# Path integral analysis of the axial anomaly in Very Special Relativity

R. Bufalo111E-mail<EMAIL_ADDRESS>1 and M. Ghasemkhani222E-mail:
<EMAIL_ADDRESS>2
1Departamento de Física, Universidade Federal de Lavras,
Caixa Postal 3037, 37200-900 Lavras, MG, Brazil
2Department of Physics, Shahid Beheshti University, 1983969411, Tehran, Iran

###### Abstract

In this note, we revise the problem of the axial anomaly in the framework of
very special relativity following Fujikawa’s path integral approach. We show
nonperturbatively that no VSR contribution is present in the path integral
measure in $(3+1)$-dimensional spacetime. Furthermore, we extend our results
to $(1+1)$ dimensions, as well as to the two-dimesional curved spacetime.

##  1 Introduction

The study of anomalies in gauge theories is extremely important in
demonstrating its consistency and physical properties, such as unitarity and
renormalizability, as they are expected to cancel. In order to verify whether
a symmetry is preserved at the quantum level, one must analyze the respective
Ward identity (or Slavnov-Taylor identity in the non-Abelian case).

Remarkably, in some special cases, the full structure of the anomaly can be
calculated exactly at the one-loop order, and in the case of the chiral
anomaly, it is given by the famous Adler-Bell- Jackiw (ABJ) anomaly,
justifying the $\pi_{0}\to\gamma\gamma$ decay [1, 2, 3, 4, 5, 6]. As an
another physical application of the anomalies, we can refer to the detected
signatures of the chiral anomaly in the magneto-transport measurements in Weyl
semimetals [7].

Naturally, the models describing the physics beyond the standard model should
fulfill such strong consistency requirements as well. In this paper, we shall
consider the problem of the anomalies in a particular class of the models
presenting Lorentz symmetry violation [8, 9, 10, 11]. Contrary to the common
lore, Lorentz violating effects are not necessarily related to Planck scale
physics, it is also possible to formulate such class of models from a
phenomenological group theory point of view.

A Lorentz violating framework, preserving all the basic elements of the
special relativity, is the Cohen and Glashow very special relativity (VSR)
[12, 13]. The main aspect of the VSR proposal is that the laws of physics are
invariant under the (kinematical) subgroups of the Poincaré group. In
$(3+1)$–dimensional spacetime, there are two VSR subgroups, SIM(2) and HOM(2),
that preserve the direction of a lightlike four-vector $n_{\mu}$ by scaling,
transforming as $n\to e^{\varphi}n$ under boost in the z direction. This
feature implies that there is a preferred direction in the Minkowski
spacetime, where the Lorentz violating terms can be constructed as ratios of
contractions of the vector $n_{\mu}$ with other kinematical vectors, for
instance $n^{\mu}/(n.p)$ [12]. The VSR approach has been extended to the gauge
theories, where many interesting theoretical and phenomenological aspects of
VSR effects have been extensively discussed [14, 15, 16, 17, 18, 19, 20, 21].

Hence, since VSR gauge theories possess very interesting features related to
the presence of gauge invariant massive modes, as well as nonlocal couplings
[14], it is a suitable Lorentz violating scenario to examine the chiral
anomaly in the VSR electrodynamics. This study could in principle signal some
modification due to VSR effects of the (effective) coupling of the process
$\pi_{0}\to\gamma\gamma$ decay. There were some works investigating the Adler-
Bell- Jackiw (ABJ) anomaly in the different Lorentz violating models [22, 23,
24, 25, 26, 27, 28], but not yet in the VSR context. Although, some aspects of
the axial anomaly have been examined in the VSR-inspired Schwinger model [29].

Recently, the reference [30] presented a perturbative and a path integral
analysis of the axial anomaly in the very special relativity (VSR) quantum
electrodynamics, showing that no VSR contribution is present in the anomaly.
However, the author assumed some superfluous conditions in the analysis, such
as “light-cone gauge”, “fixed A” and “large q”, which are actually unnecessary
within Fujikawa’s path integral approach [4, 5, 6]. Hence, in order to revise
and generalize the path integral analysis of [30], our development follows
strictly Fujikawa’s sum regularization prescription to obtain the chiral
Jacobian in a very elegant fashion [4, 5, 6]. Furthermore, contrary to [30],
our analysis is as general as it can be, since it is nonperturbative, in the
sense that no expansion is made in the nonlocal VSR terms.

We start Sec. 2 by reviewing the main features of the Fujikawa’s approach, as
well as presenting the main definitions in regard of very special relativity.
We focus great attention in presenting the regularization sum procedure since
there lies the most important analysis involving the computation of the VSR
effects into the Jacobian associated with the chiral symmetry, where we can
clearly find how the VSR terms contribute. Furthermore, we extend our results
to the two-dimensional case. In Sec. 3, we generalize the analysis of the VSR
contributions to the axial anomaly in curved spacetime, where we have
considered by simplicity a two-dimensional curved spacetime. In Sec. 4, we
summarize the results, and present our final remarks.

##  2 Fujikawa’s approach to the axial anomaly

In order to investigate the axial anomaly in the context of path integral
approach, it is necessary to establish the path integral measure accurately
[4, 5, 6]. We start with the Euclidean VSR path integral

$Z\left[A\right]=\int D\overline{\psi}D\psi\exp\left[\int
d^{4}x~{}\bar{\psi}\left(i\not{\nabla}-m_{e}\right)\psi\right],$ (2.1)

where $\nabla_{\mu}=D_{\mu}-\frac{m^{2}n_{\mu}}{2\left(n.D\right)}$ is defined
as the VSR covariant derivative with $D_{\mu}=\partial_{\mu}-ieA_{\mu}$, and
$m$ is the VSR parameter. 333 Although a gauge fixing is allowed in this
gauge-invariant approach, we shall show that contrary to [30], where the
light-cone gauge $n.A=0$ was used in order to simplify the operator $\nabla$
to $\nabla_{\mu}=D_{\mu}-\frac{m^{2}n_{\mu}}{2\left(n.\partial\right)}$
(removing all nonlocal couplings), we can perform the whole analysis in path
integral without any gauge fixing. We would like to analyze the path integral
measure behavior under local chiral transformations

$\psi\to
e^{i\beta\left(x\right)\gamma_{5}}\psi,\quad\overline{\psi}\to\overline{\psi}e^{i\beta\left(x\right)\gamma_{5}},$
(2.2)

and then determine whether VSR effects can modify the well-known axial anomaly
[4, 5, 6].

In this sense, we decompose the spinors into eigenfunctions of the Dirac
operator as follows

$\psi\left(x\right)=\sum_{n}a_{n}\varphi_{n}\left(x\right),\quad\bar{\psi}\left(x\right)=\sum_{m}\varphi_{m}^{\dagger}\left(x\right)\overline{b}_{m},$
(2.3)

where the coefficients $a_{n}$ and $\overline{b}_{m}$ are independent
Grassmann variables. The hermitian Dirac operator $\nabla$ satisfies

$\not{\nabla}\varphi_{n}\left(x\right)=\lambda_{n}\varphi_{n}\left(x\right),$
(2.4)

where $\lambda_{n}$’s are real eigenvalues of $\nabla$, the set of
eigenfunctions is orthonormal and complete

$\displaystyle\int
d^{4}x~{}\varphi_{m}^{\dagger}\left(x\right)\varphi_{n}\left(x\right)$
$\displaystyle=\delta_{mn},$
$\displaystyle\sum_{n}\varphi_{n}\left(x\right)\varphi_{n}^{\dagger}\left(y\right)$
$\displaystyle=\delta^{(4)}\left(x-y\right),$ (2.5)

The first point to emphasize is that this basis set allows us to diagonalize

$\bar{\psi}\left(i\not{\nabla}-m_{e}\right)\psi=\sum_{n}\left(i\lambda_{n}-m_{e}\right)\overline{b}_{n}a_{n}.$
(2.6)

Hence, we realize that under the decomposition (2.3) the fermionic path
integral measure can be expressed as
$D\overline{\psi}D\psi=\prod_{n}d\overline{b}_{n}da_{n}$. In the next step, we
shall consider how the fermionic fields behave under the local chiral
transformations (2.2). In particular, we are interested in analyzing how the
fermionic path integral measure behaves under these infinitesimal
transformations. Applying these considerations,we find the following
transformation law for the coefficients of (2.3)

$\left(\begin{array}[]{c}a_{m}^{\prime}\\\
\overline{b}_{m}^{\prime}\end{array}\right)=\sum_{k}C_{mk}\left(\begin{array}[]{c}a_{k}\\\
\overline{b}_{k}\end{array}\right),$ (2.7)

with the transformations matrix

$C_{mk}=\delta_{mk}+i\int
d^{4}x~{}\varphi_{m}^{\dagger}\left(x\right)\beta\left(x\right)\gamma_{5}\varphi_{k}\left(x\right).$
(2.8)

Finally, from these results, we can write the transformation for the path
integral measure as

$\displaystyle D\overline{\psi}^{\prime}D\psi^{\prime}$
$\displaystyle=\prod_{n}d\overline{b}_{n}^{\prime}da_{n}^{\prime}\equiv\pazocal{J}\left[\beta\right]D\overline{\psi}d\psi,$
(2.9)

where we have identified the Jacobian
$\pazocal{J}\left[\beta\right]=\left(\det C\right)^{-2}$. This functional can
also be cast as

$\displaystyle\pazocal{J}\left[\beta\right]=\exp\Bigg{(}-2i\int
d^{4}x~{}\beta\left(x\right)\sum_{m}\varphi_{m}^{\dagger}\left(x\right)\gamma_{5}\varphi_{m}\left(x\right)\Bigg{)},$
(2.10)

which form is valid for infinitesimal $\beta\left(x\right)$. However, this sum
is not well defined, actually

$\sum_{m}\varphi_{m}^{\dagger}\left(x\right)\gamma_{5}\varphi_{m}\left(x\right)=\text{tr}\left(\gamma_{5}\right)\delta\left(0\right),$
(2.11)

where we have used the completeness relation of the eigenfunctions (2.5).
Hence, it is necessary to regularize the sum in (2.10), and we follow
Fujikawa’s prescription [5]. As we will show below, the correct use of the
Fujikawa’s regularization approach leads to an unambiguous analysis of the
axial anomaly in the VSR setting.

In the regularization procedure, we shall consider an arbitrary function that
is smooth and decreasing sufficiently rapidly at infinity

$f\Big{(}\frac{\lambda_{m}^{2}}{\Lambda^{2}}\Big{)},$ (2.12)

with $\Lambda\to\infty$ and

$\displaystyle f\left(\infty\right)$
$\displaystyle=f^{\prime}\left(\infty\right)=f^{\prime\prime}\left(\infty\right)=\cdots=0,$
$\displaystyle f\left(0\right)$ $\displaystyle=1.$ (2.13)

Hence, since the Dirac operator is now regularized, we can express the
eigenfunctions in Fourier components, i.e. plane waves, and thus we can
compute the Jacobian (2.10) explicitly.

First, we should consider

$\displaystyle\sum_{m}\varphi_{m}^{\dagger}\left(x\right)\gamma_{5}\varphi_{m}\left(x\right)$
$\displaystyle=\lim_{\Lambda\to\infty}\int\frac{d^{4}q}{\left(2\pi\right)^{4}}\text{tr}\Big{[}e^{-iq.x}\gamma_{5}f\Big{(}\frac{\not{\nabla}^{2}}{\Lambda^{2}}\Big{)}e^{iq.x}\Big{]}$
(2.14)
$\displaystyle=\lim_{\Lambda\to\infty}\int\frac{d^{4}q}{\left(2\pi\right)^{4}}\text{tr}\Big{[}e^{-iq.x}\gamma_{5}f\Big{(}\frac{\nabla_{\mu}\nabla^{\mu}}{\Lambda^{2}}+\frac{1}{4\Lambda^{2}}\left[\gamma^{\mu},\gamma^{\nu}\right]\left[\nabla_{\mu},\nabla_{\nu}\right]\Big{)}e^{iq.x}\Big{]},$
(2.15)

where we have decomposed the Dirac operator as usual. Furthermore, the effect
of moving the plane waves is a shift in the differential operators

$e^{-iq.x}f\left(\partial_{\mu}\right)e^{iq.x}=f\left(\partial_{\mu}+iq_{\mu}\right).$
(2.16)

Hence, we observe that under this change we have $D\to D+iq$. Thus, the VSR
covariant derivative is written as

$\nabla_{\mu}^{\prime}=D_{\mu}+iq_{\mu}-\frac{m^{2}n_{\mu}}{2\left(n.D+in.q\right)}.$
(2.17)

This implies that the argument of the function $f(z)$ in (2.15) takes the form

$\displaystyle z$
$\displaystyle=-q_{\mu}q^{\mu}+\frac{2iq_{\mu}\left[D^{\mu}-\frac{m^{2}n^{\mu}}{2\left(n.D+i(n.q)\Lambda\right)}\right]}{\Lambda}+\frac{\left(D_{\mu}-\frac{m^{2}n_{\mu}}{2\left(n.D+i(n.q)\Lambda\right)}\right)\left(D^{\mu}-\frac{m^{2}n^{\mu}}{2\left(n.D+i(n.q)\Lambda\right)}\right)}{\Lambda^{2}}$
$\displaystyle+\frac{1}{4\Lambda^{2}}\left[\gamma^{\mu},\gamma^{\nu}\right]\left(\left[D_{\mu},D_{\nu}\right]-\frac{m^{2}n_{\nu}}{2}\left[D_{\mu},\frac{1}{\left(n.D+i(n.q)\Lambda\right)}\right]-\frac{m^{2}n_{\mu}}{2}\left[\frac{1}{\left(n.D+i(n.q)\Lambda\right)},D_{\nu}\right]\right).$
(2.18)

Here, it is worth to formulate some comments as follows about this expression:

  1. 1.

We have performed the rescaling in the momentum $q\to\Lambda q$, which allows
gathering the relevant contributing terms of eq.(2.15);

  2. 2.

The $m^{4}$ term vanishes, since it is proportional to the product of a
symmetric tensor $n_{\mu}n_{\nu}$ with
$\left[\gamma^{\mu},\gamma^{\nu}\right]$;

  3. 3.

Since $q$ is a fixed/constant 4-vector, its commutator with the derivative
$D_{\mu}$ is zero, i.e. $\left[D_{\mu},q_{\nu}\right]=0$;

  4. 4.

In regard to the commutator
$\left[D_{\mu},\frac{1}{\left(n.D+i(n.q)\Lambda\right)}\right]$, it vanishes
identically because in the limit $\Lambda\to\infty$ the nonlocal factor goes
as
$\frac{1}{\left(n.D+i(n.q)\Lambda\right)}\sim\frac{1}{\left(i(n.q)\Lambda\right)}$,
i.e.
$\left[D_{\mu},\frac{1}{\left(n.D+i(n.q)\Lambda\right)}\right]\sim\left[D_{\mu},\frac{1}{\left(i(n.q)\Lambda\right)}\right]=0$.

Now, we expand the content of the function $f\left(z\right)$ around the value
$y=-q_{\mu}q^{\mu}=q^{2}$. Furthermore, only the terms proportional to
$1/\Lambda^{4}$ or larger survive in the limit $\Lambda\to\infty$. Also,
looking at the trace of the $\gamma$-matrices with $\gamma_{5}$, only terms
with four or more $\gamma$-matrices have nonvanishing traces. Hence, we find

$\displaystyle\sum_{m}\varphi_{m}^{\dagger}\left(x\right)\gamma_{5}\varphi_{m}\left(x\right)$
$\displaystyle=\lim_{\Lambda\to\infty}\Lambda^{4}\int\frac{d^{4}q}{\left(2\pi\right)^{4}}\text{tr}\left[\gamma_{5}\Big{(}f\left(y\right)+wf^{\prime}\left(y\right)+\frac{w^{2}}{2!}f^{\prime\prime}\left(y\right)+\cdots\Big{)}\right],$
(2.19)

where $w$ is defined as

$\displaystyle w$
$\displaystyle=\frac{2iq_{\mu}\left[D^{\mu}-\frac{m^{2}n^{\mu}}{2\left(n.D+i(n.q)\Lambda\right)}\right]}{\Lambda}+\frac{\left(D_{\mu}-\frac{m^{2}n_{\mu}}{2\left(n.D+i(n.q)\Lambda\right)}\right)\left(D^{\mu}-\frac{m^{2}n^{\mu}}{2\left(n.D+i(n.q)\Lambda\right)}\right)}{\Lambda^{2}}+\frac{1}{2\Lambda^{2}}\gamma^{\mu}\gamma^{\nu}\left(-ie\right)F_{\mu\nu},$
(2.20)

where
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}=-\frac{1}{ie}\left[D_{\mu},D_{\nu}\right]$.
Although we have some terms of $w$ that survive in the limit
$\Lambda\to\infty$ in the Taylor’s expansion (2.19) but they all vanish due to
trace identities:
$\text{tr}(\gamma_{5})=\text{tr}(\gamma_{5}\gamma^{\mu}\gamma^{\nu})=0$ and
$\text{tr}(\gamma_{5}\gamma^{\mu}\gamma^{\nu}\gamma^{\alpha}\gamma^{\beta})=-4\epsilon^{\mu\nu\alpha\beta}$.
The only remaining term that survives after the regulator limit and taking the
trace is

$\displaystyle\sum_{m}\varphi_{m}^{\dagger}\left(x\right)\gamma_{5}\varphi_{m}\left(x\right)$
$\displaystyle=\lim_{\Lambda\to\infty}\Lambda^{4}\int\frac{d^{4}q}{\left(2\pi\right)^{4}}\text{tr}\left[\gamma_{5}\frac{1}{2!}\left(\frac{1}{2\Lambda^{2}}\gamma^{\mu}\gamma^{\nu}\left(-ie\right)F_{\mu\nu}\right)^{2}f^{\prime\prime}\left(y\right)\right]$
(2.21)
$\displaystyle=-\frac{e^{2}}{8}\text{tr}\left[\gamma_{5}\gamma^{\mu}\gamma^{\nu}\gamma^{\alpha}\gamma^{\beta}\right]F_{\mu\nu}F_{\alpha\beta}\int\frac{d^{4}q}{\left(2\pi\right)^{4}}f^{\prime\prime}\left(y\right),$
(2.22)

which corresponds exactly to the usual singlet axial anomaly expression

$\displaystyle\sum_{m}\varphi_{m}^{\dagger}\left(x\right)\gamma_{5}\varphi_{m}\left(x\right)$
$\displaystyle=\frac{e^{2}}{32\pi^{2}}\epsilon^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta},$
(2.23)

where we have made use of (obtained through integration by parts)

$\int\frac{d^{4}q}{\left(2\pi\right)^{4}}f^{\prime\prime}\left(q^{2}\right)=-\frac{1}{16\pi^{2}}\int
dq^{2}f^{\prime}\left(q^{2}\right)=\frac{1}{16\pi^{2}}.$ (2.24)

Accordingly, the Jacobian of the path integral measure (2.10) in $(3+1)$
dimensions is given by

$\pazocal{J}\left[\beta\right]=\exp\left(-\frac{ie^{2}}{16\pi^{2}}\int
d^{4}x~{}\beta\left(x\right)\epsilon^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}\right).$
(2.25)

Hence, we have shown that under the correct use of the Fujikawa’s
regularization prescription, no VSR corrections are found for the axial
anomaly in the path integral approach.

###  2.1 Two-dimensional axial anomaly in VSR

An import remark about the previous analysis in the evaluation of the operator
$\left[\gamma^{\mu},\gamma^{\nu}\right]\left[\nabla_{\mu},\nabla_{\nu}\right]$
in (2) is in order. The considerations and main results showing the absence of
VSR effects are valid for any spacetime dimension. As a matter of fact, the
only actual change is the integral measure. In the case of $(1+1)$ spacetime
we have that eq. (2.19) should be written as 444Actually, the Lorentz group in
two dimensions i.e. SO(1,1) does not have any VSR subgroup, since it has only
one generator, one boost. However, one can make use of a null-vector
$n_{\mu}=(1,1)$ and construct a VSR-like QED action [29] with the same
structure for the covariant derivative
$\nabla_{\mu}=D_{\mu}-\frac{m^{2}n_{\mu}}{2\left(n.D\right)}$.

$\displaystyle\sum_{m}\varphi_{m}^{\dagger}\left(x\right)\gamma_{5}\varphi_{m}\left(x\right)$
$\displaystyle=\lim_{\Lambda\to\infty}\Lambda^{2}\int\frac{d^{2}q}{\left(2\pi\right)^{2}}\text{tr}\left[\gamma_{5}\Big{(}f\left(y\right)+wf^{\prime}\left(y\right)+\frac{w^{2}}{2!}f^{\prime\prime}\left(y\right)+\cdots\Big{)}\right],$
(2.26)

with $w$ given by (2.20). Another difference is the result
$\text{tr}(\gamma_{5}\gamma^{\mu}\gamma^{\nu})=-2\epsilon^{\mu\nu}$, that
comes from the identity
$\gamma_{\mu}\gamma_{5}=\epsilon_{\mu\nu}\gamma^{\nu}$, with $\epsilon_{01}=1$
(here, we follow the notation of [4]). Making use of these results we find
that only the linear term in $w$ contributes in eq. (2.26), and the
regularized sum is given by

$\displaystyle\sum_{m}\varphi_{m}^{\dagger}\left(x\right)\gamma_{5}\varphi_{m}\left(x\right)$
$\displaystyle=\lim_{\Lambda\to\infty}\Lambda^{2}\int\frac{d^{2}q}{\left(2\pi\right)^{2}}\text{tr}\left[\gamma_{5}\left(\frac{1}{2\Lambda^{2}}\gamma^{\mu}\gamma^{\nu}\left(-ie\right)F_{\mu\nu}\right)f^{\prime}\left(y\right)\right]$
(2.27)
$\displaystyle=-\frac{ie}{2}\text{tr}\left[\gamma_{5}\gamma^{\mu}\gamma^{\nu}\right]F_{\mu\nu}\int\frac{d^{2}q}{\left(2\pi\right)^{2}}f^{\prime}\left(q^{2}\right),$
(2.28)

and using

$\int\frac{d^{2}q}{\left(2\pi\right)^{2}}f^{\prime}\left(q^{2}\right)=-\frac{1}{4\pi}$
(2.29)

we obtain

$\displaystyle\sum_{m}\varphi_{m}^{\dagger}\left(x\right)\gamma_{5}\varphi_{m}\left(x\right)=-\frac{ie}{4\pi}\epsilon_{\mu\nu}F^{\mu\nu},$
(2.30)

that corresponds to the usual axial anomaly in two-dimensions.

##  3 Axial anomaly in curved spacetime

As an additional analysis of the axial anomaly in VSR, we can extend our
discussion to the case of chiral anomalies associated with the (background)
gravitational field. In regard to the formal development presented above to
determine the functional Jacobian (2.10), the main change in the case of
curved spacetime is the spacetime measure
$d^{\omega}x\to\sqrt{-g}~{}d^{\omega}x$ [5].

Now, about VSR in curved spacetime, there are some formal developments and
applications found in the literature [31, 32], where the Lorentz violating
effects are studied in terms of (spinor) auxiliary fields $(\lambda,\chi)$

$\pazocal{L}=i\bar{\psi}\gamma^{\mu}\pazocal{D}_{\mu}\psi+i\bar{\chi}(n.\pazocal{D})\lambda+i\bar{\lambda}(n.\pazocal{D})\chi+im\bar{\chi}(\gamma.n)\psi+im\bar{\lambda}\psi+h.c.$
(3.1)

to avoid the presence of explicit non-local factors. Furthermore, in this
formulation $n^{\mu}$ is a recurrent null vector field satisfying
$\pazocal{D}_{\mu}n^{\nu}=0$. However, the construction of VSR invariant in
curved spacetime is not a closed matter, we shall consider as a matter of
discussion the results of [31] and assume that the functional form of the
fermionic minimal coupling is $i\bar{\psi}\gamma^{\mu}\nabla_{\mu}\psi$, with
the VSR covariant derivative defined as
$\nabla_{\mu}=\pazocal{D}_{\mu}-\frac{m^{2}n_{\mu}}{2\left(n.\pazocal{D}\right)}$,
but now with new couplings
$\pazocal{D}_{\mu}=\partial_{\mu}-ieA_{\mu}+\frac{1}{2}\omega_{\mu}^{ab}\sigma_{ab}\equiv
D_{\mu}+\frac{1}{2}\Omega_{\mu}^{ab}\sigma_{ab}$, where $\omega_{\mu}^{ab}$ is
the usual spin connection [31]. Furthermore, $\sigma_{ab}$ stands for the
representation appropriate for the field that the derivative acts on, e.g. in
the case of the usual components $\omega_{\mu}^{ab}$ acting on a fermionic
field, we have that $\sigma_{ab}=\frac{1}{4}[\gamma_{a},\gamma_{b}]$.

By simplicity, we shall consider a two-dimensional curved spacetime to develop
our analysis. Once again, the matrix structure of the regularized sum is very
similar to (2.26). Thus, we shall focus only on the evaluation of the
commutator $\left[\nabla_{\mu},\nabla_{\nu}\right]$. In this case, we have
that

$\displaystyle e^{-iq.x}\left[\nabla_{\mu},\nabla_{\nu}\right]e^{iq.x}$
$\displaystyle=\left[\pazocal{D}_{\mu},\pazocal{D}_{\nu}\right]$ (3.2)
$\displaystyle-\frac{m^{2}n_{\nu}}{2}\left[\pazocal{D}_{\mu},\frac{1}{\left(n.\pazocal{D}+i(n.q)\Lambda\right)}\right]-\frac{m^{2}n_{\mu}}{2}\left[\frac{1}{\left(n.\pazocal{D}+i(n.q)\Lambda\right)},\pazocal{D}_{\nu}\right],$
(3.3)

where the rescaling in the momentum $q\to\Lambda q$ was performed. We can make
use of the comments 3. and 4. presented after eq. (2) to show that the last
two terms of (3.2) vanish identically. Because in the limit $\Lambda\to\infty$
the nonlocal factor goes as
$\lim\limits_{\Lambda\to\infty}\frac{1}{\left(n.\pazocal{D}+i(n.q)\Lambda\right)}\sim\lim\limits_{\Lambda\to\infty}\frac{1}{\left(i(n.q)\Lambda\right)}$,
i.e.

$\lim\limits_{\Lambda\to\infty}\left[\pazocal{D}_{\mu},\frac{1}{\left(n.\pazocal{D}+i(n.q)\Lambda\right)}\right]\sim\lim\limits_{\Lambda\to\infty}\left[\pazocal{D}_{\mu},\frac{1}{\left(i(n.q)\Lambda\right)}\right]=0.$
(3.4)

We then find that

$e^{-iq.x}\left[\nabla_{\mu},\nabla_{\nu}\right]e^{iq.x}=\left[\pazocal{D}_{\mu},\pazocal{D}_{\nu}\right],$
(3.5)

showing that, once again, no VSR contribution is present in the axial anomaly
in the two-dimensional QED, even in a curved spacetime. As a matter of fact,
the chiral anomaly of the two-dimensional QED in curved spacetime is the same
as in the flat spacetime [33].

##  4 Final remarks

In this paper, we have analyzed the four-dimensional chiral anomaly for Dirac
fermions in the context of the very special relativity using the path integral
approach, where we have found no presence of VSR effects in the Jacobian
functional. Although some points were dealt in ref.[30], we believe that the
treatment in [30] is somehow limited because it makes use of unnecessary
conditions such as “light-cone gauge” (gauge-fixing), “fixed A” and “large q”.
Moreover, the main issue is the improper procedure in the regularization of
the ill-defined sum in eq.(2.10).

On the other hand, our analysis is as general as it can be since it holds
nonperturbatively. As we have shown above, the correct application of
Fujikawa’s sum regularization procedure (in terms of the limit
$\Lambda\to\infty$) is the key element to obtain the axial anomaly in the path
integral formalism. Mainly because it allows us to collect naturally the
contributing terms in the Taylor’s series (2.19).

In addition, we have extended our discussion to the case of the two-
dimensional chiral anomaly associated with the (background) gravitational
field, showing again that no VSR effect is present. In summary, as long as the
VSR effects have the “usual” nonlocal form
$\nabla_{\mu}=\pazocal{D}_{\mu}-\frac{m^{2}n_{\mu}}{2\left(n.\pazocal{D}\right)}$,
independently of the functional form of the gauge connection (electromagnetic
and gravitational part) of the covariant derivative $\pazocal{D}_{\mu}$, it
can always be removed in the evaluation of the path integral Jacobian.

Since there is a close relation among the ABJ anomaly with the
$\pi_{0}\to\gamma+\gamma$ decay [2, 3], it is a natural extension to examine
the VSR effects upon this type of anomalous processes and find how the VSR
nonlocal effects contribute to its decay width. This is currently under
development and will be reported elsewhere.

### Acknowledgements

We are grateful to M. Chaichian and A. Tureanu for their encouragements to
present this paper. R.B. acknowledges partial support from Conselho Nacional
de Desenvolvimento Científico e Tecnológico (CNPq Projects No. 305427/2019-9
and No. 421886/2018-8) and Fundação de Amparo à Pesquisa do Estado de Minas
Gerais (FAPEMIG Project No. APQ-01142-17).

## References

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  * [3] R. Jackiw, “Field theoretic investigations in current algebra,” in: Current Algebra and Anomalies, S.B. Treiman, R. Jackiw, B. Zumino and E. Witten (eds.), p.81, and p. 211, World Scientific, Singapore.
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# An Empirical Study on Arrival Rates of Limit Orders and Order Cancellation
Rates in Borsa Istanbul

Can Yilmaz Altinigne<EMAIL_ADDRESS>Harun Ozkan
<EMAIL_ADDRESS>Veli Can Kupeli<EMAIL_ADDRESS>Zehra Cataltepe<EMAIL_ADDRESS>School of Computer and Communication
Sciences, EPFL, Switzerland  Matriks Bilgi Dağıtım Hizmetleri, 34396,
Istanbul, Turkey  Department of Computer Engineering, Faculty of Computer and
Informatics Engineering, Istanbul Technical University, Istanbul, Turkey

###### Abstract

Order book dynamics play an important role in both execution time and price
formation of orders in an exchange market. In this study, we aim to model the
limit order arrival rates in the vicinity of the best bid and the best ask
price levels. We use limit order book data for Garanti Bank, which is one of
the most traded stocks in Borsa Istanbul. In order to model the daily, weekly,
and monthly arrival of limit order quantities, three different discrete
probability distributions are tested: Geometric, Beta-Binomial and Discrete
Weibull. Additionally, two theoretical models, namely, Exponential and Power
law are also tested. We aim to model the arrival rates in the first fifteen
bid and ask price levels. We use $L_{1}$ norms in order to calculate the
goodness-of-fit statistics. Furthermore, we examine the structure of weekly
and monthly mean cancellation rates in the first ten bid and ask price levels.

###### keywords:

Order arrival processes , Probability distribution fitting , Limit order book
, Queueing systems

###### JEL:

C46, C51

††journal: Borsa Istanbul Review

##  1 Introduction

One of the main research area on high-frequency financial data is to
investigate the microstructural properties of stock markets. Generally, the
research on this area contains modeling the main characteristics of the limit
order book (Cont et al. [2010], Bouchaud et al. [2002], Zovko et al. [2002])
and the behavior of traders of stocks around specific events (Mu et al.
[2010]).

Modeling some market elements such as the duration between two orders, order
volumes and order arrivals using parametric statistical distributions can help
to understand the structure of market dynamics. Exponential family
distributions are widely used in modeling these types of exchange market
elements (Cont et al. [2010], Jiang et al. [2008]). Alternatively, arrival
rates of limit orders can be modeled using a power law (Bouchaud et al.
[2002], Zovko et al. [2002]). By modeling the order dynamics in the market, we
can have some basic insight about the market microstructure. For this purpose,
we aim to model the arrival rates of limit order in the exchange market. Also,
we intend to observe the statistical features of the cancellation rates
(ratios of cancel orders to outstanding orders).

We use three well-known discrete statistical distributions for modeling the
arrivals of orders: Discrete Weibull, Geometric and Beta - Binomial. In
addition, we fit Exponential distribution and Power law on the same variables.
We use $L_{1}$ norms between probability mass functions of discrete
distributions and the true arrival rates. For continuous distributions, we
discretize the fit results by calculating the area below the probability
density function. We compare the performance scores of different fits using
Welch’s t-test.

After completing the discrete and continuous distribution fits on the arrival
rates of limit orders, we compare the best fitting discrete model which is
Discrete Weibull model with the theoretical models. We show that the
performance of Discrete Weibull model is three times better than the
Exponential model in terms of $L_{1}$ norms, and it is very competitive
against the Power law model which is suggested by Bouchaud et al. [2002]. In
this part of our research, we present that the arrival rates of limit orders
in the vicinity of the best prices (15 ticks or less) can also be represented
by discrete models.

In addition to the analysis of limit orders, we conduct a research on cancel
orders. We analyze the cancellation rates on the weekly and monthly basis. We
consider the first 10 bid and the ask price levels. We investigate whether the
behavior of order cancellation rates change with respect to different bid and
ask price levels. In our research, we observe that the hypothesis which
implies that weekly and monthly mean order cancellation rates are consistent
with Uniform distribution can not be statistically rejected.

##  2 Background and Literature Review

In exchange markets, limit orders, market orders and cancel orders constitute
the current market dynamics. Arrived limit orders in the market create a limit
order book. Buy and sell orders are placed in the limit order book according
to their price and the quantity. Until a market order or a cancel order is
executed on a particular limit order, that limit order stays in the order book
(Cont [2011]). Cancel orders delete limit orders in the table. Market orders
execute limit orders and carry out the buying and selling operation in the
market. The highest price on the buy side represents the bid price, and the
lowest price on the sell side represents the ask price. The prices on the buy
(sell) side of the limit order book are arranged in descending (ascending)
order. The mean of the bid and the ask price is the mid-price. The ask price
is always higher than the bid price only during continuous auction which is
the phase that the continuous trading occurs in the market. Difference between
them is named as the bid/ask spread (Cont et al. [2010]).

Figure 1: Example of a limit order book

An example of a limit order book is shown in Figure 1. In this limit order
book, the bid price is 12.14 and the ask price is 12.17. Limit buy (sell)
orders represent the traders that would like to buy (sell) a quantity of a
stock with the indicated price. If someone would like to buy (sell) a stock
with a price equal or higher (lower) than the ask (bid) price, the operation
is executed just after the submission of that particular limit buy (sell)
order (Cont [2011]).

The main part of our research contains the modeling of arrival rates of limit
orders. For limit buy (sell) orders, we consider the quantities on the left
(right) side of the limit order book. The price levels in the vicinity of the
best prices are named as ticks. When we consider limit buy (sell) orders, we
examine the order quantities in the ticks with respect to their distance to
the best ask (bid) price (Cont et al. [2010]). For example, the price of 12.14
(12.17) is the first tick, and the price of 12.13 (12.18) is the second tick
for the limit buy (sell) orders. As shown in this example, the tick values are
given to limit buy (sell) orders with respect to the order price’s distance to
the best ask (bid) price. Since a limit order book has a dynamic structure,
the prices always change during the day as a result of incoming limit orders,
execution of market and cancel orders. In our research, we model the order
quantities in the first 15 ticks for both limit buy and limit sell order
quantities.

The arrival of the limit orders near the best prices is dense. The price of an
order is a crucial criterion of order execution, since the distance to the
current bid and ask price is correlated with the rate of the arrival of limit
orders (Cont et al. [2010]). There are different remarks on explaining arrival
rates of limit orders mathematically. In previous research on order placement
strategies, Bouchaud et al. [2002] and Zovko et al. [2002] suggested that
arrival rates of limit orders $\Lambda(i)$ can be modeled with a power law
which can be seen below.

$\displaystyle\Lambda(i)=\frac{k}{i^{\alpha}}$ (1)

In this equation, the value i denotes the tick value. The value of
$\Lambda(i)$ can be between 0 and $\infty$. The $k$ parameter is a positive
real number and $k$ and $\alpha$ can be estimated using a least squares fit as
it is shown in Equation (2) (Cont et al. [2010]). The value of 15 is chosen
for an upper boundary in that equation, since we perform modeling in the first
15 ticks. The real arrival rates are denoted with $\hat{\Lambda}(i)$.

$\displaystyle\min_{k,\alpha}\sum_{n=1}^{15}(\hat{\Lambda}(i)-\frac{k}{i^{\alpha}})^{2}$
(2)

In another research, a stochastic model consisting of independent Poisson
processes is suggested by Cont et al. [2010]. In this model, arrival rates of
limit orders are modeled in such a way that they are distributed
exponentially.

Weibull and q-exponential distributions are used in an early work on modeling
the duration between two successive transactions (Jiang et al. [2008]). In
another work, the arrival of orders are represented as a renewal process where
the waiting times between two successive orders are distributed according to
Weibull distribution (Cincotti et al. [2006]). We can infer that the duration
between two consecutive orders would also be different in every tick, since
the arrival rates of limit orders differ with respect to the distance to the
best prices. As a result, because of its flexibility in modeling durations, we
test Weibull distribution in modeling arrival rates of limit orders. However,
we use the discrete variant of Weibull distribution proposed by Nakagawa &
Osaki [1975], since we perform an analysis on discrete distributions. Discrete
Weibull distribution has two real number parameters $q>0$ and $\beta>0$ with
integer support on [0, $\infty$). The probability mass function of Discrete
Weibull can be seen in Equation (3) (Nakagawa & Osaki [1975]). We use the
probability density function as it starts from the least tick value of 1.

$P(X=x;q,\beta)=q^{(x-1)^{\beta}}-q^{x^{\beta}},\quad x=1,2,3,\ldots$ (3)

Beta family distributions are frequently used in finance. It is also often
used in modeling the rates of recovery from debts and credit risk (Chen & Wang
[2013]). It is well-known that high skewness is frequently observed in credit
risks data (Schroeck [2002]). Similarly, arrival rates of limit orders have
positive skewness since the rates are much higher in the ticks that are close
to the best prices. Because of its good performance in highly skewed data, we
also test Beta Binomial distribution, which is a discrete member of Beta
family distributions, in modeling the arrival rates. Beta Binomial
distribution is defined by two real number parameters: $\alpha>0$ and
$\beta>0$. Both parameters have finite integer support on [0, $n$). The
probability mass function for $n$ trails in Beta Binomial can be seen in
Equation (4).

$P(X=x;\alpha,\beta)=\binom{n}{x}\frac{B(x+\alpha,n-x+\beta)}{B(\alpha,\beta)}$
(4) $\forall u,\forall v>0,B(u,v)=\int_{0}^{1}t^{u-1}(1-t)^{v-1}dt$ (5)

As indicated before, Exponential distribution is also used for modeling
arrival rates of limit orders. In an early work on modeling the market
dynamics, Cont et al. [2010] assumed that limit orders arrive at the tick i
from the best price with an exponential rate $\lambda(i)$ in his stochastic
model. As a result, Exponential distribution and Geometric distribution which
is the discrete variant of Exponential are also used in our research.
Exponential distribution has one real number parameter $\lambda>0$. The
probability density function of Exponential distribution is given below.

$P(X=x;\lambda)=\lambda e^{-\lambda x},\quad x\geq 0$ (6)

Geometric distribution has one real number parameter $0<p<1$. The probability
mass function of Geometric distribution for $x$ trails is given below. We use
the probability density function of Geometric distribution starting from the
least tick value of 1.

$P(X=x;p)=(1-p)^{x-1}p,\quad x=1,2,3,\ldots$ (7)

Since most of the continuous and discrete distributions have support for the
set $\\{x:x>0\\}$, we adjusted our first tick rate to be the zeroth tick and,
therefore, shifted the fit results to one tick right.

In an early work on cancel orders, Blanchet & Chen [2013] assumed that the
cancellation rates are relatively higher in the ticks that are close to the
best bid and ask prices than distant ticks. Cont et al. [2010] made an
assumption that the cancellation rates show an Exponential distribution with
respect to distance to the best bid and ask prices, and these rates are
proportional to the limit orders in that level. Bouchaud et al. [2018] also
suggested that the cancellation rates are proportional to the arrival rates of
limit orders with an assumption that the activity is much higher in the area
that has high arrival rates of limit orders.

##  3 Materials and method

###  3.1 Market Data

The pure market data contains network captured MoldUDP packets consisting of
ITCH® messeages. An ITCH® NASDAQ protocol for market data includes all orders
in nano-second scale (NASDAQ-OMX-Group [2015]). We use Garanti Bank stock data
in Borsa Istanbul. The data spans 40 trading days from August 1, 2017 to
September 29, 2017, and we sample 228 instances which contain the rates of
daily, weekly and monthly arrived limit orders to analyze.

###  3.2 Arrival Rates of Limit Orders

We extract the information of limit order quantities arrived at the first 15
ticks to the best prices. There are quantities arrived after the first 15
ticks, but they were few with respect to the quantities in the first 15 ticks
so we omitted those quantities. As a result, we perform discrete and
continuous fits on $\lambda^{t}(i)$ which indicates the density of quantities
arrived $Q^{t}(i)$ at the $i$th tick in time instance $t$.

$\lambda^{t}(i)=\frac{Q^{t}(i)}{\sum_{i=1}^{15}Q^{t}(i)},\quad i=1,\ldots,15$
(8)

We split the data into four different time groups. These timesteps are the
daily average of limit buy/sell quantities (40 days), the weekly average of
limit buy/sell quantities (9 weeks), the monthly average of limit buy/sell
quantities and the hourly average of limit buy/sell quantities in 9 weeks. We
consider the market working hours from 10 am to 1 pm and 2 pm to 6 pm. We
created 7 different hourly timesteps in a day.

Consequently, we obtain 40 instances for daily data, 9 instances for weekly
data, 2 instances for monthly data and 63 instances for hourly-weekly data.
Since we consider both limit buy and limit sell orders, we have 228 different
instances to perform discrete and continuous fits. Using three discrete
distributions, Discrete Weibull, Beta-Binomial and Geometric, we perform fits
on the limit buy/sell order quantities that arrived at the first 15 ticks to
the best prices. We used Exponential distribution as a continuous model
approach (Cont et al. [2010]). Also we compared the performance of the best
discrete fit with Exponential fits and Power law fits which are proposed by
Bouchaud et al. [2002] and Zovko et al. [2002] in order to examine if a
discrete approach can compete with the approaches that are suggested in
previous works.

Maximum likelihood estimation finds the parameters that maximize the joint
probability density function of data (likelihood). Since the maximization is
arduous for multiplication operation, in general, the logarithm of the
likelihood function is considered (Myung [2003]). The approach of maximum
likelihood estimation is shown in the Equation (10). In the equation $\theta$
is the parameter vector of the model, and $x_{1:n}$ is the data.

$Likelihood(\theta)=p(x_{1:n}\ |\ \theta)=\prod_{i=1}^{n}P(x_{i}\ |\ \theta)\
,\quad x_{1:n}=\left\\{x_{1},x_{2},\ldots,x_{n}\right\\}$ (9)
$\hat{\theta}=\underset{\theta}{\operatorname{arg\,max}}\ p(x_{1:n}\ |\
\theta)$ (10)

As indicated before, we did not use any functions of R to estimate parameters
of Exponential and Geometric distributions. When we take the derivative of the
logarithm of the likelihood functions and equate it to zero, we can find the
maximum likelihood estimation of the parameter $\lambda$ of Exponential
distribution and the parameter $p$ of Geometric Distribution.

$P(X=x;\lambda)=\lambda e^{-\lambda x},\quad\geq 0$ (11)
$\hat{\lambda},\hat{p}=\frac{n}{\sum_{i=1}^{n}x_{i}},\quad
x_{1:n}=\left\\{x_{1},x_{2},\ldots,x_{n}\right\\}$ (12)

The estimated parameter of Geometric distribution is also found using the same
equation. Because Geometric distribution is the discrete variant of
Exponential distribution, the only difference is that Geometric distribution
has integer $x_{1:n}$ values. Estimation of the parameters of Exponential and
Geometric distributions can be seen in Equation (12).

In order to compare the performance of the models, we consider the sum of
$L_{1}$ norms between the real values and the fit results. Sum of absolute
values of differences between observed densities and fit results are
considered as the error term. The error term at timestep $t$ is shown below.

$\text{Error = }\sum_{i=1}^{15}\
\lvert\lambda^{t}(i)-\hat{\lambda}^{t}(i)\rvert$ (13)

###  3.3 Order Cancellation Rates

We analyze the number of arrived cancel orders around the best price and the
ratio of cancel orders in the vicinity of the best prices. The ratios and
numbers of cancel orders are considered on average weekly and monthly basis.
We consider the quantity of the particular order and the total quantity in
that tick before that particular cancel order arrives. Then we sum these
ratios on the monthly and weekly basis and divide the number of cancel orders
that arrive in a particular tick on the monthly and weekly basis. We express
the cancel order ratios in tick $i$ with $k$ arrived cancel orders in timestep
$t$ with as $C_{t}(i)$.

$\displaystyle C_{t}(i)=\frac{\sum_{n=1}^{k}\frac{\text{Canceled Quantity in
tick $i$ with order $p_{n}$}}{\text{Total Quantity in tick $i$ before order
$p_{n}$}}}{\text{Number of Cancel Orders Arrived in tick $i$ in timestep
$t$}}$ (14)

We compare our experiments on the number of cancel orders arrived in the
vicinity of the best bid and ask prices and the behavior of the ratios of
cancel orders with respect to the distance to the best bid and ask prices with
previous works on cancel orders (Cont et al. [2010], Blanchet & Chen [2013],
Bouchaud et al. [2018]).

##  4 Results

###  4.1 Discrete Fits on Arrival Rates of Limit Orders

In order to find the performance of the discrete fits, we give each
distribution fit a performance score. The performance score is the ratio of
the error of a distribution fit to the minimum fit error on that instance. As
a result, this ratio is higher or equal to 1. If a distribution has the best
fit, then its score becomes 1. We consider the performance according to the
closeness of performance scores to 1. The equation of the normalized
performance score of a distribution $d$ for instance $i$ is given below.

$NPS_{d}(i)=\frac{\text{Error of distribution $d$ on instance
$i$}}{\text{Minimum error among 3 distributions on instance $i$}}$ (15)

We calculate the mean and standard deviation of performance scores of three
distributions in hourly, daily, weekly and monthly fits, and decide which
distribution has the best fit in a particular timestep. The limit buy and
limit sell fits of Geometric, Beta-Binomial and Discrete Weibull distributions
for the last 12 days (from Day 29 to Day 40) can be seen in Figure 2. There
are fits for 40 days, but showing all of them might occupy a lot of space.
Because of that, we only show fits of last 12 days.

Figure 2: The upper figure shows Daily Limit Buy orders and the lower figure
shows Daily Limit Sell orders

Day 29 and Day 30 are Thursday, September 14th and Friday, September 15th
respectively. Day 31 to Day 35 is the week starting on Monday, September 18th.
Day 36 to Day 40 is the week starting on Monday, September 25th. Since some of
the continuous and discrete probability distributions that we use have a
support from 0 to infinity, we divide the probability mass values by the sum
of all probabilities from the first tick to 15th tick for normalization. It is
striking to observe that Discrete Weibull distribution has the best fits for
75 instances out of 80. Beta Binomial outperforms the fit performance of
Discrete Weibull distribution for only 5 instances, and the fits of Geometric
distribution is 3 to 4 times worse than both Discrete Weibull and Beta-
Binomial distribution.

Normalized performance scores of each model in different time steps can be
observed in the charts in Appendix section. The performance of the model
increases as the cell color gets lighter and close to 1.

The limit buy and limit sell fits of Geometric, Beta-Binomial and Discrete
Weibull distributions from Week 1 to Week 9 can be seen in Figure 3 and Figure
4 below.

Figure 3: Discrete Fits on Weekly Limit Buy Arrival Rates Figure 4: Discrete
Fits on Weekly Limit Sell Arrival Rates

We observe that Discrete Weibull distribution has the best fits for all of the
weekly basis instances. Beta Binomial has close fit performance with respect
to Discrete Weibull distribution. Geometric distribution is 3 to 4 times worse
than both Discrete Weibull and Beta-Binomial distribution on average.

Geometric, Beta-Binomial and Discrete Weibull fits on monthly basis arrival
rate of limit orders can be observed in Figure 5. We observe that Discrete
Weibull distribution has the best fits for all of the monthly basis instances.
Geometric distribution is 3 to 4 times worse than others on average.

Figure 5: Discrete Fits on Monthly Limit Buy (above) and Sell (below) Arrival
Rates

Geometric, Beta-Binomial and Discrete Weibull fits on arrival rate of limit
orders in hourly timesteps on different weeks can be observed in Figure 6 and
Figure 7. Since there are 63 instances for this timestep, we only show the
fits on arrival rates in the first 3 weeks and the first 3 hours.

Figure 6: Discrete Fits on Hourly Limit Buy Arrival Rates Figure 7: Discrete
Fits on Hourly Limit Sell Arrival Rates

In order to compare the performance of discrete fits, we use the means of the
performance scores of three distributions in different timesteps. Welch’s
t-test is used to decide which discrete distribution has the best fits on
arrival rate of limit orders. Welch’s t-test is utilized to compare if there
is a significant difference between two samples that have different variances.
The means of performance scores are given in Table 1.

Timestep | Geometric | Discrete Weibull | Beta - Binomial
---|---|---|---
Daily Limit Buy | 4.042 +- 2.115 | 1.000 +- 0.000 | 1.262 +- 0.144
Daily Limit Sell | 3.144 +- 1.320 | 1.033 +- 0.126 | 1.214 +- 0.162
Weekly Limit Buy | 4.285 +- 0.779 | 1.000 +- 0.000 | 1.363 +- 0.106
Weekly Limit Sell | 3.183 +- 0.692 | 1.000 +- 0.000 | 1.237 +- 0.070
Monthly Limit | 3.789 +- 0.748 | 1.000 +- 0.000 | 1.312 +- 0.113
Hourly Limit Buy | 3.643 +- 1.760 | 1.012 +- 0.057 | 1.243 +- 0.160
Hourly Limit Sell | 2.895 +- 1.125 | 1.026 +- 0.083 | 1.153 +- 0.103
Table 1: Mean $NPS$ of Discrete Fits on Different Timesteps

It can be observed in Table 1 that the performance scores of Geometric fits
have 3 to 4 times higher values than Discrete Weibull and Beta Binomial fits.
As a result, we can say that Geometric fits have the worst performance among
three distribution. In order to find the best fits, we perform Welch’s t-test
between Discrete Weibull and Beta Binomial fits. The results of Welch’s
t-tests as p-values are given in Table 2.

Timestep | p-value
---|---
Daily Limit Buy | 0.011
Daily Limit Sell | 0.033
Weekly Limit Buy | 0.003
Weekly Limit Sell | 0.020
Monthly Limit | 0.061
Hourly Limit Buy | 0.092
Hourly Limit Sell | 0.039
Table 2: Welch’s t-test results between Discrete Weibull and Beta-Binomial on
Different Timesteps

We choose 95% confidence interval for t-tests. For 5 of 7 instances the
p-value is below 0.05, so we can reject the null hypothesis that indicates the
means of Discrete Weibull and Beta Binomial performance scores are not
significantly different. As a result, it can be denoted that Discrete Weibull
has significantly better fits than Beta Binomial has in Daily Limit Buy, Daily
Limit Sell, Weekly Limit Buy, Weekly Limit Sell and Hourly Limit Sell orders,
since it has smaller means. For Monthly Limit and Hourly Limit Buy orders,
there is no significant difference between Discrete Weibull fits and Beta
Binomial fits.

###  4.2 Comparison of the Best Discrete Model and Theoretical Models on
Arrival Rates of Limit Orders

We perform least squares approach for power law parameter estimation, since it
is suggested by Bouchaud et al. [2002]. Error approach, timesteps and the
comparison of performance scores are the same as in the discrete fits. We
discretize Exponential fits by using the area under the probability density
function. We divide the x-axis into 15 equal parts and we find the densities
for 15 ticks by calculating areas under the probability density function. The
limit buy and limit sell fits of Exponential, Power law and Discrete Weibull
distributions from Day 29 to Day 40 can be seen in Figure 8 and Figure 9.

Figure 8: Exponential, Power law and Discrete Weibull fits on Daily Limit Buy
Arrival Rates from Day 29 to Day 40 Figure 9: Exponential, Power law and
Discrete Weibull fits on Daily Limit Sell Arrival Rates from Day 29 to Day 40

We observe that Discrete Weibull distribution and Power law have the best fits
for most of the instances. The performance of Discrete Weibull and Power law
is very close to each other. On the other hand, Exponential distribution is 2
to 3 times worse than both Discrete Weibull distribution and Power law. The
limit buy and limit sell fits of Exponential, Power law and Discrete Weibull
distributions from Week 1 to Week 9 can be seen in Figure 10 and Figure 11
below.

Figure 10: Exponential, Power law and Discrete Weibull fits on Weekly Limit
Buy Arrival Rates Figure 11: Exponential, Power law and Discrete Weibull fits
on Weekly Limit Sell Arrival Rates

We observe that Discrete Weibull distribution has the best fits for most of
the weekly basis instances. Power law has close fit performance with respect
to Discrete Weibull distribution. Not suprisingly, Exponential distribution,
being a more parsimonious distribution in the number of parameters, performed
much worse than both Discrete Weibull and Power law on average.

Exponential, Power law and Discrete Weibull fits on monthly basis arrival rate
of limit orders can be observed in Figure 12.

(a) Limit Buy Orders

(b) Limit Sell Orders

Figure 12: Exponential, Power law and Discrete Weibull fits on Monthly Limit
Orders Arrival Rates

We observe that Discrete Weibull distribution has the best fits for 3 out of 4
monthly basis instances. Exponential distribution is 2 times worse than both
Discrete Weibull and Power law on average.

In order to compare the performance of Exponential, Power law and Discrete
Weibull, we again use the means of the performance scores of three
distributions in different timesteps. Welch’s t-test is used to decide which
discrete distribution has the best fits on arrival rate of limit orders.
Average performance scores of Discrete Weibull, Exponential and Power law fits
on different timesteps are given in Table 3.

Timestep | Exponential | Discrete Weibull | Power law
---|---|---|---
Daily Limit Buy | 2.674 +- 1.346 | 1.098 +- 0.198 | 1.154 +- 0.222
Daily Limit Sell | 2.018 +- 1.071 | 1.059 +- 0.104 | 1.128 +- 0.167
Weekly Limit Buy | 2.515 +- 0.651 | 1.165 +- 0.191 | 1.051 +- 0.066
Weekly Limit Sell | 1.771 +- 0.471 | 1.004 +- 0.014 | 1.076 +- 0.060
Monthly Limit | 2.038 +- 0.394 | 1.063 +- 0.127 | 1.048 +- 0.062
Hourly Limit Buy | 2.624 +- 1.279 | 1.094 +- 0.126 | 1.163 +- 0.268
Hourly Limit Sell | 1.918 +- 0.753 | 1.056 +- 0.074 | 1.128 +- 0.168
Table 3: Mean $NPS_{d}(Daily)$ of Proposed Distributions on Different
Timesteps

Exponential fits evidently have the worst performance among three
distribution. On the other hand, as Power law and Discrete Weibull have very
close mean performance scores, we perform a $t$-test to find if there is a
significant difference between those values.

Timestep | p-value
---|---
Daily Limit Buy | 0.363
Daily Limit Sell | 0.326
Weekly Limit Buy | 0.343
Weekly Limit Sell | 0.464
Monthly Limit | 0.973
Hourly Limit Buy | 0.743
Hourly Limit Sell | 0.242
Table 4: Welch’s t-test results between Discrete Weibull and Power law

For all of the instances, the p-values have a value that is higher than 0.05,
so we can not reject the null hypothesis that indicates the mean performance
scores of Discrete Weibull and Power Law is not significantly different. So we
can say that the performance of Power Law and Discrete Weibull fits are not
significantly different for all instances. Consequently, Discrete Weibull
models can compete with Power law models which are proposed by Bouchaud et al.
[2002] and Zovko et al. [2002]. We can use Discrete Weibull distribution to
model arrival rates of limit orders with respect to distance to the best
prices accurately.

###  4.3 Behavior of Order Cancellation Rates

We analyze the number of cancel orders and the ratio of canceled orders in the
vicinity of the best prices. We consider both cancel buy orders and cancel
sell orders on the weekly and monthly basis. In previous works, Blanchet &
Chen [2013] denoted that the cancellation activity is much higher in the close
regions to the best bid and ask prices. The number of cancel orders arrived at
the first 10 ticks on the weekly basis in our experiments are shown in Figure
13 and Figure 14.

Figure 13: Number of Cancel Buy orders arrived in the vicinity of the best ask
price on weekly basis Figure 14: Number of Cancel Sell orders arrived in the
vicinity of the best bid price on weekly basis

When we consider the cancel activity as the number of cancel orders arrived,
the results are consistent with the previous works. It can be observed that
the number of cancel orders arrived in the close regions to the best prices
are higher. We also consider the average ratio of canceled order quantity in
the ticks. We use the metric which we denote in Section 3.3 for finding the
ratios. The ratios of canceled order quantities in the first 10 ticks on
weekly basis in our experiments are shown in Figure 15 and Figure 16. The red
and blue lines in figures expected values.

Figure 15: Ratios of canceled buy order quantities in the vicinity of the best
ask price on weekly basis, Red line is the average ratio

Figure 16: Ratios of canceled sell order quantities in the vicinity of the
best bid price on weekly basis, Blue line is the average ratio

The ratios of canceled order quantities in the first 10 ticks on monthly basis
in our experiments are shown in Figure 17 and Figure 18.

Figure 17: Ratios of canceled buy order quantities in the vicinity of the best
ask price on monthly basis, Red line is the average ratio Figure 18: Ratios of
canceled sell order quantities in the vicinity of the best bid price on
monthly basis, Blue line is the average ratio

The ratios were close most of the time. Thus we perform uniformity tests on
the ratios. We use Chi-Square Test to test if the ratios are distributed
uniformly. Chi-Square test statistics formula is given in Equation 16. In the
Equation 16, $c$ means the degree of freedom. In our experiment, we have 10
different categories (because of 10 ticks). Since the degree of freedom is
equal to the number of categories minus 1, the degree of freedom value is 9
for our experiments.

$\displaystyle
X_{c}^{2}=\sum_{i=1}^{10}\frac{(Observed_{i}-Expected_{i})^{2}}{Expected_{i}}$
(16)

Chi-Square tests use count data as observed data. Therefore we convert the
cancellation ratios to integer values by multiplying them by 100. Then, we
test the null hypothesis which claims that the ratios of canceled orders are
consistent with Uniform distribution. Chi-Square test statistics and
corresponding p-values are given in Table 5 and Table 6.

Timestep | Cancel Buy | Cancel Sell
---|---|---
1st Week | 7.846 | 3.296
2nd Week | 12.197 | 8.371
3rd Week | 4.762 | 7.461
4th Week | 6.419 | 3.793
5th Week | 8.378 | 28.419
6th Week | 9.543 | 2.948
7th Week | 5.045 | 7.558
8th Week | 3.494 | 5.355
9th Week | 9.845 | 8.459
1st Month | 3.430 | 6.173
2nd Month | 4.740 | 4.539
Table 5: Comparison between the ratios of canceled orders and the uniform distribution: Chi-Square Test Statistics on Different Timesteps Timestep | Cancel Buy | Cancel Sell
---|---|---
1st Week | 0.5497 | 0.9513
2nd Week | 0.2024 | 0.4972
3rd Week | 0.8545 | 0.5892
4th Week | 0.6973 | 0.9245
5th Week | 0.4965 | 0.0008
6th Week | 0.3887 | 0.9663
7th Week | 0.8303 | 0.5792
8th Week | 0.9414 | 0.8023
9th Week | 0.3631 | 0.4886
1st Month | 0.9447 | 0.7224
2nd Month | 0.8563 | 0.8725
Table 6: Comparison between the ratios of canceled orders and the uniform
distribution: Corresponding p-values on Different Timesteps

We use 95% confidence interval for the tests. Chi-Square tests for uniformity
present p-values higher than 0.05 in all of the instances except for the
ratios of cancel sell orders in the 5th week. Consequently, we can not reject
the null hypothesis which indicates that the cancellation rates are consistent
with Uniform distribution for most of the instances.

##  5 Conclusion and Future Work

In this research, we used different statistical distributions to fit the limit
order quantities arrived in the vicinity of the best bid and ask prices. The
fits are made on Garanti Bank stock data for the period from August -
September 2017. We analyzed the daily, weekly and monthly mean limit order
quantities arrived at the first 15 levels from the best prices. Also, we
considered the weekly mean quantities of limit orders arrived in 7 different
time intervals in a day. We used total sum of $L_{1}$ norms between empirical
density and fit results in the first 15 price levels of the bid and ask prices
to evaluate the goodness of fits.

We observed that Discrete Weibull and Beta Binomial distributions are almost 4
times better at fitting the order quantity data than Geometric distribution.
We had 228 instances to fit and Discrete Weibull has the lowest $L_{1}$ norm
for 210 of them. Beta Binomial fits the data with the lowest $L_{1}$ norm for
17 of the instances and Geometric distribution has the best fit for only one
of the instances. Additionally, we used Exponential distribution to fit the
same 228 instances. We found the probability mass values by calculating the
areas of 15 bins under the Exponential probability density function using
discretization. Then we obtained the sum of $L_{1}$ norms between empirical
density and 15 probability mass values, and compared the goodness of
Exponential distribution fits with Discrete Weibull distribution fits. We
observed that Discrete Weibull fits the daily, weekly and monthly mean
quantities two times better than Exponential distribution. Also, Discrete
Weibull fits can compete with Power law fits which are proposed in early
works.

We analyzed the weekly and monthly mean ratio of cancel orders in the first 10
price levels. We conducted Chi-Square tests to test the uniformity. We
observed that we can not deny the hypothesis which claims that the
cancellation rates are consistent with Uniform distribution. As a result, we
found out that the assumption made by Cont et al. [2010] on cancellation rates
which denotes that the cancellation rates are distributed exponentially can
not be adapted to Turkish markets.

Our dataset was quite small, since it only contains the data of 2 months.
Also, we only consider Garanti Bank stock data. In future work, the same
experiments can be conducted for larger datasets such as 6 months or 1 year.
Moreover, stock data from other companies can be considered, and the relation
between different stocks would be another interesting extension. Additionally,
the relation between stock prices and arrival rates can be examined and
predicted.

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## Appendix

Figure A.1: $NPS_{d}(Daily)$ of three discrete distributions on Arrival Rates
of Daily Limit Buy/Sell Orders

Figure A.2: $NPS_{d}(Weekly)$ of three discrete distributions on Arrival Rates
of Weekly Limit Buy/Sell Orders

Figure A.3: $NPS_{d}(Hourly)$ of three discrete distributions on Arrival Rates
of Hourly Limit Buy/Sell Orders

Figure A.4: $NPS_{d}(Daily)$ of Discrete Weibull and theoretical distributions
on Arrival Rates of Daily Limit Buy/Sell Orders

Figure A.5: $NPS_{d}(Weekly)$ of Discrete Weibull and theoretical
distributions on Arrival Rates of Weekly Limit Buy/Sell Orders

Figure A.6: $NPS_{d}(Hourly)$ of Discrete Weibull and theoretical
distributions on Arrival Rates of Hourly Limit Buy/Sell Orders