Scaling laws of strategic behavior and size heterogeneity in agent dynamics

We consider the financial market as a model system and study empirically how agents strategically adjust the properties of large orders in order to meet their preference and minimize their impact. We quantify this strategic behavior by detecting scaling relations between the variables characterizing the trading activity of different institutions. We also observe power-law distributions in the investment time horizon, in the number of transactions needed to execute a large order, and in the traded value exchanged by large institutions, and we show that heterogeneity of agents is a key ingredient for the emergence of some aggregate properties characterizing this complex system.

Figure 1

(Color online) Example of an inventory time series. The series refers to a particular firm trading Santander. The vertical lines indicate the position where our algorithm predicts the boundary between two patches. Directional patches are shown as red lines. Due to their statistical nature, in each patch there are buy (with a total traded value $V_b$) and sell (with a total traded value $V_s$) trades. We consider directional patches, i.e., patches where either $V_b/V>\theta$ (buy patch) or $V_s/V>\theta$ (sell patch), where $V=V_b+V_s$. For buy patches $V_m=V_b$ whereas for sell patches $V_m=V_s$. In the present study we set $\theta =75\%$. The black patches are not directional and are not considered in the rest of the paper.

Figure 2

(Color online) Distribution of $T$, $N_m$, and $V_m$ for the stocks Banco Bilbao Vizcaya Argentaria (BBVA), Banco Santander Central Hispano (SAN), and Telefónica (TEF). In the panel showing the distribution of $T$ we plot the distribution of packages reported in the literature on packages. Specifically, empty blue circles are results from Ref. [23] for packages traded at the New York Stock Exchange and empty magenta squares are results from Ref. [24] for packages traded at the Australian Stock Exchange.

Figure 3

(Color online) Scatter plots of the variables $T$, $N_m$, and $V_m$ for Telefónica. The black lines are contour lines of the bivariate probability density function. The insets show the probability density functions of the three exponents $g_1$, $g_2$, and $g_3$ describing the scaling relations of Eq. (1) computed on the patches of individual firms with at least ten patches. The red vertical lines indicate the values of the scaling exponents computed in the pool of all firms and reported in rows 4–6 of Table . It is worth noting that the dispersion of $g_2$ is significantly larger than that for the other two exponents.

Figure 4

(Color online) Scatter plot of the mean transaction volume in a patch $E\left[v\right]$ as a function of the total volume $V_m$ of the patch. The considered stock is Telefónica. The black line shows the mean value of $E\left[v\right]$ for points in equal-sized bins of $V_m$. Error bars are one standard deviation.

Figure 5

(Color online) Probability density function of the standardized logarithm of the variables $T$, $N_m$, and $V_m$ of the firms for which the Jarque-Bera test of log-normality cannot be rejected. Specifically, for each stock and each variable we consider the firms for which the log-normal hypothesis cannot be rejected (see Table and text). For each of these firms we compute the logarithm of the variable; we subtract the mean value and divide by the standard deviation. According to the null hypothesis these normalized variables should be Gaussian distributed. In the figure we plot on a semilogarithmic scale the probability density functions for each firm (continuous lines) and we compare them with the Gaussian probability density function (dashed line). Each column refers to a stock (from left to right, BBVA, SAN, TEF) and each row refers to a variable (from top to bottom $T$, $N_m$, and $V_m$).