One-dimensional reaction-diffusion and deposition-etching models adapted for stock-market simulations František Slanina Institute of Physics, Academy of Sciences of the Czech Republic, Prague (In collaboration with: Andrej Svorenčík, Sorin Solomon) ☻Order book phenomenology ☻Early models ☻Maslov model ☻Towards more realistic models ☻Back again: schematic models

Stylized facts Return distribution scales Hurst exponent Scaling function has power-law tail return distribution exponent (``inverse cubic law'') long-time autocorr. of abs. returns or

Order book Cisco: profile relative to current price [D. Challet and R. Stinchcombe, Physica A 300, 285 (2001).]

order size market orders (line: limit orders (line: [S. Maslov and M. Mills, Physica A 299, 234 (2001).]

order position (in ticks, relative to current price) Paris bourse line: NASDAQ line: [J.-P. Bouchaud, M. Mézard. and M. Potters, Quant. Finance 2, 251 (2002); M. Potters J.-P. and Bouchaud, Physica A 324, 133 (2003).]

average order density (tail ) [J.-P. Bouchaud, M. Mézard. and M. Potters, Quant. Finance 2, 251 (2002).] [R. Weber and B. Rosenow, cond-mat/ ] D. Farmer and F. Lillo, Quant. Finance 4, C7 (2004). NYSE and London SE (line: ) price impact virtual ( ) and actual ( ) impact

Origin of power-law tails in return distribution gaps vs. returns distribution (tail: ) [J. D. Farmer et al. cond-mat/ ]

Zero intelligence Dhananjay K. Gode; Shyam Sunder The Journal of Political Economy, Vol. 101, No. 1. (Feb., 1993), pp „Individual irrationality produces collective rationality“

Bak-Paczuski-Shubik Reaction-diffusion process for 2 species: A asks B bids,

Hurst H = 1/4 unrealistic. Diffusion unrealistic. Copying placement of orders: promising. Hurst plotReturn distribution

Stigler (1964) Bids and asks placed randomly in alloved interval

„free“ Stigler model: allowed interval follows price standard ( ) and free ( ) Return distribution Absolute return correlation Return distribution unrealistic H=0 unrealistic

„Genoa market model“ [Adapted after M. Raberto, S. Cincitti, S. M. Focardi, and M. Marchesi, Physica A 299, 319 (2001) and subsequent] price

Phase transition Phase diagram volatility

Hurst plot Stigler, free Stigler, Genoa

Maslov model deposition-etching process (+ evaporation) Local deposition: new limit orders at distance 1 from current price

Hurst plot Without evaporation and with evaporation probability

Return distribution: scaling inset :

Return distribution: no scaling No evaporation ( ) evaporation probability

Autocorrelation of absolute returns With and without evaporation

Mean-field for Maslov Mean-field approx.: homogeneous upper lower order volume potential price changes: Vector performs random matrix multiplicative process [F. Slanina, Phys. Rev. E 64, (2001)]

Probability distribution of price changes (after m limit and one market order): assuming power-law tail Non-trivial solution Why the exponent differs? Assumed constant density of orders. linear

Uniform deposition (Farmer) model Global deposition: uniform rate [M. G. Daniels, J. D. Farmer, L. Gillemot, G. Iori, and E. Smith, Phys. Rev. Lett. 90, (2003) and subsequent]

Return distribution Different evap. Prob. Different densities Different lags (no scaling!)

Hurst plot No scaling regime Autocorrelation No power law

Interacting gaps model [A. Svorenčík and F. Slanina, Eur. Phys. J. B 57, (2007).] “collision” of a pair of intervals 1.collapse with probability 2.reaction with probability

Histogram of returnsAutocorrelation of absolute returns

Mean-field for IGM Master equation for gap distribution Stationary solution 1.: exponential tail special case for Stationary solution 2.: Condensate + power-law tail

Conclusions BPS, Stigler, uniform deposition Genoa Maslov model with evaporation } candidates Outlook Improvement of schematic models Market maker – merge with Minority Game