Figure 1

(Color online) Fraction of odd-valued spreads (in units of ticks) vs daily average spread for different stocks. We observe a systematic deviation from the symmetric case in which the fraction is 0.5. In the inset we plot a further average over a period of 80 days.Reuse & Permissions

Figure 2

(Color online) Conditional probability to have an odd-valued spread $s^{\prime }$ given an initial spread $s$ for real data. The plot shows clearly that with high probability an odd spread is followed by an even one (0.8) and vice versa.Reuse & Permissions

Figure 3

(Color online) Probability to have a variation $\Delta s$ of the spread given an initial $s$ for different spread values (empirical data). The most probable variation is always $\Delta s=1$. The probabilities of the other variations $\left(\Delta s=2,3,\dots \right)$ are weakly decreasing functions and, in first approximation, can be considered as constant. The dashed line is the power law with exponent 1.8 found in [24] for London Stock Exchange. In both cases $\Delta s=1$ is highly preferred.Reuse & Permissions

Figure 4

(Color online) Probability $\alpha \left(s\right)$ to place an order at the quote adjacent to the best one as a function of the spread. We have plotted this probabilities for three representative stocks and also an average over all the 20 stocks. The very different values observed can be explained by considering the different statistics of the stocks. To properly address this effect we have performed an average over the 20 stocks of our data set. The result is a value of $\alpha \left(s\right)$ approximately constant.Reuse & Permissions

Figure 5

(Color online) Comparison between the experimental data and the phenomenological model described in the text. The phenomenological probabilities of the model (dots) show a small systematic overestimation of the oscillations with respect to the experimental ones (triangles). This effect can be easily understood in terms of the approximation considered in Eq. (3).Reuse & Permissions

Figure 6

(Color online) Fraction of odd-valued spread vs average spread for experimental data and for two different Monte Carlo simulations. The first simulation (squares) is performed using the uniform order deposition mechanism. The second one (crosses) is instead performed with the nonuniform mechanism. These two simulations permit to investigate the two contributions to the asymmetry between odd and even spreads. As expected we observe that the intrinsic asymmetry does not reproduce the experimental pattern (dots) while the two combined effects fit the experimental data very well. The small discrepancy found for large spreads is originated by the approximation made in Eq. (3).Reuse & Permissions

Figure 7

(Color online) Phase diagram of the relative average spread variation for the nonuniform mechanism. The relative average spread variation $⟨\Delta s⟩/s$ is 0.5 for the uniform case. $⟨\Delta s⟩/s$ is larger than 0.5 only for spreads included in the region between $s=2$ and $s=\left(\alpha +1\right)/\alpha$. The highlighted region ($s\ge 3$ and $\alpha >0.5$) corresponds to a realistic scenario. We see that in this region the relative average variation in the spread produced by a limit order is always smaller than 0.5. Therefore the nonuniform deposition reduces the coupling action of limit orders with respect to the uniform case.Reuse & Permissions

Figure 8

(Color online) Statistical properties of the simulated order book for the uniform and nonuniform case. In panels (a) and (b) we plot the probability density functions for the returns $\left(\Delta p\right)$ and spreads, respectively, for different market order rates $\left(\pi \right)$. These plots show that the average fluctuations of spreads and returns are larger in the nonuniform case. When the order book turns to a compact regime $\left(\pi \le 0.25\right)$, the statistical properties of the model become nearly independent on the deposition details. Panel (c) reveals that the nonuniform deposition produces nontrivial fluctuations of liquidity/granularity. In panel (d) we plot the autocorrelation of the absolute values of returns and we observe the presence of persistent volatility. The decay of the autocorrelation function of the absolute values of returns is exponential except for short time lags where spurious effects take place. This persistent behavior suggests the presence of an intermittent dynamics for the order book characterized by bursts of volatility.Reuse & Permissions

Figure 9

(Color online) Snapshot of the simulated order book in the uniform case $\left(\pi =0.3\right)$. The order book is always in a compact regime in which the average spread is nearly 1 and a small and local volatility clustering is observed due to the autoregressive deposition rules. For higher values of $\pi$ small deviations from the compact regime appear but these phenomena cannot be compared to the intermittency produced when the order deposition is nonuniform.Reuse & Permissions

Figure 10

(Color online) Snapshot of the simulated order book in the nonuniform case ($\pi =0.3$ and $\alpha =0.7$). These deposition rules produce an order book which is typically quiet and compact as in the uniform case (regions I, III, and V) but exhibits bursts of activity due to large fluctuations of the spread (regions II, IV, and VI). The system gives rise to a sort of intermittency since volatility is very persistent and clustered.Reuse & Permissions

Figure 11

(Color online) Spread decay given $\Delta$ in the uniform case (solid lines) and in the nonuniform case (dashed lines). We observe a much slower decay in the latter case.Reuse & Permissions

Figure 12

(Color online) Spread decay in the nonuniform case for positive (top panel) and negative (bottom panel) values of $\Delta$. The value of $G\left(\tau \mid \Delta \right)$ is higher for higher absolute values of Δ. The patterns found in our model are very similar to the empirical results of Ponzi et al. in [24].Reuse & Permissions