Liquidity and Extreme Price Movements

Jonathan Brogaarda
Konstantin Sokolovb
Jiang Zhangb

We test the predictions of competing theories on liquidity dynamics during

extreme price movements (EPMs). Consistent with the models of the risk-
return trade-off of contrarian market making, modern liquidity providers
intensify limit order placement in the way of EPMs. This goes counter to
widespread concern that liquidity evaporates around EPMs. In contrast, a
typical EPM contains a disproportionally large pricing error that incentivizes
liquidity providers to step in. Finally, we demonstrate that limit order book
dynamics during EPMs is in line with a socially beneficial equilibrium.

We thank Andreas Park, Andriy Shkilko and Haoxiang Zhu and seminar participants at the University of Memphis
for insightful comments.
a
University of Utah, David Eccles School of Business, 1645 East Campus Center Drive, Salt Lake City, UT 84112,
USA, brogaard@eccles.utah.edu.
b
University of Memphis, Fogelman College of Business and Economics, Central Avenue, Memphis, TN 38152,
USA, ksokolov@memphis.edu and jzhang9@memphis.edu.
1. Introduction

There is concern that extreme price movements (EPMs) undermine investor confidence

and weaken the integrity of the securities markets (IOSCO, 2018). Three strands of literature

outline the incentives of liquidity provision in stressful times. The first strand emphasizes the

risks and constraints of liquidity provision implying that market makers should withdraw during

stressful times. The second strand discusses the profitability of contrarian strategies implying

that contrarian traders have the incentive to continuously accumulate inventory against the

direction of price pressures. The third strand combines the first two suggesting that market

makers have the incentive to allow for price pressures and earn higher compensation for risks

of liquidity provision through contrarian profits. Figure 1 illustrates the implications of these

literature strands for liquidity provision in EPMs. Our paper tests which of the three incentives

prevail during intraday EPMs.

[Figure 1]

According to the first strand of literature, liquidity providers may fail to accommodate

an uninformed order flow if it creates significant dislocations in their preferred holding

portfolios (Amihud and Mendelson, 1980). Large order flows may impede portfolio

diversification and set the liquidity provider’s inventory risk above the acceptable level.

According to Madhavan and Sofianos (1998), NYSE specialists maintain target inventory levels

by selectively withdrawing from liquidity provision on either the buy or sell side of the bid ask

spread. Market makers may be especially unwilling to hold a highly risky portfolio because the

stochastic nature of liquidity demand makes the portfolio holding period uncertain (Ho and Stoll,

2
1981 and Kondor, 2009). This effect may further aggravate the depletion of the limit order book

during EPMs. Comerton-Forde et al. (2010) support the abovementioned theoretical predictions

by showing that NYSE market makers provide less liquidity when their balance sheet revenues

decrease.

Price pressures, however, may incentivize liquidity providers to increase liquidity supply

in anticipation of higher contrarian profits. Although liquidity providers may not know if a

specific price movement will result in a permanent or transitory price dislocation, they may

observe that most price movements revert over time and engage in contrarian trading (Avramov,

Chordia and Goyal, 2006; Hendershott and Seasholes, 2007; Biais, Declerk and Moinas, 2016;

Gromb and Vayanos, 2018). According to Lehmann (1990) and Lo and MacKinlay (1990), the

expected reversible price component increases in the magnitude of the price movements. This

implies that large price movements should lead to high expected profits from contrarian trading

and create strong incentives for contrarian liquidity provision.

The literature suggesting that liquidity providers should withdraw during large price

movements and the literature on the profitability of contrarian trading are reconciled by papers

proposing the risk-return trade-off of contrarian liquidity provision. The trade-off between the

risks and contrarian profits of market making in volatile markets has been studied by Nagel

(2012). He estimates that contrarian trading becomes more profitable during volatile daily and

multiple-day intervals and suggests that, when uncertainty is high, liquidity providers scale back

and extract additional contrarian profits to compensate for the risks and constraints. In line with

this, So and Wang (2014) show that price reversals become more prevalent and contrarian

trading strategies become more profitable during intervals of high uncertainty around earnings

announcements. Hendershott and Menkveld (2014) develop a model predicting that the daily

3
reversible price component may become larger when risk-averse market makers rebalance their

inventory. As such, the literature on the risk-return trade-off of contrarian liquidity provision

suggests that the market is often in equilibrium where market makers have incentives to intensify

limit order placement only when the expected reversible price component is significant.

Our results indicate that liquidity providers scale back initially, but step in later on and

mitigate a typical EPM. The incentive to provide more liquidity during EPMs comes from the

uninformed nature of liquidity demand driving such events. This result is in line with the

predictions of Nagel (2012) and Hendershott and Menkveld (2014), who suggest that liquidity

providers allow for price pressures in stressful times and benefit from subsequent reversals. The

findings suggest that modern liquidity providers dynamically place limit orders consistently

with the risk-return trade-off strand of the theory literature.

We show that the liquidity dynamics during intraday EPMs is unique in several respects.

First, contrary to the widespread concerns, the total amount of depth placed in the way of the

EPMs exceeds the amount of depth cancelled. Second, EPMs typically contain an economically

and statistically large price pressure component and revert over time. Finally, net depth

placement intensifies toward the end of EPMs. As such, liquidity dynamics during intraday

EPMs is in line with the literature on the risk-return trade-off of contrarian liquidity provision.

A growing literature examines whether modern market participants can efficiently

counteract price pressures. On the one hand, several papers find that algorithmic liquidity

providers may mitigate intraday volatility spikes while making contrarian profits (Anand and

Venkataraman, 2016 and Brogaard et al., 2018). The finding that intraday algorithmic traders

act as contrarians during intraday price movements is supported by the literature suggesting that

algorithmic trading leads to more efficient intraday prices (Hendershott, Jones and Menkveld,

4
2011; Brogaard, Hendershott and Riordan, 2014; Conrad, Wahal and Xiang, 2015). Colliard

(2016) shows that although contrarian traders and market makers counteract short-term price

reversals and make prices more efficient in the short run, they may impede price discovery in

the long run. This camp suggests that the technological advancement of liquidity provision

pushes the markets to the equilibrium where intraday price movements are dominated by

competitive contrarian liquidity provision.

On the other hand, some studies show that algorithmic traders have incentives to exit the

market during large intraday price movements (van Kervel, 2015 and Kirilenko, Kyle, Samadi

and Tuzun, 2017). The finding that algorithmic traders have the potential to exacerbate price

fluctuations is consistent with the results of Korajczyk and Murphy (2018) and van Kervel and

Menkveld (2019), who show that high frequency traders switch from liquidity supply to liquidity

demand during the execution of large institutional trades. This camp suggests that the

technological advancement of liquidity provision pushes the markets to the equilibrium where

modern markets counteract price pressures inefficiently.

Our findings contribute to the evidence that overall the technological advancement of

liquidity provision allows the equilibrium where modern markets counteract the EPMs

efficiently. This paper is different from Brogaard et al. (2018), who show that high-frequency

traders provide liquidity during EPMs. We do not attempt to identify whether any specific type

of modern market participants provide liquidity in stressful times. Instead, our paper is first to

test whether during intraday EPMs the aggregate behavior of all types of modern market

participants is consistent with the theory of risk-return trade-off of contrarian liquidity provision.

The remainder of the paper is as follows. Section 2 describes the sample and outlines the

limit order book data and variables. Section 3 presents the main results. Section 4 discusses

5
alternative interpretations of the results and reports robustness tests and Section 5 concludes the

paper.

2. Data and variables

The sample comes from the NASDAQ TotalView-ITCH. This dataset covers order

submissions, cancellations, modifications and executions on the NASDAQ equity market with

nanosecond granularity. It does not, however, cover hidden order placement, which accounts

for 11.5% of the total trading volume. We address this concern in Section 4.3. The sample spans

two years from January 2017 to December 2018. Modern market makers have the highest

presence in the most active stocks. Thus, our sample includes the fifty largest stocks by market

capitalization as of December 2016.

We employ the methodology of Brogaard et al. (2018) to identify EPMs. Specifically,

for every stock, we rank all 10-second intervals between 9:35 a.m. and 3:55 p.m. by the

midquote return magnitude. Then we identify EPMs as intervals with an absolute return

exceeding the 99.99th percentile. This procedure draws the total of 5,649 EPMs with 49.27% of

them being positive. An average EPM return magnitude is almost thirty times greater than the

sample mean. Figure 2 plots the daily distribution of EPMs throughout the two-year sample

period.

[Figure 2]

6
 Academics, regulators, and market participants are often concerned that modern liquidity

providers exit when uninformed liquidity demand creates extreme price pressures. To shed some

light on this issue, we examine the dynamics of contrarian liquidity during EPMs. Our first

variable of interest is the net number of shares placed through non-marketable limit orders in

the way of a price movement.

𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 = 𝐶𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 − 𝐶𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑐𝑎𝑛𝑐𝑒𝑙𝑙𝑒𝑑,

where 𝐶𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 is the number of shares added to the limit order book layers

between the starting and ending best ask (bid) quotes of an interval with a positive (negative)

price change. 𝐶𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑐𝑎𝑛𝑐𝑒𝑙𝑙𝑒𝑑 is the number of standing shares cancelled at the

limit order book layers between the starting and ending best ask (bid) quotes of an interval with

a positive (negative) price change.1

Table 1 reports the descriptive statistics for the entire sample and for the subsample of

EPM intervals. Consistent with the reports showing that EPMs are typically triggered by large

liquidity demand, the volume traded in EPM intervals is twenty-five times greater than the

average. Yet, the average 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑, is greater during EPMs than non-

EPMs. As such, it appears that modern liquidity providers place more depth in the way of

liquidity demand as EPMs develop. In what follows, we elaborate on the significance of this

result and the economic rationale behind it.

1
We acknowledge that the 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 depends on the standing limit order book depth. If the
limit order book is depleted, then liquidity providers may have little depth to cancel. We alleviate this concern in
section 4.2.

7
 [Table 1]

3. Results

3.1 Does the limit order book depth shrink during EPMs?

The literature on contrarian trading suggests that liquidity providers have an incentive to

place more limit orders in the way of extreme price movements. In contrast, a large body of

microstructure research emphasizes the risks and constraints of counteracting large order flows

and predicts that liquidity providers may cancel standing limit orders to limit their exposure

during EPMs. In this section, we examine whether the incentives to intensify the limit order

book depth prevail over the incentives to shrink the limit order book depth during EPMs.

We begin by exploring the 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 during intraday price

movements of different magnitudes. Specifically, we sort all 10-second intervals by absolute

return magnitude and then average the 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 by return magnitude

percentile. Figure 3 illustrates the result. Consistent with the literature suggesting that

constrained liquidity providers should limit their exposure, the 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑

becomes increasingly negative with the magnitude of price movements. However, this trend

breaks at the 99.99th return magnitude percentile. The largest EPMs experience an average

inflow of liquidity of 1,766 shares per EPM. Therefore, it appears that liquidity dynamics in

modern markets is in line with the predictions of the literature on the risk-return trade-off of

contrarian liquidity provision.

8
 [Figure 3]

Next, we test the limits of liquidity provision during EPMs. Theory predicts that high

systematic volatility (Nagel, 2012, Cespa and Foucault, 2014 and Johnson, 2016) may

incentivize liquidity providers to reduce liquidity supply. This suggests that EPMs coinciding

with systematic volatility shocks should face lower contrarian liquidity provision.

We test these predictions by observing liquidity dynamics during EPMs coinciding with

volatility shocks. Specifically, we identify EPMs that coincide with the 10-second SPY ETF

returns exceeding the 99th percentile in magnitude (𝐻𝑖𝑔ℎ𝑆𝑦𝑠𝑡𝑉𝑜𝑙) and compare them with the

remaining EPMs (𝐿𝑜𝑤𝑆𝑦𝑠𝑡𝑉𝑜𝑙).2 Table 2, Panel A reports the results of such comparison. First,

the 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 increases with magnitude for both 𝐻𝑖𝑔ℎ𝑆𝑦𝑠𝑡𝑉𝑜𝑙 and

𝐿𝑜𝑤𝑆𝑦𝑠𝑡𝑉𝑜𝑙 EPMs. As such, modern liquidity providers do not leave the price unsupported,

even when the systematic volatility is intense. The results, however, suggest that liquidity

providers do scale back during high systematic volatility. In particular, the

𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 is positive for EPMs above the 99.99th return magnitude

percentile in the 𝐿𝑜𝑤𝑆𝑦𝑠𝑡𝑉𝑜𝑙 subgroup. In contrast, only EPMs exceeding the 99.9975th

percentile experience an increase in the 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 in the 𝐻𝑖𝑔ℎ𝑆𝑦𝑠𝑡𝑉𝑜𝑙

subgroup. As such, the results are consistent with the theoretical predictions that EPMs

coinciding with systematic volatility shocks should face lower contrarian liquidity provision.

Hendershott and Menkveld (2014) emphasize the relation between idiosyncratic price

pressures and market makers’ contrarian inventory. Intraday EPMs with a large idiosyncratic

2
Identification of EPM types with VXX ETF returns yield similar results.

9
component may incentivize liquidity providers to accumulate inventory in anticipation of

reversals. We explore this possibility by studying EPMs that happen on days with a large

idiosyncratic return component. First, we run the Fama and French (2015) five-factor regression

for the daily returns of every stock. Then, we extract regression residuals and identify EPMs

that happened on days with the absolute residual magnitude exceeding the 90th percentile

(𝐻𝑖𝑔ℎ𝐼𝑑𝑉𝑜𝑙) and compare them with the remaining EPMs (𝐿𝑜𝑤𝐼𝑑𝑉𝑜𝑙).3 The results in Table

2, Panel A are consistent with liquidity providers accumulating more inventory during

idiosyncratic price pressures.

[Table 2]

Although the 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 appears to be different across the EPM

types in the univariate setup, the variation of the 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 within the

EPM types can render this result statistically insignificant. We also acknowledge that the

omitted variables, such as volume, return and bid-ask spread, may bias our interpretation of the

univariate results. To address this concern, we run the following regression.

𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑𝑖𝑡 = 𝛼𝑖 + 𝛽1 𝐸𝑃𝑀𝑇𝑦𝑝𝑒𝐴,𝑖𝑡 + 𝛽2 𝐸𝑃𝑀𝑇𝑦𝑝𝑒𝐵,𝑖𝑡 + 𝐶𝑜𝑛𝑡𝑟𝑜𝑙𝑠𝑖𝑡 + 𝜀𝑖𝑡 ,

3
The overlap between 𝐻𝑖𝑔ℎ𝑆𝑦𝑠𝑡𝑉𝑜𝑙 (SPY ETF identification) and 𝐿𝑜𝑤𝐼𝑑𝑉𝑜𝑙 (five-factor model identification)
EPMs is 52%.

10
where 𝐸𝑃𝑀𝑇𝑦𝑝𝑒𝐴 and 𝐸𝑃𝑀𝑇𝑦𝑝𝑒𝐵 represent indicator variables corresponding to EPM subgroups

defined earlier in this section. 𝐶𝑜𝑛𝑡𝑟𝑜𝑙𝑠 include the volume, return and time-weighted

percentage spread. All variables are standardized by stock.

Panel B in Table 2 confirms the univariate results. The coefficients on EPM indicator

variables suggest that limit order depth dynamics during EPMs is indeed different from

dynamics during small and moderate returns. The F-tests show that 𝐿𝑜𝑤𝑆𝑦𝑠𝑡𝑉𝑜𝑙 and

𝐻𝑖𝑔ℎ𝐼𝑑𝑉𝑜𝑙 EPMs experience a significantly larger increase in the 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ

𝑝𝑙𝑎𝑐𝑒𝑑 than 𝐻𝑖𝑔ℎ𝑆𝑦𝑠𝑡𝑉𝑜𝑙 and 𝐿𝑜𝑤𝐼𝑑𝑉𝑜𝑙 EPMs. The overall results are consistent with

theories suggesting that modern liquidity providers have incentives to dynamically increase

limit order depth in the way of intraday EPMs. Moreover, shocks to systematic and idiosyncratic

volatility affect these incentives. In particular, liquidity providers accumulate smaller inventory

during systematic volatility shocks and larger inventory during idiosyncratic volatility shocks.

3.2 Why does the limit order book depth improve during EPMs?

A spike in liquidity demand may incentivize market makers to withdraw limit orders

thus shrinking the overall limit order book depth (Amihud and Mendelson, 1980; Ho and Stoll,

1981; and Madhavan and Sofianos, 1998). Indeed, the 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 is

negative for small and moderately large price movements (see Figure 3). The limit order book

dynamics is, however, drastically different during EPMs. In this section, we investigate the

properties of intraday EPMs that may incentivize modern market makers to improve limit order

depth.

11
 One potential explanation for the observed increase in the 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ

𝑝𝑙𝑎𝑐𝑒𝑑 is that EPMs typically contain a much larger pricing error component than smaller price

movements. The models of Nagel (2012), Hendershott and Menkveld (2014) and Gromb and

Vayanos (2018) show that liquidity providers have incentives to accommodate large liquidity

demand shocks if they can offload the accumulated inventory at a profit in the subsequent price

reversal.

Generally, contrarian strategies involve buying past losers and selling past winners. The

larger the magnitude of past positive (negative) returns, the larger the weight in the short (long)

portfolio the stock should have (Lehman, 1990, Lo and MacKinlay, 1990). In a dynamic market,

this implies that the weight of the stock in the portfolios of naïve contrarian traders will increase

as the stock price changes. Essentially, this means that naïve contrarian traders are more

confident that the higher-magnitude returns contain larger pricing errors than the lower-

magnitude returns.

Figure 4 illustrates this point with the notation of Hasbrouck (1991, 1993). The observed

price is a sum of two components: the unobserved price and the pricing error.

𝑞𝑡 = 𝑚𝑡 + 𝑠𝑡 ,

where 𝑚𝑡 is the true underlying price of a security and st is a transitory component (pricing

error). 𝑠𝑡 may be viewed as the size of current over-/underreaction to the past and current returns

and trades. Successful contrarian trading assumes that st is expected to be positive after positive

returns and negative after negative returns. In a competitive environment studied by Nagel

12
(2012), the willingness of a representative market maker to provide liquidity linearly depends

on the size of the pricing error. As the pricing error increases, the representative market maker

is more willing to provide liquidity to uninformed order flow. If large returns typically have a

large reversible component, then liquidity providers should be more willing to act as contrarians

against such returns.4

[Figure 4]

We explore the pricing error component of EPMs by looking at reversals surrounding

the EPM intervals. Specifically, we measure the returns ten minutes before and after the EPMs.5

Panel A in Table 3 lists the resulting statistics. Consistent with theory predictions, the EPMs

that experience the highest inflow of limit order depth contain the largest pricing error

component. EPMs with a magnitude below the 99.9925th percentile contain pricing errors of

only 14.2% of the EPM return. In contrast, EPMs with a magnitude above the 99.9975th

percentile contain pricing errors amounting to as much as 51.2% of the EPM return. To put this

in perspective, an average non-EPM contains the pricing error of 6.3%. As such, the results

suggest that large pricing errors incentivize liquidity providers to place limit orders in the way

of the EPMs despite the risks and constraints.

4
Indeed, Brogaard et al. (2018) show that latency advantage gives high-frequency traders an opportunity to update
their expectations faster and counteract EPMs before slow traders step in.
5
The results are robust to alternative estimation windows.

13
 [Table 3]

Numerous studies have shown that intraday returns have a negative autocorrelation.

Despite this, the 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 is negative for small and moderately large

price movements. The increase in the 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 during EPMs can be

attributed to disproportionally large reversals that follow the EPMs. We illustrate this point with

the following regression.

𝑅𝑒𝑣𝑒𝑟𝑠𝑎𝑙𝑃𝑜𝑠𝑡𝑡 = 𝛼𝑖 + 𝛽𝐸𝑃𝑀𝑖𝑡 + 𝐶𝑜𝑛𝑡𝑟𝑜𝑙𝑠𝑖𝑡 + 𝜀𝑖𝑡 ,

where 𝑅𝑒𝑣𝑒𝑟𝑠𝑎𝑙𝑃𝑜𝑠𝑡 is the magnitude of a return reversal within ten minutes following an EPM

interval and 𝐸𝑃𝑀𝑖𝑡 represents the indicator variable equal to one if an interval is identified as

an EPM and zero otherwise. 𝐶𝑜𝑛𝑡𝑟𝑜𝑙𝑠 include the volume, return and time-weighted percentage

spread. All variables are standardized by stock.

Table 3, Panel B reveals that EPMs contain disproportionally large pricing errors.

Consistent with the univariate results, EPMs contain a much larger reversible price component

than non-EPMs. Therefore, a naïve liquidity provider enjoys higher expected contrarian returns

by placing liquidity in the way of EPMs. This explains the rationale behind the increase in the

𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 during EPMs.

3.3 Is liquidity dynamics within EPMs socially beneficial?

14
 Thus far, we have been exploring how modern liquidity providers benefit from placing

limit orders in the way of EPMs. Nevertheless, these benefits may not necessarily improve the

aggregate welfare of market participants. In fact, the market may benefit if liquidity providers

do not accumulate large inventories at the onset of EPMs. In this section, we take one step further

and shed some light on the social value of liquidity dynamics within the EPMs.

Weill (2007) shows that social welfare may be improved if market makers do not exhaust

their capacity too early in the price movement. Instead, it is beneficial when market makers

allow liquidity demanders to consume surface layers of the limit order book with little

counteraction. Market makers should step in only after the price moves away from the

fundamental value. Thereby, liquidity providers are more likely to have enough resources to

counteract a price pressure and benefit from a subsequent reversal. In other words, the market

attains a socially beneficial outcome when liquidity providers intensify contrarian limit order

placement towards the end of EPMs.

We approach the challenge of assessing the social value of liquidity provision within

EPMs by exploring dynamics of the 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 as EPM return develops.

The socially optimal allocation of market maker capital implies that 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ

𝑝𝑙𝑎𝑐𝑒𝑑 will be low at the inception but increase towards the end of an EPM. Figure 5 shows

that indeed more of contrarian limit order book depth is cancelled then placed at the beginning

of EPMs. However, when the cumulative EPM return exceeds 23 bp, the amount of contrarian

depth cancelled breaks even with the amount of contrarian depth placed. Further on, the inflow

of liquidity exceeds the outflow until the end of a typical EPM.

15
 [Figure 5]

Next, we assess the economic and statistical significance of our findings by splitting

EPMs in thirds by return. Socially optimal allocation of liquidity providers’ capital implies that

the first one-third of an EPM return should experience less counteraction by liquidity providers

than the last one-third. The results in Table 4 show that liquidity providers do not typically step

in until the last one-third of EPM returns. On average, 221.3 shares are net cancelled in the first

one-third of EPMs and 543.7 shares are net placed in the last one-third of EPMs. This suggests

that modern liquidity providers intensify liquidity supply strategically towards the end of EPMs.

This is in line with the socially beneficial allocation suggested by Weill (2007).

[Table 4]

3.4.1 What are the alternatives to the prevailing dynamics of liquidity provision?

On average, contrarian limit order placement intensifies towards the end of EPMs. Yet,

not all EPMs cease in a price region with increased liquidity supply. Some EPMs may end as

liquidity deteriorates. Moreover, an EPM may lose momentum if liquidity demand subsides.

Finally, an EPM may stop if contrarian liquidity demanders step in and alleviate inventory

constraints of passive liquidity providers. In this section, we explore different scenarios of how

EPMs may end.

First, we compare EPMs that end with a positive and negative 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ

𝑝𝑙𝑎𝑐𝑒𝑑. A competitive liquidity supply allows investors to take positions in the market and thus

16
approach their desired portfolio allocation. If the cost of demanding liquidity is high, investors

may abstain from trading and continue to hold suboptimal portfolios (Karpoff, 1986). In fact,

Lo, Mamaysky and Wang (2004) show that a slight increase in transaction costs may lead to a

large untraded interest and lower aggregate market capitalization.

We split all EPMs into those experiencing a positive and negative 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛

𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 at the last one-third of the EPM return. Consistent with the prior results, the

EPMs with a positive 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 prevail (3,195 vs 2,454). The numbers in

Table 5, Panel A suggest that a higher amount of liquidity demand is absorbed in EPMs that end

with an increasing 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑. Consistent with the literature on trading

volume and liquidity, both the directional and total volume is 2.5 times larger in EPMs that

experience an increase in liquidity towards their end.

[Table 5]

Next, we look at EPMs that experience an increase or decline in directional liquidity

demand towards their end. In some cases, an EPM may lose momentum as the liquidity demand

is exhausted. We explore such a possibility by splitting EPMs into ones that show an increase

in directional volume in the last one-third of EPM return and the remaining ones. Panel B of

Table 5 reports the resulting statistics. The majority of EPMs show a decline in the directional

liquidity demand towards their end (3,536 vs 2,113). For both types of EPMs, the

𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 is positive for the largest of the price movements.

17
 Finally, we explore EPMs that end with an increase in contrarian liquidity demand. If

the size of the expected pricing error exceeds the bid-ask spread, then active contrarian trading

strategies become viable. Contrarian liquidity demand allows passive liquidity providers to

offload inventory thus improving their capacity to counteract a price pressure. As such, a

contrarian trader can be qualified as a liquidity supplier.6

Following this intuition, we split EPMs into two subgroups by contrarian volume. The

first subgroup includes EPMs with increasing contrarian volume in the last one-third of return.

The second subgroup includes EPMs with decreasing contrarian volume in the last one-third of

return. The statistics for both subgroups are in Table 5, Panel C. In line with the economic

intuition described above, EPMs that end with a higher contrarian volume appear to enjoy a

larger 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑. This suggests that contrarian liquidity demanders

mitigate price pressures.

Panel D of Table 5 reports the aggregate statistics across EPMs and non-EPMs. On

average, the contrarian volume constitutes 26% of the total EPM volume and 22% of the total

non-EPM volume. The mean 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 amounts to 337 shares in EPMs

and a negative 368 shares in non-EPMs. Overall, the results point to the conclusion that, despite

occasional liquidity deteriorations, contrarian liquidity supply intensifies during EPMs.

3.4.2 What affects the EPM termination?

In this subsection we take one step further and explore how different market conditions

and stock characteristics affect the way EPMs terminate. First, we look at EPMs that happen

6
https://albertjmenkveld.com/2015/02/24/who-supplies-liquidity-we-need-a-new-definition/

18
around earnings announcement days. According to So and Wang (2014), contrarian liquidity

providers generate more revenue around earnings announcements if they hold inventory for

multiple days. Modern liquidity providers, however, often avoid keeping overnight inventories.

As such, the findings of So and Wang (2014) do not necessarily apply to the market participants

who run liquidity providing strategies. To address this concern, we test whether EPM liquidity

dynamics is consistent with the implications of So and Wang (2014).

We identify the announcement EPMs as those occurring in the three-day window around

the earnings announcement in a stock. Although these days constitute only 1.6% of the entire

sample, as many as 23.9% of all EPMs happen on such days. Table 6 lays out the details of

liquidity dynamics for announcement EPMs. Overall, the results are in line with the predictions

that liquidity providers are more likely to engage in contrarian trading around earnings

announcements than on the other days. Although the 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 is positive

for both announcement and non-announcement EPMs, the former experience more intense

inflow of liquidity. Moreover, the announcement EPMs are more likely to end with increasing

liquidity supply, decreasing liquidity demand in the direction of EPMs and increasing liquidity

demand against the direction of EPMs. The results are robust in both univariate and regression

settings.

[Table 6]

Next, we explore the dynamics of EPM termination in stocks with different

characteristics. Specifically, we ask whether relative tick size affects market making incentives

19
during EPMs. The empirical literature provides mixed evidence on whether market makers

improve liquidity in tick-constrained stocks (Angel, 1997 and Schultz, 2000). We take one step

further and shed some light on liquidity provision to EPMs in tick-constrained stocks.

An expensive stock with a quoted spread of several ticks will be less constrained and

experience smoother price path than a cheap stock with a one-tick spread. The following

procedure identifies stocks that plausibly experience tick-constrained price movements. First,

we sort all stock-month observations by the percentage of time a stock had one-cent spread

within the given month. Then, we identify the stocks with quoted spread equal to one cent more

than 80% of the time within a given month as tick-constrained stocks. Finally, we split EPMs

into those occurring in the tick-constrained stocks (tick-constrained EPMs) and the remaining

stocks (not tick-constrained EPMs).

Our findings in the Table 6 indicate that tick-constrained EPMs experience more

significant liquidity improvement than not tick-constrained EPMs.7 Tick-constrained EPMs are

more likely to end with decreasing liquidity supply, increasing liquidity demand in the direction

of EPMs and decreasing liquidity demand against the direction of EPMs. As such, our results

indicate that in stressful times market makers have lower incentives to supply liquidity in more

tick-constrained stocks.

3.5 Can a slow trader anticipate EPMs?

7
𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 appears to be larger in tick-constrained EPMs because tick-constrained stocks
are relatively cheaper. The average monthly stock price is $53.8 in the tick-constrained stocks and $196.2 in the
not tick-constrained ones. This bias is corrected in the regression with standardized variables and stock fixed
effects.

20
 Regulators are often concerned that extreme volatility events undermine investor

confidence (IOSCO, 2018). In particular, an investor may face unexpectedly large transaction

costs if an EPM happens at the time of order execution. Consistent with this, Korajczyk and

Murphy (2018) and van Kervel and Menkveld (2019) show that some low-latency liquidity

providers may exacerbate the implementation shortfall of institutional trades. We take one step

further and examine whether a relatively slow liquidity demander can foresee an EPM by

observing the limit order book depth dynamics.

According to Brogaard, Hendershott and Riordan (2019), between 15 and 30% of price

discovery is attributable to limit order placement and cancellation. As such, the updates in the

limit order book depth are informative of upcoming price changes. We explore whether a

relatively unsophisticated and slow liquidity demander can anticipate the size of the coming

EPM. Therefore, in the following analysis, we intentionally avoid any advanced statistical

techniques and apply a rather intuitive approach. Specifically, we estimate the ex-ante

probability that the coming EPM ends at the price level where the limit order book depth has

been building up in advance.

We start by identifying EPM ending quotes. For instance, for a positive EPM, this is the

best ask quote at the end of the EPM interval. Then we estimate the 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ

𝑝𝑙𝑎𝑐𝑒𝑑 for the EPM ending quotes. Next, we assign the value one if the EPM ends within a

certain range8 from the EPM ending quote and zero otherwise. Averaging these values gives us

the ex-ante probability that the EPM will end at the price region with the quote with the highest

8
The ranges include 0.01, 0.05 and 0.1% of the price.

21
𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑. We perform these calculations for each of the twenty seconds

before the end of the EPM intervals.

The results reveal that past limit order book depth dynamics is informative of the size of

the upcoming price movements. In the ten seconds before the onset of an EPM, there is a roughly

30% probability that the EPM will end within 0.1% of the limit order book layer with the highest

𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 (Table 7). This suggests that a liquidity demander with a ten-

second latency can anticipate an EPM and time their trades accordingly.

[Table 7]

To put this in perspective, we estimate the frequency of instances when the quote with

the highest 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 falls within the region close to the ending price of

an average EPM. The mean EPM return amounts to 0.382%. To calculate the unconditional

frequency of the limit order book depth buildup around the EPM ending price, we estimate the

percentage of all price movements when the quote with the highest 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ

𝑝𝑙𝑎𝑐𝑒𝑑 is within 0.1, 0.05 and 0.01% of 0.382%. The resulting estimates differ from what we

observe around EPMs. The probability of the quote with the highest 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ

𝑝𝑙𝑎𝑐𝑒𝑑 falling within the region close to the ending price of an average EPM is 10 to 15 times

higher than average in the 10 seconds before the onset of an EPM.

4. Alternative interpretations and robustness

4.1 The role of trade suspensions in EPM liquidity dynamics

22
 In this section, we examine the role of trade suspensions (halts) in liquidity dynamics

during EPMs. Greenwald and Stein (1991) theoretically show that trading halts can mitigate

EPMs. In line with the theory, Corwin and Lipson (2000) find that halts incentivize liquidity

providers to replenish the limit order book depth. Therefore, the increase in the

𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 documented in our paper can be a result of trading halts

triggered by EPMs rather than a natural liquidity providers’ behavior. In what follows, we

explore this alternative interpretation of our findings.

In our sample stocks, a trading halt is triggered when a price moves by more than 5%

from the average execution price within the past five minutes.9 Although EPMs observed over

a ten-second window do not typically exceed 0.5%, they can still trigger a halt if the price moves

in the EPM direction in the five minutes preceding the EPM. When such a halt happens, the

TotalView-ITCH data feed reports the corresponding message. We use these messages to

identify EPMs that coincide with trading halts. According to Jiang, McInish and Upson (2009),

trading halts have a spillover liquidity effect across the market. Thus, we split our sample into

EPMs that coincided with a trading halt in at least one of the sample stocks (halt EPMs) and the

remaining EPMs (non-halt EPMs). We report the result of this split in Table 8.

[Table 8]

9
www.nasdaqtrader.com/content/MarketRegulation/LULD_FAQ.pdf describes the detailed protocol for
transition into a trading pause mode.

23
 Although EPMs occasionally coincide with trading halts, such events amount to only

18% of the sample. The results indicate that halt EPMs experience more significant

improvement in liquidity than non-halt EPMs. Nevertheless, the dynamics of the

𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 suggests liquidity improvement in both halt and non-halt

EPMs. Hence, the overall liquidity dynamics in EPMs is unlikely to be solely driven by trading

halts.

4.2 Reservations about Net contrarian depth placed

We estimate the 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 as the difference between the number

of shares placed and cancelled between the starting and ending best quote in the direction of a

price movement. As such, the 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 depends on the standing limit

order book depth before the onset of an EPM. If liquidity demand consumes many layers of the

limit order book, it may become depleted. As a result, few standing limit orders remain and

market participants may only add liquidity.

We address this concern by exploring the limit order book depth right before the start of

the EPMs. In section 3.5, we document that liquidity builds up around the future ending quotes

of EPMs. Table 9 further explores the limit order book in advance of EPMs. Specifically, we

trace the limit order book depth on the side of oncoming EPMs within 1, 0.5 and 0.01% of the

price over the five-minute interval.

[Table 9]

24
 The numbers indicate that a significant amount of depth builds up before the onset of an

average EPM. Specifically, 8,683 shares rest at the layers up to 0.5% deep and 14,798 shares up

to 1% deep. The average EPM magnitude is 0.38% and the largest of EPMs do not typically

exceed 0.55%. Therefore, liquidity demand during EPMs does not leave the limit order book

depleted. There is typically a large amount of resting depth that could be cancelled. As such, the

increase in the 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 during EPMs is not caused by the lack of

standing limit orders.

4.3 Does hidden volume shrink during EPMs?

Modern liquidity providers may choose to place hidden liquidity to avoid revealing their

trading intentions. As a result, liquidity demanders face uncertain execution costs. In our sample,

11.5% of trading volume comes from executions against unobserved liquidity. In this section,

we estimate whether hidden liquidity shrinks during EPMs.

TotalView-ITCH does not report hidden order placement. As such, the data does not

allow us to compute the 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 for hidden liquidity. Instead, we adopt

an alternative measure that shows whether a liquidity demander ends up enjoying a relatively

larger amount of hidden depth during EPMs. Specifically, we estimate the ratio of volume

executed against hidden liquidity to the total volume traded in the direction of a price movement.

𝐶𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛𝐻𝑖𝑑𝑑𝑒𝑛𝐷𝑒𝑝𝑡ℎ
𝐻𝑖𝑑𝑑𝑒𝑛𝑅𝑎𝑡𝑖𝑜 = ,
𝑇𝑜𝑡𝑎𝑙𝐶𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛𝐷𝑒𝑝𝑡ℎ

25
where 𝐶𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛𝐻𝑖𝑑𝑑𝑒𝑛𝐷𝑒𝑝𝑡ℎ is the number of shares that hit hidden limit orders placed in

the direction of return and 𝑇𝑜𝑡𝑎𝑙𝐶𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛𝐷𝑒𝑝𝑡ℎ is the total number of shares that hit both

hidden and lit limit orders placed in the direction of return.

The average 𝐻𝑖𝑑𝑑𝑒𝑛𝑅𝑎𝑡𝑖𝑜 in the sample amounts to 7.78%. However, during EPMs,

the proportion of executions against contrarian hidden orders surges to 17.61%. Thus, liquidity

demanders enjoy improved hidden liquidity during EPMs. We report the detailed statistics for

EPM subgroups in Table 10, Panel A.

[Table 10]

Finally, we test the results in a regression setting and run the following model.

𝐻𝑖𝑑𝑑𝑒𝑛𝑅𝑎𝑡𝑖𝑜𝑡 = 𝛼𝑖 + 𝛽𝐸𝑃𝑀𝑖𝑡 + 𝐶𝑜𝑛𝑡𝑟𝑜𝑙𝑠𝑖𝑡 + 𝜀𝑖𝑡 ,

where all variables have been previously defined and 𝐶𝑜𝑛𝑡𝑟𝑜𝑙𝑠 include the return, volume and

spread.

The results in Table 10, Panel B confirm the univariate statistics. The probability that a

market order submitted in the direction of a price movement hits a hidden limit order is

significantly higher during EPMs. As such, liquidity demanders discover more hidden liquidity

during EPMs.

26
 5. Conclusion

When a price moves in a certain direction, liquidity providers face two opposing

incentives. The first is to accumulate inventory in anticipation of a price reversal. The second is

to withdraw due to capital constraints, inventory and adverse selection risks. We find that the

incentive to accumulate inventory is stronger during intraday extreme price fluctuations. This

finding alleviates concerns that prices are subject to periods of extreme volatility due to

systematic liquidity withdrawals. Contrary to these concerns, liquidity providers appear

sufficiently incentivized to dampen intraday volatility.

Academic researchers and regulators often voice concerns that liquidity providers may

manage their capital in a socially inefficient way. These concerns are fostered by the literature

that suggests that market makers tend to overcommit their resources at the inception of price

pressures and lack the resources to counteract such pressures as they develop further. We shed

some light on this issue by looking at limit order book depth dynamics during extreme price

movements. We find that modern liquidity providers place limit orders strategically and step in

only after uninformed liquidity demand has created substantial price pressures. Therefore,

liquidity provision in modern markets is consistent with the socially beneficial equilibrium.

Although contrarian liquidity provision increases in the magnitude of intraday price

movements, we find some evidence consistent with the scaling back of liquidity providers

proposed by the literature on evaporating liquidity. Specifically, extreme price fluctuations face

lower counteraction by contrarian liquidity providers when they coincide with high systematic

volatility events. In contrast, liquidity providers accumulate larger inventories during high

systematic volatility.

27
 Finally, we show that limit order book dynamics around extreme price movements make

the magnitude of these movements predictable. The literature on high-frequency trading shows

that fast algorithms learn about the trading intentions of slow investors and exacerbate their

execution costs. In line with this, regulators are often concerned that slow investors may end up

facing excessive execution costs if they trade in the direction of an extreme price movement.

We approach this issue from a new angle and show that a relatively slow and unsophisticated

market participant can anticipate oncoming extreme price movements and time their trades

accordingly.

28
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31
Table 1: Summary statistics

The table reports descriptive statistics for the sample of 50 largest stocks by market capitalization traded on
NASDAQ. The sample spans January 2017 to December 2018. Panel A reports statistics for the full sample of 10-
second intervals, while Panel B reports statistics for the subsample of intervals with a return magnitude exceeding
the 99.99 percentile. Return is the absolute midquote-to-midquote return, Contrarian depth placed is the number of
shares added to the limit order book layers between the starting and ending best ask (bid) quotes of an interval with
a positive (negative) price change, Contrarian depth cancelled is the number of standing shares cancelled at the limit
order book layers between the starting and ending best ask (bid) quotes of an interval with a positive (negative) price
change. Net contrarian depth placed is the difference between the Contrarian depth placed and the Contrarian depth
cancelled. Directional volume is the total number of shares traded in the direction of the interval return. Contrarian
volume is the total number of shares traded against the direction of the interval return. Total volume is the sum of the
Directional volume and Contrarian volume. Spread is the time-weighted quoted spread.
Panel A: All price movements Mean Median Std
Return, bp 1.28 0.57 1.98
Contrarian depth placed 1,536.91 100.00 7,658.95
Contrarian depth cancelled 1,904.87 100.00 8,214.15
Net contrarian depth placed -367.96 0.00 1,602.50
Directional volume 358.42 0.00 1,517.99
Contrarian volume 99.88 0.00 587.82
Total volume 458.31 0.00 1,859.11
Spread, bp 2.72 2.16 2.15
N 56.6M
Panel B: EPMs
Return, bp 38.18 32.94 20.09
Contrarian depth placed 23,109.48 6,810.00 57,208.97
Contrarian depth cancelled 22,772.47 6,921.00 55,452.16
Net contrarian depth placed 337.01 -112.00 7,204.76
Directional volume 8,375.82 2,369.00 21,738.42
Contrarian volume 2,963.99 630.00 7,968.07
Total volume 11,339.81 3,178.00 27,945.80
Spread, bp 12.96 10.12 11.21
N 5,649

32
Table 2: Net liquidity placed during EPMs
Panel A reports univariate statistics for the Net contrarian depth placed (defined in Table 1) during EPMs. The column 𝐴𝑙𝑙 𝐸𝑃𝑀𝑠
contains the entire sample of EPMs. 𝐻𝑖𝑔ℎ𝐼𝑑𝑉𝑜𝑙 contains the EPMs occurring on the days with the residual of Fama and French
(2015) 5-factor model exceeding the 90th percentile in magnitude, while 𝐿𝑜𝑤𝐼𝑑𝑉𝑜𝑙 contains the remaining EPMs. 𝐻𝑖𝑔ℎ𝑆𝑦𝑠𝑡𝑉𝑜𝑙
contains the EPMs that coincide with the SPY ETF returns exceeding the 99th percentile in magnitude and 𝐿𝑜𝑤𝑆𝑦𝑠𝑡𝑉𝑜𝑙 contains the
remaining EPMs. 𝑟99.9925, 𝑟99.9950, 𝑟99.9975 and 𝑟100 denote EPMs with return magnitude percentiles between 99.99 and
99.9925, 99.9925 and 99.9950, 99.9950 and 99.9975 and 99.9975 and 100, respectively. Return magnitude subgroups contain roughly
1,400 EPMs each. Panel B reports the 𝛽1 and 𝛽2 coefficients of the following model: 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑𝑖𝑡 = 𝛼𝑖 +
𝛽1 𝐸𝑃𝑀𝑇𝑦𝑝𝑒𝐴,𝑖𝑡 + 𝛽2 𝐸𝑃𝑀𝑇𝑦𝑝𝑒𝐵,𝑖𝑡 + 𝐶𝑜𝑛𝑡𝑟𝑜𝑙𝑠𝑖𝑡 + 𝜀𝑖𝑡 , where 𝐸𝑃𝑀𝑇𝑦𝑝𝑒𝐴 and 𝐸𝑃𝑀𝑇𝑦𝑝𝑒𝐵 represent indicator variables corresponding to
the EPM subgroups defined above. 𝐶𝑜𝑛𝑡𝑟𝑜𝑙𝑠 include the volume, return and time-weighted percentage spread. All variables are
standardized by stock. P-values corresponding to double-clustered standard errors are in parentheses. *, ** and *** indicate
significance at the 90th, 95th and 99th confidence levels.
𝐴𝑙𝑙 𝐸𝑃𝑀𝑠 𝐻𝑖𝑔ℎ𝐼𝑑𝑉𝑜𝑙 𝐿𝑜𝑤𝐼𝑑𝑉𝑜𝑙 𝐻𝑖𝑔ℎ𝑆𝑦𝑠𝑡𝑉𝑜𝑙 𝐿𝑜𝑤𝑆𝑦𝑠𝑡𝑉𝑜𝑙
Panel A: Univariate statistics
𝑟99.9925 -303.8* 153.5 -633.4** -1,354.9*** 458.3*
(0.08) (0.51) (0.01) (0.00) (0.07)
𝑟99.9950 -318.7** 286.6 -787.7*** -1,108.6*** 238.7
(0.01) (0.13) (0.00) (0.00) (0.19)
𝑟99.9975 227.9 471.4 53.7 -719.7*** 890.0***
(0.17) (0.10) (0.79) (0.00) (0.00)
𝑟100 1,765.9*** 1,943.8*** 1,629.7*** 688.5** 2,581.0***
(0.00) (0.00) (0.00) (0.04) (0.00)
𝑁 5,649 2,409 3,240 2,367 3,282
Panel B: Regressions
𝐸𝑃𝑀𝑎𝑙𝑙 9.48***
(0.00)
𝐸𝑃𝑀𝐻𝑖𝑔ℎ𝐼𝑑𝑉𝑜𝑙 10.00***
(0.00)
𝐸𝑃𝑀𝐿𝑜𝑤𝐼𝑑𝑉𝑜𝑙 9.09***
(0.00)
𝐸𝑃𝑀𝐻𝑖𝑔ℎ𝑆𝑦𝑠𝑡𝑉𝑜𝑙 8.62***
(0.00)
𝐸𝑃𝑀𝐿𝑜𝑤𝑆𝑦𝑠𝑡𝑉𝑜𝑙 10.10***
(0.00)
𝐻: 𝐸𝑃𝑀𝑇𝑦𝑝𝑒𝐴 = 𝐸𝑃𝑀𝑇𝑦𝑝𝑒𝐵 𝑅𝑒𝑗𝑒𝑐𝑡𝑒𝑑*** 𝑅𝑒𝑗𝑒𝑐𝑡𝑒𝑑***
(0.00) (0.00)
2
𝐴𝑑𝑗. 𝑅 0.239 0.239 0.239

33
Table 3: EPMs and reversals

Panel A reports the average absolute return magnitude in the EPM subgroups (EPM subgroups are defined in Table
2) and the magnitude of reversals within 10 minutes before (𝑅𝑒𝑣𝑒𝑟𝑠𝑎𝑙𝑃𝑟𝑒) and after (𝑅𝑒𝑣𝑒𝑟𝑠𝑎𝑙𝑃𝑜𝑠𝑡) the EPM
intervals. Panel B reports the 𝛽 coefficient of the following model: 𝑅𝑒𝑣𝑒𝑟𝑠𝑎𝑙𝑃𝑜𝑠𝑡𝑡 = 𝛼𝑖 + 𝛽𝐸𝑃𝑀𝑖𝑡 +
𝐶𝑜𝑛𝑡𝑟𝑜𝑙𝑠𝑖𝑡 + 𝜀𝑖𝑡 , where 𝐸𝑃𝑀𝑖𝑡 represents the indicator variable equal to one if an interval is identified as an EPM
and zero otherwise. 𝐶𝑜𝑛𝑡𝑟𝑜𝑙𝑠 include the volume, return and time-weighted percentage spread. All variables are
standardized. P-values corresponding to double-clustered standard errors are in parentheses. *, ** and *** indicate
significance at the 90th, 95th and 99th confidence levels.

Panel A: Univariate statistics Panel B: Regression

𝑅𝑒𝑡𝑢𝑟𝑛, % 𝑅𝑒𝑣𝑒𝑟𝑠𝑎𝑙𝑃𝑟𝑒, % 𝑅𝑒𝑣𝑒𝑟𝑠𝑎𝑙𝑃𝑜𝑠𝑡, %
𝑟99.9925 0.295 -0.006 0.048** 𝐸𝑃𝑀 0.388***
(0.88) (0.03) (0.00)
2
𝑟99.9950 0.321 0.090* 0.065** 𝐴𝑑𝑗. 𝑅 0.0001
(0.06) (0.01)
𝑟99.9975 0.365 0.087 0.065**
(0.10) (0.03)
𝑟100 0.551 0.165** 0.117***
(0.01) (0.00)

34
Table 4: Liquidity dynamics within EPMs
This table splits individual EPMs into thirds of the price change and compute the Net contrarian depth placed (defined in Table
1) for each third. P-values are reported in parentheses. *, ** and *** indicate significance at the 90th, 95th and 99th confidence
levels.
𝐴𝑙𝑙 𝑟99.9925 𝑟99.9950 𝑟99.9975 𝑟100
First one-third of an EPM -221.3*** -240.4** -362.3*** -329.4*** 47.6
(0.00) (0.02) (0.00) (0.00) (0.66)
Middle of an EPM 14.7 -196.0*** -198.5*** -33.3 494.1***
(0.71) (0.00) (0.00) (0.66) (0.00)
Last one-third of an EPM 543.7*** 132.7* 242.0*** 590.6*** 1224.2***
(0.00) (0.07) (0.00) (0.00) (0.00)

35
Table 5: Different scenarios of EPM termination
Panel A compares EPMs that experience a positive and negative 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 (defined in Table 1) at the last
one-third of the EPM return. Panel B compares EPMs that experience increasing and decreasing liquidity demand in the direction
of return at the last one-third of the EPM return. Panel C compares EPMs that experience increasing and decreasing contrarian
liquidity demand at the last one-third of the EPM return. Panel D reports aggregate statistics for EPMs and non-EPMs. 𝑁𝐶𝐷𝑃
stands for the 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑, 𝐷𝑖𝑟𝑉𝑜𝑙 is the total number of shares traded in the direction of the EPM return,
𝐶𝑜𝑛𝑡𝑉𝑜𝑙 is the total number of shares traded against the direction of the EPM return and 𝑇𝑜𝑡𝑉𝑜𝑙 is the total mean number of
shares traded during EPMs. *, ** and *** indicate significance at the 90th, 95th and 99th confidence levels.
𝑁𝐶𝐷𝑃 𝐷𝑖𝑟𝑉𝑜𝑙 𝐶𝑜𝑛𝑡𝑉𝑜𝑙 𝑇𝑜𝑡𝑉𝑜𝑙 𝑁 𝑁𝐶𝐷𝑃 𝐷𝑖𝑟𝑉𝑜𝑙 𝐶𝑜𝑛𝑡𝑉𝑜𝑙 𝑇𝑜𝑡𝑉𝑜𝑙 𝑁
EPMs ending with positive EPMs ending with negative
Panel A 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑
𝑟99.9925 1,571*** 8,339 2,735 11,075 718 -2,145*** 4,110 1,518 5,628 731
𝑟𝑃99.9950 1,245*** 8,822 3,106 11,928 749 -2,113*** 4,344 1,572 5,915 653
𝑟𝑃99.9975 1,871*** 11,316 4,059 15,375 811 -2,042*** 4,604 1,227 5,830 587
𝑟𝑃100 3,727*** 15,722 5,985 21,708 917 -1,957*** 5,346 1,692 7,038 483
EPMs ending with increasing liquidity demand in the EPMs ending with decreasing liquidity demand in
Panel B direction of return the direction of return
𝑟99.9925 148 9,106 3,226 12,331 493 -537*** 4,710 1,552 6,262 956
𝑟𝑃99.9950 -23 8,710 3,410 12,120 517 -492*** 5,583 1,796 7,379 885
𝑟𝑃99.9975 1,156*** 10,901 4,133 15,034 541 -358* 6,981 2,072 9,053 857
𝑟𝑃100 3,058*** 15,953 6,482 22,435 562 899*** 9,587 3,177 12,765 838
EPMs ending with increasing contrarian liquidity EPMs ending with decreasing contrarian liquidity
Panel C demand demand
𝑟99.9925 -272 6,067 2,141 8,208 625 -328 6,311 2,106 8,417 824
𝑟𝑃99.9950 -196 6,805 2,585 9,390 648 -424** 6,677 2,225 8,902 754
𝑟𝑃99.9975 377 7,581 2,772 10,353 628 106 9,245 2,950 12,195 770
𝑟𝑃100 2,215*** 13,953 5,248 19,201 679 1,343*** 10,438 3,804 14,241 721
Panel D All EPMs All non-EPMs
337*** 8,376 2,964 11,340 5,649 -368*** 358 100 457 56.6M

36
Table 6: Trading conditions and EPM termination
The table explores how different trading conditions and stock characteristics affect EPM termination. First, we split all EPMs into those occurring in the three-day window around
earnings announcements (Announcement day) and the EPMs occurring on the remaining days (Non-announcement day). Second, we split EPMs into those occurring in the stocks with
quoted spread equal to one cent more than 80% of the time within a given month (Tick-constrained) and the remaining EPMs (Not tick-constrained). Then, we further dissect the EPM
subgroups by their termination. The definitions of EPM termination subgroups are in the Table 5. 𝑁𝐶𝐷𝑃 stands for the 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑.
We report 𝛽 coefficients of the following model: 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑𝑖𝑡 = 𝛼𝑖 + 𝛽1 𝐸𝑃𝑀𝑇𝑦𝑝𝑒𝐴,𝑖𝑡 + 𝛽2 𝐸𝑃𝑀𝑇𝑦𝑝𝑒𝐵,𝑖𝑡 + 𝐶𝑜𝑛𝑡𝑟𝑜𝑙𝑠𝑖𝑡 + 𝜀𝑖𝑡 , where 𝐸𝑃𝑀𝑇𝑦𝑝𝑒𝐴 and 𝐸𝑃𝑀𝑇𝑦𝑝𝑒𝐵
represent indicator variables corresponding to the EPM subgroups defined above. We run this model for every EPM termination subgroup. 𝐶𝑜𝑛𝑡𝑟𝑜𝑙𝑠 include the volume, return and time-
weighted percentage spread. All variables are standardized by stock. P-values corresponding to double-clustered standard errors are in parentheses. *, ** and *** indicate significance at
the 90th, 95th and 99th confidence levels.
Announcement day Non-announcement day Tick-constrained Not tick-constrained
Panel A 𝑁𝐶𝐷𝑃 𝛽𝐴 𝑁 𝑁𝐶𝐷𝑃 𝛽𝑛𝐴 𝑁 𝛽𝐴 = 𝛽𝑛𝐴 𝑁𝐶𝐷𝑃 𝛽𝑇𝐶 𝑁 𝑁𝐶𝐷𝑃 𝛽𝑛𝑇𝐶 𝑁 𝛽𝑇𝐶 = 𝛽𝑛𝑇𝐶
EPMs ending with positive
1,545 10.43 800 2,405 11.44 2,395 Rejected*** 4,946 11.20 888 1,129 11.18 2,307 NotRejected
𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑
(0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.56)
EPMs ending with negative
-1,028 7.82 551 -2,378 6.62 1,903 Rejected*** -5,331 5.71 717 -731 7.38 1,737 Rejected***
𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑
(0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)
Panel B 𝑁𝐶𝐷𝑃 𝛽𝐴 𝑁 𝑁𝐶𝐷𝑃 𝛽𝑛𝐴 𝑁 𝛽𝐴 = 𝛽𝑛𝐴 𝑁𝐶𝐷𝑃 𝛽𝑇𝐶 𝑁 𝑁𝐶𝐷𝑃 𝛽𝑛𝑇𝐶 𝑁 𝛽𝑇𝐶 = 𝛽𝑛𝑇𝐶
EPMs ending with increasing liquidity
1,088 9.97 480 1,153 10.05 1,633 NotRejected 1,644 9.19 706 885 10.45 1,407 Rejected***
demand in the direction of return
(0.00) (0.00) (0.00) (0.00) (0.12) (0.00) (0.00) (0.00) (0.00) (0.00)
EPMs ending with decreasing liquidity
168 9.02 871 -243 8.83 2,665 Rejected*** -657 8.36 899 34 9.05 2,637 Rejected***
demand in the direction of return
(0.18) (0.00) (0.07) (0.00) (0.00) (0.09) (0.00) (0.50) (0.00) (0.00)
Panel C 𝑁𝐶𝐷𝑃 𝛽𝐴 𝑁 𝑁𝐶𝐷𝑃 𝛽𝑛𝐴 𝑁 𝛽𝐴 = 𝛽𝑛𝐴 𝑁𝐶𝐷𝑃 𝛽𝑇𝐶 𝑁 𝑁𝐶𝐷𝑃 𝛽𝑛𝑇𝐶 𝑁 𝛽𝑇𝐶 = 𝛽𝑛𝑇𝐶
EPMs ending with increasing contrarian
792 9.76 624 486 9.55 1,956 Rejected*** 828 8.83 725 455 9.91 1,855 Rejected***
liquidity demand
(0.00) (0.00) (0.01) (0.00) (0.00) (0.10) (0.00) (0.00) (0.00) (0.00)
EPMs ending with decreasing contrarian
240 8.99 727 122 9.06 2,342 Rejected* -34 8.64 880 224 9.21 2,189 Rejected***
liquidity demand
(0.12) (0.00) (0.42) (0.00) (0.06) (0.93) (0.00) (0.00) (0.00) (0.00)
Panel D 𝑁𝐶𝐷𝑃 𝛽𝐴 𝑁 𝑁𝐶𝐷𝑃 𝛽𝑛𝐴 𝑁 𝛽𝐴 = 𝛽𝑛𝐴 𝑁𝐶𝐷𝑃 𝛽𝑇𝐶 𝑁 𝑁𝐶𝐷𝑃 𝛽𝑛𝑇𝐶 𝑁 𝛽𝑇𝐶 = 𝛽𝑛𝑇𝐶
All EPMs 495 9.53 1,351 287 9.46 4,298 Rejected** 355 8.89 1,605 330 9.71 4,044 Rejected***
(0.00) (0.00) (0.01) (0.00) (0.02) (0.26) (0.00) (0.00) (0.00) (0.00)

37
Table 7: Predictability of EPMs
This table reports the ex-ante probabilities that the EPM will end at the price region with the highest Net
contrarian depth placed. The first column shows the countdown until the end of individual EPM intervals. Net
within 0.01%, 0.05% and 0.1% correspond to statistics computed with 0.01%, 0.05% and 0.1% price intervals,
respectively.
𝑆𝑒𝑐𝑜𝑛𝑑𝑠 𝑡𝑖𝑙𝑙 𝑒𝑛𝑑 𝑜𝑓 𝐸𝑃𝑀 𝑁𝑒𝑡 𝑤𝑖𝑡ℎ𝑖𝑛 0.01% 𝑁𝑒𝑡 𝑤𝑖𝑡ℎ𝑖𝑛 0.05% 𝑁𝑒𝑡 𝑤𝑖𝑡ℎ𝑖𝑛 0.1%
20 2.11% 12.61% 33.03%
19 2.27% 12.49% 32.35%
18 2.34% 12.43% 32.16%
17 1.76% 11.94% 31.15%
16 1.97% 11.76% 31.75%
15 2.09% 12.26% 33.13%
14 2.47% 12.61% 32.32%
13 2.00% 12.66% 32.62%
12 2.11% 12.40% 32.74%
11 2.52% 12.95% 33.04%
10 2.50% 13.66% 33.10%
9 3.34% 16.37% 38.01%
8 3.74% 18.89% 42.04%
7 3.88% 20.38% 45.87%
6 5.11% 24.26% 50.23%
5 6.07% 25.20% 53.16%
4 6.95% 27.94% 57.68%
3 8.59% 30.83% 61.92%
2 10.66% 33.36% 64.23%
1 18.29% 39.25% 69.49%
Unconditional % 0.20% 1.08% 2.12%

38
Table 8: Halts vs non-halts
This table compares EPMs that happened on the days when at least one of the sample stocks experienced a trading halt
(𝐻𝑎𝑙𝑡) with the rest of the EPMs (𝑛𝑜𝑛𝐻𝑎𝑙𝑡). Panel A reports the Net contrarian depth placed (defined in Table 1) during
EPMs. Panel B reports the 𝛽1 and 𝛽2 coefficients of the following model: 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑𝑖𝑡 = 𝛼𝑖 +
𝛽1 𝐸𝑃𝑀𝐻𝑎𝑙𝑡,𝑖𝑡 + 𝛽2 𝐸𝑃𝑀𝑛𝑜𝑛𝐻𝑎𝑙𝑡,𝑖𝑡 + 𝐶𝑜𝑛𝑡𝑟𝑜𝑙𝑠𝑖𝑡 + 𝜀𝑖𝑡 , where 𝐸𝑃𝑀𝐻𝑎𝑙𝑡 and 𝐸𝑃𝑀𝑛𝑜𝑛𝐻𝑎𝑙𝑡 represent the indicator variable
corresponding EPM subgroups defined above. 𝐶𝑜𝑛𝑡𝑟𝑜𝑙𝑠 include the return, volume and spread. All variables are
standardized. P-values corresponding to double-clustered standard errors are in parentheses. *, ** and *** indicate
significance at the 90th, 95th and 99th confidence levels.
Panel A: Univariate statistics Panel B: Regression
𝐻𝑎𝑙𝑡 𝑛𝑜𝑛𝐻𝑎𝑙𝑡
𝑟99.9925 466.7* -430.8** 𝐸𝑃𝑀𝐻𝑎𝑙𝑡 12.11***
(0.08) (0.03) (0.00)
𝑟99.9950 626.7*** -484.4*** 𝐸𝑃𝑀𝑛𝑜𝑛𝐻𝑎𝑙𝑡 8.91***
(0.00) (0.00) (0.00)
𝑟99.9975 1,039.4*** 64.7 𝐻: 𝐸𝑃𝑀𝐻𝑎𝑙𝑡 = 𝐸𝑃𝑀𝑛𝑜𝑛𝐻𝑎𝑙𝑡 𝑅𝑒𝑗𝑒𝑐𝑡𝑒𝑑***
(0.00) (0.73) 𝐴𝑑𝑗. 𝑅2 0.239
𝑟100 2,185.9*** 1,618.3***
(0.00) (0.00)
𝑁 1,012 4,637

39
Table 9: Ex-ante limit order book depth
This table reports the limit order book dynamics for a period of 300 seconds (five minutes) before the EPM intervals.
𝐷𝑒𝑝𝑡ℎ 1%, 𝐷𝑒𝑝𝑡ℎ 0.5% and 𝐷𝑒𝑝𝑡ℎ 0.1% is the limit order book depth in the direction of the EPM return within 1, 0.5
and 0.1% of the current price.
Seconds before an EPM 𝐷𝑒𝑝𝑡ℎ 1% 𝐷𝑒𝑝𝑡ℎ 0.5% 𝐷𝑒𝑝𝑡ℎ 0.1%
0 14,798 8,683 2,080
1 14,780 8,642 2,074
5 14,556 8,366 1,964
10 14,312 8,078 1,871
30 13,718 7,468 1,708
60 13,061 6,872 1,547
300 11,373 5,752 1,243

40
Table 10: Discovery of hidden liquidity during EPMs
Panel A reports the univariate statistics for the HiddenRatio. HiddenRatio is the number of shares traded against the
hidden liquidity in the direction of return divided by the total number of shares traded in the direction of return. We run
a t-test comparing HiddenRatio during EPMs to the average across all price movements and report the p-values in
parentheses. Panel B reports the 𝛽 coefficient of the following model: 𝐻𝑖𝑑𝑑𝑒𝑛𝑅𝑎𝑡𝑖𝑜𝑡 = 𝛼𝑖 + 𝛽𝐸𝑃𝑀𝑖𝑡 + 𝐶𝑜𝑛𝑡𝑟𝑜𝑙𝑠𝑖𝑡 +
𝜀𝑖𝑡 , where 𝐸𝑃𝑀𝑖𝑡 represents the indicator variable equal to one if an interval is identified as an EPM and zero otherwise.
𝐶𝑜𝑛𝑡𝑟𝑜𝑙𝑠 include the volume, return and time-weighted percentage spread. All variables are standardized. P-values
corresponding to double-clustered standard errors are in parentheses. *, ** and *** indicate significance at the 90th, 95th
and 99th confidence levels.
Panel A: Univariate statistics Panel B: Regression
𝑟99.9925 18.99%*** 𝐸𝑃𝑀 0.667***
(0.00) (0.00)
2
𝑟99.9950 20.79%*** 𝐴𝑑𝑗. 𝑅 0.019
(0.00)
𝑟99.9975 21.52%***
(0.00)
𝑟100 24.27%***
(0.00)
𝐴𝑙𝑙𝑃𝑟𝑖𝑐𝑒𝑀𝑜𝑣𝑒𝑚𝑒𝑛𝑡𝑠 7.78%

41
 a) Implication of the literature on risks and constraints of liquidity provision
Added
liquidity

Return

b) Implication of the literature on contrarian trading

Added
liquidity

Return

c) Implication of the literature on risk-return trade-off of contrarian liquidity provision

Added
liquidity

Return

Figure 1: Theories of liquidity provision in stressful times.

42
Figure 2: Distribution of price movements with an absolute return exceeding the 99.99th percentile.

43
Figure 3: Net contrarian liquidity placed in the way of price movements of different magnitudes.

44
price price

𝑞𝑡
𝑠𝑡
𝑚𝑡
𝑚𝑡
𝑠𝑡
𝑞𝑡

t t
Figure 4: The figure shows expected transitory components of large positive (left) and negative (right) returns. The
solid black line is the current return, the dashed line is the size of the contemporary reversible component and the
dotted line is the price movement consistent with the expectations of contrarian traders.

45
Figure 5: Dynamics of 𝑁𝑒𝑡 𝑐𝑜𝑛𝑡𝑟𝑎𝑟𝑖𝑎𝑛 𝑑𝑒𝑝𝑡ℎ 𝑝𝑙𝑎𝑐𝑒𝑑 within EPMs.

46