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Modeling financial markets- cellular automata thinking Financial markets foundations Agent-based computational finance heterogeneous agents agents interacting locally : Cont-Bouchaud model (modeling an emerging market)


  1. Modeling financial markets- cellular automata thinking • Financial markets foundations • Agent-based computational finance – heterogeneous agents – agents interacting locally : •Cont-Bouchaud model (modeling an emerging market) •cellular automata models of local interaction 1

  2. Financial markets foundations: Three pilars: Efficient Market Hypothesis (Fama, Samulsen) ����������� ���������� ������������� ������ information MARKET price 2

  3. Stock markets 14000 30000 DJ Industrial London FT-SE 12000 DAX 25000 WIG 10000 20000 DJ, FTSE, DAX WIG values 8000 15000 6000 10000 4000 If 5000 Agents are rational 2000 Market is efficient 0 0 Then 95 96 97 98 99 00 01 02 03 04 05 06 Price is represented by a random walk Financial markets foundations: Three pilars: Efficient Market Hypothesis (Fama, Samulsen ) Capital Asset Pricing Model ( Sharpe, Lintner) 3

  4. Agents have homogeneous expectations: All investors hold the same portfolio: stocks bonds Financial markets foundations: Three pilars: Efficient Market Hypothesis (Fama, Samulsen) Capital Asset Pricing Model ( Sharpe, Lintner) Black-Scholes option pricing formula ( +Merton) 4

  5. Financial markets foundations: Three pilars: Efficient Market Hypothesis (Fama, Samulsen) Capital Asset Pricing Model ( Sharpe, Lintner) Black-Scholes option pricing formula ( +Merton) �������� �������� ����������������� ��� �������� ������������ q ��������� ���� ������ ������� q • Return( t, time-horizon ) = log Price( t+ time-horizon ) – log Price( t ) • Volatility ( t , time-horizon ) = | Return( t , time-horizon ) | • Volume (t ) WIG Electrim zywiec 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 5

  6. 1 0 0 0 too many W IG 1 0 0 ����������� too little ������������ 1 0 too many 1 1 (| | ) > ∝ P return x 3 x -4 -2 0 2 4 ���� !�"� ##$� %������������ &���������� �� &������ ���������� ��� &������ '�������� (���� �� )���*(+#, 6

  7. Complex system modeling -a population of different elements with well defined microscopic attributes and interacting. -show emergent macroscopic phenomena: - self-organization ( functional-organization ) - unpredictability - evolution through punctuated equilibriums ( step-wise response ) Independent Agents models: '����� ���������������� ������ �"��� �"� ���������������� ������ ��� ������������� 7

  8. Independent Agents models: Strong simplifications: agents share the same information agents act independently -.) ������ ������ ������������� �"������� *������ ������� , ��������������� *��������� ������� , Independent Agents models: Strong simplifications: agents share the same information agents act independently -.) ������ ������ ������������� �"������� *������ ������� , ��������������� *��������� ������� , -Bak, Paczuski, Shubik ( 97 ) -Gardina, Bouchaud ( 2002) 8

  9. Bak, Paczuski, Shubik ( 97 ) Sell(3,t) Sell(4,t) Buy(5,t) Buy(1,t) Buy(6,t) Sell(2,t) Sell(8,t) Buy(7,t) �� ������������������� ���� ����������� B 1 B 2 B 7 B 5 B 3 B 4 B 8 B 6 -Gardina, Bouchaud ( 2002) 9

  10. B 1 B 2 B 7 B 5 B 3 B 4 B 8 B 6 -Gardina, Bouchaud ( 2002) Agents interacting locally ������ ���� ��������� ��������� ������ ��� ����� ����������� &��� / -���"��� �������� "������ R.Cont and J.P.Bouchaud, Macroeconomic Dynamics (2000) 10

  11. B 1 B 2 B 7 B 5 B 3 B 4 B 8 B 6 B 1 B 2 B 7 B 5 B 3 B 4 B 8 B 6 ∆ P(t)= Σ demand(i) – Σ supply(i) 11

  12. �������������� ���� ����������� � ���� ���� ����� �� Percolation Probability 1 0 p c p 10 9 log L=200 power-law with 1.77 exponent 10 8 p=0.1 p=0.2 p=0.3 10 7 p=0.4 p=0.5 frequency in 20000 experiments p=0.6 10 6 p=0.7 p=0.8 p=0.9 10 5 10 4 10 3 10 2 10 1 10 0 10 0 10 1 10 2 10 3 10 4 log cluster size 12

  13. D.Stauffer and T.J.P.Penna (1998) CA approach to Count- Bouchaud model Large Small investor investor p c =0.592746….. D.Stauffer and T.J.P.Penna (1998) CA approach to Count- Bouchaud model ∆ P(t)= Σ demand(i) – Σ supply(i) Large Small investor investor p c =0.592746 13

  14. D.Makowiec, Acta Phys. Pol B 0.60 (2004) DFA exponents 0.55 ���������� 0.50 ����������� ��������� 0.45 0.40 0.35 0.30 first 10 days exponents after a month exponents 0.25 0.20 0 10 20 30 40 50 60 70 Daily ����������� returns ������������� distribution 10 7 40 10 7 10 6 350 volume 10 6 price 30 10 5 10 5 300 10 4 Volume price 10 4 Volume 20 price 250 10 3 10 3 10 2 200 10 10 2 10 1 volume 10 1 KGHM Polska Miedz price BPH 150 10 0 Bank Przemyslowo-Handlowy 0 10 0 1998 1999 2000 2001 2002 2003 2004 1998 1999 2000 2001 2002 2003 2004 date histogram of histogram of daily returns daily returns 0.1 0.1 probability probability 0.01 0.01 0.001 0.001 -4 -2 0 2 4 -4 -2 0 2 4 normalized daily returns normalized daily returns 14

  15. ��������"������� A 1 B 2 B 1 A 2 B 7 B 3 A 3 B 5 B 4 B 6 B 8 �������� price -����� �0 ����������������������������� ������ ���12���23.&�)23 price ���������� -������� ������ )"���������"�������� �4#�+5�6 5�#�##+ a=0.1 b=0.0 0.1 b=0.1 b=0.2 b=0.3 b=0.4 b=0.5 probability 0.01 0.001 -2 -1 0 1 2 normalized returns 15

  16. D.Makowiec, P.Gnacinski and W.Miklaszewski, Physica A331 (2004) 269 ��������"�������� Assumption: B 2 Information B 1 is available to some agents B 5 only A 1 ��������������������� B 4 B 3 0��������� �������������5���� ������"�������� A 2 B 8 B 7 -���������������� 0��������� ����������"��� B 6 �0 ����������� ������������������������������ ��������"�������� -������ ��������� �� ������� B 2 B 1 �"���� �� ������� �������� ���������� B 5 A 1 B 4 B 3 B 6 A 2 B 8 B 7 16

  17. �������-��� ���"����������-������� �����������"�������� B 1 B 2 B 3 B 4 B 5 A 1 B 6 A 2 B 7 B 8 17

  18. �4#�##+ �4#�##+ 18

  19. �4#�##+ �4#�##+ 19

  20. Examples of CA models: q Lattice gas dynamics persistency Stauffer, Oliveira and Bernardes(1999) q Evolving network tail decay with exponent 2.5 Equiluz&Zimmermann (2000) q Spin ferromagnetic interaction- tail decay with exponent 4 Chowdhury, Stauffer(1999), Bornholdt (2001) q Forest fire rule – multifractal properties Bartolozzi& Thomas (2004) q Local trend rule – less locality then more stable market Bandini, Manzoni,Naimzada& Pave (2 004) 20

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