Agent-based models for latent liquidity and concave price impact

We revisit the “$\varepsilon$-intelligence” model of Tóth et al. [Phys. Rev. X 1, 021006 (2011)], which was proposed as a minimal framework to understand the square-root dependence of the impact of meta-orders on volume in financial markets. The basic idea is that most of the daily liquidity is “latent” and furthermore vanishes linearly around the current price, as a consequence of the diffusion of the price itself. However, the numerical implementation of Tóth et al. (2011) was criticized as being unrealistic, in particular because all the “intelligence” was conferred to market orders, while limit orders were passive and random. In this work, we study various alternative specifications of the model, for example, allowing limit orders to react to the order flow or changing the execution protocols. By and large, our study lends strong support to the idea that the square-root impact law is a very generic and robust property that requires very few ingredients to be valid. We also show that the transition from superdiffusion to subdiffusion reported in Tóth et al. (2011) is in fact a crossover but that the original model can be slightly altered in order to give rise to a genuine phase transition, which is of interest on its own. We finally propose a general theoretical framework to understand how a nonlinear impact may appear even in the limit where the bias in the order flow is vanishingly small.

Figure 1

Impact of CFM trades on the GBP futures market, obtained by averaging over $3\times {10}^4$ meta-orders executed during the period 2008–2012. The full lines with symbols in the main plot correspond to two styles of execution of the meta-order (either with a mix of limit and market orders or exclusively with limit orders). The average impact in the two cases appears to be the same. The soft dashed lines plotted for comparison show power laws with exponents 0.5 and 1. The thick dashed line indicates the result of a power-law fit, with exponent $\delta =0.44$. In the inset we plot the intraday sign autocorrelation function for the same product averaged over all the trades of year 2012, exhibiting a power-law decay with exponent $\gamma \approx 0.76$.

Figure 2

Plot of the impact exponent $\delta$ against the persistence exponent $\gamma$ characterizing the autocorrelation of trade signs for a representative set of futures contracts. The impact exponent $\delta$ has been computed by using a data set consisting of approximately ${10}^6$ meta-orders, executed during the period 2008–2012. Buy orders and sell orders lead to very similar results. The exponent $\gamma$ is calculated by using intraday data for the year 2012, consisting of roughly ${10}^4$ trades per day per contract (marketwide). Note that there not seem to be any significant correlation between $\delta$ and $\gamma$, in disagreement with the model of Ref. [9], which predicts $\delta =\gamma$ (dashed line).

Figure 3

(Left) Phase diagram for the model in the regime $\mu =0.1{\text{s}}^{-1}$, $\lambda w=5\times {10}^{-3}{\text{s}}^{-1}$, $\nu ={10}^{-7}{\text{s}}^{-1}$. The diffusive nature of the model is assessed by considering the quantity $S=ln\left[\frac{1}{100}\frac{D(t={10}^3)}{D(t={10}^1)}\right]$, so the perfectly diffusive regime diffusion corresponds to $S=0$ (thick black line). (Right) The same quantity is plotted for the modified model in which the order consumption is controlled by the $\psi$ exponent for the same set of parameters. In both cases, for any value of $\gamma$ one can find a critical $\zeta$ beyond which the behavior of the model passes from superdiffusion to (apparent) subdiffusion.

Figure 4

(Left) Signature plots ${\sigma }_t^2$ for the parameter choice $\mu =0.1$ s${}^{-1}$, $\lambda w=5\times {10}^{-3}$ s${}^{-1}$, $\nu ={10}^{-7}$ s${}^{-1}$, $\gamma =0.5$, and different values of $\zeta$. Decreasing values of $\zeta$ lead to more strongly diffusive behavior. Note, however, that the long-time behavior is always superdiffusive in this case. (Right) Signature plots for the modified model with power-law volume consumption for various values of the $\psi$ exponent. The parameters set is the same one adopted in the left plot, except for the value of $\gamma =0.4$. The light gray lines correspond to the limiting cases $\psi =0$ and $\psi =1$. Note that, in this case, the long-time behavior is subdiffusive when $\psi \lesssim 0.8$, superdiffusive for larger values of $\psi$, and exactly diffusive for $\psi ={\psi }_c\approx 0.8$.

Figure 5

Histogram of the volume distribution for the best bid and ask prices, computed for various values of $\zeta$ (left plot) and of $\gamma$ (right plot). The curves on the left plot are almost exactly superposed, while the ones on the right plot share approximatively the same slope. The gray points on the right plot correspond to the case of independent market orders. The lines plotted for comparison show a power-law decay with exponent $-1.5$. The parameters used to obtain this figures correspond to the ones used to generate Fig. 3, so the maximum depth is given by $\lambda w/\nu =5\times {10}^4$.

Figure 6

(Top) Temporary impact for the execution of a meta-order in the case $\gamma =0.5$, ${\zeta }^{\prime }=\zeta =0.95$, for the set of parameters $\mu =0.1$ s${}^{-1}$, $\lambda w=5\times {10}^{-3}$ s${}^{-1}$, $\nu ={10}^{-7}$ s${}^{-1}$. The right plot shows the fitted exponent for the impact function under this particular execution schedule (solid line), compared with the ones corresponding to different execution protocols (dashed lines). Note that except for unit execution where concavity is weaker, the value of the impact exponent $\delta$ is compatible with empirical data. (Bottom) Temporary impact for the modified model in which the $\psi$ parameter controls the order consumption mechanism. We considered the case ${\psi }^{\prime }=\psi =0.75$ and $\gamma =0.4$ for which the model is approximately diffusive. The other parameters are set to $\mu =0.1$ s${}^{-1}$, $\lambda w=5\times {10}^{-3}$ s${}^{-1}$, $\nu ={10}^{-7}$ s${}^{-1}$. The right plot shows the fitted impact exponent. The results that we obtain for this model are very close to the ones reported above for the $\varepsilon$-intelligence model. The soft dashed lines in the top and bottom left panel are plotted for reference and indicate the scalings $\mathcal{I}\propto Q^{1/2}$ and $\mathcal{I}\propto Q$.

Figure 7

Dynamics of price impact during the execution of a trade of fixed duration $T=300$ ${\mu }^{-1}$ and stochastic executed quantity $Q$ for different participation rates. We have used $\gamma =0.5$, $\zeta ={\zeta }^{\prime }=0.95$, $\mu =0.1$ s${}^{-1}$, $\lambda w=5\times {10}^{-3}$ s${}^{-1}$, $\nu ={10}^{-4}$ s${}^{-1}$, while the simulation consisted of $3\times {10}^4$ realizations of the submission process. Even though the impact is concave at small times, it finally crosses over to a linear (in time) regime. The relaxation part is represented in semilogarithmic scale.

Figure 8

Average shape of the book at equilibrium before (thick solid line) and after (dashed lines) a long buy meta-order executed at a frequency $\varphi =0.0316$ with ${\zeta }^{\prime }=\infty$. We considered the same parameters as in Fig. 6, except for a larger value of $\nu ={10}^{-3}\mu$. The soft dashed line indicates the prediction of Eq. (8) for the unperturbed state of the book. The plot shows that after a long buy meta-order the ask levels are on average more populated than the bid ones. The presence of an offset between bid and ask volume in the perturbed case also evidences that the bulk of the book can store information about the past order flow.

Figure 9

Temporary impact induced by a meta-order executed through the deposition of limit orders. We have considered submissions of volumes equal to a fraction $f=1/2$ of the best and fixed the other parameters to $\mu =0.1$ s${}^{-1}$, $\lambda w=5\times {10}^{-3}$ s${}^{-1}$, $\nu ={10}^{-4}$ s${}^{-1}$. As in Fig. 6, soft dashed lines indicate the reference scalings $\mathcal{I}\propto Q^{1/2}$ and $\mathcal{I}\propto Q$.

Figure 10

Phase diagram for the model with correlated limit orders in the plane $\gamma ,\alpha$. We used the parameters $\zeta =0.4$, $\mu =0.1$ s${}^{-1}$, $\lambda w=5\times {10}^{-3}$ s${}^{-1}$, $\nu ={10}^{-4}$ s${}^{-1}$. The diffusion behavior is estimated as in Fig. 3. We again find, within this setting, a crossover line that separates sub- and superdiffusion regimes.

Figure 11

Temporary impact for the execution of a meta-order in a model with correlated limit orders. We have considered $\alpha =0.85$, $\gamma =0.5$, and ${\zeta }^{\prime }=\zeta =0.4$, corresponding to an approximately diffusive price dynamics. The choice of the other parameters is $\mu =0.1$ s${}^{-1}$, $\lambda w=5\times {10}^{-3}{\text{s}}^{-1}$, and $\nu ={10}^{-4}{\text{s}}^{-1}$. As in Fig. 6, we use soft dashed lines in order to display the reference scalings $\mathcal{I}\propto Q^{1/2}$ and $\mathcal{I}\propto Q$. Impact is strongly concave, with a weak dependence of the impact exponent on the participation rate. The inset shows the dependence of the impact exponent $\delta$ upon the participation rate $\varphi$.

Figure 12

Average price change during and after the execution of a meta-order for the same choice of parameters as in Fig. 7 with $T=3000{\mu }^{-1}$ (left plot) and with $T=300{\mu }^{-1}$ (right plot). The area under the curve represents the price impact of the meta-order. The figures show the transition to the asymptotic linear regime occurring for extremely long orders ($T\sim 1000{\mu }^{-1}$), as opposed to the initial transient regime which causes concave impact. Also notice that the dynamics always tends to decrease the average price change, as it is expected in a market in which arbitrage opportunities are reduced as a consequence of trading itself.