# Modeling bid and ask price dynamics with an extended Hawkes process and its
empirical applications for high-frequency stock market data

Kyungsub Lee<EMAIL_ADDRESS>Department of Statistics, Yeungnam
University, Gyeongsan, Gyeongbuk 38541, Korea. Byoung Ki Seo 222(corresponding
author<EMAIL_ADDRESS>School of Business Administration, Ulsan National
Institute of Science and Technology, Ulsan 44919, Korea.

###### Abstract

This study proposes a versatile model for the dynamics of the best bid and ask
prices using an extended Hawkes process. The model incorporates the zero
intensities of the spread-narrowing processes at the minimum bid-ask spread,
spread-dependent intensities, possible negative excitement, and nonnegative
intensities. We apply the model to high-frequency best bid and ask price data
from US stock markets. The empirical findings demonstrate a spread-narrowing
tendency, excitations of the intensities caused by previous events, the impact
of flash crashes, characteristic trends in fast trading over time, and the
different features of market participants in the various exchanges.

##  1 Introduction

A considerable volume of research on high-frequency trading, quotes, and
financial data is available. High-frequency quotes and trading, which are the
main sources of high-frequency financial data, are considered to be
(automated) trading generally based on a (mathematical) algorithm that can
generate a large number of quotes and trades over a short time horizon.
However, the precise definition varies across studies and regulatory entities.
Numerous studies consider the impact of high-frequency trading on market
quality in terms of liquidity, transaction costs, volatility, price discovery,
social benefits, and its role in market crashes.

Studies of market microstructures and high-frequency data rely on statistical
models that can precisely analyze market data such as order duration,
transaction time, and price changes. One effective method is based on the
Hawkes process (Hawkes, 1971a, b; Hawkes and Oakes, 1974), a type of point
process for estimating the parameters related to the impacts of events.
Researchers first applied Hawkes models to seismology, but presently apply
them extensively throughout the natural and social sciences, including
finance. For example, the earliest studies applying the Hawkes process to
financial markets include Hewlett (2006), who introduces a symmetric bivariate
Hawkes process to examine the arrival times of trades and price impacts; Large
(2007), who examines market resilience; and Bowsher (2007), who analyzes the
relationship between trading times and mid-price changes. Since then, the
model has often been applied to financial data. Bacry et al. (2015), Law and
Viens (2015), and Hawkes (2018) summarize recent developments in the
application of the Hawkes process to finance.

The basic Hawkes model captures several interesting properties of price
dynamics at the micro level (Chavez-Demoulin and McGill, 2012; Bacry and Muzy,
2014; Da Fonseca and Zaatour, 2014) such as order clustering, microstructure
noise, and the impacts of past events. The Hawkes process is suitable for
modeling the intensities of a series of events such as price changes in tick
structure and order arrivals. In addition, other studies aim to clarify the
complex microstructure of the limit order book and price dynamics using
extended models (Zheng et al., 2014; Morariu-Patrichi and Pakkanen, 2018;
Hainaut and Goutte, 2019; Jang et al., 2020). Our approach is also based on
modifying the existing Hawkes model to allow applications to bid and ask price
dynamics. Owing to its versatility, this extended Hawkes process allows us to
infer useful information from high-frequency financial data and understand the
market microstructure and high-frequency activities.

Specifically, we develop a rigorous and novel probability model that describes
the nature of bid-ask price dynamics. The proposed model incorporates
characteristics that cannot be described by the basic Hawkes model. Bid-ask
spreads are positive and our model guarantees this property. If the spread
widens, competition among market makers to replenish liquidity increases under
typical market conditions and thus the proposed model has spread-level-
dependent intensities.

In stock market microstructures, a bid or ask price change is expected to
increase the likelihood of events such as new order arrivals and successive
price changes occurring. The basic Hawkes model can describe these excitation
properties. However, if price changes occur such that spreads decline,
intensity may not increase. For example, if the spread reduces to a tick size
(i.e., it reaches the minimum level), then no further events that reduce the
spread occur; that is, the associated intensities that narrow the spread
become zeros. Since the basic Hawkes model always has positive intensities, it
needs to be modified using, for example, negative excitements. To add a
negative excitement term into the model, the model design must ensure that
intensity is nonnegative to satisfy mathematical rigor.

Related ideas can be found in the past literature. An extended Hawkes process
that partially describes non-exciting relationships was proposed in Brémaud
and Massoulié (1996). Along that line, Hansen et al. (2015) considered
multivariate counting processes with inhibitory effects in a non-linear
setting whose importance is also mentioned in Bacry et al. (2016).

We apply this model to high-frequency financial data and make the following
interesting findings. First, we find evidence of a growing number of high-
frequency traders even though not all stocks show the same monotone increasing
growth pattern over time. Second, liquidity provision and removal behaviors
have different responsiveness. In all the stocks in our sample, we find higher
instantaneous responsiveness in liquidity provision than in removal. Third, we
also observe different features from different exchanges under market-making
activity. Being a highly fragmented market, stock market exchanges have
different fee structures, policies, and technology, which leads their
participants to have different characteristics.

We also report the dramatic changes in market conditions under special
circumstances such as the Flash Crash of 2010. Although we perform most of our
analyses on a daily basis, because of the nature of high-frequency financial
data, sufficient data are available for a few minutes, thereby allowing us to
conduct the analysis on a near real-time basis. Lee and Seo (2017a) also show
this property. Using sophisticatedly designed parameters, we can thus observe
how the behaviors of high-frequency traders change during the crash, recovery
period, and next day.

The remainder of the paper proceeds as follows. Section 2 reviews the related
literature. Section 3 presents the extended Hawkes model of the dynamics of
the best bid and ask prices. The proposed model incorporates the zero
intensities of the spread-narrowing processes at the minimum bid-ask spread,
spread-dependent intensities, possible negative excitement, and nonnegative
intensities. Section 5 discusses the findings from our empirical study
performed by applying the presented model to high-frequency financial data.
The results demonstrate increasing trends in low-latency quotes and trading,
the characteristics of stocks and exchanges, a comparison between liquidity
provision and removal, and the interesting behaviors of high-frequency traders
during and after the Flash Crash. Section 6 concludes.

##  2 Literature review

This section reviews studies analyzing high-frequency trading and high-
frequency financial data. The main advantage of high-frequency trading,
especially quotes, is the supply of liquidity. Many studies demonstrate that
the liquidity provided by active high-frequency quotes and large numbers of
outstanding orders reduces the bid-ask spread and that transaction costs
decrease accordingly (Bershova and Rakhlin, 2013; Chaboud et al., 2014; Stoll,
2014). In addition, Hagströmer and Nordén (2013) and Menkveld (2013) report
that many high-frequency traders are liquidity providers. However, a
proportion of this liquidity is an illusion in the abundance of fleeting
orders (Hasbrouck and Saar, 2009), and some researchers express skepticism
about whether it is sustainable during a market crisis (Cvitanic and
Kirilenko, 2010), even though high-frequency trading is not generally
considered to be the primary originator of the Flash Crash (Kirilenko et al.,
2017).

The rapidity of high-frequency trading is based on high-end, state-of-the-art
computing, network equipment, and costly financial services such as
colocation, which involves installing traders’ servers as close to trade
matching engines as possible. However, while this rapidity can help provide
liquidity (Brogaard et al., 2015), high-frequency traders armed with new
equipment may take advantage of low-frequency traders (Jarrow and Protter,
2012; van Kervel, 2015; Biais et al., 2015). Theoretical and empirical
research thus suggests that the rapidity of high-frequency trading can be
harmful as well as beneficial (Foucault and Moinas, 2018). The general
assumption is that the speed of high-frequency trading is a positive aspect of
price discovery as it allows the market to rapidly process new information and
reflect it in new prices (Manahov and Hudson, 2014). However, whether
information processing and quote updates that occur just a few seconds or
milliseconds earlier are of practical utility remains an open question
(Brogaard et al., 2014).

Because volatility is an important indicator of the market conditions,
measuring the volatility of prices or returns more accurately using high-
frequency financial data has long been a topic of study (Zhou, 1996;
Barndorff-Nielsen and Shephard, 2002; Andersen et al., 2003). In addition, the
effect of high-frequency trading on volatility is under debate (Bollen and
Whaley, 2015; Kelejian and Mukerji, 2016; Hasbrouck, 2018). In some ways,
high-frequency activities such as the exploitation of arbitrage opportunities
in high-frequency trading, crowding effect, flickering quotes, and high order
cancellation rates are sources of higher volatility, but some studies find no
significant statistical relations between high-frequency activities and
volatility. Chung and Lee (2016), Menkveld (2016), and Virgilio (2019) provide
excellent summaries of the arguments in the literature above.

The research interest in high-frequency trading is not limited to the above
discussion; it extends to the optimal strategies of market makers and
investors based on high-frequency trading (Avellaneda and Stoikov, 2008;
Guilbaud and Pham, 2013; Cartea and Jaimungal, 2013; Cartea et al., 2018; Choi
et al., 2021) and limit order book modeling (Cont and de Larrard, 2013; Toke,
2015). Bouchaud et al. (2002), Hollifield et al. (2004), and Huang et al.
(2015) describe the statistical properties of limit order books.

##  3 The model

###  3.1 Definition

In this section, we model the best bid and ask price dynamics of financial
assets using point processes. The best bid and ask price processes consist of

$\displaystyle A^{u}\textrm{: the number of upward movements in the best ask
process,}$ $\displaystyle A^{d}\textrm{: the number of downward movements in
the best ask process,}$ $\displaystyle B^{u}\textrm{: the number of upward
movements in the best bid process, and}$ $\displaystyle B^{d}\textrm{: the
number of downward movements in the best bid process,}$

defined on the probability space
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geq 0},\mathbb{P})$ satisfying
the usual conditions. Each process is a random counting measure that counts
the number of events in any Borel set in the timeline, $\mathbb{R}$. We can
represent each process using a stochastic process (i.e., nondecreasing
integer-valued random step functions) by denoting
$A^{u}(t)=A^{u}(\omega,(0,t])$ for $\omega\in\Omega$ and similarly denoting
the other counting measures. We assume the point processes are simple, i.e.,

$\mathbb{P}\\{N(\\{t\\})=0\textrm{ or }1\textrm{ for all }t\\}=1$

for $N=A^{u},A^{d},B^{u}$ or $B^{d}$. We can categorize the counting processes
further as

$\displaystyle A^{u},B^{d}\textrm{ : the bid-ask spread-widening processes}$
$\displaystyle A^{d},B^{u}\textrm{ : the bid-ask spread-narrowing processes}.$

We establish a system of counting processes with the corresponding conditional
intensities:

$\bm{N}_{t}=\begin{bmatrix}N_{1}(t)\\\ N_{2}(t)\\\ N_{3}(t)\\\
N_{4}(t)\end{bmatrix}=\begin{bmatrix}A^{u}(t)\\\ A^{d}(t)\\\ B^{u}(t)\\\
B^{d}(t)\end{bmatrix},\quad\bm{\lambda}_{t}=\begin{bmatrix}\lambda_{1}(t)\\\
\lambda_{2}(t)\\\ \lambda_{3}(t)\\\
\lambda_{4}(t)\end{bmatrix}=\begin{bmatrix}\lambda_{A}^{u}(t)\\\
\lambda_{A}^{d}(t)\\\ \lambda_{B}^{u}(t)\\\ \lambda_{B}^{d}(t)\end{bmatrix}.$

We can express the conditional intensities as the expected number of events
over an infinitesimal interval $\mathrm{d}t$:

$\lambda_{i}(t)\mathrm{d}t\approx\mathbb{E}[N_{i}(t+\mathrm{d}t)-N_{i}(t)|\mathcal{F}_{t-}],$

where $\mathcal{F}_{t-}$ is the $\sigma$-field generated by $\bm{N}$ at times
up to but not including $t$. Thus, the intensities are left continuous and the
counting processes are right continuous. We have two types of notations,
number-based, such as $N_{i}$ and $\lambda_{i}$, and Roman-based, $A^{u}$ and
$\lambda_{A}^{u}$. We use them interchangeably depending on the context.

The following model is used in our main empirical study in Section 5. Slightly
different versions of the model are also examined in Section 4. The motivation
and property of the model is further explained in Subsection 3.2.

######  Definition 1.

The intensities of the up and down movements of the best bid and ask prices
are assumed to be

$\bm{\lambda}_{t}=\bm{\mu}_{t}+\int_{-\infty}^{t}\bm{h}(t,u)\mathrm{d}\bm{N}_{u},$
(1)

where

$\bm{\mu}_{t}=\begin{bmatrix}\mu\\\ f(\ell(t-))\\\ f(\ell(t-))\\\
\mu\end{bmatrix},$

where $\ell$ denotes the level of the bid-ask spread relative to the mid-price
and

$f(\ell(t-))=\eta\ell(t-)$ (2)

with a constant parameter, $\eta>0$. The relative level is defined by

$\ell(t)=\frac{L(t)}{p(t)},$

where $p(t)$ is the mid-price and $L(t)\in\\{0,1,2,\cdots\\}$ is the absolute
level of the bid-ask spread with $L(t)=0$ implying the minimum level. The
matrix of excitement terms is based on an exponential kernel:

$\bm{h}(t,u)=\mathrm{e}^{-\beta(t-u)}\begin{bmatrix}\alpha_{s1}&\alpha_{m}&\alpha_{s2}&0\\\
\alpha_{w1}&\alpha_{n1}(u)&\alpha_{n1}(u)&\alpha_{w2}\\\
\alpha_{w2}&\alpha_{n2}(u)&\alpha_{n2}(u)&\alpha_{w1}\\\
0&\alpha_{s2}&\alpha_{m}&\alpha_{s1}\\\ \end{bmatrix},$

where $\alpha_{s1},\alpha_{s2},\alpha_{m},\alpha_{w1},$ and $\alpha_{w2}$ are
parameters under the condition that

$\alpha_{s1}+\alpha_{s2}+\alpha_{m}<\beta$ (3)

and $\alpha_{n1}$ and $\alpha_{n2}$ are stochastic terms with

$\displaystyle\alpha_{n1}(u)=-\sum_{j=1}^{4}\lambda_{2j}(u)+\xi\ell(u),\quad\alpha_{n2}(u)=-\sum_{j=1}^{4}\lambda_{3j}(u)+\xi\ell(u),$
(4)

for constant $\xi\geq 0$ and $\lambda_{ij}$ is a component of $\lambda_{i}$
such that

$\lambda_{ij}(t)=\int_{-\infty}^{t}h_{ij}(t,u)\mathrm{d}N_{j}(u).$

We use relative level rather than absolute level in the modeling of $\bm{\mu}$
and $\bm{h}$ for empirical research. In the US stock market, the tick size is
fixed at $0.01, while the price of a stock varies from stock to stock.
However, for mathematical analysis, the absolute level model is more
tractable, which will be discussed in Subsection 4.3.

In the definition, we use the left continuous version of $\ell(t-)$ to define
$\bm{\mu}$ to ensure that the intensity process is left continuous with a
right limit. The base intensities of the spread-narrowing processes, $A^{d}$
and $B^{u}$, increase as the spread increases. When $\ell(t-)=0$ (i.e., the
spread is at its minimum), the base intensities of $A^{d}$ and $B^{u}$ are
equal to zero.

The stochastic terms $\alpha_{n1}$ and $\alpha_{n2}$ control the amounts of
changes in the spread-narrowing intensities right after the spread narrows. We
introduce these terms as stochastic to ensure mathematical rigor which will be
discussed in the next subsection. We cannot rule out the possibility that
$\xi$ has different values for $\alpha_{n1}$ and $\alpha_{n2}$ in practice,
but we assume that $\xi$ takes the same value in both for model parsimony.

The coefficients in the top right and left bottom corners of the matrix
$\bm{h}$ are intentionally set equal to zero for model parsimony, since the
interaction between upward movement in the best ask and downward movement in
the best bid can negligible especially in ultra-high-frequency. The empirical
evidence will be discussed in Subsection 5.3.1.

An example of intensity changes by the events under the model in Definition 1
is shown in Figure 1. An hypothetical situation where the bid-ask spread
widens first and then decreases to the minimum level is illustrated. Note that
the spread-narrowing intensities $\lambda_{b}^{u}$ and $\lambda_{a}^{d}$
become zeros right after the spread hits the minimum.

Figure 1: Illustration of the intensity processes in the proposed model

###  3.2 Discussion of the model

Previous studies often assume that the mid-price process follows a self and
mutually excited Hawkes process, which has empirical support (Bacry et al.,
2013; Da Fonseca and Zaatour, 2014; Lee and Seo, 2017b). Self-excitement means
that upward or downward movements of mid-prices are excited by past events in
their own processes and are often considered to be due to the order splitting.
Mutual excitement means that downward movement excites upward movement and
vice versa. Because the mid-price is the mean of the best bid and ask prices,
it is natural to assume that we can use a Hawkes-type model to represent the
bid and ask price processes in this context. The intensities in the Hawkes
model consist of the base intensities and excitations by past events. The
typical intensity process in the Hawkes model takes the form of Eq. (1) with
constant base intensity $\bm{\mu}$ and constant excitation kernel $\bm{h}$.

Our goal of modeling is to find the latent information in the level-one limit
order book by modeling the base intensities and excitation matrix
appropriately. To define a model as in Definition 1, the followings are
considered. The model is proposed to incorporate the following properties in
addition to Hawkes-type features in the mid-price process.

Zero intensity with minimum spread

When the spread between the best bid and ask prices reaches the minimum tick
size, the ask price does not fall and the bid price does not rise before the
spread widens. Mathematically, the intensities of the spread-narrowing
processes, $A^{d}(t)$ and $B^{u}(t)$, should be zero at the minimum level of
the bid-ask spread until a new event occurs. Thus, $\lambda_{A}^{d}(t)$ and
$\lambda_{B}^{u}(t)$ do not follow the intensities of the classical Hawkes
model in which the intensities can be close to zero, but are always greater
than zero. Hence, it is necessary to tweak the model to set both the
background intensities and the excitations equal to zero when the spread
reaches the minimum.

Spread-dependent intensity

When the bid-ask spread is wide, we assume that it tends to narrow depending
on its level. This property may be due to market makers who quote limit orders
with aggressive prices between the widened bid and ask orders, providing
market liquidity. As the spread increases, the potential profit for market
makers increases, and market making hence becomes more active. We can observe
this tendency in two ways. One is the increase in the base intensities,
regardless of previous events, and the other comes from the excitations due to
past events. Our model includes both aspects. The base intensity rates of the
spread-narrowing processes, $A^{d}$ and $B^{u}$, increase as the spread
widens. In addition, the overall intensities of the processes are excited
immediately after the spread-widening events by the Hawkes-type kernel.

Possible negative excitement when the spread narrows

If spread-narrowing events occur, we can assume that these activities can slow
the rate of the additional arrivals of aggressive quotes. In other words, the
intensities of the spread-narrowing processes, $A^{d}(t)$ and $B^{u}(t)$, may
diminish, or even vanish, when the spread becomes minimal. We can express
decreases in intensities due to a series of activities as a negative term in
the Hawkes-type kernel.

Avoiding negative intensity

As mentioned above, negative excitements (i.e., diminishing intensity) can
occur, but we should control this amount to ensure that intensity is
nonnegative. Thus, the amount of excitement in the spread-narrowing processes
after a spread-narrowing event depends on the intensities of the narrowing
processes.

Simultaneous movement of bid and ask prices

Theoretically, bid and ask prices can move simultaneously in rare
circumstances. For example, assume 100 volumes of the best ask price in a
limit order book. When a marketable limit order to buy 120 shares at the best
ask price arrives, it consumes the existing best ask orders and the previous
second-best ask order becomes the best. In addition, the remaining market
order of 20 volumes converts to the bid order, which becomes the best bid.
This feature is difficult to interpret as a mathematical model, since we
assume simple processes in general, where only one jump occurs in an
infinitesimal time interval. In practice, we assume a small time interval
between two events. We can model this using the network, computational, and
transaction delays in a trading system, and hence from raw data, or by data
preprocessing, as we explain in Section 5.2.

The parameters in the excitation kernel,
$\alpha_{s1},\alpha_{s2},\alpha_{m},\alpha_{w1},$ and $\alpha_{w2}$, increase
the intensities after certain events. Increased intensities induce successive
arrivals of corresponding events. Our empirical study shows that the parameter
estimates are large owing to the abundant high-frequency trading in the highly
liquid US equity markets. Next, we discuss the role of each constant
parameter.

  * •

$\alpha_{w1}$: This parameter is related to liquidity provision. If
$\alpha_{w1}$ is large and the other parameters are small, then the bid and
ask prices are likely to follow the series of movements in Figure 3(a) when a
spread-widening event occurs. From the market-making perspective, this
parameter is associated with the replenishment speed for limit orders. When a
large market order sweeps the existing best limit orders or all the best
orders are canceled, the bid-ask spread widens. If $\alpha_{w1}$ is
sufficiently large, then liquidity providers tend to replenish the depleted
limit orders immediately. These activities may occur through (automated)
market makers or rebate-seekers in maker-taker exchanges to supply liquidity.

  * •

$\alpha_{w2}$: This parameter is also related to liquidity provision and is
similar to $\alpha_{w1}$. The relevant bid and ask prices behaviors with large
$\alpha_{w2}$ is presented in Figure 3(b) when a spread-widening event occurs.
With large $\alpha_{w2}$, the widened bid-ask gap tends to narrow immediately,
but by a different type of limit order from a depleted limit order. If these
events occur consecutively in the same direction, then significant mid-price
changes may eventually result.

  * •

$\alpha_{s1}$: This parameter is related to removing liquidity. If
$\alpha_{s1}$ is large, then we can observe increments in the ask price or
decrements in the bid price by successive limit order depletions, as in Figure
3(c). If the order depletion in the same direction happens in a very short
time, it may be related to anticipatory order cancellation, which means that
when a large market order arrives, the market maker considers it to be an
informed trader’s action and immediately cancels its limit orders to prevent
adverse selection (Fox et al., 2019). If the market is sufficiently liquid, as
the US equity market then successive depletions and the corresponding widened
spread are likely to be replenished immediately with large $\alpha_{w}$s.

  * •

$\alpha_{s2}$: This parameter is also related to liquidity removal and related
movements are described in Figure 3(d) when a spread-narrowing event occurs.
As in the case of $\alpha_{w2}$, if these events occur consecutively, then
price changes may eventually result.

  * •

$\alpha_{m}$: This parameter is also related to removing liquidity. When
$\alpha_{m}$ is large, if a better limit order than the current best price
arrives, then the new limit order is executed or canceled immediately, and the
price rebounds to its original status, as in Figure 3(e). This result may
occur because market participants may consider the new order to be good and
high-frequency traders execute the market order immediately. Alternatively, a
trader may issue a limit order aggressively and cancel the order instantly to
avoid adverse selection, especially in an uncertain market. Order
cancellations are abundant in stock markets. Scholtus and van Dijk (2015)
report that 60% of all orders are canceled within one second in the S&P 500
exchange-traded funds traded on the Nasdaq from 2009 to 2011. Hasbrouck and
Saar (2009) refer to limit orders canceled immediately as fleeting orders.
Repeated order submissions and cancellations are also known as flickering
quotes, a concept introduced more than two decades ago (Harris, 1999).

The terms $\alpha_{n}$ can be negative in contrast to the classical Hawkes
model in which t he $\alpha$ parameters are nonnegative. Although
$\alpha_{n}(t)$s may be negative, however, they are bounded by
$-\sum_{j=1}^{4}\lambda_{ij}(t)$ below, and hence $\lambda$s are always
nonnegative. More precisely, for $i=3$ or 4, and assume that there is a spread
narrowing event at time $u$, then

$\lambda_{i}(u+)=\lambda_{i}(u)+\alpha_{nm}(u)\geq\lambda_{i}(u)-\sum_{j=1}^{4}\lambda_{ij}(u)=\eta\ell(u)\geq
0$

where $m=1$ for $i=2$ and $m=2$ for $i=3$. In addition, under the symmetric
setting, i.e., $\xi$ is the same in $\alpha_{n1}$ and $\alpha_{n2}$, we have
$\lambda_{2}=\lambda_{3}$ and $\alpha_{n1}=\alpha_{n2}$.

Figure 2 shows the impact of the spread-narrowing intensities when the spread
narrows. The figure shows that the impact can be positive or negative
depending on the level. If the level is high, even if the spread narrows, the
speed at which it narrows can still increase. If the level is sufficiently
low, then the spread-narrowing event reduces the rate at which the spread
decreases. The $y$-intercept of the line is $-\sum_{j=1}^{4}\lambda_{ij}(t)$,
a $\mathcal{F}_{t-}$-measurable random variable, and the slope $\xi$ is a
constant parameter. This stochastic $y$-intercept ensures that the negative
excitement does not make the intensities less than zero.

Figure 2: $\alpha_{ni}(t)$ versus the relative spread in a linear model

We can rewrite the intensity processes as

$\displaystyle\lambda_{A}^{u}(t)={}$
$\displaystyle\mu+\int_{-\infty}^{t}\alpha_{s1}\mathrm{e}^{-\beta(t-u)}\mathrm{d}N_{A}^{u}(u)+\int_{-\infty}^{t}\alpha_{m}\mathrm{e}^{-\beta(t-u)}\mathrm{d}N_{A}^{d}(u)+\int_{-\infty}^{t}\alpha_{s2}\mathrm{e}^{-\beta(t-u)}\mathrm{d}N_{B}^{u}(u)$
$\displaystyle\lambda_{A}^{d}(t)={}$ $\displaystyle
f(\ell(t-))+\int_{-\infty}^{t}\alpha_{w1}\mathrm{e}^{-\beta(t-u)}\mathrm{d}N_{A}^{u}(u)+\int_{-\infty}^{t}\alpha_{w2}\mathrm{e}^{-\beta(t-u)}\mathrm{d}N_{B}^{d}(u)$
$\displaystyle\phantom{f(\ell(t-))}+\int_{-\infty}^{t}\alpha_{n1}(u)\mathrm{e}^{-\beta(t-u)}(\mathrm{d}N_{A}^{d}(u)+\mathrm{d}N_{B}^{u}(u))$
$\displaystyle\lambda_{B}^{u}(t)={}$ $\displaystyle
f(\ell(t-))+\int_{-\infty}^{t}\alpha_{w2}\mathrm{e}^{-\beta(t-u)}\mathrm{d}N_{A}^{u}(u)+\int_{-\infty}^{t}\alpha_{w1}\mathrm{e}^{-\beta(t-u)}\mathrm{d}N_{B}^{d}(u)$
$\displaystyle\phantom{f(\ell(t-))}+\int_{-\infty}^{t}\alpha_{n2}(u)\mathrm{e}^{-\beta(t-u)}(\mathrm{d}N_{A}^{d}(u)+\mathrm{d}N_{B}^{u}(u))$
$\displaystyle\lambda_{B}^{d}(t)={}$
$\displaystyle\mu+\int_{-\infty}^{t}\alpha_{s2}\mathrm{e}^{-\beta(t-u)}\mathrm{d}N_{A}^{d}(u)+\int_{-\infty}^{t}\alpha_{m}\mathrm{e}^{-\beta(t-u)}\mathrm{d}N_{B}^{u}(u)+\int_{-\infty}^{t}\alpha_{s1}\mathrm{e}^{-\beta(t-u)}\mathrm{d}N_{B}^{d}(u).$

In the classical Hawkes model, the parameter $\alpha$s are restricted by
$\beta$ for nonexplosive solutions. Intuitively, a large $\alpha$ causes
intensity to increase rapidly. For example, if $\alpha$ is greater than
$\beta$, which describes the decrease in intensity over time, then the
increment in intensity caused by $\alpha$ is much larger than the decrease
caused by $\beta$, and intensity explodes. By contrast, in our model,
$\alpha_{w}$, which relates to the spread-narrowing processes, can have a
larger number than $\beta$. If $\lambda_{A}^{d}$ or $\lambda_{B}^{u}$ rapidly
increases owing to a large $\alpha_{w}$, then the spread goes to zero and
$\lambda_{A}^{d}$ or $\lambda_{B}^{u}$ vanishes according to the model
definition. This feature makes it possible to estimate $\alpha_{w}$ without
restriction which represents market resiliency.

(a) $\alpha_{w1}$: after a widening event, the spread decreases by the process
that caused the widening

(b) $\alpha_{w2}$: after a widening event, the spread decreases by the process
that did not cause the widening

(c) $\alpha_{s1}$: self-excitement in the mid-price by one type of process

(d) $\alpha_{s2}$: self-excitement in the mid-price by different types of
processes

(e) $\alpha_{m}$: mutual excitement in the mid-price by one type of process

Figure 3: Possible bid and ask price movements when the corresponding $\alpha$
is large and the other $\alpha$s are close to zero

Note that to fully characterize the mid price and spread processes, we need
additional information about the size of movement in the price processes. The
mid-price process is

$p(t)=p(0)+\int_{0}^{t}\delta_{A}^{u}(u)\mathrm{d}N_{A}^{u}(u)+\int_{0}^{t}\delta_{B}^{u}(u)\mathrm{d}N_{B}^{u}(u)-\int_{0}^{t}\delta_{A}^{d}(u)\mathrm{d}N_{A}^{d}(u)-\int_{0}^{t}\delta_{B}^{d}(u)\mathrm{d}N_{B}^{d}(u)$

where $\delta_{A}^{u},\delta_{B}^{u},\delta_{A}^{d}$ and $\delta_{B}^{d}$
denote the corresponding jump sizes. Similarly, the spread process is

$L(t)=L(0)+\int_{0}^{t}\delta_{A}^{u}(u)\mathrm{d}N_{A}^{u}(u)+\int_{0}^{t}\delta_{B}^{d}(u)\mathrm{d}N_{B}^{d}(u)-\int_{0}^{t}\delta_{A}^{d}(u)\mathrm{d}N_{A}^{d}(u)-\int_{0}^{t}\delta_{B}^{u}(u)\mathrm{d}N_{B}^{u}(u).$

However, the inclusion of the jump size to the model make introduce marked
point processes, which is beyond the scope of our paper. We assume that the
jump sizes at time $t$ are exogenously given, i.e., $\delta(t)$s are
$\mathcal{F}_{t}$-measurable but not measurable to the internal history of
$\bm{N}$. In our model, only if the jump sizes are all constant unity, then
the mid-price and spread processes are endogenously determined.

##  4 Mathematical analysis

In this section, we investigate the mathematical properties of the model.
However, since the proposed model in Definition 1 is too complicated, slightly
different versions are investigated instead in Subsections 4.2 and 4.3. This
does not provide a complete proof of the stability of the proposed model, but
will indirectly provide insight.

###  4.1 Reset property

The stochastic terms $\alpha_{n}$s reset the excitement terms in the spread-
narrowing intensities, $\lambda_{2}$ and $\lambda_{3}$, to $\xi\ell$ right
after a spread-narrowing event. Let $\\{\tau_{k}\\}$ be a sequence of the
spread-narrowing event times. For $i=2$ and $3$,

$\lambda_{i}(\tau_{k}+)=f(\tau_{k})+\xi\ell(\tau_{k})$

where the excitement term is reset to $\xi\ell(\tau_{k})$ which only depends
on the current level of spread. In addition, for $\tau_{k}<t<\tau_{k+1}$,

$\lambda_{i}(t)=f(\ell(t-))+\xi\ell(\tau_{k})\mathrm{e}^{-\beta(t-\tau_{k})}+\int_{\tau_{k}}^{t}\alpha_{w1}\mathrm{e}^{-\beta(t-u)}\mathrm{d}N_{1}(u)+\int_{\tau_{k}}^{t}\alpha_{w2}\mathrm{e}^{-\beta(t-u)}\mathrm{d}N_{4}(u).$
(5)

The above equation implies that whenever a spread-narrowing event happens, the
excitement term reset to $\xi\ell$ with decreasing rate $\beta$ and up to the
next spread-narrowing event, typical constant Hawkes mutually excitement
terms, $\alpha_{w}$s, are added with decreasing rate $\beta$ on the
occurrences of $N_{1}$ and $N_{4}$. This is because the excited parts of the
spread-narrowing intensities before time $\tau_{k}$ are canceled out by

$-\sum_{j=1}^{4}\lambda_{ij}(\tau_{k}),\textrm{ for }i=2\textrm{ and }3.$

Especially, when the spread hits the minimum by a spread-narrowing event, the
reset term $\xi\ell$ becomes zero and hence the excitements of intensities of
the spread-narrowing processes also become zero.

###  4.2 Mid-price process under constant model

In this subsection, we examine the properties of the mid-price process from a
slightly different version of our model. In the linear model of $f$ and
$\alpha_{n}$ in Definition 1, there is a rather complicated aspect of the
analysis, so let’s simplify the model a little bit. Because the mid-price is
the average of the bid and ask prices, we can derive the mid-price process
from the bid and ask price processes. Let $N^{u}$ and $N^{d}$ be the counting
processes for the upward and downward movements of the mid-price process;
then,

$N^{u}=N^{u}_{A}+N^{u}_{B},\quad N^{d}=N^{d}_{A}+N^{d}_{B}$

and

$\lambda^{u}=\lambda^{u}_{A}+\lambda^{u}_{B},\quad\lambda^{d}=\lambda^{d}_{A}+\lambda^{d}_{B}.$

Thus, in the symmetric setting (i.e., $\alpha_{n}:=\alpha_{n1}=\alpha_{n2}$),

$\displaystyle\lambda^{u}(t)=\mu+f(\ell(t-))$
$\displaystyle+\int_{-\infty}^{t}(\alpha_{s1}+\alpha_{w2})\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{u}_{A}(u)+\int_{-\infty}^{t}(\alpha_{s2}+\alpha_{n}(u))\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{u}_{B}(u)$
$\displaystyle+\int_{-\infty}^{t}(\alpha_{m}+\alpha_{n}(u))\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{d}_{A}(u)+\int_{-\infty}^{t}\alpha_{w1}\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{d}_{B}(u)$
$\displaystyle\lambda^{d}(t)=\mu+f(\ell(t-))$
$\displaystyle+\int_{-\infty}^{t}\alpha_{w1}\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{u}_{A}(u)+\int_{-\infty}^{t}(\alpha_{m}+\alpha_{n}(u))\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{u}_{B}(u)$
$\displaystyle+\int_{-\infty}^{t}(\alpha_{s2}+\alpha_{n}(u))\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{d}_{A}(u)+\int_{-\infty}^{t}(\alpha_{s1}+\alpha_{w2})\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{d}_{B}(u).$

As mentioned before, we adopt a simpler model in this subsection, a constant
model for $f$ and $\alpha_{n}$; that is,

$f(\ell(t-))=\bf{1}_{\\{\ell(t-)>0\\}}\eta$

and

$\displaystyle\alpha_{n1}(t)=-\bf{1}_{\\{\ell(t)=0\\}}\sum_{j=1}^{4}\lambda_{2j}(t)+\bf{1}_{\\{\ell(t)>0\\}}\xi,\quad\alpha_{n2}(t)=-\bf{1}_{\\{\ell(t)=0\\}}\sum_{j=1}^{4}\lambda_{3j}(t)+\bf{1}_{\\{\ell(t)>0\\}}\xi,$

for some constants $\eta>0$ and $\xi>0$. This model is simpler than the linear
model, but it consistent with the general concept in Subsection 3.1. This
model guarantees zero intensities of the spread-narrowing processes at the
minimum spread as in the linear model. The base intensities for the spread-
narrowing processes become zero right after the spread hit the minimum due to
the fact that $f(0)=0$, and the excitation parts are canceled out by

$-\sum_{j=1}^{4}\lambda_{2j}(t).$

By canceling the exact amount of the existing excitation parts, it guarantees
the nonnegative intensities. Once the spread becomes larger than zero, the
model acts like a basic Hawkes model with constant parameters $\eta$ and
$\xi$.

Under the special circumstances, the mid-price process of the constant model
is similar to that of the classical Hawkes model. We assume that the spread is
sufficiently large; that is, $\mathbb{P}(\ell(t)=0)$ is close to zero under
the constant model. Then,

$\displaystyle\lambda^{u}(t)\approx\mu+\eta$
$\displaystyle+\int_{-\infty}^{t}(\alpha_{s1}+\alpha_{w2})\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{u}_{A}(u)+\int_{-\infty}^{t}(\alpha_{s2}+\xi)\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{u}_{B}(u)$
$\displaystyle+\int_{-\infty}^{t}(\alpha_{m}+\xi)\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{d}_{A}(u)+\int_{-\infty}^{t}\alpha_{w1}\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{d}_{B}(u)$
$\displaystyle\lambda^{d}(t)\approx\mu+\eta$
$\displaystyle+\int_{-\infty}^{t}\alpha_{w1}\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{u}_{A}(u)+\int_{-\infty}^{t}(\alpha_{m}+\xi)\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{u}_{B}(u)$
$\displaystyle+\int_{-\infty}^{t}(\alpha_{s2}+\xi)\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{d}_{A}(u)+\int_{-\infty}^{t}(\alpha_{s1}+\alpha_{w2})\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{d}_{B}(u).$

For further simplification, suppose that
$\alpha_{s1}+\alpha_{w2}=\alpha_{s2}+\xi$ and $\alpha_{w1}=\alpha_{m}+\xi$,
which is used for demonstration purposes. Then,

$\displaystyle\lambda^{u}(t)$
$\displaystyle\approx\mu+\eta+\int_{-\infty}^{t}(\alpha_{s1}+\alpha_{w2})\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{u}(u)+\int_{-\infty}^{t}\alpha_{w1}\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{d}(u)$
$\displaystyle\lambda^{d}(t)$
$\displaystyle\approx\mu+\eta+\int_{-\infty}^{t}\alpha_{w1}\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{u}(u)+\int_{-\infty}^{t}(\alpha_{s1}+\alpha_{w2})\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{d}(u),$

which is a typical symmetric self- and mutually excited Hawkes process for the
mid-price dynamics. To describe the full dynamics of the mid-price process, we
require a jump size distribution.

###  4.3 Spread process under absolute level model

This subsection investigate the stability in mean of the spread process under
a slightly different version. The spread is the difference between the bid and
ask prices; thus, the bid and ask price processes can induce counting
processes for the number of upward or downward movements of the spread
process. Let $N_{s}^{u}$ and $N_{s}^{d}$ be the counting processes for the
upward and downward movements of the spread process, respectively. Then,

$N_{s}^{u}=N^{u}_{A}+N^{d}_{B},\quad N_{s}^{d}=N^{d}_{A}+N^{u}_{B}$

and

$\lambda_{s}^{u}=\lambda^{u}_{A}+\lambda^{d}_{B},\quad\lambda_{s}^{d}=\lambda^{d}_{A}+\lambda^{u}_{B}.$

Thus, in the symmetric setting,

$\displaystyle\lambda_{s}^{u}(t)=2\mu+\int_{-\infty}^{t}\alpha_{s1}\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{u}_{s}(u)+\int_{-\infty}^{t}(\alpha_{s2}+\alpha_{m})\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{d}_{s}(u)$
(6)
$\displaystyle\lambda_{s}^{d}(t)=2f(L(t-))+\int_{-\infty}^{t}(\alpha_{w1}+\alpha_{w2})\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{u}_{s}(u)+\int_{-\infty}^{t}\alpha_{n}(u)\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{d}_{s}(u).$
(7)

The intensity of the upward movements of the spread process is the classical
Hawkes model and that of the downward movements is the Hawkes model with
stochastic excitement. As mentioned previously, to describe the full dynamics
of the spread process, we require a jump size distribution of up and down
price movements, which is beyond the scope of this study. In addition, for
mathematical analysis, it is more tractable when modeling with absolute level
of spread (especially in the base intensity) in Eq. (7).

Assume that the jump size is one so that the spread process is represented by
the difference between $N_{s}^{u}$ and $N_{s}^{d}$. In addition, let us start
with a simple case such that

$\displaystyle\lambda_{s}^{u}(t)=2\mu+\int_{-\infty}^{t}\alpha_{s1}\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{u}_{s}(u)+\int_{-\infty}^{t}(\alpha_{s2}+\alpha_{m})\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{d}_{s}(u)$
(8) $\displaystyle\lambda_{s}^{d}(t)=2\eta L(t-)$ (9)

i.e., without the excitement terms in $\lambda_{s}^{d}(t)$. Since the
differential form of $\lambda_{s}^{u}(t)$ can be written as

$\mathrm{d}\lambda_{s}^{u}(t)=\beta(2\mu-\lambda_{s}^{u}(t))\mathrm{d}t+\alpha_{s1}\mathrm{d}N_{s}^{u}(t)+(\alpha_{s2}+\alpha_{m})\mathrm{d}N_{s}^{d}(u),$

we have

$\mathbb{E}[\lambda_{s}^{u}(t)|\mathcal{F}_{0}]=\lambda_{s}^{u}(0)+\int_{0}^{t}2\\{\beta\mu+(\alpha_{s1}-\beta)\mathbb{E}[\lambda_{s}^{u}(u)]+(\alpha_{s2}+\alpha_{m})\mathbb{E}[\lambda_{s}^{d}(u)]\\}\mathrm{d}u.$

In addition,

$\lambda_{s}^{d}(t)=2\eta L(t-)=2\eta(L(0)+N_{s}^{u}(t-)-N_{s}^{d}(t-))$

and

$\mathbb{E}[\lambda_{s}^{d}(t)|\mathcal{F}_{0}]=2\eta
L(0)+\int_{0}^{t}2\eta(\mathbb{E}[\lambda_{s}^{u}(u)]-\mathbb{E}[\lambda_{s}^{d}(u)])\mathrm{d}u.$

Thus, we construct a system of differential equation,

$\displaystyle\frac{\mathrm{d}\mathbb{E}[\lambda_{s}^{u}(t)|\mathcal{F}_{0}]}{\mathrm{d}t}=2\beta\mu+(\alpha_{s1}-\beta)\mathbb{E}[\lambda_{s}^{u}(t)]+(\alpha_{s2}+\alpha_{m})\mathbb{E}[\lambda_{s}^{d}(t)]$
$\displaystyle\frac{\mathrm{d}\mathbb{E}[\lambda_{s}^{d}(t)|\mathcal{F}_{0}]}{\mathrm{d}t}=2\eta(\mathbb{E}[\lambda_{s}^{u}(t)]-\mathbb{E}[\lambda_{s}^{d}(t)]).$

The matrix of the system of differential equation is

$A=\begin{bmatrix}\alpha_{s1}-\beta&\alpha_{s2}+\alpha_{m}\\\
2\eta&-2\eta\end{bmatrix}.$

It is known that the stability conditions of the system is that the trace of
$A$ is negative and the determinant is positive, i.e.,

$\mathrm{tr}(A)=\alpha_{s1}-\beta-2\eta<0,\quad\mathrm{det}(A)=2\eta(\beta-\alpha_{s1}-\alpha_{s2}-\alpha_{m})>0.$

Since $\eta>0$, if Eq. (3) holds, then this stability condition satisfies. In
addition, under the steady-state of the differential equation system,

$\begin{bmatrix}\mathbb{E}[\lambda_{s}^{u}(t)]\\\
\mathbb{E}[\lambda_{s}^{d}(t)]\end{bmatrix}=-A^{-1}\begin{bmatrix}2\beta\mu\\\
0\end{bmatrix}$

and this yields that the steady-state average intensity is

$\mathbb{E}[\lambda_{s}^{u}(t)]=\mathbb{E}[\lambda_{s}^{d}(t)]=\frac{2\beta\mu}{\beta-\alpha_{s1}-\alpha_{s_{2}}-\alpha_{m}}$

and

$\mathbb{E}[L(t)]=\frac{\mathbb{E}[\lambda_{s}^{d}(t)]}{2\eta}=\frac{\beta\mu}{\eta(\beta-\alpha_{s1}-\alpha_{s_{2}}-\alpha_{m})}.$

Therefore, under the Eqs. (8) and (9), the expected intensities and the
absolute level of the spread are finite.

Adding exciting terms to $\lambda_{s}^{d}$ in Eq. (9) does not change much of
the discussion above. As in the previous, let $\\{\tau_{k}\\}$ be a sequence
of the spread-narrowing event times. For $\tau_{k}<t<\tau_{k+1}$,

$\lambda_{s}^{u}(t)=2\mu+(\lambda_{s}^{u}(\tau_{k})-2\mu)\mathrm{e}^{-\beta(t-\tau_{k})}+\int_{\tau_{k}}^{t}\alpha_{s1}\mathrm{e}^{-\beta(t-u)}\mathrm{d}N_{s}^{u}(u)$

and $\lambda_{s}^{u}(t)$ only has a self-exciting feature, and hence if Eq.
(3) holds, there are only finite number of events by $N_{s}^{u}$ during the
period $[\tau_{k},\tau_{k+1}]$ almost surely and $\lambda_{s}^{u}(t)$ will not
explode to infinity. In addition, for $\tau_{k}<t<\tau_{k+1}$, by recalling
the reset property for $\lambda_{s}^{d}=\lambda_{2}+\lambda_{3}$ in Eq (5),

$\lambda_{s}^{d}(t)=2\eta L(t-)+\xi
L(\tau_{k})\mathrm{e}^{-\beta(t-\tau_{k})}+\int_{\tau_{k}}^{t}(\alpha_{w1}+\alpha_{w2})\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{u}_{s}(u)$

is also finite almost surely, because there are only finite events by
$N_{s}^{u}$ (even when $\alpha_{w1}+\alpha_{w2}>\beta$).

Since the processes are stable for $\tau_{k}<t<\tau_{k+1}$, it is sufficient
to check the stability at each time of $\\{\tau_{k}\\}$. By recalling the
reset property, the intensities right after a spread-narrowing event can be
represented by

$\displaystyle\lambda_{s}^{u}(\tau_{k}+)$
$\displaystyle=2\mu+\int_{-\infty}^{\tau_{k}}\alpha_{s1}\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{u}_{s}(u)+\int_{-\infty}^{\tau_{k}}(\alpha_{s2}+\alpha_{m})\mathrm{e}^{-\beta(t-u)}\mathrm{d}N^{d}_{s}(u)$
(10) $\displaystyle\lambda_{s}^{d}(\tau_{k}+)$ $\displaystyle=2\eta
L(\tau_{k}).$ (11)

These are discrete-time versions of Eqs. (8) and (9) and as long as Eq. (3)
holds, we can expect the stability at times of $\\{\tau_{k}\\}$.

##  5 Empirical study

Our model allows us to examine various aspects of the first level order book
in high-frequency. This section includes the following topics.

  * •

Since the intensities are functions of the bid-ask spread level, the model
captures the narrowing tendency of the spread when the spread is large, see
Subsection 5.3.2.

  * •

The parameter $\alpha$s measure the responsiveness rates of the market to the
previous events, see also Subsection 5.3.2.

  * •

For each stock, by comparing liquidity providing parameters, $\alpha_{w}$s,
and removing parameters, $\alpha_{m}$ and $\alpha_{s}$s, we can examine the
market-making speed in various stocks, see Subsection 5.3.3.

  * •

We also found some stocks received particular attentions from ultra-high-
frequency liquidity providers, see Subsection 5.3.4.

  * •

The estimation results also show that the propensities of participants differ
for each stock exchange, see Subsection 5.4.

  * •

The estimation result of $\eta$ in an almost continuous-time provides another
perspective during the Flash Crash, see Subsection 5.5.

Before the main empirical analysis results, we explain the method, data and
model selection.

###  5.1 Likelihood estimation and simulation study

We estimate the parameters of the model using the maximum log-likelihood
method. The method to calculate the log-likelihood of our model is similar to
that for calculating the log-likelihood of the classical Hawkes process. For
discussions on maximum likelihood estimations for the stationary point
process, see Ogata (1978) and Ozaki (1979). The log-likelihood of the model up
to time $T$ is

$\log\mathcal{L}(\theta,T)=\sum_{i=1}^{4}\left(\int_{0}^{T}\log\lambda_{i}(t)\mathrm{d}N_{i}(t)-\int_{0}^{T}\lambda_{i}(t)\mathrm{d}t\right),$

where the intensity functions should be considered to be left continuous and
$\theta$ denotes the set of model parameters. We calculate the location of the
maximum log-likelihood using a numerical optimizer in R; for more details, see
Nash (2014).

We conduct simulations to check whether the numerical optimizer works well in
our model. We generate 500 simulated sample paths using presumed parameter
values and then estimate the parameters for each sample path. For the
simulation of the Hawkes process, we can apply the thinning algorithm of Ogata
(1981) or the exact method of Dassios et al. (2013). Thus, we obtain 500
estimation results for each parameter. Each sample path has 10,000
observations of bid or ask price changes. The presumed parameter values in the
simulation are the estimates from the real stock data in the later empirical
study. Likewise, the distributions of the jump sizes in the price processes
are based on empirical distributions. Table 1 summarizes the results. The row
labeled “mean” is the average of the 500 estimates calculated from the 500
sample paths and the row labeled “std.” is the standard deviation of the
estimates. The results show that the numerical estimator is unbiased with
sufficiently small standard deviations.

Table 1: Simulation study with 500 sample paths | $\mu$ | $\eta$ | $\alpha_{s1}$ | $\alpha_{s2}$ | $\alpha_{m}$ | $\alpha_{w1}$ | $\alpha_{w2}$ | $\beta$ | $\xi$
---|---|---|---|---|---|---|---|---|---
True | 0.080 | 0.100 | 4.000 | 26.00 | 5.000 | 11.00 | 7.000 | 50.00 | 2.700
Mean | 0.080 | 0.100 | 3.993 | 26.02 | 5.003 | 11.02 | 6.973 | 50.03 | 2.705
Std. | 0.003 | 0.002 | 0.360 | 0.658 | 0.277 | 0.477 | 0.404 | 0.796 | 0.085
True | 0.170 | 0.140 | 200.0 | 250.0 | 150.0 | 300.0 | 330.0 | 1200 | 50.00
Mean | 0.170 | 0.140 | 199.6 | 252.3 | 151.3 | 301.5 | 332.0 | 1207 | 50.04
Std. | 0.003 | 0.003 | 8.369 | 8.362 | 7.327 | 9.371 | 9.210 | 7.570 | 2.441

Since the log-likelihood function is not convex, it may be questionable
whether the numerical optimizer can find the maximum well. The following
simulation study demonstrates that if true $\beta$ is large and the initial
value of $\beta$ is also large, then sometimes the appropriate convergence for
$\alpha$s, $\xi$, and $\beta$ might not occur. The difficulty can be mitigated
with large sample sizes and small initial points of optimization.

Figure 4 is for simulation result testing convergence rate with true
$\beta=400$ in the left and true $\beta=1,600$ in the right. For other
parameters, we use $\mu=0.1,\eta=0,\xi=\beta/10$ and all $\alpha$s are a
quarter of $\beta$, for simplicity. With various fixed initial $\beta_{0}$,
represented in the x-axis, the rate of successful convergences is examined
with 500 randomly generated sample paths. The convergence is defined as
success when the root mean squared error between true parameter values and
estimates is less than 0.2 The initial values of $\mu_{0}$ and $\eta_{0}$ are
randomly chosen between 0 and 10, and $\alpha$s and $\xi$ are also randomly
chosen between 0 and $\beta_{0}$.

The left panel of the figure shows that when $\beta$ is relatively small such
as 400, in both sample sizes 5,000 and 10,000, the optimization converges well
even with relatively large $\beta_{0}$. For relatively large $\beta$ as in the
right panel, we recommend relatively small initial points for $\beta$ and
$\alpha$s. Numerical optimization with a rather small initial value of $\beta$
leads to a successful result in most cases. With sample size 10,000 and true
$\beta=1,600$, with reasonable choices of the initial values of $\beta$ such
as around from 10 to 1,000, the optimizations converge well.

Figure 4: The success rate of estimation to find the global maximum with
various initial points (x-axis) with true $\beta=400$ (left) and $\beta=1,600$
(right)

###  5.2 Data

The high-frequency data for the analysis consist of the arrival time stamps
and prices of the best bid and ask quotes from various stock exchanges in the
US managed by the Consolidated Tape Association or Unlisted Trading Privileges
(UTP). We obtain the limit order information for NYSE-listed stocks from the
Consolidated Tape System (CTS) with time resolutions in milliseconds up to
July 2015, microseconds up to around September 2018, and nanoseconds for the
rest. The series of time stamps of the Nasdaq-listed securities’ limit order
events managed by the UTP Plan are recorded in milliseconds up to July 2015,
microseconds up to around October 2016, and nanoseconds for the rest. The
sample for the empirical study ranges from March 2009 to December 2019. We use
data from 10:00 to 15:30 to reduce the intraday seasonal effects of market
opening and closing.

We focus on two types of best bid and ask price processes: the national best
bid and offer (NBBO) price processes and the best bid and offer (BBO) price
processes per stock exchange, as the US stock market is highly fragmented. See
Table 2 for more details. More precisely, the NBBO price data consist of the
official NBBO data from the NYSE and the quotes completed by Wharton Research
Data Services (WRDS). The quotes supplemented by the WRDS are those that
contain both the national best bid and ask prices that the NYSE does not
add333https://wrds-www.wharton.upenn.edu/pages/support/data-overview/wrds-
overview-taq/. Because only the displayed limit orders generate the quote
data, we base our analysis solely on these values. In addition, since our
model depends on the spread, we use samples with a sufficient spread (i.e.,
with prices sufficiently large, around $100 or higher, relative to the tick
size $0.01).

Table 2: US stock market exchanges Abbreviation | Exchange
---|---
NYSE | New York Stock Exchange
NSDQ | Nasdaq Stock Market
Arca | NYSE Arca
BZX | Cboe BZX exchange, formerly Bats Global Markets
BYX | Cboe BYX exchange, formerly Bats-Y Global Markets
EDGA | Cboe EDGA exchange, formerly Direct Edge Exchange A
EDGX | Cboe EDGX exchange, formerly Direct Edge Exchange X
BSTN | Nasdaq OMX BX, formerly Boston Stock Exchange
PSX | Nasdaq PSX
NSX | National Cincinnati Stock Exchange
IEX | Investors Exchange

Table 3 shows the percentage of NBBO contributions by exchange for the IBM and
AMZN stocks. The NYSE, Arca, NSDQ, and BZX mainly determine the NBBO for the
NYSE-listed stock IBM. The Arca, NSDQ, BZX, and, recently, PSX determine the
NBBO for the Nasdaq-listed stock AMZN. A higher percentage indicates more
substantial liquidity provided by the corresponding exchange.

Table 3: Percentage of BBOs from each exchange in the NBBO | IBM
---|---
Year | NYSE | Arca | NSDQ | BZX | BYX | EDGA | EDGX | BSTN | PSX | NSX
2009 | 20.1 | 20.3 | 27.6 | 22.9 | 0.0 | 0.0 | 0.0 | 1.8 | 0.0 | 2.8
2010 | 45.8 | 14.6 | 26.6 | 6.6 | 0.0 | 0.8 | 1.0 | 1.3 | 0.0 | 1.0
2011 | 20.2 | 11.5 | 20.7 | 8.0 | 10.4 | 7.1 | 3.3 | 11.4 | 2.0 | 5.3
2012 | 16.2 | 12.1 | 23.2 | 7.8 | 8.8 | 2.1 | 5.3 | 10.7 | 0.4 | 13.0
2013 | 22.5 | 13.0 | 18.8 | 11.1 | 5.8 | 3.8 | 7.2 | 7.6 | 2.3 | 6.8
2014 | 16.0 | 10.5 | 25.8 | 8.1 | 11.7 | 7.0 | 6.6 | 6.4 | 4.3 | 2.9
2015 | 25.3 | 10.3 | 25.6 | 7.7 | 9.7 | 5.8 | 5.9 | 6.0 | 3.6 | 0.0
2016 | 50.1 | 8.8 | 23.4 | 4.8 | 1.9 | 0.6 | 4.8 | 2.1 | 1.1 | 0.0
2017 | 35.5 | 12.0 | 23.7 | 8.2 | 2.8 | 1.0 | 8.8 | 3.0 | 1.7 | 0.0
2018 | 27.5 | 6.5 | 18.9 | 16.3 | 2.8 | 1.1 | 6.5 | 3.5 | 2.8 | 2.1
| AMZN
2009 | 0.0 | 17.5 | 43.7 | 27.0 | 0.0 | 0.0 | 0.0 | 3.6 | 0.0 | 1.9
2010 | 0.0 | 18.3 | 52.3 | 11.1 | 0.0 | 0.8 | 1.7 | 1.2 | 0.0 | 10.8
2011 | 0.0 | 17.7 | 23.3 | 6.6 | 8.2 | 7.3 | 3.2 | 11.9 | 2.0 | 18.1
2012 | 0.0 | 15.4 | 21.1 | 5.7 | 8.9 | 2.9 | 4.2 | 31.0 | 1.5 | 9.0
2013 | 0.0 | 9.6 | 21.2 | 5.4 | 7.0 | 6.2 | 6.8 | 21.6 | 8.8 | 12.7
2014 | 0.0 | 10.4 | 31.1 | 5.8 | 11.6 | 10.0 | 9.8 | 10.6 | 7.8 | 0.7
2015 | 0.0 | 14.2 | 33.2 | 8.0 | 9.8 | 6.3 | 8.2 | 13.9 | 6.5 | 0.0
2016 | 0.0 | 15.3 | 31.3 | 8.7 | 3.6 | 1.1 | 7.1 | 1.4 | 8.7 | 0.0
2017 | 0.0 | 7.3 | 24.9 | 6.1 | 3.2 | 0.4 | 4.0 | 2.4 | 33.8 | 0.0
2018 | 0.0 | 3.2 | 21.0 | 7.5 | 0.8 | 0.2 | 2.6 | 0.4 | 22.4 | 19.1

Because it is not possible to apply some of the original data directly to the
Hawkes estimation, we apply data preprocessing. For example, some records of
different quotes have the same time stamps. Because the probability of two or
more events occurring at the same time is almost surely zero in the Hawkes-
type model, these time records require modification. We assume that the same
time quotes are nearly simultaneously released limit orders and we relocate
the times of these records to equidistant arrivals within the time resolution.

In some cases, the best bid and ask prices change at the same time. Similar to
the previous case, we separate the orders into two successive events that we
assume occur sequentially within the time resolution. Whether we assume that
the bid or ask price changes first depends on the spread. For example, if we
assume that the bid quote occurs first and the bid price becomes greater than
or equal to the ask price as a result, then the assumption is false. In this
case, it is reasonable to assume that the ask quote occurs first. If it does
not matter which quote occurs first, then we randomly select the order of the
quotes.

Before March 2009, the percentage of data with these issues is as high as 20%.
From March 2009 (more precisely, February 25, 2009), the quality of the data
improves dramatically. For example, from March 2009, around 2% of the data
have the above issues, and in 2018, less than 0.1% of the data do. Because the
proportion of abnormal data is small, we assume that the data preprocessing
described above does not have a significant impact on our statistical
analysis.

In addition, we need to modify the NBBO data before the estimation. The
national best bid and ask prices can be equal or sometimes reversed for
reasons such as network delays. The following is an example of IBM’s NBBO on
January 3, 2018. At time 10:03:09.057956, the national best bid price was
$159.33, disclosed by exchange X (Nasdaq PSX). After about a second, the new
national best ask price was $159.33, disclosed by exchange P (NYSE Arca).

Time | Bid | Ask | Exchange
---|---|---|---
10:03:09.057956 | 159.33 |  | X
$\vdots$
10:03:10.107464 |  | 159.33 | P
10:03:10.107586 | 159.32 |  | X

To conserve space, we do not show the best ask prices for all the exchanges,
but $159.33 was the prevailing ask price across all the exchanges at this
moment. However, because the best bid price was still $159.33, this best ask
price was not logically correct. This result may have been due to network or
computational delays, and we consider the following scenario in this case.

The following table lists the transaction details near the time of the event
above. Using these details, we can presume the following scenario, although it
is not possible to know the actual process exactly.

Time | Price | Exchange
---|---|---
10:03:09.060686 | 159.33 | P
10:03:09.150237 | 159.33 | K
10:03:10.107383 | 159.33 | K
10:03:10.107446 | 159.33 | K
10:03:10.107650 | 159.33 | X

A large sell order arrived at exchange P at time 10:03:09.060686. It consumed
all existing bid limit orders at exchange P, but was still not filled
completely. This order was routed to other exchanges, and the trading system
noticed that the volume was still insufficient across all the exchanges. For
the existing matched quantity, the transactions proceeded in parallel on
exchanges K and X, and the insufficient quantity was posted as an ask limit
order on exchange P at time 10:03:10.107464, as in the previous table. The
transaction processes took more time and were eventually finalized in exchange
X at time 10:03:10.107586. The best bid price was updated when all the limit
bid orders were consumed at $159.33. Transaction reporting ended slightly
later and finally finished at 10:03.10.107650 (combined with the previous
transactions in exchange K). For physical reasons, this type of delay seems to
be inevitable. For example, Ding et al. (2014) report that the discrepancy
between the NBBOs and synthetic NBBOs calculated using data from the exchanges
is around one to two milliseconds.

Events that violate the assumptions of our model constitute about 1–2% of all
events. Because the data set is so large, it is impossible to identify the
cause of these anomalies in each case. We assume that these data do not
significantly change the statistical properties and proceed with the
estimation after removing these records.

###  5.3 Estimation results with the NBBO

####  5.3.1 Model selection

This subsection covers the results of various model selection issues. First,
we discuss the estimation results from the basic Hawkes model for the bid and
ask price processes, even though the model is unsuitable because it does not
guarantee the positivity and reverting property of the spread process.
Consider the basic Hawkes process with constant parameters:

$\textrm{Basic Hawkes model: }\bm{\mu}_{t}=\begin{bmatrix}\mu\\\ \mu\\\ \mu\\\
\mu\end{bmatrix},\quad\bm{h}(t,u)=\mathrm{e}^{-\beta(t-u)}\begin{bmatrix}\alpha_{11}&\alpha_{12}&\alpha_{13}&\alpha_{14}\\\
\alpha_{21}&\alpha_{22}&\alpha_{23}&\alpha_{24}\\\
\alpha_{31}&\alpha_{32}&\alpha_{33}&\alpha_{34}\\\
\alpha_{41}&\alpha_{42}&\alpha_{43}&\alpha_{44}\\\ \end{bmatrix}.$

We can obtain some intuitions by fitting the data to the basic Hawkes model.

Figure 5(a) plots the daily estimated $\alpha_{21}$ and $\alpha_{34}$, while
Figure 5(b) plots $\alpha_{24}$ and $\alpha_{31}$. These terms represent the
excitation terms of the upward movements of the ask price and downward
movements of the bid price caused by events that widen the spread (i.e.,
$A^{u}$ and $B^{d}$). To improve the clarity of the graph, we omit the
standard errors, but note that these values are significantly greater than
zero. The estimates of $\alpha_{24}$ and $\alpha_{31}$ are similar, as are
those of $\alpha_{21}$ and $\alpha_{34}$. The parameters $\alpha_{21}$ and
$\alpha_{34}$ are related to $\alpha_{w1}$ in our model, whereas $\alpha_{24}$
and $\alpha_{31}$ are related to $\alpha_{w2}$.

Figures 5(c) and 5(d) present the estimates of
$\alpha_{11},\alpha_{44},\alpha_{12},$ and $\alpha_{43}$. The parameters
$\alpha_{11}$ and $\alpha_{44}$ are typical self-exciting terms in the Hawkes
model and their values are similar. The parameters $\alpha_{12}$ and
$\alpha_{43}$ are typical mutually exciting terms in the Hawkes model and,
again, they have similar values. Figure 5(e) illustrates the other mutually
exciting terms $\alpha_{13}$ and $\alpha_{42}$, showing that these values are
also positive and similar.

Figure 5(f) shows the estimates of $\alpha_{14}$ and $\alpha_{41}$. These
values are much lower than those of the other parameters. Thus, if the ask
price is up, then the tendency of the bid price to decline is relatively less
significant. We can apply the same argument to the case in which the bid price
is down. Because these values are relatively small, we assume that the
corresponding parameters in our proposed model are zeros for model parsimony.
We discuss more details in the next model comparison part.

(a) $\alpha_{21}$ and $\alpha_{34}$

(b) $\alpha_{24}$ and $\alpha_{31}$

(c) $\alpha_{11}$ and $\alpha_{44}$

(d) $\alpha_{12}$ and $\alpha_{43}$

(e) $\alpha_{13}$ and $\alpha_{42}$

(f) $\alpha_{14}$ and $\alpha_{41}$

Figure 5: Daily estimates for the NBBO of IBM under the basic Hawkes model
from 2009 to 2018

To highlight the difference between the basic Hawkes model and proposed model,
Figure 6 shows the sample paths for the bid and ask processes. For the
simulation, the parameter values are assumed to be one of the estimates
obtained in the empirical analysis. As shown in Figure 6(a), the basic Hawkes
model has negative or unrealistically wide spreads. The proposed model, on the
contrary, incorporates the basic nature of the bid-ask spread.

(a) basic Hawkes

(b) proposed model

Figure 6: Comparison of the bid and ask processes between the basic Hawkes
model and proposed model

Figure 7 shows the Akaike information criterion (AIC) and Bayesian information
criterion (BIC) using IBM’s NBBO data for the first quarter of 2017. The
proposed model (solid black line) is preferred, as it has lower values than
the basic Hawkes model (dashed blue line) in this period.

(a) AIC

(b) BIC

Figure 7: Comparison of the AIC and BIC between the proposed model and basic
Hawkes model

Second, we examine the model assuming that the coefficients of the top right
and bottom left of $\mathbf{h}$ are not zeros.

$\textrm{Extended model I:
}\quad\bm{h}(t,u)=\mathrm{e}^{-\beta(t-u)}\begin{bmatrix}\alpha_{s1}&\alpha_{m}&\alpha_{s2}&\alpha_{14}\\\
\alpha_{w_{1}}&\alpha_{n1}(u)&\alpha_{n1}(u)&\alpha_{w_{2}}\\\
\alpha_{w_{2}}&\alpha_{n2}(u)&\alpha_{n2}(u)&\alpha_{w_{1}}\\\
\alpha_{41}&\alpha_{s2}&\alpha_{m}&\alpha_{s1}\\\ \end{bmatrix}$

and the other parts are as in Definition 1. The assumption in our model that
the corner coefficients of $\mathbf{h}$ are zeros does not imply that these
coefficients are always zeros. Based on the empirical results analyzed daily,
on some of the days, the null hypothesis that the corner coefficients are
zeros is not rejected while on other days this hypothesis is rejected.
However, even on the day that this hypothesis is rejected, the coefficients
have smaller values than the other coefficients in the matrix $\bm{h}$, as
shown in Figure 8(a). The estimates of $\alpha_{14}$ and $\alpha{41}$ are
shown as the red lines in the figure. These parameters may only be significant
on certain days when unexpected news causes a liquidity shock with sudden
spread widening (Jiang et al., 2011; Christensen et al., 2014) and can reflect
the bilateral spread-widening event. However, the inclusion of these two
parameters does not significantly improve the model from the AIC point of
view, as shown in Figure 8(b) on typical days.

(a) $\alpha$s

(b) AIC

Figure 8: Estimates of the $\alpha$s under extended model I (left) and
comparison of the AIC between the proposed model and extended model I (right)

Third, we examine the model with different $\mu$s and $\eta$s in
$\mathbf{\mu}$:

$\textrm{Extended model II:
}\quad\bm{\mu}_{t}=\begin{bmatrix}\mu_{1}+\eta_{1}\ell(t-)\\\
\eta_{2}\ell(t-)\\\ \eta_{3}\ell(t-)\\\
\mu_{4}+\eta_{4}\ell(t-)\end{bmatrix}\quad$

and the other parts are as in Definition1. One may think that when the spreads
increase and market makers provide liquidity more actively, the intensities of
bid-ask narrowing $A_{d}$ and $B_{u}$ increase and the intensities of bid-ask
widening $A_{u}$ and $B_{d}$ decrease. Thus, one can assume that the base
intensities of $A_{d}$ and $B_{u}$ are also functions of the spread. Indeed,
this negative relationship between the spread and intensities of the spread-
widening processes is observed to some extent, as shown in Figure 9(a), but is
close to zero compared with the other coefficients. It also does not
significantly improve the model from the perspective of the AIC, as shown in
Figure 9(b). We do not show a graph of $\mu_{1}$ and $\mu_{4}$ because of a
lack of space, but their estimates are found to be similar.

(a) $\alpha$s

(b) AIC

Figure 9: Estimates of the $\alpha$s under extended model II (left) and
comparison of the AIC between the proposed model and extended model II (right)

Fourth, we examine the model with different $\xi$s. To see if it can be
justified to assume that the $\alpha_{n}$s in the middle of the matrix
$\bm{h}$ are equal, we extend the model so that the $\alpha_{n}$s have
different $\xi$s:

$\textrm{Extended model III:
}\quad\bm{h}(t,u)=\mathrm{e}^{-\beta(t-u)}\begin{bmatrix}\alpha_{s1}&\alpha_{m}&\alpha_{s2}&0\\\
\alpha_{w_{1}}&\alpha_{n1}(u)&\alpha_{n2}(u)&\alpha_{w_{2}}\\\
\alpha_{w_{2}}&\alpha_{n3}(u)&\alpha_{n4}(u)&\alpha_{w_{1}}\\\
0&\alpha_{s2}&\alpha_{m}&\alpha_{s1}\\\ \end{bmatrix}$

where

$\alpha_{ni}(t)=-\sum_{j=1}^{4}\lambda_{kj}(t)+\xi_{i}\ell(t)$

and the other parts are as in Definition 3.1. As in Figure 10(a),
$\xi_{1}\approx\xi_{4}$ and $\xi_{2}\approx\xi_{3}$ and hence the model
extension can serve as an alternative. However, no significant benefit can be
obtained from the perspective of the AIC, as shown in Figure 10(b).

(a) $\xi$s

(b) AIC

Figure 10: Estimates of the $\xi$s under extended model III (left) and
comparison of the AIC between the proposed model and extended model III
(right)

Fifth, one may think that the decaying parameter $\beta$ does not have to be
the same and we compare the proposed model with an extended model with a
column-wise different $\beta$:

$\textrm{Extended model IV:
}\quad\bm{h}(t,u)=\mathbf{B}(t-u)\circ\begin{bmatrix}\alpha_{s1}&\alpha_{m}&\alpha_{s2}&0\\\
\alpha_{w_{1}}&\alpha_{n1}(u)&\alpha_{n2}(u)&\alpha_{w_{2}}\\\
\alpha_{w_{2}}&\alpha_{n3}(u)&\alpha_{n4}(u)&\alpha_{w_{1}}\\\
0&\alpha_{s2}&\alpha_{m}&\alpha_{s1}\\\ \end{bmatrix}$

where $\circ$ denotes the element-wise multiplication (the Hadamard product),

$\mathbf{B}(t-u)=\begin{bmatrix}\mathrm{e}^{-\beta_{1}(t-u)}&\mathrm{e}^{-\beta_{2}(t-u)}&\mathrm{e}^{-\beta_{3}(t-u)}&\mathrm{e}^{-\beta_{4}(t-u)}\\\
\mathrm{e}^{-\beta_{1}(t-u)}&\mathrm{e}^{-\beta_{2}(t-u)}&\mathrm{e}^{-\beta_{3}(t-u)}&\mathrm{e}^{-\beta_{4}(t-u)}\\\
\mathrm{e}^{-\beta_{1}(t-u)}&\mathrm{e}^{-\beta_{2}(t-u)}&\mathrm{e}^{-\beta_{3}(t-u)}&\mathrm{e}^{-\beta_{4}(t-u)}\\\
\mathrm{e}^{-\beta_{1}(t-u)}&\mathrm{e}^{-\beta_{2}(t-u)}&\mathrm{e}^{-\beta_{3}(t-u)}&\mathrm{e}^{-\beta_{4}(t-u)}\\\
\end{bmatrix}$

and the other parts are as in Definition 3.1. As shown in Figure 11, the
column-wise $\beta$s seem to have the same value and there is no improvement
in terms of the AIC.

(a) $\beta$s

(b) AIC

Figure 11: Estimates of the $\beta$s under extended model IV (left) and
comparison of the AIC between the proposed model and extended model IV

Sixth, we compare the proposed model with an extended model with a row-wise
different $\beta$:

$\textrm{Extended model V:
}\quad\bm{h}(t,u)=\mathbf{B}(t-u)\circ\begin{bmatrix}\alpha_{s1}&\alpha_{m}&\alpha_{s2}&0\\\
\alpha_{w_{1}}&\alpha_{n1}(u)&\alpha_{n2}(u)&\alpha_{w_{2}}\\\
\alpha_{w_{2}}&\alpha_{n3}(u)&\alpha_{n4}(u)&\alpha_{w_{1}}\\\
0&\alpha_{s2}&\alpha_{m}&\alpha_{s1}\\\ \end{bmatrix}$

where

$\mathbf{B}(t-u)=\begin{bmatrix}\mathrm{e}^{-\beta_{1}(t-u)}&\mathrm{e}^{-\beta_{1}(t-u)}&\mathrm{e}^{-\beta_{1}(t-u)}&\mathrm{e}^{-\beta_{1}(t-u)}\\\
\mathrm{e}^{-\beta_{2}(t-u)}&\mathrm{e}^{-\beta_{2}(t-u)}&\mathrm{e}^{-\beta_{2}(t-u)}&\mathrm{e}^{-\beta_{2}(t-u)}\\\
\mathrm{e}^{-\beta_{3}(t-u)}&\mathrm{e}^{-\beta_{3}(t-u)}&\mathrm{e}^{-\beta_{3}(t-u)}&\mathrm{e}^{-\beta_{3}(t-u)}\\\
\mathrm{e}^{-\beta_{4}(t-u)}&\mathrm{e}^{-\beta_{4}(t-u)}&\mathrm{e}^{-\beta_{4}(t-u)}&\mathrm{e}^{-\beta_{4}(t-u)}\\\
\end{bmatrix}$

and the other parts are as in Definition 3.1. Extended model V also shows
slightly improved performance in terms of the AIC, even though
$\beta_{1}\approx\beta_{4}$ and $\beta_{2}\approx\beta_{3}$, which is
consistent with the results reported by Lee and Seo (2017b). This slight
improvement is insufficient to use the extended model at the expense of
increased model complexity. Therefore, the empirical studies presented
hereafter are conducted based on the proposed model in Definition 1.

(a) $\beta$s

(b) AIC

Figure 12: Estimates of the $\beta$s under extended model V (left) and
comparison of the AIC between the proposed model and extended model V

In Figure 13, quantile-quantile plots of the residuals versus the unit
exponential distribution are presented. The residuals are defined by

$\bigcup_{1\leq i\leq
4}\left\\{\int_{t_{i,j}}^{t_{i,j+1}}\hat{\lambda}_{i}(u)\mathrm{d}u\right\\}$

where $\hat{\lambda}_{i}$ are fitted intensities of our model and $t_{i,j}$
are event times. The plot shows that the residuals follow a slightly fatter
tail distribution. In order to obtain a better fitting result, it is necessary
to introduce a multi-kernel with different $\beta$s, which leads more complex
model. This is beyond our scope. Instead we focus on describing the activities
at the ultra-high-frequency level, so the extension to multiple kernels is
left as a future work.

(a) IBM

(b) MCD

Figure 13: Q-Q plot of residual process of our model

####  5.3.2 Basic results

This section reports the basic results obtained by the NBBO. Some of these
results may seem obvious, but they are worth checking, while other results are
new. Figure 14 illustrates the dynamics of the estimates from the NBBO of the
IBM stock under the model defined in Definition 3.1 from 2014 to 2019. We
compute all the estimates on a daily basis, and hence have around 250
estimates for each parameter in each year. The time unit for the intensities
is one second. For the detailed results of the estimations with standard
errors, see Appendix A.

(a) $\alpha_{s1}$

(b) $\alpha_{s2}$

(c) $\alpha_{w1}$

(d) $\alpha_{w2}$

(e) $\alpha_{m}$

(f) $\beta$

(g) $\eta$

(h) $\xi$

Figure 14: Dynamics of the daily estimated parameters for the NBBO of IBM from
2014 to 2019

First, the results show that the narrowing tendency of the bid-ask spread
depends on the spread. The estimates of liquidity parameter $\eta$ are
significantly positive throughout the period, as Figure 14(g) shows. Recall
that we use $\eta\ell(t-)$ in the base intensity of the spread-narrowing
processes (i.e., downward movements of ask prices and upward movements of bid
prices). A positive $\eta$ implies that with a larger bid-ask spread, it is
more likely that the spread will narrow as the base intensities of the
narrowing movements increase. This finding is in line with our intuition
because the larger the spread is, the more likely it is that market makers
will earn profits and will submit aggressive limit orders. Note that $\eta$
exists only in the base intensity, but not in the kernel; hence, is not
related to the previous events, but only to the current spread. In addition,
this parameter changes little over time and has a stable value in general.

Not every day shows a strong spread-narrowing tendency. If liquidity providers
are concerned about adverse selection (i.e., they expect the realized spread
to be negative), then they will not try to reduce the spread. For example, on
January 5, 2017, IBM’s liquidity parameter $\eta$ is 0.025, which is less than
half of the average $\eta$s estimated in 2017, and we can assume that market
makers were particularly careful about adverse selection on this day. In
addition, when market disruption occurs, liquidity parameter $\eta$ can come
close to zero during the day, which we discuss in Subsection 5.5.

Second, the figure shows a large excitement parameter $\alpha$, which implies
an abundance of automated high-frequency quotes and trading. In our model,
some limit order arrivals depend on the spread described by $\eta$, whereas
the rest relate to the excitations caused by past events, as the parameter
$\alpha$s also affect the arrivals or removals of limit orders, which changes
the best bid or ask prices. The estimates of the $\alpha$s range in magnitude
from tens to hundreds in general, which means that in the event of a limit
order arrival or removal that changes the best bid or ask price, the
intensities increase by tens to hundreds, and hence the probability of
subsequent events dramatically increases. Because the decaying parameter
$\beta$ is also large, the increased probability also decreases sharply over
time. For example, if an event increases intensity to 500, then approximately
five successive events can also occur in the next 0.01 seconds. Intensity also
tends to converge to the background rate $\mu$ owing to the large decaying
parameter $\beta$, provided that no additional events occur.

This instantaneous responsiveness is high; as human behavior is unlikely to be
the cause, it may be closely related to high-frequency automated algorithmic
trading. Many trading venues ensure that transactions are fast, especially in
quote revisions. Chan (2017) demonstrates that better financial services and
technologies such as colocation systems in which the trading servers are in
the same facilities as the execution venues can reduce the latency to
microseconds.

Third, the excitement parameters $\alpha$s show an increasing trend (similar
to the decaying parameter) in general. One reason for this increasing trend is
the more accurate time resolution. As mentioned before, from August 2015 to
around September 2018, the data are recorded in microseconds compared with the
previous frequency of milliseconds for the stocks reported by the CTS. This
change is noticeable on the graph, as we observe peaks around August 2015 for
the excitement parameters. However, even when excluding the effect of the
improved resolution, the overall estimates increase, which means lower
latencies for order submissions and the data feed in high-frequency trading
over time. In addition, when the time resolution improves to nanoseconds
(around September 2018), we find no notable abrupt changes. This result
implies that the resolution of microseconds is sufficient for this analysis.

Because the daily estimates of the excitement parameter $\alpha$s are too
variable over time, Figure 15 presents each $\alpha$ estimated from IBM’s NBBO
as a moving average. We compute the moving average using 20 successive days
from 2009 to 2019. The increasing trends for all the parameters are clear.
Overall, $\alpha_{w2}$ is the largest, $\alpha_{s1}$ is the smallest, and the
rest are similar. The large $\alpha_{w}$s mean that liquidity-providing
actions occur with fast reactions in response to spread-widening events,
indicating that market making is active.

Figure 15: Estimates of the $\alpha$s for IBM’s NBBO, smoothed by the moving
average of 20 successive days

We define

$\bar{\alpha}=\frac{\alpha_{s1}+\alpha_{s2}+\alpha_{m}+\alpha_{w1}+\alpha_{w2}}{5}$

and consider this value to be a measure of the overall instantaneous
responsiveness of each stock. Of course, we cannot assert that successive
events occur solely based on some decision making or algorithm depending on
preceding events, but this instantaneous responsiveness can measure how fast a
series of successive potentially related events occurs. Figure 16 plots the
moving average of $\bar{\alpha}$ from 2009 to 2019 for the NBBO of Accenture,
Berkshire Hathaway Inc. Class B, and IBM. Instantaneous responsiveness
generally increases over time owing to technological innovations such as
enhanced and cheaper hardware, faster communication speeds, new software, and
better financial services. Including these stocks, most stocks show a typical
increasing pattern, but not all equities show monotone increases over time. We
discuss other examples in the next subsection.

Figure 16: Moving average of $\bar{\alpha}$ for the NBBOs of Accenture,
Berkshire Hathaway Inc. Class B, and IBM

####  5.3.3 Liquidity provision versus depletion (removal)

Figure 17 shows the estimation results of the ratio between the parameters
related to liquidity provision and those related to liquidity depletion for
various stocks. The parameter for the former is the mean of $\alpha_{w1}$ and
$\alpha_{w2}$, whereas that for the latter is the mean of $\alpha_{s1}$,
$\alpha_{s2}$, and $\alpha_{m}$. The figure shows the average values of the
parameter estimates in 2016 and 2017. In all the examined stocks, the average
estimates of the liquidity provision parameters are larger than the average
liquidity depletion parameters. This result implies that actions to provide
liquidity generally occur faster than do actions that remove liquidity in
response to previous events.

(a) 2016

(b) 2017

Figure 17: Responsiveness ratio measured using the liquidity provision and
depletion parameters for various stocks, 2016 and 2017

Among those, Amazon in 2017 has the smallest ratio of liquidity provision, at
slightly greater than 0.5, and Visa has a value close to 1 in 2016. Market
markers’ speed competition in providing liquidity can be well understood,
since the fastest provider of limit orders preempts the advantageous position
in the limit order queue. For brevity, we illustrate only the estimates for
2016 and 2017, but the ratio changes over time. We can also clearly see that
the ratios of liquidity provision are generally larger in 2016.

Some stocks show an interesting pattern that peaks around 2016 and diminishes
in the liquidity provision parameter in Figure 18. This figure shows the
smoothed dynamics of the liquidity provision (blue dashed line) and depletion
$\alpha$ parameters (red solid line). For example, the largest liquidity
provision parameter for Visa is over 4,000, which is large compared with the
$\alpha$ values in Figure 16, suggesting that ultra-high-frequency market
makers paid much attention to these stocks during this period. However, market
makers might conclude that such ultra-fast reaction times do not necessarily
improve profits. Indeed, to reduce costs, the liquidity provision parameters
converge to much lower values after 2018, which are similar to the levels of
$\alpha$ in Figure 16. We discuss the reaction speed for liquidity provision,
which is particularly high in 2016, in Section 5.4.2 in more detail.

(a) Johnson & Johnson

(b) Lockheed Martin

(c) Mastercard

(d) Visa

Figure 18: The 2016 peak and diminishing pattern in the liquidity provision
parameter

As we show, for most stocks, $\bar{\alpha}$, or the liquidity-providing
$\alpha$, which represents the average reaction time, reaches levels of around
tens to hundreds by 2019. In some cases, it increases steadily to that level
in Figure 16; on the contrary, in other cases, it reaches this level after
several years of volatile liquidity provision speeds, as in Figure 18. The
behavior of high-frequency traders on most stocks appears to eventually reach
similar levels over time in terms of speed and risk attitude. However, not all
the stocks show the same pattern. In 2018 and 2019, a few stocks have fierce
market-making competition in ultra-high-frequency trading with strong
confidence in adverse selection.

####  5.3.4 Outliers

Some of the stocks we analyze seem to have received particular attention from
ultra-high-frequency liquidity providers in terms of the liquidity parameter
$\eta$. These are IT giants such as Microsoft, Apple, and Facebook. The larger
the $\eta$ is, the stronger market makers tend to narrow the spread through
aggressive limit orders when spreads widen. This happens when market makers
are confident and less concerned about adverse selection. Figure 19 shows bar
graphs of the $\eta$s estimated in 2018 and 2019 on average for 20 stocks. The
$\eta$ of Microsoft is large, followed by those of Apple and Facebook. For the
other stocks, the $\eta$s are at a similar level, implying a similar
assessment of adverse selection risk on average.

(a) 2018

(b) 2019

Figure 19: The average $\eta$ for various stocks, 2018 and 2019

The competition for these stocks, particularly for Microsoft, to occupy the
top priority of the limit order books is intense, as Figure 20 shows. This
figure presents the yearly averages of the liquidity-providing $\alpha$s,
namely, $\alpha_{w1}$ and $\alpha_{w2}$, with green bars and the yearly
averages of the liquidity-depleting $\alpha$s with red bars. Microsoft, Apple,
and Facebook has the top liquidity provision speed in 2018, but responsiveness
decreases from 2018 to 2019. Especially for Apple and Facebook, by 2019, the
relative gap between the liquidity provision speed, measured by the average
$\alpha_{w}$s, and the other stocks narrows. However, Microsoft is still the
only outlier in 2019, at least in our sample, and seems to attract more
attention from high-frequency liquidity providers than the other stocks do.
Although we do not know the reason for Microsoft’s popularity among high-
frequency traders, this result means that not all the stocks receive the same
interest from high-frequency traders.

(a) 2018

(b) 2019

Figure 20: The average liquidity-providing and -depleting $\alpha$s for
various stocks, 2018 and 2019

###  5.4 Estimation results by exchange

####  5.4.1 NYSE versus BZX

Using our model, we can capture the varying characteristics of the different
exchanges by analyzing the quote data for each trading venue.

The left-hand side of Figure 21 plots the dynamics of the moving average
$\bar{\alpha}$ for the BBO of IBM in several of the stock exchanges. In 2010
and 2011, the responsiveness of the Nasdaq, shown by the red dotted line, is
the fastest, whereas the responsiveness of the Nasdaq, NYSE (black solid
line), and BZX (blue dashed line) is similar. Although omitted, the Arca has
similar $\bar{\alpha}$ values to the NYSE, which is not surprising as it
operates as a subsidiary of the NYSE Group. Indeed, the Arca, formerly the
Archipelago, merged with the NYSE in 2006. The Archipelago had electronic
stock trading technology, which the NYSE did not at the time. However, from
2009, the starting point of our data analysis, we see no substantial
performance difference in the speeds of the NYSE and Arca.

The Nasdaq’s electronic stock trading technology originated from one of the
first electronic communication networks, Island, which had a good reputation
for fast electronic trading in the early 2000s (Patterson, 2012). By adopting
this technology, the Nasdaq shows relatively large $\bar{\alpha}$ values in
the early 2010s. We also include a graph of the Cboe BZX Exchange, which
claims to be designed for cheap and fast transactions after consecutive
mergers between traditional stock exchanges and electronic trading systems.
The Cboe BZX Exchange had a 6% market share in 2017 and is considered to be
among top-tier exchanges such as the NYSE, NYSE Arca, and Nasdaq (Wah et al.,
2018).

The right-hand side of Figure 21 shows the time-varying shape of IBM’s overall
responsiveness $\bar{\alpha}$ for the NYSE and Cboe BZX Exchange. We do not
smooth the estimates in this figure. The black solid line represents the
$\bar{\alpha}$ estimated daily by the best bid and ask prices quoted in the
NYSE and the red dashed line represents the $\bar{\alpha}$ estimated by the
prices in the BZX. When comparing the two exchanges, the NYSE has relatively
little variation in $\bar{\alpha}$, but $\bar{\alpha}$ changes more
drastically over time for the BZX. The maximum $\bar{\alpha}$ of the BZX is
almost twice that of the NYSE, implying that the BZX can operate under very
low latency compared with the NYSE and can attract high-frequency traders in
this respect. This result is in line with the BZX’s original claim of high-
speed technology.

Figure 21: Smoothed $\bar{\alpha}$ for IBM from the NYSE, NSDQ, and BZX (left)
and nonsmoothed $\bar{\alpha}$ for IBM from the NYSE and BZX (right)

We can also see the different behaviors between NYSE and BZX traders in
volatile markets in Figure 22 based on data for IBM from 2017 to 2018. The
figure presents the scatterplots of $\alpha_{w1}$ and $\alpha_{w2}$, the
liquidity provision parameters, versus daily log-returns. The other excitement
parameters have similar patterns. In the NYSE (left), when the absolute value
of the log-return is large (i.e., on days when the market is volatile),
liquidity provision parameters are relatively small, and the absolute return
and excitement parameters have a negative correlation. Thus, high-frequency
traders in the NYSE, especially liquidity providers, are less active in
volatile conditions, perhaps concerned about adverse selection. Conversely, we
can observe the opposite situation in the BZX. High-frequency quotes,
indicated by the high liquidity provision parameter $\alpha_{w}$ values,
remain active and become even more active in a volatile market in the BZX. The
absolute return and liquidity provision parameters are positively correlated.
Thus, high-frequency traders in the BZX tend to take aggressive positions,
even in volatile markets.

Figure 22: Scatterplot of the $\alpha$s versus the daily log-return, IBM,
2017–2018, from the NYSE (left) and BZX (right)

####  5.4.2 The BZX versus BYX and the EDGA versus EDGX

In this subsection, we compare instantaneous responsiveness depending on the
fee structure. The right-hand side of Figure 21 shows that the averaged
excitement parameters, $\bar{\alpha}$, of the BZX approaching 2016 is close to
zero; thus, the left-hand side of Figure 23 plots the $\bar{\alpha}$s of the
BZX and BYX. The reason for the low averaged excitement parameter values in
this period is unknown, but we suppose that the high-frequency traders,
especially liquidity providers, active in the BZX before 2016 moved to the BYX
at this time. The averaged excitement parameters of the BYX rises sharply
during the period when the ones of the BZX is small. Conversely, after 2017,
the averaged excitement parameters of the BZX rises steadily, whereas the ones
of the BYX remains close to zero. The BZX provides some rebates when traders
provide liquidity through a limit order and takes fees when transacting market
orders. Conversely, the BYX uses an inverted fee schedule, providing rebates
to traders removing liquidity through market orders and taking fees for
quoting limit orders.

We can see a similar relationship for the EDGA and EDGX. The EDGX’s averaged
excitement parameters increases from the second half of 2018, whereas the
EDGA’s parameter is close to zero in later years, as shown on the right-hand
side of Figure 23. In addition, the EDGA’s averaged excitement parameters
rises sharply in 2016. The EDGA provides rebates for removing liquidity,
whereas the EDGX provides rebates for adding liquidity.

Figure 23: Smoothed dynamics of the $\bar{\alpha}$s of the BZX and BYX (left)
and the EDGA and EDGX (right)

The BYX adopted an inverted fee schedule, and the EDGA also had an inverted
fee structure until June 1, 2017. These exchanges have very high excitement
parameter values for about one year or a few months around 2016. Thus, we
discuss whether we can assume that limit order quotes and trading were more
active in both exchanges during this period. Indeed, not all the excitement
parameters of the BYX and EDGA are large. The estimated $\alpha_{w1}$
dominates all the other values on the left-hand side of Figure 24 for the BYX;
the case is similar for the EDGA on the right-hand side of the figure. Recall
that $\alpha_{w1}$ is related to the replenishment rate when the best limit
orders are removed, which we can consider to be the basic job of market
makers. We assume that market making that provides liquidity was active, with
low latency in the BYX (and EDGA) around 2016. Thus, we can assume that market
participants actively tried to profit through market making during this
period. While it may seem strange to see markets on exchanges with inverted
fee schedules that charge a fee to provide liquidity, inverted structures
attract market orders, and hence the limit orders in these exchanges are
likely to be concluded first. However, these activities were discouraged after
2017, or at least market making slowed. Although we do not show the results in
detail, the $\alpha_{w1}$ for IBM in the BYX is still profound in 2018, with
an average around 15; however, it is much smaller than the value in 2016 is.
Perhaps the BYX’s market makers determined that market making at a higher
latency than before would be sufficiently competitive given that low latency
is costly. Conversely, the $\alpha_{w1}$ of the EDGA is small in 2018 (average
around 0.7), meaning that high-frequency market making should not have
occurred in that year.

Figure 24: Smoothed dynamics of the $\alpha$s of IBM from the BYX (left) and
EDGA (right)

Around 2016, the best bid and ask price processes of the BYX (and EDGA) differ
from the typical dynamics of liquid exchanges. The left-hand side of Figure25
shows the dynamics of the best bid and ask prices in the NYSE and the right-
hand side shows the best bid and ask prices in the BYX. Although we do not
show the NBBO’s movements, the NYSE’s movements closely resemble them.
Conversely, the bid-ask spread in the BYX changes rapidly in jump size and the
overall movement is different from the NBBO dynamics. We observe similar
movements in the EDGA. On the right-hand side of the figure, when a
transaction occurs and the existing best quote is removed, we momentarily
observe a rather large spread. However, responsible market makers seem to
supplement quotes quickly and the spread narrows. Except for these quotes, it
appears that limit orders lack sufficient depth. The best bid and ask prices
are far from the mid-price when the market maker does not quote the limit
orders sufficiently near the mid-price. This shallow depth is consistent with
the fact that the BYX has a maker-taker fee schedule.

Figure 25: Comparison of IBM’s tick dynamics in the NYSE (left) and BYX
(right)

Figure 18 shows the large liquidity-providing $\alpha$s (the mean of
$\alpha_{w1}$ and $\alpha_{w2}$) around 2016. We strongly suspect that the
main source of this phenomenon is an exchange such as the BYX. For the
Lockheed Martin and Mastercard stocks, Figures 26(a) and 26(b) plot the time-
varying liquidity-providing and -depleting excitement parameter, respectively,
using the bid and ask prices published by the BYX. The liquidity-providing
$\alpha$s in both stocks from the BYX remain high until the second half of
2016, but suddenly fall close to zero after 2017. On the contrary, Figures
26(c) and 26(d) show the results estimated using the data from the NYSE (the
main exchange for these stocks), indicating that the $\alpha$s remain
relatively stable.

These two results suggest that the peak of the liquidity-providing excitement
parameters around 2016 observed from the NBBO in Figure 18 did not exist in
all the exchanges, but particularly occurred in exchanges such as the BYX.
This result is consistent with the previous discussion. High-frequency traders
thought that they would profit from ultra-high-frequency trading on specific
stocks such as Lockheed Martin and Mastercard on specific exchanges such as
the BYX. The exact reason is unknown; however, after 2017, it seems that high-
frequency traders stopped making these types of trades.

(a) Lockheed Martin from the BYX

(b) Mastercard from the BYX

(c) Lockheed Martin from the NYSE

(d) Mastercard from the NYSE

Figure 26: Smoothed dynamics of the liquidity-providing and -depleting
$\alpha$s from 2015 to 2018

####  5.4.3 The IEX

Recently, the IEX introduced the speed bump technique, which intentionally
slows order processing to 350 microseconds to protect low-frequency traders
(Bishop, 2017). Exchanges can use this technique to slow ultra-fast market
orders to prevent orders from snipping stale quotes that do not yet reflect
the state of the national bid and offer prices. The IEX can also use it to
help protect market orders executed with the rest of the orders to complete on
other exchanges from phantom liquidity by delaying the execution and
transmission of the results. In addition, with the crumbling quote detection
algorithm and discretionary pegged orders, in which orders aim to access
liquidity more aggressively in a stable market and become passive when quotes
crumble, traders in the IEX are less concerned about speed.

Unfortunately, we cannot obtain information on the invisible pegged orders of
the IEX. Instead, we perform the estimation using visible limit orders, which
represents about 20% of all the orders of the IEX (Aldrich and Friedman,
2019). The responsiveness we measure using the excitement parameter $\alpha$s
in this exchange is lower than that in the other exchanges, as Figure 27
shows, for both liquidity depletion and liquidity provision. For a better
visualization, we use the moving averages of the excitement parameter
$\alpha$s’ estimates. We can interpret these results in two ways. The IEX’s
liquidity providers are relatively unconcerned about improving responsiveness
because they expect the speed bump technique to protect their stale quotes.
Alternatively, the IEX’s meticulous traders already use invisible pegged
orders, meaning that we cannot observe their exact behaviors and can only
present the behaviors of those traders relatively insensitive to increasing
the speed of quote processing.

Figure 27: Estimates of IBM’s $\alpha$s from the IEX, smoothed by a moving
average of 20 successive days

###  5.5 The Flash Crash

On May 6, 2010, the Flash Crash occurred during which stock prices fell
sharply due to large sell orders and then recovered quickly. Our model can
investigate the details of the behavior of market participants on this day.
The left-hand side of Figure 28 plots the IBM stock price (black solid line)
and dynamics of the liquidity parameter $\eta$ (blue dashed line). Recall that
$\eta$ represents the tendency to narrow when the bid-ask spread widens.

Unlike the previous cases, to examine the intraday changes, we perform the
estimations every minute and base them on three-minute overlapping intervals
using IBM’s NBBO data on bid and ask prices on May 6, 2010. The Flash Crash
occurred around 14:32. As the figure shows, the high $\eta$ value indicates
that liquidity providers continued to provide liquidity confidently for a few
minutes immediately after the start of the price collapse. However, $\eta$
then drops to almost zero, implying that liquidity providers became seriously
concerned about adverse selection due to the unexpectedly large price drop
(i.e., they did not try to narrow the spread). The value of $\eta$ begins to
rise almost at the same time as the price recovery and then returns to its
original level with synchronized price fluctuations.

The right-hand side of the figure depicts the liquidity-providing $\alpha$
(blue dashed line) and -depleting $\alpha$ (red solid line) with price changes
(black solid line). Interestingly, both the excitement parameters start to
rise at the point the price reaches its lowest value. In addition, the two
parameters are almost the same near to and immediately after the time of the
lowest price. In a stable market, the liquidity-providing parameter $\alpha$
is greater than the liquidity-depleting $\alpha$, as shown in the previous
figures. Immediately after the Flash Crash, these parameter values rise; that
is, market participants are more likely to respond faster to market changes in
both liquidity provision and liquidity depletion, which continued and
strengthened until the next day.

Figure 28: Stock price dynamics during the Flash Crash and intraday changes in
$\eta$ and $\alpha$

Returning to Figure 21, we can see that the average of excitement parameters,
$\bar{\alpha}$, of the NYSE reaches its maximum value in May 2010. Figure 29
provides an enlarged picture of this part of the graph together with the
$\bar{\alpha}$ of AMZN. Interestingly, the peaked day with the largest
excitement parameters (i.e., May 7, 2010) is the day after the Flash Crash. On
this day, the excitement parameter values estimated for the NYSE and Arca are
much higher than those on the other days; that is, it seems to be the busiest
day of 2010 for the high-frequency traders on these exchanges. Additionally,
when the market closes on that day without any particular event, their
responsiveness returns to their original state after May 7, 2010. However, the
BZX data indicate no such busy activities on May 7, 2010.

Figure 29: Dynamics of $\bar{\alpha}$ around the Flash Crash for IBM (left)
and AMZN (right). The peak occurs the day after the Flash Crash.

The reason behind the above estimation results becomes clearer when we observe
the best bid and ask price processes on this day. The left-hand side of Figure
30 shows the best bid and ask price processes reported by the NYSE the day
after the Flash Crash, May 7, 2010. The price process moves much more
irregularly in the NYSE than in the BZX, as the right-hand side of the figure
shows. The NYSE best bid and ask processes on this day are more irregular than
on the other days, although we do not show them to save space. The liquidity
providers on the NYSE were extra careful to post limit orders on this day
based on the previous Flash Crash experience, and they tried to avoid adverse
selection by repeating the process of submitting and then canceling limit
orders. The two largest estimates of excitement parameters on this day are
$\alpha_{w1}=883$ and $\alpha_{m}=554$, representing the fast repetitions of
liquidity supply and removal. Note that $\alpha_{w1}$ is related to
supplementing quotes and $\alpha_{m}$ is related to submitting and immediately
canceling (or executing) limit orders.

Figure 30: IBM stock tick structure the day after the Flash Crash on the NYSE
(left) and BZX (right)

##  6 Conclusion

We propose an extended Hawkes model to describe the best bid and ask price
processes. The model aims to rigorously reflect the nature of the price
processes and capture the various aspects of high-frequency stock price data
observed in micro- or milliseconds.

The empirical analysis shows that larger spreads tend to reduce spreads in
general, but that this tendency decreases greatly when adverse selection is a
concern. In addition, the movements of the bid and ask price processes not
only depend on the spread, but also are excited by recent events that cause
price changes. With the abundance of automated high-frequency trading, the
estimated responsiveness and corresponding decay rate have become high in
recent years. For most stocks, we observe increasing responsiveness over the
past decade and find that some stocks show unique behavior. Specifically,
responsiveness to liquidity provision increased dramatically around 2016 and
then declined. Moreover, some stocks, namely Microsoft, Apple, and Facebook,
have become of particular interest to high-frequency traders in recent years.
These stocks, particularly Microsoft, show high responsiveness in terms of
liquidity provision and a high tendency to replenish when spreads widen.

Exchanges have different responsiveness patterns. For example, the
responsiveness of the NYSE is more stable than that of the Cboe BZX, which
changes more drastically. The proposed model also allows us to observe the
different characteristics of high-frequency traders across exchanges. We find
that the speed of liquidity provision peaked around 2016, stemming from the
speed of stock exchanges such as the BYX. However, the new IEX has low
responsiveness relative to other exchanges.

Focusing on intraday changes in the parameters during the Flash Crash in 2010,
we observe a change in confidence related to adverse selection among market
makers along with a sharp decline and recovery in a stock price. We also find
that responsiveness peaked on the day after the Flash Crash in response to the
previous market anomaly, as traders repeatedly submitted and canceled limit
orders.

The application of the proposed model is not limited to those presented here.
Owing to its nature, which can incorporate a variety of properties, our
proposed model, or variations of it, is able to reveal some of the information
hidden in high-frequency financial data. Hence, the presented findings should
provide a better understanding of the market and aid decision making and
policymaking.

##  Appendix A Estimation result: Example

Table 4 presents the estimates and standard errors in parentheses from our
model for IBM for January 2018. We calculate the standard error numerically.
To save space, we report only some of the results. However, the other cases
show similar standard errors.

Table 4: Estimates and standard errors (in parentheses) from our model for IBM, January 2018 Date | $\mu$ | $\eta$ | $\alpha_{s1}$ | $\alpha_{s2}$ | $\alpha_{m}$ | $\alpha_{w1}$ | $\alpha_{w2}$ | $\beta$ | $\xi$
---|---|---|---|---|---|---|---|---|---
2018-01-02 | 0.067 | 0.0693 | 96.32 | 193.2 | 219.5 | 271.6 | 459.6 | 991.1 | 61.44
| (0.0013) | (0.0014) | (0.3570) | (0.3306) | (0.3929) | (0.2521) | (0.2169) | (0.0668) | (0.1012)
2018-01-03 | 0.1004 | 0.1184 | 69.34 | 192.6 | 173.5 | 210.1 | 381.2 | 707.0 | 71.82
| (0.0016) | (0.0018) | (0.0642) | (0.0605) | (0.0818) | (0.1111) | (0.0990) | (0.0444) | (0.1146)
2018-01-04 | 0.0787 | 0.1165 | 66.79 | 158.3 | 193.5 | 201.4 | 479.0 | 718.3 | 89.27
| (0.0014) | (0.0022) | (0.1439) | (0.1829) | (0.2113) | (0.1049) | (0.1261) | (0.0410) | (0.1658)
2018-01-05 | 0.0736 | 0.0758 | 63.81 | 185.0 | 142.2 | 126.2 | 371.6 | 739.4 | 48.41
| (0.0014) | (0.0014) | (0.3090) | (0.7946) | (0.5339) | (0.3335) | (0.0165) | (0.0719) | (0.1475)
2018-01-08 | 0.0757 | 0.0886 | 69.77 | 150.3 | 158.9 | 157.6 | 395.6 | 699.0 | 47.31
| (0.0014) | (0.0016) | (0.1229) | (0.1692) | (0.3506) | (0.4053) | (0.1474) | (0.0543) | (0.2686)
2018-01-09 | 0.0768 | 0.0911 | 50.70 | 178.1 | 169.1 | 174.7 | 514.5 | 881.1 | 71.06
| (0.0014) | (0.0019) | (0.1341) | (0.1719) | (0.2671) | (1.1020) | (1.1630) | (0.0869) | (0.1191)
2018-01-10 | 0.0447 | 0.0369 | 71.22 | 201.6 | 119.3 | 135.6 | 281.6 | 831.4 | 43.54
| (0.0011) | (0.0008) | (0.4306) | (0.4451) | (0.6818) | (0.9216) | (0.4001) | (0.1445) | (0.0683)
2018-01-11 | 0.0513 | 0.0635 | 50.83 | 190.2 | 150.2 | 160.2 | 344.6 | 683.5 | 60.97
| (0.0012) | (0.0013) | (0.2936) | (0.2478) | (0.1626) | (0.3738) | (0.0587) | (0.1526) | (0.4125)
2018-01-12 | 0.0625 | 0.0669 | 64.45 | 139.4 | 137.8 | 166.9 | 362.0 | 595.0 | 54.68
| (0.0013) | (0.0014) | (0.2930) | (0.1787) | (0.1425) | (0.3041) | (0.1897) | (0.0838) | (0.1723)
2018-01-16 | 0.1099 | 0.1124 | 51.86 | 113.7 | 150.6 | 142.8 | 265.1 | 576.0 | 42.94
| (0.0017) | (0.0017) | (0.3374) | (0.2102) | (0.2304) | (0.2500) | (0.6593) | (0.1097) | (0.4373)
2018-01-17 | 0.1051 | 0.0931 | 55.83 | 110.5 | 135.9 | 141.3 | 256.4 | 541.7 | 40.56
| (0.0017) | (0.0014) | (0.0950) | (0.0968) | (0.1000) | (0.1193) | (0.1199) | (0.0337) | (0.2273)
2018-01-18 | 0.1211 | 0.135 | 97.30 | 166.6 | 159.9 | 186.7 | 386.7 | 750.5 | 55.04
| (0.0018) | (0.0019) | (0.0918) | (0.1867) | (0.0113) | (0.0626) | (0.0604) | (0.1009) | (0.0402)
2018-01-19 | 0.1791 | 0.1977 | 124.8 | 251.1 | 331.5 | 334.2 | 466.9 | 1091 | 97.98
| (0.0021) | (0.0022) | (0.1484) | (0.1087) | (0.0670) | (0.1072) | (0.0495) | (0.0112) | (0.0487)
2018-01-22 | 0.0835 | 0.1042 | 80.68 | 276.9 | 342.1 | 280.2 | 529.5 | 1098 | 94.90
| (0.0015) | (0.0016) | (0.0517) | (0.0723) | (0.0333) | (0.0955) | (0.0500) | (0.0508) | (0.0647)
2018-01-23 | 0.0781 | 0.0645 | 72.98 | 173.7 | 139.6 | 141.8 | 264.7 | 760.0 | 37.03
| (0.0014) | (0.0011) | (0.0704) | (0.0602) | (0.1765) | (0.1796) | (0.2232) | (0.0419) | (0.1726)
2018-01-24 | 0.0736 | 0.0621 | 12.04 | 20.31 | 36.15 | 19.56 | 37.39 | 99.01 | 4.355
| (0.0014) | (0.0008) | (0.2121) | (0.0651) | (0.0849) | (0.1409) | (0.1630) | (0.2076) | (0.0836)
2018-01-25 | 0.0544 | 0.0565 | 5.861 | 15.96 | 33.48 | 12.35 | 27.12 | 79.13 | 3.944
| (0.0012) | (0.0009) | (0.2745) | (1.1480) | (1.4180) | (0.4565) | (0.5676) | (0.1170) | (0.6511)
2018-01-26 | 0.0534 | 0.0407 | 35.69 | 117.1 | 104.9 | 81.64 | 247.1 | 550.5 | 39.55
| (0.0012) | (0.0009) | (0.4864) | (0.3846) | (0.4970) | (1.1400) | (0.7538) | (0.1820) | (0.2034)
2018-01-29 | 0.046 | 0.0519 | 4.861 | 15.46 | 31.30 | 8.930 | 20.29 | 68.35 | 2.682
| (0.0011) | (0.0008) | (0.4999) | (1.772) | (1.883) | (0.2002) | (0.5845) | (0.1414) | (0.2195)
2018-01-30 | 0.0567 | 0.0575 | 8.437 | 22.42 | 34.29 | 14.94 | 35.35 | 90.93 | 5.069
| (0.0012) | (0.0009) | (0.2398) | (0.4787) | (0.6678) | (0.4286) | (0.1209) | (0.0723) | (0.1531)
2018-01-31 | 0.0839 | 0.0453 | 70.94 | 146.2 | 98.07 | 94.21 | 253.4 | 649.5 | 32.98
| (0.0015) | (0.0008) | (0.2563) | (0.2819) | (0.3680) | (0.1556) | (0.2291) | (0.0819) | (0.2457)

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