 287, 374-382 (2000)
9
The portfolio selection problem

• How to select stocks that their specific returns
  ri are uncorrelated?
• Problems
• no explicit knowledge of system dynamics
• neglects higher-order correlations
• time dependence of correlation matrix
• finite sample

V. Plerou et al. Physica A 287, 374-382 (2000)
10
Why random matrix theory?

• Wigner (1950) interactions in complex nuclei

V. Plerou et al. Physica A 287, 374-382 (2000)
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Why random matrix theory?

• Wigner (1950) interactions in complex nuclei
• Random matrix Hamiltonian

V. Plerou et al. Physica A 287, 374-382 (2000)
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Why random matrix theory?

• Wigner (1950) interactions in complex nuclei
• Random matrix Hamiltonian
• Real, symmetric, Gaussian elem.

V. Plerou et al. Physica A 287, 374-382 (2000)
13
Financial cross-correlations

• comparison of random and financial matrix
  results
• is there information?
• how to extract it?

V. Plerou et al. Physica A 287, 374-382 (2000)
14
Financial cross-correlations

• comparison of random and financial matrix
  results
• is there information?
• how to extract it?
• eigenvalues
• random noise
• correlated groups

V. Plerou et al. Physica A 287, 374-382 (2000)
15
Financial cross-correlations

• for any large eigenvalue ?i

eigenvector correlated group of stocks
components 1..N weight of the stock in the group
V. Plerou et al. Physica A 287, 374-382 (2000)
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Financial cross-correlations

• the largest eigenvalue ?1000 index

V. Plerou et al. Physica A 287, 374-382 (2000)
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Financial cross-correlations

• other large eigenvalues

B. Rosenow, AKSOE Winter School 2004, Konstanz
18
Sectors important correlations
B. Rosenow, AKSOE Winter School 2004, Konstanz
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J.-P. Onnela, K. Kaski, J. Kertész Eur. Phys. J.
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J.-P. Onnela, K. Kaski, J. Kertész Eur. Phys. J.
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Sectors important correlations

• to have a well-diversified portfolio
• rPORTFOLIO(t) ßrMARKET(t) Siwiri(t)
• choose stocks from many sectors

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The MST approach

• The distance matrix

J.-P. Onnela et al. Eur. Phys. J. B 30, 285-288
(2002)
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The MST approach

• The distance matrix
• The Minimum Spanning Tree
• tree (N-1 edges)
• includes all nodes (stocks)
• sum of distances minimal

J.-P. Onnela et al. Eur. Phys. J. B 30, 285-288
(2002)
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Asset tree and clusters
dij
J.-P. Onnela et al. Eur. Phys. J. B 30, 285-288
(2002)
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Asset tree topology change

• normal market topology crash topology

J.-P. Onnela et al. Physica A 324, 247 (2003)
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Why NOT physics?

• Physics close to equilibrium
• time reversal symmetry (TRS)
• detailed balance
• symmetric correlation functions
• fluctuation-dissipation theorem, etc.

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Why not physics?

• Physics close to equilibrium
• time reversal symmetry (TRS)
• detailed balance
• symmetric correlation functions
• fluctuation-dissipation theorem, etc.
• No fundamental reason to force TRS
• a deal is advantageous for both parties (or at
  least one ?)

J. Kertész et al. Physica A 324, 74-80 (2003)
44
Why not physics?

• No fundamental reason to force TRS
• a deal is advantageous for both parties (or at
  least one ?)
• Possibility of
• asymmetric cross-correlation functions
• different response to spontaneous fluctuations
  and external perturbations (A.G. Zawadowski, R.
  Karádi, J. Kertész, Physica A 316, 403-413 (2002))

J. Kertész et al. Physica A 324, 74-80 (2003)
45
Time-dependent cross-correlations

• time dependent cross-correlations of two assets

L. Kullmann, J. Kertész, K. Kaski Phys. Rev. E
66, 26125 (2002)
46
Time-dependent cross-correlations

• time dependent cross-correlations of two assets
• is it asymmetric?
• difficulties trades not syncronized, low
  signal/noise ratio, etc.

L. Kullmann, J. Kertész, K. Kaski Phys. Rev. E
66, 26125 (2002)
47
Toy model

• Persistent random walk

L. Kullmann, J. Kertész, K. Kaski Phys. Rev. E
66, 26125 (2002)
48
Toy model

• Persistent random walk
• ?0200, ? 1000, ? 0.99, then we remove 99

L. Kullmann, J. Kertész, K. Kaski Phys. Rev. E
66, 26125 (2002)
49
Results on toy model

• results depend on ?t (averaging time)
• ?0200, ? 1000, ? 0.99, then we removed 99

L. Kullmann, et al. Phys. Rev. E 66, 26125 (2002)
50
Results on real data

• NYSE Trade And Quote database
• all trades of 10000 companies tick by tick
• 195 companies traded more than 15000 times in 54
  days (01/12/1997 09/03/1998)
• ?t 100 sec, but results checked for 50-500 sec

L. Kullmann, J. Kertész, K. Kaski Phys. Rev. E
66, 26125 (2002)
51
Results on real data

• typical values for significant results
• SNR gt 6
• tmax gt 100 sec
• C(tmax) gt 0.04

L. Kullmann, J. Kertész, K. Kaski Phys. Rev. E
66, 26125 (2002)
52
Results on real data

• typical values for
  significant results
• SNR gt 6
• tmax gt 100 sec
• C(tmax) gt 0.04
• XOM Exxon mobile
  (major oil company)
  capitalization 306.15 Bn
• ESV Ensco INTL
  (offshore contract drilling company)
  capitalization 4.52 Bn

L. Kullmann, J. Kertész, K. Kaski Phys. Rev. E
66, 26125 (2002)
53
Results on real data

• not all pairs show
  the effect
• peak not only shifted
  but also asymmetric
• large, frequently
  traded companies
  pull small ones
• weak effect, short
  characteristic times (few
  minutes)

L. Kullmann, J. Kertész, K. Kaski Phys. Rev. E
66, 26125 (2002)
54
Directed network of influence
L. Kullmann, et al. Phys. Rev. E 66, 26125 (2002)
55
No circles Many leaders for a follower Many
followers for a leader Disconnected graph
Directed network of influence
L. Kullmann, et al. Phys. Rev. E 66, 26125 (2002)
56
The Lux-Marchesi model
traders
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The Lux-Marchesi model
expect prices to...
go up
chartists
go down
fluctuate around the fundamental value
SELL
pf
BUY
58
The Lux-Marchesi model
market
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The Lux-Marchesi model

• make deals with the market maker
• evaluate the present price
• sell or buy the quantity desired

• buy or sell the quantities specified by all
  traders
• set the new price according to supply and demand

60
The Lux-Marchesi model
transitions
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The Lux-Marchesi model
trading transitions
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The role of the chartists

• Destabilize prices
• Herding behavior
• Bubbles are formed

p
pf
t
t
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The role of the fundamentalists
p

• Stabilize prices
• only minor fluctuations around the fundamental
  price

pf
t
64
T. Lux, AKSOE Winter School 2004, Konstanz
65
T. Lux, AKSOE Winter School 2004, Konstanz
66
Externally induced crashes

• some news or announcement causes price(s) to drop
  abruptly
• source is identifiable
• internally induced
• no identifiable cause
• due to internal dynamics (bubbles, etc.)

Budapest Stock Exchange (BUX), 12 Nov.
2001, after crash of AA587 (rumors of terror
attack)
A.G. Zawadowski, R. Karádi, J. Kertész Physica A
316, 403-413 (2002)
67
Externally induced crashes

• after a sudden drop there is a correction
  (overshoot)
• drop in pf

Simulation of the Lux-Marchesi model
A.G. Zawadowski, R. Karádi, J. Kertész Physica A
316, 403-413 (2002)
68
Externally induced crashes

• after a sudden drop there is a correction
  (overshoot)
• drop in pf

A.G. Zawadowski, R. Karádi, J. Kertész Physica A
316, 403-413 (2002)
69
Externally induced crashes

• after a sudden drop there is a correction
  (overshoot)
• drop in pf
• mechanism
• fundamentalists sell
• pessimists become extremely successful

• high number of pessimists even after reaching new
  pf
• overshoot purely speculative

A.G. Zawadowski, R. Karádi, J. Kertész Physica A
316, 403-413 (2002)
70
Real market crashes

• after a sudden drop there is a correction
  (overshoot)
• profitability
• ask price lowest offer to sell
• bid price highest offer to buy

average of 222 drops (exceeding 4) at NYSE
in 2000-2002
Á.G. Zawadowski, Gy. Andor, J. Kertész
cond-mat/0406696, subm. to Quant. Fin.
71
Real market crashes

• after a sudden drop there is a correction
  (overshoot)
• profitability
• ask price lowest offer to sell
• bid price highest offer to buy

average of 215 drops (exceeding 4) at
NASDAQ in 2000-2002
Á.G. Zawadowski, Gy. Andor, J. Kertész
cond-mat/0406696, subm. to Quant. Fin.
72
Real market crashes

• after a sudden drop there is a correction
  (overshoot)
• profitability
• ask price lowest offer to sell
• bid price highest offer to buy
• predictability
• precursor phenomena

NYSE NASDAQ
Á.G. Zawadowski, Gy. Andor, J. Kertész
cond-mat/0406696, subm. to Quant. Fin.