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Attività del nodo di Alessandria Enrico Scalas www.econophysics.org www.mfn.unipmn.it/~scalas/fisr.html ENOC 05 Eindhoven, The Netherlands, 7-12 August 2005

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Riassunto A4.1 e A4.4 Risultati A4.1 e A4.4 Il futuro A4.2 e A4.3 Nuovi risultati Due presentazioni (ENOC’05 e WEHIA 2005)

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A Lévy-noise generator Matteo Leccardi and Enrico Scalas www.econophysics.org www.mfn.unipmn.it/~scalas/fisr.html ENOC 05 Eindhoven, The Netherlands, 7-12 August 2005

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Summary Theory (and motivation) Algorithm Demo Conclusions

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Theory

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Theory (I): Continuous-time random walk (basic quantities, physical and financial intepretation) : price of an asset at time t : log price or position of a particle : joint probability density of jumps and of waiting times : probability density function of finding log-price or position x at time t

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Theory (II): Master equation (pure jump process) Marginal jump pdf Marginal waiting-time pdf Permanence in 0 Jump into x,t In case of independence: Survival probability

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This is the characteristic function of the log-price process subordinated to a generalised Poisson process. Theory (III): Limit theorem, uncoupled case (I) (Scalas, Mainardi, Gorenflo, PRE, 69, 011107, 2004) Mittag-Leffler function Subordination: see Clark, Econometrica, 41, 135-156 (1973).

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Theory (IV): Limit theorem, uncoupled case (II) (Scalas, Gorenflo, Mainardi, PRE, 69, 011107, 2004) This is the characteristic function for the Green function of the fractional diffusion equation. Scaling of probability density functions Asymptotic behaviour

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Theory (V): Fractional diffusion (Scalas, Gorenflo, Mainardi, PRE, 69, 011107, 2004) Green function of the pseudo-differential equation (fractional diffusion equation): Normal diffusion for =2, =1.

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Algorithm

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The Ziggurat Algorithm Marsaglia, G., Tsang, W.W. (2000). The ziggurat method for generating random variables. In Journal of Statistical Software, Vol. 5, Issue 8, pp.1-7.

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stable symmetric density (I) = 1 Cauchy distribution = 2 Gauss distribution

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Power-law tails: stable symmetric density (II)

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Simplified Ziggurat Algorithm (for hardware implementation) Truncated density

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FPGA (I) Field Programmable Gate Array Introduced in 1985 Regular modular structure with interconnections Up to 10 7 logical gates

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FPGA (II) Reprogrammable Two kinds OTP Reprogrammable: –Based on SRAM –Logic defined in LUT OTP –Anti-fuse technology –Logic defined with traditional logical gates

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HDL (I) Hardware Description Language It describes the behaviour of a circuit and not its structure Faster development stage Projects easier to modify

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HDL (II) Verilog Two main dialects VHDL Verilog is close to C

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Verilog code for the generator module levy( input clk, // clock input [3:0] alpha, // the stable index is a=(alpha+3)/10 input signed [31:0] j, // uniform random variable output reg signed [15:0] lev // levy random variable ); reg [15:0] x; reg signed [47:0] p; always @(alpha, j) begin case({alpha,j[6:0]}) 0: x = 1; 1: x = 1; 2: x = 1;.................. 2047: x = 65535; endcase end always @(posedge clk) begin p = x*j; lev = p>>32; end endmodule

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Algorithm test (I) =1.7, =1350

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Algorithm test (II) =1.7, =1350

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Algorithm test (III) =0.9, =56

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Algorithm test (IV) =0.9, =56

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Kolmogorov-Smirnov Test dd(1) 5%d(2) 15% 0.60.126750.0181440.014423FALSE 0.70.038910.0170840.013581FALSE 0.80.015120.0162120.012888TRUEFALSE 0.90.007280.0152790.012145TRUE 10.0065970.0145650.011578TRUE 1.10.0057820.0138930.011044TRUE 1.20.0041810.0132070.010499TRUE 1.30.0040860.0127590.010143TRUE 1.40.0051210.0122350.009726TRUE 1.50.0044730.0115920.009215TRUE 1.60.0033020.0111230.008842TRUE 1.70.0055850.0105310.008372TRUE 1.80.0043170.0099080.007876TRUE 1.90.0038980.0092260.007334TRUE 1.950.001660.0075580.006008TRUE 1.990.001780.0067870.005395TRUE

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Conclusions

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A Lévy white-noise generator has been implemented It generalizes Gaussian white-noise generators (GWNG) Large fluctuations are much more frequent than in GWNG It is cheap and versatile Various applications are envisaged: to finance to tests of materials to basic research …

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Enrico Scalas (DISTA Università del Piemonte Orientale) www.econophysics.org www.fracalmo.org WEHIA 2005 Colchester, Essex, UK – 13 -15 June 2005 Waiting times between orders and trades in double-auction markets

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In collaboration with: Jürgen Huber (Innsbruck) Taisei Kaizoji (Tokyo) Michael Kirchler (Innsbruck) Alessandra Tedeschi (Rome)

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Summary The continuous double auction Experiments Empirical results Discussion and conclusions

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The continuous double auction

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Orders and trades time order process trade process selection: market and limit orders double auction as thinning of a point process n: number of events from time origin up to time t ( ): probability density of waiting times between two events

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The trade process I The Poisson distribution is equivalent to exponentially distributed waiting times; in this case, the survival function is: The trade process is non-exponential; what about the order process?

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Interval 1 (9-11): 16063 data; 0 = 7 s Interval 2 (11-14): 20214 data; 0 = 11.3 s Interval 3 (14-17): 19372 data; 0 =7.9 s where 1 2 … n A 1 2 = 352; A 2 2 = 285; A 3 2 = 446 >> 1.957 (1% significance) The trade process II Scalas et al., Quantitative Finance (2004)

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Experiments

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Experiments I Experiments generalize Hellwig’s model (1982) Generalization based on Schredelseker (2002) Hellwig: traders do or do not know future 1-period dividends Schredelseker: there are n discrete information levels Kirchler/Huber/Sutter: n traders compete in a continuous double auction market; trader i knows future dividends D i for i periods; i = 1,…, n. She also knows the net present value of the stock given her information: I j,k : information in period k, for agent j; r e : risk adjusted interest rate

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Experiments II Beginning with I9, the functions in the Figure are shifted for each information level Ij by ( 9-j ) periods to the right, showing a main characteristic of the model, namely that better informed agents receive information earlier than less informed traders. So, information on the intrinsic value of the company that trader I9 sees in one period is seen by trader I8 one period later, and by trader I1 eight periods later, giving the better informed an informational advantage. For more details on the design, see Kirchler and Huber (2005). n = 9

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Experiments III Waiting-time survival functions of orders (dots) and trades (crosses) in five cases out of six, the measured data do not follow the exponential law both for orders and trades!

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Empirical results

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Empirical results I Order data from the LSE Full order book available for the electronic market Glaxo Smith Kline (GSK) and Vodafone (VOD) March, June and October 2002 Nearly 800,000 orders and 540,000 trades analyzed Both limit and market orders have been included

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Empirical results II There is always excess standard deviation; the null hypothesis of exponentially distributed data is always rejected.

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Empirical results III Waiting-time survival functions for orders (dots) and trades (crosses) in seconds for GSK, March 2002. The solid lines represent the corresponding standard exponential survival function

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Discussion and conclusions

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Discussions and conclusions Why should we bother? This has to do with the market price formation mechanism and with the order process. If the order process is modeled by means of a Poisson distribution (exponential survival function), its random thinning should yield another Poisson distribution. This is not the case! Moreover, experiments and empirical analyses show that already the order process cannot be described by a Poisson process. Researchers working in the field of agent-based market models are warned! Simple explanation: variable human activity (see Scalas et al., Quantitative Finance 4, 695-702, 2004).

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