Price Impact Without Averaging

Figures & data

Table 1. LOBSTER labels of order book events.

1:	Submission of a new limit order
2:	Cancellation (partial deletion of a limit order)
3:	Deletion (total deletion of a limit order)
4:	Execution of a visible limit order
5:	Execution of a hidden limit order
6:	Indicates a cross trade, e.g., auction trade
7:	Trading halt indicator

Table 2. Mapping of LOBSTER labels to event types.

LOBSTER event label	Bid/Ask	Event type
1	ask	3
1	bid	4
2	ask	4
2	bid	3
3	ask	4
3	bid	3
4	ask	2
4	bid	1
5	ask	2
5	bid	1

Table 3. Ten time stamps from the filtered message file and order book file.

Time	Bid/Ask	LOBSTER event label	Event type	price	size
35400.092452	ask	4	2	460700	300
35400.092533	ask	1	3	460700	200
35400.092768	ask	4	2	460700	100
35400.113748	ask	4	2	460700	400
35400.113776	bid	1	4	460700	100
35400.121175	bid	4	1	460700	253
35400.121258	ask	1	3	460700	3300
35400.123294	ask	4	2	460700	100
35400.123334	ask	4	2	460700	6
35400.125010	ask	4	2	460700	14
Time	State	Ask price 1	Ask volume 1	Bid price 1	Bid volume 1
35400.092452	(1, 1)	460700	200	460600	900
35400.092533	(−1, 1)	460700	200	460600	1578
35400.092768	(0, 1)	460700	200	460600	1553
35400.113748	(1, 1)	460700	300	460600	1029
35400.113776	(1, −1)	460800	1600	460700	100
35400.121175	(−1, −1)	460800	1800	460700	153
35400.121258	(−1, 0)	460700	3300	460600	1029
35400.123294	(0, 1)	460700	352	460600	900
35400.123334	(0, 1)	460700	346	460600	900
35400.125010	(0, 1)	460700	332	460600	900

We show the first level of the limit order book for ease of exposition.

Table 4. Transition probabilities ${\varphi }_e$ calibrated on INTC as of 25 January 2019.

Probabilities of mid-price movements $X_1(T)\in \{-1,0,+1\}$ when event of type either 1 or 2 occurs at time T and queue imbalance $X_2(T-)$ is negative ($-1$), neutral (0) or positive ($+1$):
	$X_1$
event	$X_2$	−1	0	1
1	−1	0.476657	0.523343	0.000000
	0	0.410729	0.589271	0.000000
	1	0.346157	0.653843	0.000000
2	−1	0.000000	0.702554	0.297446
	0	0.000000	0.617759	0.382241
	1	0.000000	0.533823	0.466177

Full transition probability from $X(T-)=(X_1(T-),X_2(T-))$ to $X(T)=(X_1(T),X_2(T))$ when an execution happens on the bid side at time T.
		$X_1$	−1			0			1
		$X_2$	−1	0	1	−1	0	1	−1	0	1
event	$X_1$	$X_2$
1	−1	−1	0.176622	0.175752	0.000000	0.359614	0.283382	0.004630	0.0	0.0	0.0
		0	0.059468	0.279633	0.005352	0.077036	0.550193	0.028318	0.0	0.0	0.0
		1	0.009081	0.217948	0.071566	0.019519	0.433721	0.248165	0.0	0.0	0.0
	0	−1	0.287448	0.245331	0.003610	0.414895	0.048716	0.000000	0.0	0.0	0.0
		0	0.065844	0.350749	0.021704	0.083769	0.473489	0.004445	0.0	0.0	0.0
		1	0.005626	0.162936	0.130160	0.008709	0.242360	0.450209	0.0	0.0	0.0
	1	−1	0.257898	0.279307	0.004003	0.327795	0.130997	0.000000	0.0	0.0	0.0
		0	0.066807	0.357431	0.025199	0.091674	0.447274	0.011614	0.0	0.0	0.0
		1	0.013147	0.224193	0.203817	0.033174	0.353798	0.171872	0.0	0.0	0.0

Table 5. Hawkes parameters ${\nu }_e$, ${\alpha }_{e^{\mathrm{\prime }},x^{\mathrm{\prime }},e}$, and ${\beta }_{e^{\mathrm{\prime }},x^{\mathrm{\prime }},e}$ calibrated on INTC as of 25 January 2019.

	event 1	event 2	event 3	event 4
Base rates ${\nu }_e$, $e=1,\dots ,4$.
base rate	0.040201	0.050182	0.000735	0.000608
${\alpha }_{e^{\mathrm{\prime }},x^{\mathrm{\prime }},e}$ for $e=1,\dots ,4$ and $e^{\mathrm{\prime }}=1,\dots ,4$ when queue imbalance $X_2=0$ and $X_1=0$.
event 1	9.25	0.08	2.74	0.75
event 2	0.03	11.15	0.25	2.85
event 3	0.04	0.21	0.21	0.25
event 4	0.45	0.01	0.22	0.16
${\beta }_{e^{\mathrm{\prime }},x^{\mathrm{\prime }},e}$ for $e=1,\dots ,4$ and $e^{\mathrm{\prime }}=1,\dots ,4$ when queue imbalance $X_2=0$ and $X_1=0$.
event 1	14.71	5.32	7.22	5.64
event 2	4.61	16.92	4.92	7.92
event 3	1.57	1.92	3.41	2.84
event 4	3.53	2.70	2.84	3.56

Figure 1. Goodness-of-fit diagnostics for the model calibrated on INTC as of 25 January 2019. We test that the time-changed inter-arrival times of Equation (4(4) ${\tau }_e^j:=\mathrm{\Lambda }(T_j^e)-\mathrm{\Lambda }(T_{j-1}^e)$(4) ) are i.i.d. samples from a unit rate exponential distribution. Empirical quantiles of the time-changed inter-arrival times on the y-axis and theoretical quantiles of unit-rate exponential distribution on the x-axis.

Figure 2. Mid-price trajectories on two time scales. Origin of time is set at 9.55am 25 January 2019. Time is measured in seconds. Prices are in ${10}^{-4}$USD.

Table 6. Parameters of the liquidation schedule.

Name	Symbol	Formula
base rate	${\nu }_0$	Zero-th component of $\nu =({\nu }_0,{\nu }_1,\dots ,{\nu }_4)$
clustering rate	a	${\alpha }_{e^{\mathrm{\prime }},x,0}=a\cdot {\alpha }_{e^{\mathrm{\prime }},x,1}$
order size	c	$q_{M,j}=c\cdot \sum _{i=1}^n{V}_{T_j^0-}^{b,i}$

Figure 3. Price impact with low rate of walking the book and no clustering. Initial inventory $Q_0=10$, base rate ${\nu }_0=0.03$, clustering rate a = 0, order size c = 0.075, start time $t_0=0$, termination time $\tau =9250.2$, and price impact score is 0.0346.

Figure 4. Simulated trajectories of the intensity of the price impact profile, i.e., the map $t\mapsto \mathrm{Dir}(t)+\mathrm{Indir}(t)$ (see Definition 5.8) for the scenario considered in Figure 3. The black solid line is the median trajectory of the impact profile across simulations and the red dotted lines are the 25% and 75% quantile trajectories.

Figure 5. Left panel: box plots of the distribution of $\mathrm{Dir}(\tau )+\mathrm{Indir}(\tau )$ when stressing the order size parameter c. Right panel: box plots of the distribution of the price impact score as a function of the base rate ${\nu }_0$.

Figure 6. Price impact with high rate of walking the book and no clustering. Initial inventory $Q_0=10.0$, base rate ${\nu }_0=0.03$, clustering rate a = 0, order size c = 0.5, start time $t_0=0$, termination time $\tau =1060.7$, and price impact score is 0.0426.

Figure 7. Price impact with clustering of liquidator's orders and low rate of walking the book. We show the first 300 s. Initial inventory $Q_0=10.0$, base rate ${\nu }_0=0$, clustering rate a = 0.25, order size c = 0.015, start time $t_0=0$, termination time $\tau =722.3$, and price impact score is 0.999.