1. Multifractal analysis and multiagent simulation for market crash prediction V. Romanov, V.Slepov, M. Badrina, A. Federyakov Russian Plekhanov Academy of Economics Computational Finance 2008 27 – 29 May 2008 Cadiz, Spain April 16, 2011

3. April 16, 2011 Examples of outputs market model Non-linear oscillation The strange attractor This output looks like head and shoulder pattern Artificial time series generation

6. Mandelbrot Set April 16, 2011 Mandelbrotset, rendered with Evercat 's program.

7. Dynamic systems fractals April 16, 2011

11. Multifractal time series (1) April 16, 2011 The process is multifractal if: where c(q) – predictor, E – expectation operator, scaling function, which expresses mutifractality properties of time series In case of monofractal For scaling function estimation we will construct partition function

12. Multifractal time series (2) April 16, 2011

15. Partition functions April 16, 2011 For each preprocessed time series compute partition function for different N and q values :

16. Scaling functions (see main fractal property) April 16, 2011

19. Experimental results (multifractal analysis) April 16, 2011 The method of multifractal analyses, described above, has been applied also for October 1987 USA financial crises, using Dow Jones index Fig. 1. Figure 1: Dow Jones industrial average data for period 01.02.1985 – 31.12.87. Axis X contains serial numbers of readings.

20. Fractal spectrum estimation April 16, 2011 Figure 2: Fractal dimension spectrum F2 (  ) for DJ industrial average series for period 10.10.85-19.10.87. Fractal dimension spectrum for 18.11.96-30.11.98 time period ( Russian default currency exchanging data ) Fractal dimension spectrum for 09.07.96-21.07.98 time period ( Russian default currency exchanging data )

21. Multifractal spectrum width before and after crisis April 16, 2011 Figure 3: Fractal dimension spectrum width F1 (  ) changing before and after crises.

22. Multifractal spectrum width before and after crisis (continued) April 16, 2011 Figure 4: Fractal dimension spectrum width F2 (  ) changing before and after crises.

24. Wavelet analysis of multifractal time series April 16, 2011 where   ,  (t) – function with zero mean centered around zero with time scale  and time horizon  . Family of wavelet vectors is created from mother function by displacement and scaling

25. Time series f(t) representation as linear combination of wavelet functions April 16, 2011 where j o – a constant, representing the highest level of resolution for which the most acute details are extracted .

26. Experimental results (wavelet analysis) April 16, 2011 Figure 5: The plot of changing maximum values detail coefficients Daubichies -12 expansion . Figure 6: The plot of maximum differences.

28. Virtual market program interface April 16, 2011

30. Program realization April 16, 2011 Minimum, maximum and average price changes Price time series Real price and fundamental price distributions Minimum, maximum and average price distributions

32. Price time series. Experiment 2 April 16, 2011

33. Experiment 3: April 16, 2011 Overall parameters: FUNDAMENTAL_TRADER_ MARKET_MAKER_TRADER_COUNT = 2; RANDOM_TRADER_COUNT = 250; COUNT = 250; BROKER_COUNT = 5; MARKET_COUNT = 1; COMPANY_COUNT = 10; CLASSIFICATORS_COUNT = 31; Companies: COMPANY_MAX_ASSETS = 50000000; // 50Mbyte COMPANY_MIN_ASSETS = 1000000; // 1Mbyte Brokers: MIN_BROKER_MARKET_ACCOUNT_MONEY = 100000; // 100k. MAX_BROKER_MARKET_ACCOUNT_MONEY = 300000; // 300k. BROKER_MONEY = 10000; // 10k. Broker and market: MAX_COMMISION_PLANS = 3; Market maker trader parameters: MIN_MM_TRADER_CHANGE_PERCENT = 0.1; MAX_MM_TRADER_CHANGE_PERCENT = 0.5; Random Trader parameters: MIN_RANDOM_TRADER_PORTFOLIOS = 0; MAX_RANDOM_TRADER_PORTFOLIOS = 2; MIN_RANDOM_TRADER_MONEY = 500; MAX_RANDOM_TRADER_MONEY = 5000; MIN_RANDOM_TRADER_ACCOUNT_MONEY = 2000; MAX_RANDOM_TRADER_ACCOUNT_MONEY = 7000; MIN_RANDOM_TRADER_PORTF_ITEM_PRICE = 2000; MAX_RANDOM_TRADER_PORTF_ITEM_PRICE = 4000; MIN_RANDOM_TRADER_RISK_AMOUNT = 0.01; MAX_RANDOM_TRADER_RISK_AMOUNT = 0.10;

34. Price time series. Experiment 3 April 16, 2011