# Electron cooling in graphene enhanced by plasmon-hydron resonance

Xiaoqing Yu1, Alessandro Principi2, Klaas-Jan Tielrooij3,4, Mischa Bonn1 and
Nikita Kavokine1,5<EMAIL_ADDRESS>1Max Planck Institute for
Polymer Research, Ackermannweg 10, Mainz 55128, Germany  2School of Physics
and Astronomy, University of Manchester, M13 9PL Manchester, U.K.  3Catalan
Institute of Nanoscience and Nanotechnology (ICN2), BIST and CSIC, Campus UAB,
Bellaterra, Barcelona, 08193, Spain  4Department of Applied Physics, TU
Eindhoven, Den Dolech 2, 5612 AZ, Eindhoven, The Netherlands  5Center for
Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York,
NY 10010, USA

###### Abstract

Evidence is accumulating for the crucial role of a solid’s free electrons in
the dynamics of solid-liquid interfaces. Liquids induce electronic
polarization and drive electric currents as they flow; electronic excitations,
in turn, participate in hydrodynamic friction. Yet, the underlying solid-
liquid interactions have been lacking a direct experimental probe. Here, we
study the energy transfer across liquid-graphene interfaces using ultrafast
spectroscopy. The graphene electrons are heated up quasi-instantaneously by a
visible excitation pulse, and the time evolution of the electronic temperature
is then monitored with a terahertz pulse. We observe that water accelerates
the cooling of the graphene electrons, whereas other polar liquids leave the
cooling dynamics largely unaffected. A quantum theory of solid-liquid heat
transfer accounts for the water-specific cooling enhancement through a
resonance between the graphene surface plasmon mode and the so-called hydrons
– water charge fluctuations –, particularly the water libration modes, that
allows for efficient energy transfer. Our results provide direct experimental
evidence of a solid-liquid interaction mediated by collective modes and
support the theoretically proposed mechanism for quantum friction. They
further reveal a particularly large thermal boundary conductance for the
water-graphene interface and suggest strategies for enhancing the thermal
conductivity in graphene-based nanostructures.

Free electrons in graphene exhibit rather unique dynamics in the terahertz
(THz) frequency range, including a highly non-linear response to
photoexcitation by THz pulses Hwang2013 ; Hafez2018 . Graphene’s distinctive
dynamical properties on picosecond timescales have found several applications
in, e.g., ultrafast photodetectors, modulators, and receivers Liu2011 ;
Romagnoli2018 ; Muench2019 . The THz frequency range acquires particular
importance at room temperature $T$, where it corresponds to the typical
frequency of thermal fluctuations: $k_{\rm B}T/\hbar\sim 6~{}\rm THz$, with
$k_{\rm B}$ Boltzmann’s constant and $\hbar$ Planck’s constant. One may
therefore expect non-trivial couplings between the graphene electrons and the
thermal fluctuations of their environment. These couplings have been
intensively studied in the case of a solid environment: for instance, non-
adiabatic effects have been shown to arise in the graphene electron-phonon
interaction Pisana2007 , and plasmon-phonon coupling between graphene and a
polar substrate has been demonstrated Hwang2010 ; Dai2015 ; Koch2016 . More
recently, it has been theoretically proposed that similar effects are at play
when graphene has a liquid environment: then, the interaction between the
liquid’s charge fluctuations – dubbed hydrons – and graphene’s electronic
excitations tunes the hydrodynamic friction at the carbon surface Kavokine2022
; Bui2022 . This "quantum friction" mechanism holds the potential of entirely
new strategies for controlling liquid flows at nanometer scales Coquinot2022 ;
Marcotte2022 ; it is therefore of interest to experimentally probe the
underlying electron-hydron interaction.

In this Article, we probe solid-liquid interactions by measuring energy
transfer at the solid-liquid interface. Specifically, we use a femtosecond
visible pulse to introduce a quasi-instantaneous temperature difference
between the electrons of a graphene sample and their environment. The cooling
rate of the electronic system is followed in real-time using terahertz pulses.
Such Optical Pump - Terahertz Probe (OPTP) spectroscopy is a well-established
tool for probing electron relaxation in 2D materials George2008 ; Kar2015 ;
Mihnev2015 ; Mihnev2016 ; Pogna2021 ; Zheng2022 . In high-quality graphene, it
has been used to identify the interaction of hot carriers with optical phonons
Mihnev2016 ; Pogna2021  and with substrate phonons as the main electron
cooling mechanisms Tielrooij2018 ; it has also identified the role of Coulomb
interactions in the interlayer thermal conductivity of graphene stacks
Mihnev2015 . Here, we measure the electron relaxation time in the presence of
different polar liquids to probe the electron-hydron interaction, which we
find to be comparable to the electron - optical phonon interaction only when
the liquid is water. A complete theoretical analysis shows that this
specificity of water is explained by the strong coupling of its THz
(libration) modes to the graphene surface plasmon, with the electron-electron
interactions in graphene playing a crucial role.

Figure 1: Heat transfer and friction at the solid-liquid interface. a.
Schematics of the system under study: the interface between water and a
graphene sheet. The picture emphasizes the electron cloud and its wave-like
plasmon excitation. b. Momentum transfer processes at the solid-liquid
interface. A flowing liquid (the flow profile is shown by the thin blue
arrows) may not only transfer momentum to the crystal lattice (exciting phonon
vibrations) through classical hydrodynamic friction, but also directly to the
electrons through quantum friction. c. Energy transfer (ET) processes at the
solid-liquid interface. In the typically-assumed "classical" pathway, hot
electrons first transfer energy to the phonons, which transfer energy to the
liquid. An alternative "quantum" pathway consists in the electrons
transferring energy directly to the liquid through Coulomb coupling.

Solid-liquid heat transfer
The energy transfer between a solid and a liquid is usually considered to be
mediated by molecular vibrations at the interface, as most of a solid’s heat
capacity is contained in its phonon modes Phillpot2005 . Even if an optical
excitation of the solid’s electrons is used to create the temperature
difference, the electrons are typically assumed to thermalize with phonons on
a very short time scale, so that the solid’s phonons ultimately mediate the
energy transfer to the liquid’s vibrational modes Gutierrez2022 ; Herrero2022
. However, if the electrons were to transfer energy to the liquid faster than
to the phonons, the interfacial thermal conductivity would contain a non-
negligible contribution from near-field radiative heat transfer Volokitin2007
; Biehs2021  (Fig. 1c). Such an electronic or "quantum" contribution to heat
transfer is in close analogy with the quantum contribution to hydrodynamic
friction. Quantum hydrodynamic friction relies on momentum being transferred
directly between the solid’s and the liquid’s charge fluctuation modes,
coupled by Coulomb forces (Fig. 1b): the two processes are mediated by the
same solid-liquid interaction.

In order to probe this interaction through a hydrodynamic friction
measurement, one needs to ensure that quantum friction dominates over the
classical surface roughness contribution: this imposes stringent constraints
on the sample’s surface state, in already technically-difficult experiments
Maali2008 ; Secchi2016 ; Xie2018 . Similarly, in the case of energy transfer,
the quantum contribution needs to be comparable to the classical phonon-based
contribution in order to become measurable; yet, this condition is easier to
satisfy since it is insensitive to the sample’s surface roughness. We show
that this condition is met upon optically exciting a graphene-water interface,
owing, in particular, to graphene’s weak electron-phonon coupling
Bistritzer2009 ; Betz2012 .

Figure 2: Measurement of picosecond hot electron relaxation in graphene. a.
Schematic of the experimental setup. A graphene sample (Fermi level in the
range $\rm 100-180~{}meV$, see SI Sec. 1.4) is placed in contact with a liquid
inside a fused silica flow cell. An optical excitation pulse impulsively heats
up the graphene electrons, and the electron temperature dynamics are then
monitored with a THz probe. b. Normalized electron temperature as a function
of time after photoexcitation. The dotted lines represent raw data and the
full lines are exponential fits. c. Electron cooling time obtained through
exponential fitting (see b) for the different liquids that have been placed in
the flow cell and different initial electron temperatures, set by the
excitation laser fluence. Faster cooling is observed in the presence of water
and heavy water. Error bars represent 95% confidence intervals of the
exponential fits, and the centre point corresponds to the result of the least-
squares fitting procedure.

Time-resolved electron cooling
Our experimental setup is schematically represented in Fig. 2a. A monolayer
graphene sample grown by chemical vapor deposition (CVD) was transferred onto
a fused silica flow cell, filled with either nitrogen gas or a liquid of our
choice (SI Sec. 1.1). The graphene chemical potential was in the range $\rm
100-180~{}meV$ as determined from Raman measurements (SI Sec. 1.4). In a
typical experiment, the graphene electrons were excited using a $\sim 50~{}\rm
fs$ laser pulse with 800 nm central wavelength. Then, the attenuation of a
$\sim 1~{}\rm ps$ THz probe pulse (precisely, the modulation of the peak
electric field) was monitored as a function of the pump-probe delay (SI Sec.
1.2). After absorption of the exciting pump pulse, the non-equilibrium
electron distribution typically thermalizes over a sub-100 fs timescale
through electron-electron scattering Brida2013 : it can then be described as a
Fermi-Dirac distribution at a given temperature. A hotter electron
distribution results in a lower THz photoconductivity, since hotter electrons
are less efficient at screening charged impurities Tomadin2018 ;
Massicotte2021 . The pump-probe measurement thus gives access to the electron
temperature dynamics after photoexcitation (Fig. 2b).

Regardless of the medium that the graphene is in contact with, the electronic
temperature $T(t)$ exhibits a relaxation that can be approximated by an
exponential function : $\Delta T(t)=T(t)-T_{0}=\Delta T_{0}e^{-t/\tau}$. This
allows us to extract the cooling times $\tau$ for the different liquids and
different initial electronic temperatures (determined by the excitation laser
fluence), displayed in Fig. 2c. We observe that the cooling time is longer for
an initially hotter electron distribution, in agreement with previous reports
Pogna2021 . Now, for all initial temperatures, we consistently observe the
same dependence of the cooling time on the sample’s liquid environment. In the
presence of water ($\rm H_{2}O$) and heavy water ($\rm D_{2}O$), the graphene
electrons cool faster than they do intrinsically, in an inert nitrogen
atmosphere. Conversely, methanol and ethanol have almost no effect on the
electron cooling time. Interestingly, we observe an isotope effect in the
electron cooling process: there is a difference in the cooling times in the
presence of $\rm H_{2}O$ and $\rm D_{2}O$ that well exceeds experimental
uncertainties.

We are thus led to hypothesize, as anticipated above, that the liquid provides
the electrons with a supplementary cooling pathway, which, in the case of
water, has an efficiency comparable to the intrinsic cooling pathway via
phonons. We then interpret the faster cooling as a signature of "quantum"
electron-liquid energy transfer. We assess the pertinence of this hypothesis
by developing a complete theory of quantum energy transfer at the solid-liquid
interface.

Theoretical framework
In order to tackle the interaction between a classical liquid and an
electronic system whose behavior is intrinsically quantum, we describe the
liquid in a formally quantum way. Following ref. Kavokine2022 , we represent
the liquid’s charge density as a free fluctuating field with prescribed
correlation functions. This naturally leads to a Fourier-space description of
the solid-liquid interface in terms of its collective modes, rather than the
usual molecular scale interactions. Within this description, the quantum
solid-liquid energy transfer amounts to electron relaxation upon coupling to a
bosonic bath, a problem that has been extensively studied in condensed matter
systems Mahan . Interestingly, in the case of graphene, many of these studies
are carried out within a single-particle Boltzmann formalism, which may
incorporate multiple screening effects only in an _ad hoc_ fashion
Bistritzer2009 ; Principi2017 ; Pogna2021 . These effects turn out to be
crucial for the solid-liquid system under consideration: we have therefore
developed an _ab initio_ theory of solid-liquid heat transfer based on the
non-equilibrium Keldysh formalism Rammer1986 , which has only very recently
been considered for problems of interfacial heat transfer Wise2022 . Our
computation, detailed in the SI Sec. 2.2, is closely analogous to the one
carried out for quantum friction in ref. Kavokine2022 . The theoretical
framework can formally apply to fully non-equilibrium situations and take
interactions into account to arbitrary order. However, to obtain a closed-form
result, assume that the liquid and the solid internally equilibrated at
temperatures $T_{\ell}$ and $T_{\rm e}$ respectively. Furthermore, we take
electron-electron and electron-liquid Coulomb interactions into account at the
Random Phase Approximation (RPA) level. With these assumptions, we obtain the
electron-liquid energy transfer rate as

$\mathcal{Q}_{\rm
Q}=\frac{1}{2\pi^{3}}\int\mathrm{d}\mathbf{q}\int_{0}^{+\infty}\mathrm{d}\omega\,\hbar\omega[n_{\rm
B}(\omega,T_{\rm e})-n_{\rm B}(\omega,T_{\ell})]\frac{\mathrm{Im}\,[g_{\rm
e}(\mathbf{q},\omega)]\mathrm{Im}\,[g_{\rm\ell}(\mathbf{q},\omega)]}{|1-g_{\rm
e}(\mathbf{q},\omega)g_{\rm\ell}(\mathbf{q},\omega)|^{2}},$ (1)

Here, $n_{\rm B}(\omega,T)=1/(e^{\hbar\omega/T}-1)$ is the Bose distribution
and the $g_{\rm e,\ell}$ are surface response functions of the solid and the
liquid, respectively. These are analogues of the dielectric function for semi-
infinite media, whose precise definition is given in the SI, Sec. 2.3. For the
liquids under consideration, it will be sufficient to use the long-wavelength-
limit expression of the surface response function:

$g_{\rm\ell}(q\to
0,\omega)=\frac{\epsilon_{\ell}(\omega)-1}{\epsilon_{\ell}(\omega)+1},$ (2)

where $\epsilon_{\ell}(\omega)$ is the liquid’s bulk dielectric function. For
two-dimensional graphene, we show in the SI (Sec. 2.3) that the surface
response function can be expressed as

$g_{\rm e}(q,\omega)=-\frac{e^{2}}{2\epsilon_{0}q}\chi(q,\omega),$ (3)

where $\chi(q,\omega)$ is graphene’s charge susceptibility.

The result in Eq. (1) has been derived for two solids separated by a vacuum
gap in the framework of fluctuation-induced electromagnetic phenomena
Pendry1999 ; Volokitin2001 ; Volokitin2007 ; our non-equilibrium framework,
however, is better suited to the solid-liquid system under consideration. We
note that Eq. (1) takes the form of a Landauer formula for the transport of
bosonic quasiparticles – elementary excitations of the solid’s and the
liquid’s charge fluctuations modes Biehs2021 . It involves indeed the
difference of Bose distribution functions between the solid and the liquid,
and the product of surface response functions plays the role of a transmission
coefficient for the quasiparticles. One may count either the energy or the
momentum transported by the quasiparticles: the former corresponds to near-
field heat transfer, the latter to quantum friction. This quasiparticle
picture thus makes explicit the fundamental connection between the two
processes.

Figure 3: Mechanism of electron-liquid heat transfer. a. Surface excitation
spectra $\mathrm{Im}\,[g_{\ell}(\omega)]$ of the different liquids under study
obtained according to Eq. (2) from the experimentally-measured bulk dielectric
permittivities. The arrows indicate the libration modes of $\rm H_{2}O$ and
$\rm D_{2}O$. b. Graphene surface excitation spectrum $\mathrm{Im}\,[g_{\rm
e}(q,\omega)]$, calculated at a chemical potential $\mu=100~{}\rm meV$ and
temperature $T_{\rm e}=623~{}\rm K$. The main feature is the collective
plasmon mode. c. Theoretical prediction for the graphene-water energy transfer
rate resolved in frequency-wavevector space. The main contribution originates
from a resonance between the graphene plasmon mode and the water libration
mode. d. Experimentally-measured electron cooling rate in the presence of the
various liquids. Error bars represent 95% confidence intervals of the
exponential fits to the temperature decay curves. e. Theoretical prediction
for the liquid contribution to the electron cooling rate, reproducing the
experimentally-observed trend in terms of the nature of the liquid. The symbol
size in the vertical direction represents the variation in the theoretical
prediction when the graphene chemical potential spans the range $\rm
100-180~{}meV$. f. Schematic of the water-mediated electron cooling mechanism
inferred from the combination of theoretical and experimental results. The
cooling proceeds through the Coulomb interaction between the graphene plasmon
mode and the hindered molecular rotations (librations) in water.

Plasmon-hydron resonance
The graphene electrons may relax either through direct interaction with the
liquid, or through emission of optical phonons. The latter process has been
well-studied, both theoretically and experimentally Bistritzer2009 ; Pogna2021
. Our non-equilibrium formalism applies in principle to any electron-boson
system: when applied to the electron-phonon system, it recovers the result for
the energy transfer rate $\mathcal{Q}_{\rm ph}$ (from electrons to phonons)
obtained in ref. Pogna2021  (see SI, Sec. 2.3.2). Then, within a three-
temperature model, where the electrons, liquid and phonons are assumed to be
internally-equilibrated at temperatures $T_{\rm e}$, $T_{\ell}$ and $T_{\rm
ph}$, respectively, we may determine the evolution of the electron temperature
according to

$C(T_{\rm e})\frac{\mathrm{d}T_{\rm e}(t)}{\mathrm{d}t}=-\mathcal{Q}_{\rm
Q}(T_{\rm e},T_{\ell})-\mathcal{Q}_{\rm ph}(T_{\rm e},T_{\rm ph}),$ (4)

where $C(T_{\rm e})$ is the graphene electronic heat capacity at temperature
$T_{\rm e}$. We focus in the following on the liquid contribution to the
electron cooling rate, defined as $1/\tau=\mathcal{Q}_{\rm Q}(T_{\rm
e},T_{\ell})/(C(T_{\rm e})\times(T_{\rm e}-T_{\ell}))$, which may be compared
with the experimental results. The quantitative evaluation of $\tau$ requires
the surface response functions of graphene and of the various liquids. We
compute the graphene surface response function according to Eq. (3) by
numerical integration Wunsch2006 , at the chemical potential determined for
our samples by Raman spectroscopy (SI Sec. 1.4). For the liquids, we use the
expression in Eq. (2), with the bulk dielectric function determined by
infrared absorption spectroscopy (Fig. 3a and SI Sec. 1.3).

Our theoretical prediction for the various liquids’ contribution to the
electron cooling rate is shown in Fig. 3e. Quantitatively, we obtain cooling
rates of the order of $1~{}\rm ps^{-1}$, in excellent agreement with the
experimentally observed range (Fig. 3d) : our theory indicates that the
quantum electron-liquid cooling is a sufficiently efficient process to compete
with the intrinsic phonon contribution, estimated at around $0.6~{}\rm
ps^{-1}$ from the cooling rate in the absence of liquid. Moreover, our theory
reproduces the experimentally observed trend in cooling rates, with a
significant liquid contribution arising only for water and heavy water; the
dependence of the cooling rate on initial electron temperature is also well-
reproduced (Fig. S7). Finally, the theory reproduces the isotope effect, that
is, the slightly slower cooling observed with $\rm D_{2}O$ as compared to $\rm
H_{2}O$.

We may now exploit the theory to gain insight into the microscopic mechanism
of the liquid-mediated cooling process. In Eq. (1), the difference of Bose
distributions decreases exponentially at frequencies above $T_{\rm
e}/\hbar\sim 100~{}\rm meV$. At frequencies below 100 meV, the graphene
spectrum is dominated by a plasmon mode, that corresponds to the collective
oscillation of electrons in the plane of the graphene layer Wunsch2006  (Fig.
3b). In this same frequency range, water and heavy water have a high spectral
density due to their libration mode, that corresponds to hindered molecular
rotations Carlson2020  (Fig. 3a). As a result, the energy transfer rate
resolved in frequency-momentum space (the integrand in Eq. (1), plotted in
Fig. 3c) has its main contribution from the spectral region where the two
modes overlap. We conclude that the particularly efficient electron-water
cooling is due to a resonance between the graphene plasmon mode and the water
libration mode. This conclusion is further supported by the isotope effect.
Indeed, the libration of the heavier $\rm D_{2}O$ is at slightly lower
frequency than that of the lighter $\rm H_{2}O$, and a higher frequency mode
makes a larger contribution to the cooling rate due to the factor
$\hbar\omega$ in Eq. (1). In the Landauer picture, the quasiparticle transport
rates are almost the same for the graphene-$\rm H_{2}O$ and graphene-$\rm
D_{2}O$ systems, but in the case of $\rm H_{2}O$ each quasiparticle carries
more energy. Overall, our experiments evidence a direct interaction between
the graphene plasmon and water libration, as shown schematically in Fig. 3f.
We note that plasmons have been shown to play a role in the energy transfer
between two graphene sheets Ying2020 ; however, to our knowledge, a plasmon-
hydron interaction has so far not been suggested as a possible electron
relaxation mechanism.

Figure 4: Strong plasmon-hydron coupling. a. Theoretical prediction for the
graphene electron cooling rate in contact with different liquids, within
different treatments of interactions. The cooling rate is strongly
overestimated if no electron-electron interactions are taken into account
(blue symbols), and underestimated if the electron-liquid interactions are
considered only to first order (orange symbols). b. Graphene surface
excitation spectrum $\mathrm{Im}\,[g_{\rm e}(q,\omega)]$, calculated at a
chemical potential $\mu=180~{}\rm meV$ and temperature $T_{\rm e}=623~{}\rm
K$, renormalized by the presence of water according to Eq. (5). The white
dashed lines are guides to the eye showing the strongly-coupled plasmon-hydron
mode. Inset: bare and renormalized graphene spectra at fixed wavevector
$q_{0}=0.15~{}\rm nm^{-1}$. c. Comparison between the spectrally resolved
energy transfer rates obtained to first order and to arbitrary order in the
solid-liquid interaction. Higher-order effects enhance the energy transfer
rate at low frequencies.

Interactions and strong coupling
The combination of theory and experiment allows us to identify the key
physical ingredients that are required to account for energy transfer at the
water-graphene interface. First, our results reveal that electron-electron
interactions are crucial, since they produce the plasmon mode that is
instrumental to the energy transfer mechanism. Indeed, applying our theory to
non-interacting graphene would result in a strongly overestimated liquid
contribution to the cooling rate (Fig. 4a). This precludes single-particle
Boltzmann approaches – such as those that have been used for the electron-
phonon interaction in graphene Bistritzer2009 ; Pogna2021  – for accurately
describing the water-graphene interaction.

Furthermore, the detailed examination of our theoretical result reveals that
the efficiency of the electron-water cooling is enhanced by the formation of a
strongly-coupled plasmon-hydron mode. Indeed, the result in Eq. (1) involves
bare surface response functions, without any renormalization due to the
presence of the other medium. However, the denominator $|1-g_{\rm
e}g_{\rm\ell}|^{2}$ accounts for solid-liquid interactions to arbitrary order
(at the RPA level) and contains the signature of any potential strong coupling
effects. We find that these effects are indeed important, as removing the
denominator in Eq. (1) (that is, treating the electron-liquid interactions
only to first order) results in under-estimation of the liquid-mediated
cooling rate by about 30% (Fig. 4a) . In order to gain physical insight into
the nature of these higher order effects, we may compute the graphene surface
response function renormalized by the presence of water, which is given by
(see SI Sec. 2.3)

$\tilde{g}_{\rm e}(q,\omega)=\frac{g_{\rm e}(q,\omega)}{1-g_{\rm
e}(q,\omega)g_{\ell}(q,\omega)}.$ (5)

The renormalized surface excitation spectrum $\mathrm{Im}\,[\tilde{g}_{\rm
e}(q,\omega)]$ is plotted in Fig. 4b, for a chemical potential $\mu=180~{}\rm
meV$. We observe that the graphene plasmon now splits into two modes, which
are both a mixture of the the bare plasmon and water libration. These are in
fact analogous to the coupled plasmon-phonon modes that have been predicted
Hwang2010  and measured Dai2015 ; Koch2016  for graphene on a polar substrate.
It can be seen in the inset of Fig. 4b that coupling to the water modes also
increases the spectral density at low frequencies (below the plasmon peak),
compared to the bare graphene response function. This is in fact the higher-
order effect that is mainly responsible for the enhancement of the electron
cooling rate. As shown in Fig. 4c, taking into account solid-liquid
interactions to arbitrary order mainly enhances the contribution of low
frequencies to the energy transfer.

Conclusions
We have carried out ultrafast measurements of electron relaxation in graphene,
revealing signatures of direct energy transfer between the graphene electrons
and the surrounding liquid. These results speak to the importance of
electronic degrees of freedom in the dynamics of solid-liquid interfaces,
particularly interfaces between water and carbon-based materials. Despite
conventional theories and simulations that describe the interface in terms of
atomic-scale Lennard-Jones potentials Gutierrez2022 ; Herrero2022 , or with
electronic degrees of freedom in the Born-Oppenheimer approximation Tocci2014
; Tocci2020 , here we demonstrate experimentally that the dynamics of the
water-graphene interface need to be considered at the level of collective
modes in the terahertz frequency range. In particular, our semi-quantitative
theoretical analysis attributes the observed cooling dynamics to the strong
coupling between the graphene plasmon and water libration modes.

The experimental observation of such a collective mode interaction supports
the proposed mechanism for quantum friction at the water-carbon interface,
which is precisely based on momentum transfer between collective modes
Kavokine2022 . The near-quantitative agreement between the experiment and
theory obtained for energy transfer suggests that a similar agreement should
be achieved for momentum transfer. The water-graphene quantum friction force
is small if the graphene electrons are at rest, but becomes important if they
are driven at a high velocity by a phonon wind or an applied voltage
Coquinot2022 . The quantum-friction-based driving of water flows by graphene
electronic currents appears as a promising avenue in light of our findings.
The electric circuit configuration would furthermore allow for noise
thermometry Yang2018 ; Baudin2020  to be used as a supplementary probe of the
electron relaxation mechanisms.

Our results provide yet another example of the water-carbon interface
outperforming other solid-liquid systems Bocquet2020 . Indeed, the electronic
contribution to the graphene-water thermal boundary conductance is as high as
$\lambda=0.25~{}\rm MW\cdot m^{-2}\cdot K^{-1}$, exceeding the value obtained
with the other investigated liquids by at least a factor of 2. This even
exceeds the thermal boundary conductance obtained for the graphene-hBN
interface, at which particularly fast "super-Planckian" energy transfer was
observed Principi2017 ; Tielrooij2018 . Our investigation thus suggests that
the density of modes in the terahertz frequency range is a key determinant for
the thermal conductivity of graphene-containing composite materials.

Acknowledgements
We acknowledge financial support from the MaxWater initiative of the Max
Planck Society. We thank Xiaoyu Jia and Hai Wang for carrying out preliminary
experiments, Maksim Grechko and Detlev-Walter Scholdei for assisting with the
FTIR measurements, Marc-Jan van Zadel and Florian Gericke for constructing the
sample holder. X.Y. is grateful for support from the China Scholarship
Council. K.J.T. acknowledges funding from the European Union’s Horizon 2020
research and innovation program under Grant Agreement No. 804349 (ERC StG
CUHL), RYC fellowship No. RYC-2017-22330, and IAE project
PID2019-111673GB-I00. A.P. acknowledges support from the European Commission
under the EU Horizon 2020 MSCA-RISE-2019 programme (project 873028
HYDROTRONICS) and from the Leverhulme Trust under the grant RPG-2019-363. N.K.
acknowledges support from a Humboldt fellowship. The Flatiron Institute is a
division of the Simons Foundation. We thank Lucy Reading-Ikkanda (Simons
Foundation) for help with figure preparation.

Author contributions
M.B., K.-J.T. and N.K. conceived the project. X.Y. carried out the experiments
and analyzed the data. N.K. developed the theoretical model and wrote the
paper. All authors discussed the results and commented on the manuscript.

Competing interests
The authors declare no competing interests.

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Data availability
The experimental data supporting the findings is available on Zenodo. DOI:
10.5281/zenodo.7738429


# Empirical Analysis of Indirect Internal Conversions in Cryptocurrency
Exchanges

Paz Grimberg, Tobias Lauinger, Damon McCoy
New York University

##  I Abstract

Algorithmic trading is well studied in traditional financial markets. However,
it has received less attention in centralized cryptocurrency exchanges. The
Commodity Futures Trading Commission (CFTC) attributed the $2010$ flash crash,
one of the most turbulent periods in the history of financial markets that saw
the Dow Jones Industrial Average lose $9\%$ of its value within minutes, to
automated order “spoofing” algorithms. In this paper, we build a set of
methodologies to characterize and empirically measure different algorithmic
trading strategies in Binance, a large centralized cryptocurrency exchange,
using a complete data set of historical trades. We find that a sub-strategy of
triangular arbitrage is widespread, where bots convert between two coins
through an intermediary coin, and obtain a favorable exchange rate compared to
the direct one. We measure the profitability of this strategy, characterize
its risks, and outline two strategies that algorithmic trading bots use to
mitigate their losses. We find that this strategy yields an exchange ratio
that is $0.144\%$, or $14.4$ basis points (bps) better than the direct
exchange ratio. $2.71\%$ of all trades on Binance are attributable to this
strategy.

##  II Introduction

Cryptocurrency exchanges today handle more than $50b in daily trade volume
[5]. Most of it occurs on centralized exchanges, which hold assets and settle
trades on behalf of their customers. Traders on these exchanges can convert
different coins at certain exchange ratios, just like traditional foreign
exchange (FX) markets. The ability to convert between coins creates potential
arbitrage opportunities, where traders can make a profit through a series of
conversions. The case involving three conversions, coin $c_{1}$ converted to
coin $c_{2}$, which is then converted to coin $c_{3}$ and back to $c_{1}$, is
called triangular arbitrage if the proceeds of the conversions are greater
than the initial quantity. The existence of triangular arbitrage in foreign
exchange markets is well documented [14] [19] [18] [26]. The characteristics
of cryptocurrency exchanges and their relationship to traditional foreign
exchange markets have been studied as well [17]. However, to the best of our
knowledge, triangular arbitrage has never been studied within the context of a
centralized cryptocurrency exchange.

In this paper, we measure arbitrage activity in Binance, a centralized
cryptocurrency exchanges, by empirically exploring its complete historical
trade data. Since different traders cannot be identified from trade data, we
cluster sequences of consecutive trades that match in their quantities and
timing and attribute them to the same trader. To the best of our knowledge, we
are the firsts to employ a clustering methodology for identifying triangular
arbitrage traders, based on trade-by-trade data.

We find that triangular arbitrage is rarely accomplished. Participants
predominantly engage in an alternative strategy, which we call indirect
internal conversions, where coin $A$ is converted to coin $B$ through an
intermediary coin $x$, at a favorable exchange ratio compared to directly
converting $A$ to $B$. This activity accounts for $2.71$% of the total daily
volume, and offers an exchange ratio that is $14.4$ bps better on average.

We believe that the fee structure in cryptocurrency exchanges makes it
unprofitable for participants to engage in triangular arbitrage. Instead,
participants turn to indirect conversions as an efficient way to rebalance
their holdings.

##  III Background

###  III-A Exchanges

An exchange is an organized market where tradable securities, commodities,
foreign exchange/cryptocurrencies (or “coins”) and derivatives are sold and
bought (collectively referred to as instruments). In a centralized exchange,
the deposits and assets of participants are held and settled by the exchange.
In decentralized exchanges (or “DEXes”), a smart contract (a program executing
on a blockchain) or other form of peer-to-peer network executes exchange
functionality. In DEXes, funds cannot be stolen by the exchange operator,
because their custody and exchange logic is processed and guaranteed by the
smart contract.

In centralized cryptocurrency exchanges, different cryptocurrencies can be
exchanged to others, such as Bitcoin and Ethereum. In addition, some exchanges
list ERC20 tokens, or simply “tokens,” that can also be exchanged to
cryptocurrencies. Tokens are essentially smart contracts that make use of the
Ethereum blockchain [13].

Participants can place orders on an exchange. An order is an instruction to
buy or sell some traded instrument. These instructions can be simple or
complicated, and can be sent to either a broker or directly to an exchange via
direct market access. There are some standard instructions for such orders.
For example, a market order is a buy or sell order to be executed immediately
at the current market prices, i.e., buy at the lowest asking price or sell to
the highest bidding price. Market orders are typically used when certainty of
execution is a priority over the price of execution. A limit order is an order
to buy an instrument at no more than a specific price, or to sell at no less
than a specific price (called “or better” for either direction). This gives
the trader control over the price at which the trade is executed; however, the
order may never be executed (“filled”). Limit orders are typically used when
the trader wishes to control price rather than certainty of execution.

Each instrument traded on an exchange has an order book. The order book refers
to an electronic list of buy and sell orders for a specific security or
financial instrument organized by price level. An order book lists the number
of shares being bid on or offered at each price point, or market depth [7].

###  III-B Arbitrage

The traditional financial industry has settled on three main types of
quantitative strategies that are sustainable because they provide real
economic value to the market: arbitrage, market-making and market-taking [8].
In market-taking strategies, traders post both buy and sell orders in the same
financial instrument, hoping to make a profit on the bid-ask spread [25]. In
market-taking strategies, traders engage in longer-term trading, subject to
some rules-based investment methodology. Market-taking strategies on
centralized cryptocurrency exchanges have been studied in [23].

Arbitrage and its economic benefits have been well understood for quite some
time and documented by academia [27] [15]. Competitive arbitrageurs on
centralized exchanges have at least one of three advantages.

  1. 1.

Scale: participants who trade large volumes are often compensated in the form
of kickbacks, rebates, and low (or zero) trading fees, which provide such
participants the opportunity to make profits in cases where others with less
scale cannot.

  2. 2.

Speed: the ability to access order book information and trade faster than
others provides an opportunity to gain from mispricings. For example,
triangular arbitrage on FX products is a major impetus for the Go West and
Hibernia microwave telecommunications projects [9] [1], where multi-million
dollar network infrastructure was developed for the purpose of shaving off
milliseconds in electronic trading latency.

  3. 3.

Queue position: being able to enter one leg of an arbitrage trade by placing
orders ahead of time, without crossing the spreads, i.e., placing orders that
execute immediately without being queued in the order book, significantly
reduces fees, which enables profitable completion of triangular arbitrage
trades. Participants are compensated for the risk of queuing orders in the
order book.

Arbitrage strategies involving multiple centralized cryptocurrency exchanges,
exploiting different prices of same assets, have been studied [22] [21] [24]
[20].

Arbitrage strategies on decentralized exchanges are executed by bots who pay
high transaction fees and optimize their network latency to front-run ordinary
users’ trades [16].

At time of writing, the vast majority of cryptocurrencies trading volume (over
$99\%$) is done in centralized exchanges [5], therefore in this paper, we
focus on triangular arbitrage in a single centralized cryptocurrency exchange
(Binance). Triangular arbitrage is an arbitrage strategy resulting from a
discrepancy between three coins that occurs when their exchange rates give
rise to a profitable sequence of trades, i.e., trades of the form
$c_{1}\mapsto c_{2}\mapsto c_{3}\mapsto c_{1}$, where $c_{1},c_{2},c_{3}$ are
coins, and the ending balance of $c_{1}$ is greater than the initial balance.
We call $c_{1}$ the base coin, $c_{2}$ and $c_{3}$ intermediary coins, and the
sequence a cycle [10].

To the best of our knowledge, we are the firsts to employ a clustering
methodology for identifying triangular arbitrage traders.

##  IV Dataset

We use historical cryptocurrency trade data and order book data provided by
Kaiko from Nov $9$th, $2017$ through Aug $6$th, $2019$ [6]. We conducted our
measurements on Binance’s exchange data, as it is the largest centralized
exchange by daily volume and has the highest number of listed symbols (the
“ticker” of an exchange pair). Kaiko’s data consists of trade-by-trade data
and market depth data for all cryptocurrencies and ERC20 tokens traded on
Binance. There are $1,964,461,269$ trades. Figure 1 shows the monthly number
of trades executed on Binance.

Figure 1: Number of monthly trades, in millions, executed on Binance, based on
Kaiko’s data set.

Every coin in Binance is denominated by at least one of the following coins:
BNB (Binance’s native token), Bitcoin, ALTS (Etheruem, Ripple, Tron) and
stable coins. Stable coins are fiat-pegged cryptocurrencies designed to
minimize the volatility of price fluctuations. We call these coins anchor
coins, as these coins can be directly exchanged to many other coins and have
historically had the greatest market value (in US dollar terms) [3].

There are $204$ different coins and $704$ different possible conversions. We
denote such conversions by $c_{1}\Leftrightarrow c_{2}$, where $c_{1}$ and
$c_{2}$ are two different coins. We call $c_{1}\Leftrightarrow c_{2}$, a pair.
For example, if $c_{1}=\text{BTC}$ and $c_{2}$ = ETH, then the pair is denoted
$\text{BTC}\Leftrightarrow\text{ETH}$. Traders can exchange their BTC to ETH
(or vice versa) using this conversion. We write $c_{1}\mapsto c_{2}$ when
considering the side of the $c_{1}$ holder.

There are 6,534 possible cycles in Binance, i.e., sequences of the form
$c_{1}\mapsto c_{2}\mapsto c_{3}\mapsto c_{1}$. In $3,900$ of them, one of
$c_{2}$ and $c_{3}$ is an anchor. In $2,634$ of them, both $c_{2}$ and $c_{3}$
are anchors. There are no cases where both $c_{2}$ and $c_{3}$ are non-
anchors. $c_{1}$ can be anchor or non-anchor.

Cycle statistics in Binance are summarized in Table I. There are $1,982$
cycles with a non-anchor coin as the base coin. These cycles have the
potential to create arbitrage gains in non-anchor coins. However, anchor coins
represent over $92\%$ of the total market value of all listed coins and
historically had lower volatility, compared to non-anchor coins. Therefore, we
focus on cycles with anchor base coins. As future work, we could explore
cycles with no-anchor base coins. However, we note that there is inherent risk
in operating in non-anchor coins, due to volatility.

Base coin | Number of cycles
---|---
BTC | 986
BNB | 818
ALTS | 908
Stable coins | 1,840
Other coins | 1,982
Total | 6,534
Table I: Number of cycles in Binance by base coin. BTC,BNB,ALTS and stable
coins represent over $92\%$ of the total market value of all listed coins and
historically had lower volatility. Therefore, we omit other cycles in this
study.

###  IV-A Data Fields

Kaiko’s historical trade data fields are described in Table II. The
granularity of the date field is in ms. Since multiple trades can execute
within the same ms, we use the monotonically increasing trade ids as a way to
order the trades.

Field | Description
---|---
id | Incremental trade id
exchange | Exchange id
symbol | Pair symbol
date | Time (in epoch) when the trade took place
price | Trade price
amount | Trade quantity (quoted in base asset)
sell |  | Binary. If TRUE, then the
---
holder of $c_{1}$, swapping to $c_{2}$,
was a liquidity taker, and $c_{1}$ is the
base asset in the pair
listed on the exchange
Table II: Kaiko’s trade data fields.

Kaiko’s historical order book data is given on a 60-second basis, that is, for
every pair listed, its entire order book is given in 60 second increments of
the date field. Note that the quotes’ depth are not given and need to be
reconstructed separately. The data fields are described in Table III.

Field | Description
---|---
symbol | Pair symbol
date | Time (in epoch) when the order was placed
type | Bid or ask
amount | Order quantity (quoted in base asset)
Table III: Kaiko’s order book data fields.

###  IV-B Data Limitations

Kaiko’s order book data is given on a $60$-second interval basis and Binance’s
API does not provide historical order-book data [4]. Complete order book data
could reveal all arbitrage opportunities, not just the ones executed with
actual trades, as it would accurately paint the traders’ view of the market at
any point in time. When merging Kaiko’s historical trade data with its order
book, only about $5\%$ of the trade data has matching order book data
available within a $100ms$ interval. (During our measurements, we used smaller
time intervals than $100ms$.) For profitability measurements, about $0.5\%$ of
the time, we were able to use the order book’s bid/ask directly. For the
remainder, we approximated the bid/ask price with the volume-weighted average
price of that time interval, based on the trade data. In addition, Kaiko does
not include trade fees or historical fee schedules, so we had to reconstruct
Binance’s historical fee time series.

###  IV-C Binance Trading Fees

An important data point for our analysis are the trading fees collected by
Binance [2]. For every trade executed, Binance collects a fee from the
proceeds of the trade. The fees are collected either in the currency of the
proceeds or in Binance’s native ERC20 token, BNB, depending on the trader’s
BNB balance. If the trader has enough BNB balance, Binance will calculate the
conversion rate between the proceeds and BNB, and withdraw the fees from the
BNB balance. In general, Binance rewards participants who trade large volumes
by charging them a lower fee. In addition, for high-volume traders, Binance
distinguishes between liquidity-taking trades, i.e., market order trades or
limit order trades that were matched immediately at the best bid/ask level,
and market-making trades, which are trades that were placed somewhere in the
order book but were not executed immediately. Binance charges less fees for
market-making trades, as they wish to encourage participants to “fill up” the
order book, thereby narrowing the bid/ask spread and increasing liqudity. This
is a common practice; however, in traditional financial exchanges, market-
makers pay zero or even negative fees (rebates, kickbacks).

We assume that arbitrageurs are operating at the lowest fee level. To track
Binance’s fee schedule, we used the Wayback Machine [12], a digital archive of
the World Wide Web, to view Binance’s fee web page historically. In our
analysis time span, Binance’s fee web page changed $46$ times. However, the
lowest fee level remained constant at $1.2$bps for makers and $2.4$bps for
takers.

##  V Methodology

In this section, we describe our methodology for discovering triangular
arbitrage trade sequences in Binance by examining the trade data. During our
analysis, we discovered that a sub-strategy of triangular arbitrage, involving
only the first two conversions, is widely deployed. We refined our original
methodology to identify such conversions. These conversions exhibit some
risks, one of them is the scenario where multiple traders compete for the same
intermediary coin, potentially harming laggards’ profitability. We identify
these risks and outline a methodology for clustering competing events. Lastly,
we observe different strategies traders take to mitigate this risk and
describe a methodology to detect it.

###  V-A Discovering Triangular Arbitrage Trading Sequences

To discern triangular arbitrage trading sequences, we design a methodology to
identify likely arbitrage trading sequences. We look for trading sequences of
the form $c_{1}\mapsto c_{2}\mapsto c_{3}\mapsto c_{1}$, where $c_{1}$ is an
anchor coin (BTC, BNB, ALTS, Stable coins). Arbitrageurs start with some
quantity $Q_{1}$ of $c_{1}$, exchange it to $c_{2}$ at price $p_{12}$,
resulting in $Q_{2}$ units of $c_{2}$. In practice, Binance deducts the fee
upon conversion, therefore $Q_{2}$ will be slightly lower than the conversion
price. Next, $Q_{2}$ units of $c_{2}$ are converted to $c_{3}$, resulting in
$Q_{3}$ units, minus fees. Lastly, $Q_{3}$ units of $c_{3}$ are converted back
to $c_{1}$, minus fees, resulting in $Q_{1}^{\prime}$ units of $c_{1}$. If
$Q_{1}^{\prime}>Q_{1}$, then the arbitrage sequence is profitable.

To successfully profit from this opportunity, arbitrageurs need to execute $3$
trades, $c_{1}\mapsto c_{2}$, $c_{2}\mapsto c_{3}$ and $c_{3}\mapsto c_{1}$.
First, an arbitrageur needs to calculate the initial quantity $Q_{1}$ of
$c_{1}$, such that during conversions, fee payments and other trading
constraints will leave no or minimal residue in the intermediary coins.
Second, since Binance does not support batch trading, i.e., grouping multiple
orders in a single request, arbitrageurs need to ensure correct timing of
their trades. For example, if the order $c_{2}\mapsto c_{3}$ arrives before
$c_{1}\mapsto c_{2}$, then it will fail as the arbitrageur does not hold
$c_{2}$ yet. Furthermore, the arbitrageur competes with other arbitrageurs for
these trades, so speed is an important factor.

In order to identify triangular arbitrage sequences, we first need to ensure
that the same quantity is flowing through different coins, ending at the base
coin. Binance quotes quantities based on how the pair is listed. Different
pairs have different quantity/price restrictions, namely, minimum, maximum and
increment size. Since quantities and prices change in discrete steps, a small
quantity might not be converted, leaving a residue. These small residues need
to be accounted for when identifying trades with equal quantities. While these
residues have a (small) value, in practice, they cannot be converted
immediately to anchor coins, due to minimum size restrictions. As residue
accumulates beyond the exchange threshold, arbitrageurs can convert it back to
anchor coins. We found that less than $0.01\%$ of the time, the profitability
was decided by the residue. Therefore, in our detection process we ignore the
residue value. To illustrate coin exchanges, we follow an actual trade example
from Binance.

####  V-A1 BTC$\Leftrightarrow$ETH Exchange Example

Consider the conversion between BTC and ETH. It is listed as ETH/BTC in
Binance. ETH is called base asset and BTC is called the quote asset. The
quantity is always given in base asset terms. This pair has a $0.001$ minimum
quantity (base asset), $100,000$ maximum quantity, $0.001$ quantity
increments, $0.000001$ minimum price (quote asset), $100,000$ maximum price
and $10^{-6}$ price increments. At the time of writing, the last trade of
ETH/BTC was executed at a price of $0.018876$ and quantity $0.153$. The “sell”
flag was on, meaning that the holder of ETH was the one to initiate the trade
and met the buyer at the buyer’s bid price. Assume both participants are at
the cheapest fee level, currently at $2.4$bps for takers and $1.2$bps for
makers.

From the ETH holder’s perspective: Since ETH is the base asset and the “sell”
flag is on, this means the ETH holder initiated the trade and met the buyer at
the buyer’s bid price, therefore paying a liquidity taking fee of $2.4$bps.
The holder exchanged $0.153$ units of his ETH at a price of $0.018876$ BTC per
$1$ ETH. This means the holder ended with $0.153\cdot 0.018876\cdot
0.99976=0.002887334873$ units of BTC. Note that if $0.002887334873$ BTC are to
be exchanged for some other coin, only $0.00288733$ could be converted,
leaving $0.000000004873$ BTC residue.

From the BTC holder’s perspective: Since BTC is the quote asset and the “sell”
flag is on, this means the BTC holder provided the liquidity for the trade and
his price was met by the seller, thus paying a liquidity making fee of
$1.2$bps. The BTC holder exchanged $0.153\cdot 0.018876=0.002888028$ units of
BTC at a price of $0.018876$ BTC per $1$ ETH, while paying $1.2$bps fee,
resulting in $(0.002888028/0.018876)\cdot 0.99988=0.15298164$ ETH.

From Binance’s perspective: They collect $2.4$bps of the BTC proceeds from the
ETH holder, i.e., $0.00024\cdot 0.002888028\approx 6.93\cdot 10^{-7}$ units of
BTC. If the ETH holder also holds BNB, then this amount is actually converted
to BNB terms (using an average of recent exchange ratios between BTC and BNB),
and deducted from the BNB balance. If the ETH holder does not own BNB, then
$6.93\cdot 10^{-7}$ BTC are deducted from the proceeds. In addition, Binance
also collect $1.2$bps of the ETH proceeds from the BTC holder, i.e.,
$0.153\cdot 0.00012\approx 18.36\cdot 10^{-6}$ units of ETH. If the BTC holder
also holds BNB, this amount is collected in BNB terms as well. Note that if
the seller/buyer did not hold BNB, the fee would have been higher. In our
analysis, we assumed arbitrageurs operate at the lowest fee level.

####  V-A2 Identifying Equal Quantities in Triangular Arbitrage Sequences

We identify sequences of trades, $c_{1}\mapsto c_{2}\mapsto c_{3}\mapsto
c_{1}$, where the same quantity is passed. The quantity is quoted in the base
asset and depends on whether $c_{1}/c_{2}$ or $c_{2}/c_{1}$ is listed, whether
$c_{2}/c_{3}$ or $c_{3}/c_{2}$ is listed and whether $c_{1}/c_{3}$ or
$c_{3}/c_{1}$ is listed. When matching quantities between trades of two pairs,
having a common coin, $c_{1}\mapsto c_{2}$ and $c_{2}\mapsto c_{3}$, we
translate the quantities to the common coin’s terms, $c_{2}$, and check if
they are equal, up to fees and residue resulting from the trade. In Table IV,
we describe the translation process, based on different listing scenarios in
Binance.

Binance listing | Matching condition
---|---
$c_{2}/c_{3}$, $c_{1}/c_{2}$ | $q_{12}p_{12}-q_{23}=\text{fee + residue}$
$c_{2}/c_{3}$, $c_{2}/c_{1}$ | $q_{12}-q_{23}=\text{fee + residue}$
$c_{3}/c_{2}$, $c_{1}/c_{2}$ | $q_{12}p_{12}-q_{23}p_{23}=\text{fee + residue}$
$c_{3}/c_{2}$, $c_{2}/c_{1}$ | $q_{12}-q_{23}p_{23}=\text{fee + residue}$
Table IV: Matching equal quantities in conversions with common coin. $q_{ij}$
and $p_{ij}$ are the quantities and prices quoted in the trade data.

####  V-A3 Identifying Trade Latency in Triangular Arbitrage Sequences

When a triangular arbitrage opportunity presents itself, arbitrageurs compete
amongst each other to be the first to execute the trade sequences. There are
three trades to be executed, $c_{1}\mapsto c_{2}$, $c_{2}\mapsto c_{3}$ and
$c_{3}\mapsto c_{1}$. Since Binance does not support batch trading, an
arbitrageur will have to send three limit orders with prices
$p_{12},p_{23},p_{31}$ and quantities $q_{12},q_{23},q_{31}$. These quantities
are equal in the conversion process, in the sense explained previously and the
prices match the prices quoted in the order book. To ensure that trades are
received in their original order, the arbitrageur will take into account
network latency and wait a small amount of time between sending consecutive
orders. Analysis of the trade data suggests that the average latency between
consecutive trades is $20$ ms with a standard deviation of $15$ ms. Around
$95\%$ of trades with matching quantities are executed within $10$ms-$30$ms of
each other. Therefore, clustering trades using longer latency does not have
material impact on the number of trades identified. We find that using $\Delta
t\approx 50$ms as an approximation for arbitrage latency between consecutive
trades maximizes precision and minimizes recall.

We found that less than $0.01\%$ of triangular arbitrage sequences contain a
liquidity-making trade. We believe this behavior is caused by Binance’s fee
model, which charges traders a commission even for liquidity making trades. If
traders used liquidity-making orders, they would need to pay the fee in case
they were filled. At that point, it is not guaranteed that an arbitrage
opportunity will exist, while the fee is already paid and exposure to what is
mostly an illiquid coin is already established. To further enhance our
precision, we require arbitrage trade sequences to be all liquidity-taking.

Furthermore, we found a surprisingly low number of triangular arbitrage
trades, 1,381,928 in total, accounting for $0.24\%$ of total number of trades.
However, in the course of clustering triangular arbitrage trades, we witnessed
a much larger number of partial arbitrage sequences, 20,892,825, or $2.71\%$
of total trades, where traders executed the first two trades, $c_{1}\mapsto
c_{2}$ and $c_{2}\mapsto c_{3}$, but did not execute the third trade,
$c_{3}\mapsto c_{1}$. Executing the first two trades effectively converts
$c_{1}$ to $c_{3}$ at an exchange ratio of $p_{12}\cdot p_{23}$, minus fees
and residue. Interestingly, $95\%$ of the time these trades resulted in an
exchange ratio that was favorable to the exchange ratio of the direct trade
$c_{1}\mapsto c_{3}$. We call this trading strategy indirect internal
conversions and explain below in more details what is means to have a
“favorable” rate.

We refine our methodology to identify indirect internal conversions and study
the root cause of unprofitable instances. We elaborate on the risks associated
with this trading strategy in the discussion section.

###  V-B Discovering Indirect Internal Conversion Attempts

We refined our original methodology for discovering triangular arbitrage
sequences by relaxing the constraint for the third trade to be executed. We
identify equal quantities in the first two trades in the same way as before.
To determine the trade latency, we empirically explored different time
constraints. $93\%$ of trades with equal quantities have a latency between
$10$ms and $100$ms. Latency lower than $10$ms gives poor recall, with less
than $5\%$ of total attempts. Latency greater than $100$ms only accounts for
$2\%$ of all attempts. The average latency is $22$ms with a standard deviation
of $15$ms. We find that, as before, $\Delta t\approx 50$ms is an approximation
that likely provides both high precision and recall for identifying indirect
conversion trading sequences. However, we lack ground truth to definitively
evaluate this part of our methodology.

####  V-B1 Determining Profitability of Internal Conversions

The first two trades of a triangular arbitrage sequence, $c_{1}\mapsto
c_{2}\mapsto c_{3}\mapsto c_{1}$ are $c_{1}\mapsto c_{2}$ and $c_{2}\mapsto
c_{3}$. Executing these two trades gives an indirect exchange ratio between
$c_{1}$ and $c_{3}$. If this exchange ratio, net of fees and residue, is
greater than the exchange ratio of $c_{1}\mapsto c_{3}$, then this conversion
offers a favorable exchange ratio to the direct one. We call such favorable
conversions indirect internal conversions. Since the exchange ratio between
$c_{1}$ and $c_{3}$ fluctuates, it is important to define the time it is
taken. We wish to approximate the trader’s view of the market upon executing
the internal conversion. Therefore, we take the exchange ratio at a short
period of time prior to the first trade’s execution time. As discussed above,
$50$ms is a good delay approximation for the trader’s view. Kaiko’s order book
data is given on a $60$-second basis, so it is unlikely that this $50$ms time
window falls within Kaiko’s data intervals. To approximate the order book’s
best bid/ask price at the time, we take the volume weighted average price
(VWAP) [11] of $c_{1}\mapsto c_{3}$, over the period of $50$ms, prior to the
execution time of the first trade. These trades tell us the best bid/ask level
at that time, and taking an average, weighted by volume, is a good
approximation of the trader’s view of the order book. However, it is possible
that within that time period, other participants posted bids and asks and then
cancelled them, giving the trader a completely different view of $c_{1}\mapsto
c_{3}$. In our analysis, we assume that this did not occur. If there were no
trades during that period and order book data is unavailable, we cannot
determine if the conversion is profitable and did not include such conversions
in our profitability analysis.

We found $26,603,038$ indirect conversions, which is $2.71\%$ of the total
number of trades. $95\%$ of the indirect conversions resulted in a favorable
exchange ratio. We hypothesized that unprofitable conversions occur when
multiple traders obtain the same intermediary coin $c_{2}$, and simultaneously
attempt to convert it to $c_{3}$. For example, one trader is looking to
convert $c_{1}\mapsto c_{2}\mapsto c_{3}$ for a favorable exchange ratio to
$c_{1}\mapsto c_{3}$, and a second trader is looking to convert $y\mapsto
c_{2}\mapsto c_{3}$ at a favorable exchange ratio to $y\mapsto c_{3}$ ($y$
could be the same coin as $c_{1}$). When both traders obtain $c_{2}$, there
could be a scenario where only one trader is able to convert $c_{2}$ to
$c_{3}$ at the best bid/ask level. This happens when the first one to execute
$c_{2}\mapsto c_{3}$ consumes the entire quantity of the best bid/ask and
causes the order book to change. The laggard in this case engages in a loss
mitigating strategy. One option is to convert $c_{2}$ to $c_{3}$ at a worse
exchange ratio than originally intended, potentially resulting in a conversion
that is worse than the direct one. Another option is to convert $c_{2}$ to
several other coins, with the goal of minimizing the potential losses from
unfavorable exchange ratios. We call the former strategy a full-exit strategy
and the latter partial-exit strategy.

To corroborate our hypothesis with the trade data, we refine our methodology
to cluster competing conversions and identify loss-mitigating exit strategies.

###  V-C Clustering Competing Indirect Internal Conversion Attempts

Using our methodology to discover indirect conversions, we identified
conversions that were initiated around the same time and had an overlapping
intermediary coin. This is because competing conversions try to complete the
same second trade. Therefore, for a given second trade $c_{2}\mapsto c_{3}$,
we look at indirect conversions attempts of the form $x\mapsto c_{2}\mapsto
c_{3}$ that started within $100$ms of each other.

Every cluster of competing conversions can have winning conversions, i.e.,
indirect conversions that were completed at a favorable rate to the direct
rate and losing conversions, conversions that completed at an unfavorable rate
to the direct rate.

Losing conversions can have many causes, such as mistiming or an inaccurate
view of the order book. We believe that one of those reasons is that traders
who successfully completed the first trade, and failed to complete the second
trade, are unloading their $c_{2}$ coin at an unfavorable rate to avoid having
directional exposure to $c_{2}$.

####  V-C1 Identifying Loss-Mitigating Trading Strategies

For losing conversions, we wanted to see if the loss was the result of a
competing winning conversion that utilized all the capacity of $c_{2}\mapsto
c_{3}$. Determining whether a conversion utilized all capacity is impossible
without knowing the order book at that time. However, we can approximate it by
observing the next trade of $c_{2}\mapsto c_{3}$. Since we have the trade id,
which is monotonically increasing with the trades, we can tell whether the
next trade of $c_{2}\mapsto c_{3}$ was completed at the same price or not. If
it was not completed at the same price, it is likely that the previous trade
used up the capacity of the previous level. This is only a heuristic, as
traders can post and cancel orders at any time.

Loss-mitigating conversions are losing conversions where a winning conversion
in the same cluster used up the intermediary’s coin capacity.

By analyzing the trade data, we found that $17.7\%$ of all losing conversions
corresponded to loss-mitigating traders.

We identified two sub-strategies of loss-mitigating traders.

####  V-C2 Full-Exit Loss Mitigating Strategy

These are conversions that converted an equal quantity $Q_{1}$ from $x\mapsto
c_{2}$ and $c_{2}\mapsto c_{3}$, i.e., these are traders who converted all
their $c_{2}$ into one coin.

####  V-C3 Partial-Exit Loss Mitigating Strategy

These are conversions that converted a quantity $Q_{1}$ from $x\mapsto c_{2}$
at price $p_{12}$, but converted a lower quantity $Q_{2}<Q_{1}$ from
$c_{2}\mapsto c_{3}$, i.e., these are traders who unloaded some, but not all
of their $c_{2}$ into one coin. This strategy can be attributed to traders
solving the following minimization problem: Find a set of $k$ coins
$\\{d_{1},d_{2},\ldots d_{k}\\}$ having exchange ratios $\\{p_{1},p_{2},\ldots
p_{k}\\}$, and a set of $k$ quantities $\\{q_{1},q_{2},\ldots q_{k}\\}$, where
$\sum_{j=1}^{k}q_{j}=Q_{1}$ such that the following loss function is
minimized:

$\displaystyle
Loss=\sum_{j=1}^{k}q_{j}\left(\frac{p_{13}}{p_{12}p_{j}}\right)$

where $p_{13}$ is the direct exchange ratio of $x\mapsto c_{3}$. To detect
partial exits, we iterate over all different combinations of trades from
$c_{2}\mapsto c_{3}$ having quantities less than $Q_{1}$, such that they sum
exactly to $Q_{1}$, up to fees and residue.

We found that $85\%$ of loss-mitigating strategies are partial exits and
$15\%$ are full exits. It also turns out that solving a loss minimization
problems is effective, as the average partial-exit loss is $21$bps while the
full-exit average loss is $25$bps.

##  VI Results

###  VI-A Volume

We found $26,603,038$ indirect conversions, which is $2.71\%$ of total trades.
We found $1,381,928$ triangular arbitrage sequences, accounting for $0.24\%$
of total trades.

The time series of the number of favorable indirect conversion attempts as a
percentage of number of direct conversions, on a daily basis, is shown in
Figure 2. The number of favorable indirect conversion attempts, on a daily
basis, in shown in Figure 3. The number of triangular arbitrage attempts, on a
daily basis, in shown in Figure 4.

Figure 2: Percentage of daily volume of direct conversions trades that is also
done indirectly. Typically, $1\%-3\%$ of the daily volume of a direct
conversion pair is also executed via indirect conversion sequences. Figure 3:
Daily number of indirect conversion attempts. From October $2017$ \- October
$2018$, the number of indirect conversions has been trending down and since
October $2018$ has been trending up. Figure 4: Daily number of triangular
arbitrage attempts. From October $2017$ \- April $2019$, the number of
triangular arbitrage attempts has been trending down and since April $2019$
has been trending up, with a sharp spike in July $2019$

###  VI-B Latency

We found that indirect conversion sequences have, on average, $21.43$ms of
delay between the first trade and the second, with a standard deviation of
$11.04$ms. $78.94\%$ have a latency of $30$ms or less. Triangular arbitrage
sequences are faster than indirect conversions. This could be an indication
that this space is more competitive. We found that triangular arbitrage
sequences have, on average, $20.7$ms of delay between consecutive trades, with
a standard deviation of $11.04$ms. $85.43\%$ have a latency of $30$ms or less.
The latency statistics of triangular arbitrage sequences and indirect
conversions are shown in Table V.

Type | Metric | Value
---|---|---
Triangular Arbitrage | Latency Average |  $20.7$ms
Triangular Arbitrage | Latency Stdev |  $11.04$ms
Triangular Arbitrage | % Below $30$ms | $85.43\%$
Indirect Conversion | Latency Average |  $21.43$ms
Indirect Conversion | Latency Stdev |  $13.7$ms
Indirect Conversion | % Below $30$ms | $78.94\%$
Table V: Latency statistics for triangular arbitrage and indirect conversions.
The triangular arbitrage strategy exhibits lower latency than indirect
conversions, possibly a sign that this strategy is more competitive

The latency distribution, in ms, of indirect conversions and triangular
arbitrage is shown in Figure 5.

Figure 5: Left: Latency distribution, in ms, of indirect conversion trade
sequences. Right: Latency distribution, in ms, of triangular arbitrage trade
sequences. Indirect conversion trades exhibit higher latency with a fatter
tail, indicating that triangular arbitrage is more competitive.

###  VI-C Profitability

We calculate the equal-weighted return and the return on capital for
triangular arbitrage trades and indirect conversions. The equal-weighted
return is the average of returns. The return on capital is the arithmetic
average of returns, weighted by quantity of base coin committed to the trade,
i.e., larger trades receive higher weight.

Indirect conversion attempts were profitable $93.92\%$ of the time with an
equal-weighted net return of $11.8$bps. The return on capital for indirect
conversions is $22\%$ greater, at $14.4$bps, which means that traders
efficiently commit more coins to more profitable opportunities.

Triangular arbitrage trades were profitable $94.97\%$ of the time with an
equal-weighted net return of $9.3$bps. The return on capital for triangular
arbitrage sequences is $5\%$ greater, at $9.8$bps, which is slightly higher
than the equal-weighted return, but not significantly.

It appears that triangular arbitrage traders are unable to significantly
allocate more coins to more profitable opportunities. One potential
explanation is that the space is more crowded, or “arb-ed out,” than the
indirect conversions strategy.

###  VI-D Loss-Mitigating Strategies

We saw $1,617,464$ instances of indirect conversion completed at an
unfavorable rate compared to the direct rate, or $6.08\%$ of the total number
of indirect conversions. $17.7\%$ of such conversions were loss-mitigating
trades, triggered by a competing indirect conversion that used all the
intermediary coin’s capacity. When this happened, there were $1.06$ competing
conversions on average. The highest number of competing conversions we saw in
one cluster was $5$.

In $15\%$ of such cases, the entire base quantity was unloaded into another
coin, i.e., they were full exits. The return on capital for the full exit
strategy was $-27.4$bps. $85\%$ of the time, the base coin quantity was
unloaded to several other coins, i.e., they were partial exits. On average,
the number of coins used to exit was $2.7$, and the return on capital was
$-23.8$bps. Intuitively, this shows that solving a loss minimization problem
mitigates losses by $3.6$bps compared to a full exit. Note, however, that a
full exit could potentially be the solution to the minimization problem as
well.

##  VII Discussion

The existence of triangular arbitrage in traditional foreign exchange markets
is well documented and studied in academia. In centralized cryptocurrency
exchanges such as Binance, triangular arbitrage happens at a small scale.
However, we found that a different strategy is taking place, which converts
two coins through an intermediary coin, at a favorable rate to the direct
conversion rate. This strategy comes with a risk, however, that multiple
competing conversions occur at the same time, preventing slower ones from
completing their conversions. We saw that when this happens, traders engage in
a loss-mitigating strategy. We believe that the rationale for this strategy
stems from Binance’s fee structure, where there is a fee for market-making,
thus creating frictions for the triangular arbitrage strategy. We believe
arbitrageurs adapted to the fee structure by executing the indirect
conversions strategy. While triangular arbitrage increases the arbitrageur’s
holding in the base asset, indirect conversions simply provide a more
favorable ratio to the direct one. Participants who already have a stake in
the cryptocurrency ecosystem and wish to rebalance their portfolio might
choose to engage in this strategy.

##  VIII Conclusion

We found that $0.24\%$ of daily trades on Binance can be attributed to
triangular arbitrage trades. We found that a different strategy is $11$ times
more prevalent, at $2.71\%$ of daily trades, which involves exchanging coins
through an intermediary coin at a favorable rate to the direct exchange rate.
We designed a methodology to measure such events and discovered that $93.92\%$
of the time the rate obtained from these conversions is $14.4$bps better than
the direct one. When it is not, $17.7\%$ of the time traders engage in loss-
mitigating strategies, and we identified two of them.

## Acknowledgements

The authors would like to thank Andrew Papanicolaou and David Yermack for
their feedback and suggestions to improve the quality of our manuscript. We
acknowledge funding support under National Science Foundation award 1844753.

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##  IX Appendix

In this appendix, we define notations that are specific to the market
microstructure of Binance and formally define triangular arbitrage and
indirect conversions.

Let $\mathbb{P}$ be the set of all pairs $x/y$ traded on Binance. Every pair
facilitates two conversions; from $x$ to $y$ and from $y$ to $x$, denoted by
$(x\mapsto y)$ and $(y\mapsto x)$, respectively.

######  Definition IX.1.

(exchange ratio) Let $\psi_{ij}$ be the proceeds of converting 1 unit of coin
$c_{i}$ to $c_{j}$. We call $\psi_{ij}$ the exchange ratio and it is given by

$\displaystyle\psi_{ij}=\begin{cases}\frac{1}{p_{c_{j}/c_{i}}^{ask}}&\text{if
$c_{j}/c_{i}\in\mathbb{P}$}\\\ p_{c_{i}/c_{j}}^{bid}&\text{if
$c_{i}/c_{j}\in\mathbb{P}$}\\\ \end{cases}$

######  Definition IX.2.

(pair capacity) Denote $\eta_{ij}$ the maximum quantity $c_{i}$ that can be
converted to $c_{j}$. We call $\eta_{ij}$ the capacity of $(c_{i}\mapsto
c_{j})$ and it is given by

$\displaystyle\eta_{ij}=\begin{cases}p_{c_{j}/c_{i}}^{ask}q_{c_{j}/c_{i}}^{ask}&\text{if
$c_{j}/c_{i}\in\mathbb{P}$}\\\ q_{c_{i}/c_{j}}^{bid}&\text{if
$c_{i}/c_{j}\in\mathbb{P}$}\\\ \end{cases}$

######  Definition IX.3.

(bid-ask spread) We write

$\displaystyle\Delta_{ij}=\begin{cases}p_{c_{j}/c_{i}}^{ask}-p_{c_{j}/c_{i}}^{bid}&\text{if
$c_{j}/c_{i}\in\mathbb{P}$}\\\
p_{c_{i}/c_{j}}^{ask}-p_{c_{i}/c_{j}}^{bid}&\text{if
$c_{i}/c_{j}\in\mathbb{P}$}\\\ \end{cases}$

the bid ask spread of pair $(c_{i}\mapsto c_{j})$

######  Definition IX.4.

(trade quantity) suppose we wish to convert $q$ units of $c_{i}$ to $c_{j}$.
The trade quantity passed to the exchange as a parameter is denoted $T(q)$ and
is given by

$\displaystyle T_{ij}(q)=\begin{cases}\frac{q}{p_{c_{j}/c_{i}}^{ask}}&\text{if
$c_{j}/c_{i}\in\mathbb{P}$}\\\ q&\text{if $c_{i}/c_{j}\in\mathbb{P}$}\\\
\end{cases}$

######  Definition IX.5.

(minimum trade lot size) Let $m_{ij}$ be the minimum amount of $c_{i}$ that
must be converted to $c_{j}$ in
$\left(\text{$c_{i}$}\mapsto\text{$c_{j}$}\right)$, i.e., when converting $q$
units of $c_{i}$ to $c_{j}$, $m_{ij}\leq q$

######  Definition IX.6.

(high-level parameters) the $h$-th order book exchange ratio and capacity are
denoted $\psi_{ij}^{h}$ and $\eta_{ij}^{h}$, respectively.

######  Definition IX.7.

(minimum price increment) Denote $dx_{ij}$ the the minimum order book price
increments of $(c_{i}\mapsto c_{j})$.

######  Remark.

It holds that $dx_{ij}=dx_{ji}$.

In practice, when converting $q$ units of $c_{i}$ to $c_{j}$, the trade
quantity passed to the exchange as a parameter has to be a multiple of
$dx_{ij}$, therefore only $\left\lfloor\frac{T(q)}{dx_{ij}}\right\rfloor
dx_{ij}$ units are passed. Denote $\left\lfloor
x\right\rfloor_{y}\mathrel{\mathop{:}}=\left\lfloor\frac{x}{y}\right\rfloor y$
the rounding operation of $x$ to the nearest multiple of $y$.

######  Definition IX.8.

(residuals) Let $r_{ij}(q)$ be the quantity of $c_{i}$ left when converting
$q$ units of $c_{i}$ to $c_{j}$. We write

$\displaystyle r_{ij}(q)=\begin{cases}\left(T(q)-\left\lfloor
T(q)\right\rfloor_{dx_{ij}}\right)p_{c_{j}/c_{i}}^{ask}&\text{if
$c_{j}/c_{i}\in\mathbb{P}$}\\\ T(q)-\left\lfloor
T(q)\right\rfloor_{dx_{ij}}&\text{if $c_{i}/c_{j}\in\mathbb{P}$}\\\
\end{cases}$

Therefore, converting $x$ units of $c_{i}$ yields
$\psi_{ij}\left[x-r_{ij}(x)\right]$ units of $c_{j}$ and $r_{ij}(x)$ residual
units of $c_{i}$

###  IX-A Cycle Arbitrage

The set of pairs $C=\\{(c_{1}\mapsto c_{2}),(c_{2}\mapsto
c_{3}),\ldots,(c_{n}\mapsto c_{n+1})\\}$ is called a cycle, if $c_{n+1}=c_{1}$
and $c_{j}\not\in\\{c_{1},\ldots\,c_{j-1}\\}$ for $1<j\leq n$. In the context
of cycles, we use the shortened notation $\psi_{i},\eta_{i},dx_{i},r_{i}$ when
referring to pairs of the form $(c_{i}\mapsto c_{i+1})$. A cycle with $n=3$ is
called a triangular sequence.

######  Definition IX.9.

The capacity of cycle $C=\\{(c_{1}\mapsto c_{2}),\ldots,(c_{n}\mapsto
c_{n+1})\\}$ is the maximum quantity of $c_{1}$ that can be converted through
its pairs.

The capacity of a cycle is given by

$\displaystyle Q=\min_{1\leq i\leq
n}\left\\{\eta_{i}\left(\prod_{k=1}^{i-1}\psi_{k}\right)^{-1}\right\\}$ (1)

The balance of $c_{i}$, denoted $q_{i}$, is given by the recurrence relation

$\displaystyle q_{1}$ $\displaystyle=Q-r_{1}(Q)$ $\displaystyle q_{i+1}$
$\displaystyle=\psi_{i}\left[q_{i}-r_{i}(q_{i})\right],\qquad 1\leq i\leq n$

######  Remark.

$r_{i}\Big{(}x-r_{i}(x)\Big{)}=0$

###  IX-B Binance Fee Structure

The fees paid on $(c_{i}\mapsto c_{i+1})$ are $fq_{i+1}b_{i+1}$ where
$f=5\cdot 10^{-4}$ and $b_{i+1}$ is the last traded price of
$(c_{i+1}\mapsto\text{BNB})$

$\displaystyle b_{i}=\begin{cases}\frac{1}{p_{\text{BNB}/c_{i}}}&\text{if
$\text{BNB}/c_{i}\in\mathbb{P}$}\\\ p_{c_{i}/\text{BNB}}&\text{if
$c_{i}/\text{BNB}\in\mathbb{P}$}\\\ \end{cases}$

The gain/loss in $c_{1}$ terms is given by

$\displaystyle
G=(q_{n+1}-q_{1})+\underbrace{\sum_{i=1}^{n}r_{i}(q_{i})\psi_{i1}}_{\text{residuals
$c_{1}$
value}}-\underbrace{f\sum_{i=1}^{n}q_{i+1}b_{i+1}\psi_{B1}}_{\text{fees
$c_{1}$ value}}$ (2)

Note that $G$ is a function of
$\\{\psi_{21},\ldots,\psi_{n1},\psi_{B1},b_{2},\ldots,b_{n+1}\\}$ as well as
$\psi_{1},\ldots\psi_{n}$.

######  Definition IX.10.

a cycle is called an arbitrage free cycle if $G\leq 0$

######  Definition IX.11.

a cycle is called an open cycle if $G>0$

######  Remark.

One can argue that there are additional costs for converting the residuals to
$c_{1}$ and purchasing BNB tokens ahead of time to be able to pay fees, which
are not accounted for in (2). However, this can be done in infrequent bulk
trades. We make the assumption that small directional exposure to BNB and
residuals is negligible compared to the accumulated gains over a period of
time.

######  Remark 1.

If we assume zero residuals, i.e., $\forall i:r_{i}=0,$ then
$q_{i+1}=Q\prod_{j=1}^{i-1}\psi_{j}$ and (2) reduces to
$G=Q\left(\prod_{i=1}^{n}\psi_{i}-1\right)-f\sum_{i=1}^{n}Q\left(\prod_{j=1}^{i}\psi_{j}\right)b_{i+1}\psi_{B1}$

###  IX-C Indirect Internal Conversions

The set of pairs $V=\\{(c_{1}\mapsto c_{2}),(c_{2}\mapsto
c_{3}),\ldots,(c_{n}\mapsto c_{n+1})\\}$ is called a conversion if
$c_{j}\not\in\\{c_{1},\ldots\,c_{j-1}\\}$ for $1<j\leq n+1$. In the context of
conversions, we use notation $\psi_{i},\eta_{i},dx_{i},r_{i}$ when referring
to pairs of the form $(c_{i}\mapsto c_{i+1})$. When the intermediate pairs are
clear, we use the shortened notation $c_{1}\stackrel{{\scriptstyle
V}}{{\leadsto}}c_{n+1}$

######  Definition IX.12.

The capacity of conversion $V=\\{(c_{1}\mapsto c_{2}),\ldots,(c_{n}\mapsto
c_{n+1})\\}$ is the maximum quantity of $c_{1}$ that can be converted through
its pairs and is defined similarly to the capacity of a cycle.

######  Definition IX.13.

(conversion proceeds) Let $c_{1}\stackrel{{\scriptstyle
V}}{{\leadsto}}c_{n+1}$ be a conversion from $c_{1}$ to $c_{n+1}$ with
capacity $Q$. The conversion proceeds of $q\leq Q$ units of $c_{1}$ is the
quantity of $c_{n+1}$ that results from converting through the pairs of $V$,
accounting for residuals and fees, i.e.,

$\displaystyle
Pr(q,V)=q_{n+1}+\underbrace{\sum_{i=1}^{n}r_{i}(q_{i})\psi_{i(n+1)}}_{\text{residuals
$c_{n+1}$
value}}-\underbrace{f\sum_{i=1}^{n}q_{i+1}b_{i+1}\psi_{B(n+1)}}_{\text{fees
$c_{n+1}$ value}}$

######  Definition IX.14.

(profitable conversions) Let $x\stackrel{{\scriptstyle V_{1}}}{{\leadsto}}y$
and $x\stackrel{{\scriptstyle V_{2}}}{{\leadsto}}y$ be two conversions from
$x$ to $y$, with capacities $Q_{1}$ and $Q_{2}$, respectively. Let
$Q=\min\\{Q_{1},Q_{2}\\}$. We say that $V_{1}$ is a profitable conversion
w.r.t. $V_{2}$ if the proceeds of conversion $V_{1}$ are greater than the
proceeds of conversion $V_{2}$, i.e., $Pr(Q,V_{1})>Pr(Q,V_{2})$

###  IX-D Binance Order Types

Binance supports the following order types:

  1. 1.

LIMIT \- an order to buy/sell a pair at a specified price and can only be
executed at that price (or better). Not guaranteed to execute

  2. 2.

MARKET \- an order to buy/sell a pair at the best current market price, i.e.,
lowest ask or highest bid.

  3. 3.

STOP_LOSS_LIMIT \- an order to buy (sell) a pair, once its price exceeds
(drops below) the specified price. In contrast to a LIMIT order, the price
should be above (below) the lowest ask (highest bid). The execution price is
guaranteed to be the specified price.

  4. 4.

STOP_LOSS \- same as STOP_LOSS_LIMIT, but when the price threshold is
breached, a MARKET order is executed and the execution price is not
guaranteed.

  5. 5.

TAKE_PROFIT_LIMIT \- equivalent to a LIMIT order

  6. 6.

TAKE_PROFIT \- automatically places a MARKET order when the specified price
level is met

  7. 7.

LIMIT_MAKER \- LIMIT orders that will be rejected if they would immediately
match and trade as a taker.

  8. 8.

ICEBERG \- an order used for large quantities as it automatically breaks down
to multiple LIMIT orders with different prices. The goal is to hide the actual
order quantity and prevent trend-followers from unfavorably moving the price.