# Stock price fluctuations and the mimetic behaviors of traders

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4 J.-i. Maskawa / Physica A 382 (2007) PDF Relative Limit price Fig. 2. Probability distribution function of the relative limit price. The results are shown for the three cases with p ¼ 0:3 (dotted line), p ¼ 0:4 (dashed line), p ¼ 0:5 (dot-dash line) and p ¼ 0:6 (solid line) PDF 10-5 Price shift Fig. 3. Probability distribution function of price shift of the surrogate data. The original data is generated by 100 times runs of 10,000 step iterations with p ¼ 0:5. The comparison with that of the original data (dashed line) is also given. of the tails have minima in the interval 0:45opo0:5, and p the ffiffi values are near 3 at p ¼ as given in the caption of Fig. 1. We use the Hill estimator of the largest n data, where n is the size of sample. We see from Fig. 2 that the relative limit price, namely the distance at which new limit orders are placed away from the current best price, is broadly distributed when p becomes large. We want to demonstrate,

5 176 ARTICLE IN PRESS J.-i. Maskawa / Physica A 382 (2007) however, that the broadness itself of the distribution of the relative limit price does not create the fat tail of the price shift distribution. First of all, for the purpose, we collect the data of the limit order by a numerical simulation of the model. Then we shuffle the order of arrivals of limit orders, and perform a simulation using the surrogate data instead of the limit price generated by the original rule of our model. The comparison of the resultant probability distribution of price shift of the surrogate data with that of the original data is shown in Fig. 3. The tail of the distribution does not show a power-law behavior, though the original data do. This experiment reveals that the information of orderbook plays a essential role in the decision of the price at which new orders are placed in our model. A similar role of the orderbook will be expected even in real markets, though the style of the reference to the orderbook is possibly different from that assumed here. 4. Autocorrelation functions of the model We derive the autocorrelation functions of price shift and of the absolute value of the price shift obtained by the numerical simulations of the model. The results are given in the panels of Fig. 4 including comparison with the autocorrelation functions of the surrogate data mentioned in the previous section. In those panels, the unit of time increment corresponds to a market order. The autocorrelation function of price shift almost vanishes except the value of time lag t ¼ 1 for both data. The values of the autocorrelation at time lag t ¼ 1 are 0:41 and 0:46, respectively. Those values are close to 0:5, and are explainable by the mean field approximation of the autocorrelation function as follows: let d 1 and d 2 denote the mean square root of price shift normalized by the standard deviation. The value d 1 measures the price shift across the spread, corresponding to the case that the side of the trade changes from bid to ask or from ask to bid. The value d 2 corresponds to the case that the side of the trade remains the same. If we assume that the four cases occurs with the same probability 1 4, the mean field approximation of autocorrelation functions gives the equation r i ¼hdp t dp tþi i=s 2 ¼ 1=4ðd 2 2 d2 1 Þd 1i. From the normalization condition d 2 1 þ d2 2 ¼ 2 and the inequality d 1 bd 2 (because spread always exists, while the trade successively occurred on the same side do not necessarily move the price), we have the result r i 0:5d 1i. The profile of the autocorrelation ACF ACF Time lag Time lag Fig. 4. Autocorrelation functions of price shift and of the absolute value of price shift obtained by the numerical simulations of the model. In both panels, the unit of time increment corresponds to a market order. Empty circle () represents the results for the original data, and filled circle () for the surrogate data mentioned in the previous section. (a) The autocorrelation function of price shift. (b) The autocorrelation of the absolute value of price shift with the power law fittings (solid lines). The exponents of the power law fittings are estimated by linear regression of the data plotted in log log plain. The result is 0:40 (R 2 ¼ 0:99) for the original data, and 0:60 (R 2 ¼ 0:79) for the surrogate data.

6 J.-i. Maskawa / Physica A 382 (2007) Var (p t -p 0 )/σ Free diffusion 1.0 (1+2A)t-2B Time (trades) Fig. 5. Empirical study of the price diffusion. We analyzed about 45 millions transaction data from November 1999 through October 2000 of active 5 IT or e-commerce companies (Intel, Microsoft, Amazon, Oracle, Cisco) listed on Nasdaq using TAQ Database. The theoretical line is also given, where A ¼ P t i¼1 r i and B ¼ P t i¼1 ir i. function is responsible for the value of the Hurst exponent H through the equation Varðp t p 0 Þ=s 2 ¼ Varð P t i¼1 dp iþ=s 2 ¼ t þ P t i¼1 ðt iþr it 2H. In such case of short memory as our model, we have the equationvarðp t p 0 Þ=s 2 ¼ Dtt 2H for tb1. In our case, the diffusion constant D ¼ 1 þ 2 P t i¼1 r i is quite small owing to the equation r i 0:5d 1i, and H ¼ 1 2 for large t. An empirical study of the price diffusion is presented in Fig. 5. We see from the panel (b) of Fig. 4 that the autocorrelation functions of the absolute value of price shift (empirical volatility) have long memory. Both data plotted there are well fitted by power laws. The original data, however, hold the memory of volatility stronger than the surrogate data do. 5. Conclusions Taking the strategy leaving the decision of the limit price to the others in the stochastic model of financial markets driven by continuous double auction, the virtual market shows the power-law tail of the distribution of returns with the exponent near 3 according to the parameter which determines the ratio of the mimetic limit order. The short memory of returns and the long memory of volatilities are also reproduced by the model. The Hurst exponent H of our model is asymptotically 1 2. The mean field approximation explains the profile of the autocorrelation function, which is responsible for the value of the Hurst exponent H. The strategy assumed here are effective in holding the memory of market volatility strong. Acknowledgment The author thanks D. Challet for attracting his notice to his paper. He learns a lot from him. References [1] X. Gabaix, P. Gopikrishnan, V. Plerou, H.E. Stanley, A theory of power law distributions in financial market fluctuations, Nature 423 (2003) 267. [2] J.D. Farmer, L. Gillemot, F. Lillo, S. Mike, A. Sen, What really causes large price change?, Quant. Finance 4 (2004) 383. [3] S. Maslov, Simple model of a limit order-driven market, Physica A 278 (2000) 571.

7 178 ARTICLE IN PRESS J.-i. Maskawa / Physica A 382 (2007) [4] P. Gopikrishnan, M. Meyer, L.A.N. Amaral, H.E. Stanley, Inverse cubic law for the distribution of stock price variations, Eur. Phys. J. B 3 (1998) 139. [5] V. Plerou, P. Gopikrishnan, L.A.N. Amaral, M. Meyer, H.E. Stanley, Scaling of the distribution of price fluctuations of individual companies, Phys. Rev. E 60 (1999) [6] D. Challet, R. Stinchcombe, Non-constant rates and over-diffusive prices in a simple model of limit order market, Quant. Finance 3 (2003) 155. [7] Y. Liu, P. Gopikrishnan, Cizeau, Meyer, Peng, H.E. Stanley, Statistical properties of the volatility of price fluctuations, Phys. Rev. E 60 (1999) [8] E. Smith, J.D. Farmer, L. Gillemot, S. Krishnamurthy, Statistical theory of the continuous double auction, Quant. Finance 3 (2003) 481. [9] A.-L. Barabasi, R. Albert, Emergence of scaling in random networks, Science 286 (1999) 509. [10] R. Albert, A.-L. Barabasi, Statistical mechanics of complex networks, Rev. Mod. Phys. 74 (2002) 47. [11] F. Slanina, Mean-field approximation for a limit order driven market model, Phys. Rev. E 64 (2001)

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