# Hawkes-based cryptocurrency forecasting via Limit Order Book data

Raffaele Giuseppe Cestari  Filippo Barchi  Riccardo Busetto  Daniele Marazzina
Simone Formentin  Department of Electronics, Information, and Bioengineering,
Politecnico di Milano, Milano, Italy.  Department of Mathematics, Politecnico
di Milano, Italy

###### Abstract

Accurately forecasting the direction of financial returns poses a formidable
challenge, given the inherent unpredictability of financial time series. The
task becomes even more arduous when applied to cryptocurrency returns, given
the chaotic and intricately complex nature of crypto markets. In this study,
we present a novel prediction algorithm using limit order book (LOB) data
rooted in the Hawkes model, a category of point processes. Coupled with a
continuous output error (COE) model, our approach offers a precise forecast of
return signs by leveraging predictions of future financial interactions.
Capitalizing on the non-uniformly sampled structure of the original time
series, our strategy surpasses benchmark models in both prediction accuracy
and cumulative profit when implemented in a trading environment. The efficacy
of our approach is validated through Monte Carlo simulations across 50
scenarios. The research draws on LOB measurements from a centralized
cryptocurrency exchange where the stablecoin Tether is exchanged against the
U.S. dollar.

###### keywords:

financial systems, time series forecasting, machine learning in finance,
cryptocurrencies

##  1 Introduction

Predicting with reliable accuracy the sign of returns in high-frequency
trading is still an open problem when developing automatic algorithms.
Increasing the accuracy of the prediction means significantly increasing the
size of the revenues. In this framework, one main source of information to
achieve this goal is exploiting at best the informative content carried in the
limit order book (LOB). The LOB is an electronic real-time record where market
participants save their buying or selling intentions. Limit orders are
instructions to buy or sell a specific quantity of a security with the
flexibility to specify the maximum price one is willing to pay when buying or
the minimum at which sell. When the bid price $P^{bid}$ exceeds or matches the
ask price $P^{ask}$, a trade takes place between the buyer and the seller. All
not executed orders, whether they are buy orders (bids) or sell orders (asks),
are contained in the LOB. Each record includes prices and quantities specified
by traders. As market conditions fluctuate, orders in the LOB are continuously
updated, reflecting the evolving supply and demand dynamics of the market.
Given this wealth of information, it is recognized in literature as a valuable
asset for attempting to predict a stock’s price, see Cao et al. (2009). There
exists several studies leveraging order book data for stock price prediction.
They encompass a variety of models, from simple logistic regression as in
Zheng et al. (2012) to sophisticated approaches, like convolutional neural
networks in Zhang et al. (2019). Our reference application is Busetto and
Formentin (2023), where the authors explore the correlation between a new
variable defined as base imbalance (BI) and future returns and construct a
continuous output error model (COE) that establishes a connection between BI
and future returns. The study yields promising results when predicting return
sign. However, it relies on the assumption of knowing the timing of the next
event to compute future returns. In this paper, we eliminate this assumption.
The idea is to extract valuable information also from the timing on which the
orders are placed on the LOB. In this way, we can dig out additional resources
which could improve model accuracy and, simultaneously, maintain the original
not uniformly sampled in time data structure. We achieve this by exploiting
point processes models, in particular Hawkes processes. Point and Hawkes
processes find application in several fields such as epidemiology and disease
spread (Garetto et al. (2021)), neuroscience (Reynaud-Bouret et al. (2013)),
earthquakes (Kwon et al. (2023)), social media user interaction (Alvari and
Shakarian (2019)), and finance. One of the pioneers utilizing Hawkes process
to model financial data is Cao et al. (2009). The author employed a Hawkes
process to demonstrate the existence of bidirectional relationships between
trade timing and mid-quote fluctuations. In Cartea et al. (2014), the authors
used a Hawkes process to simulate the arrival of market orders, but they did
not take into account the complete order flow, including limit orders and
cancellations. Sjogren (2022) employs a specific class of Hawkes processes,
demonstrating their usefulness not only as short-term indicators for
forecasting market price movements but also as a reliable resource for longer-
term predictions. Chen (2017) introduce a 4-dimensional Hawkes process to
model the LOB and to forecast mid-price movement probabilities using Monte
Carlo simulations. Rambaldi et al. (2017) uses a multivariate Hawkes process
to analyze the interactions between the orders arrival time and their size.
Morariu-Patrichi and Pakkanen (2022) explores state-dependent Hawkes
processes, a combination of Hawkes processes and Markov chains. This study
applies these processes to high-frequency limit order book data, revealing the
dynamic interplay between order flow and the state of the limit order book,
explaining how these factors mutually influence each other. In this paper, we
integrate a Hawkes-based predictive model, which demonstrates a good level of
accuracy in forecasting the timing of the next price change, with the COE
model described in Busetto and Formentin (2023). To assess the quality of the
proposed architecture, we validate the performances in a real-time high-
frequency trading problem with $50$ samples of $2$ minutes of actual LOB
Tether cryptocurrency data. We assess the accuracy of return sign prediction
using various alternative strategies compared to the Hawkes model for
predicting the next event time. These alternatives include the Oracle (perfect
event-time knowledge), which serves as our ideal reference, and two benchmark
strategies: Naive (next event time is the minimum system resolution of $1$
[s]), and Moving Average (next event time occurs at the average event time
occurrence of the events happened in the past $W=60$ [s]). We show that the
Hawkes-based strategy outperforms the benchmarks and gets close performances
to the ideal Oracle. We provide also a simulation, on the same $50$ validation
scenarios, of the strategies if inserted into a high-frequency trading
paradigm. We show that, also in this case, the Hawkes-based strategy surpasses
the benchmarks guaranteeing a higher average profit. The main paper
contributions are as follows:

  * •

For the first time, as far as we are aware, we develop a return sign
prediction model based on Hawkes processes to estimate the next LOB order time
and we couple the predicted information with a COE model to predict the return
sign. We show that this paradigm outperforms benchmark strategies both in term
of return sign accuracy and profit when used with the same trading framework.

  * •

We apply this prediction strategies on real cryptocurrency data collected in a
LOB dataset.

  * •

We prove that, the Base Imbalance regressor introduced by Busetto and
Formentin (2023) is a suitable input also when considering LOB cryptocurrency
data, which differ from LOB stock market data on which it was developed.

The remainder of the paper is as follows. In Section 2 the mathematical
notation of the variables of interest is shown. In Section 3 we show the
exploratory data analysis of the LOB. In Section 3.1 we show the correlation
analysis between base imbalance and returns. Section 4 shows the mathematical
formulation. In particular, Section 4.1 introduces point processeses and
Section 4.2 the Hawkes ones. Section 4.3 shows the COE model and Section 4.4
shows the entire algorithm workflow. In Section 5 the numerical simulations
and results are shown. In particular, in Section 5.1 we show the return sign
prediction accuracy, and in 5.2 the trading simulation outcomes. The paper
ends with model limitations, conclusions and future work in Section 6.

##  2 Notation

In this section, we briefly introduce the notation used in the rest of the
paper. Table 1 shows the description and symbols of the main variables. The
definitions of mid-price, return, and base imbalance follow, preparatory for
the subsequent discussion.

$k\in N^{+}$: LOB event index |  $\theta_{COE}$: COE parameters
---|---
$K\in N^{+}$: # predicted LOB events |  $t_{0}$: initial time
$t_{k}$: k-th event time |  $T_{\text{train}}^{\text{H}}$: Hawkes training time
$i\in[1,L]$: LOB level index |  $T_{\text{train}}^{\text{COE}}$: COE training time
$L$: LOB levels |  $T_{\text{sim}}$: simulation time
$\lambda(t)$: intensity function |  $T_{\text{warm}}$: Hawkes warm-up time
$\mu$: $\lambda(t)$ baseline |  $\Delta T$: Hawkes forecast window
$\alpha$: $\lambda(t)$ self-excitation rate |  $P$: mid-price
$\beta$: $\lambda(t)$ decay rate |  $R$: return
$\theta=[\mu,\alpha,\beta]$: Hawkes parameters |  $BI$: base imbalance
Table 1: Notation.

We adopt the same notation introduced in Busetto and Formentin (2023):

  * •

$P_{k}=\frac{p^{ask}_{k,1}+p^{bid}_{k,1}}{2}$: Mid-price value at LOB event
$k$.

  * •

$R_{k}=\frac{P_{k+1}-P_{k}}{P_{k}}$: Future financial return.

  * •

$BI_{k}=\frac{(P^{Bid}_{k,1}-P^{Bid}_{k,10})-(P^{Ask}_{k,10}-P^{Ask}_{k,1})}{(P^{Bid}_{k,1}-P^{Bid}_{k,10})+(P^{Ask}_{k,10}-P^{Ask}_{k,1})}$
Base imbalance.

##  3 Exploratory Data Analysis

In this section we introduce the characteristics of the financial
cryptocurrency LOB data at hand. Tether (USDT) is a stable cryptocurrency
(stablecoin), designed to maintain a value close to one U.S. dollar (USD). Its
primary purpose is to facilitate cryptocurrency transactions and mitigate the
volatility associated with other cryptocurrencies, like Bitcoin or Ether, see
Barucci et al. (2022, 2023). It is based on the blockchain technology which
ensure safe transactions and prevents fraud. In particular, we focus on a
market on a centralized exchange where USDT is traded against the USD; the
USDT-USD price represents the amount of USD necessary to buy/sell a USDT. More
precisely, we used the provider CryptoTick to obtain the LOB data from the
Bitfinex exchange. Data availability for this work ranges from April 13, 2019,
to May 7, 2019, for a total of $1,615,982$ LOB records. LOB depth goes up to
the 50th level, timestamps are provided for each record. Event resolution is
of at least $1$ [s].

Days | $25$
---|---
Date format | hh:mm:ss:fffffff
Records | $1,615,982$
Levels $L$ | $50$
Level Headers ($4$) | Ask price/size, Bid price/size
Table 2: LOB Tether Cryptocurrency Dataset.

Table 2 summarizes LOB Tether headers and characteristics.

Figure 1: Candlestick chart of day-to-day Tether (USDT) mid price $P_{k}$.

Figure 1 shows the candlestick chart of day-to-day USDT mid price $P_{k}$.
Down label represents a day where the closing price is lower than the opening
price, while up label stands for a day where the closing is higher than the
opening. Candlestick charts are used to identify trading opportunities, and
eventually trend and/or seasonalities by visual inspection. In this case, it
shows the presence of a changepoint in the average price of the cryptocurrency
in the time horizon available.

Figure 2:  LOB visualization of the Tether (USDT) dataset ($n=10$) at a fixed
time instant.

In Figure 2 we show the first $10$ levels of the LOB, on both the ask and bid
sides. Each level is associated with a specific price and depth, representing
the accumulation of orders at that price point. These levels may consist of
various orders placed by different market participants. Describing the details
of the LOB is out of this paper scope. Our interests focus on the timing with
which events are placed on the LOB, and at the same time, capturing the impact
of the spread of the LOB through the base imbalance regressor.

###  3.1 Correlation Analysis

We exclude all records with a return $R_{k}$ equal to zero, as they lack
informativeness. This corresponds to 80$\%$ of the data. Among the remaining
records, 34$\%$ have missing data for one minute. As Goldstein et al. (2021),
we divide the non-zero return dataset into deciles. For each decile, we
compute the average $\overline{BI}$ and the average return $\overline{R}$ and
evaluate the decile Pearson correlation coefficient $\rho=-0.96$. This result,
shows the promising relation between $\overline{BI}$ and $\overline{R}$,
confirming the intuition of Busetto and Formentin (2023) of the relevance of
base imbalance feature as meaningful regressor for return sign prediction,
also with different data sources (not only stock market but also
cryptocurrencies). This result allows us to apply the same COE model as
Busetto and Formentin (2023).

Figure 3: Correlation between $\overline{BI}$ and $\overline{R}$

##  4 mathematical formulation

In this section we introduce the mathematical formulation of the proposed
architecture. We describe the point process models in general and Hawkes
process in particular, and their role in predicting non-uniformly sampled
event times. We recall the COE model and we describe its usage, together with
the Hawkes predictions, to forecast financial return sign.

###  4.1 Point process

Point processes are stochastic processes used to predict the occurrence of
events over time. They can be described as counting processes, where we define
the counting process $N(t)$ as a cumulative count of events up to time $t$,
indicating the number of events within the interval $(0,t]$. We introduce
$\Delta N_{(t_{1},t_{2})}$, e.g., the total number of events within the
interval $(t_{1},t_{2}]$, defined as $\Delta
N_{(t_{1},t_{2})}=N(t_{2})-N(t_{1})$ (see Laub et al. (2015)). The most
important parameter common to all point processes is the intensity function,
quantifying the expected number of events occurring within a time window $T$.
In general, it is defined as:

$\lambda(t|H_{t})=\lim_{{\Delta t\to 0}}\frac{Pr(\Delta N_{(t,t+\Delta
t]}=1|H_{t})}{\Delta t}$ (1)

where $H_{t}$ is the history of the events up to time $t$ and $Pr(\Delta
N_{(t,t+\Delta t]}=1|H_{t})$ is the instantaneous conditional probability of
an event. Different classes of point processes are distinguished through
intensity functions. The most common class is the Poisson one, where the
intensity function is history-independent (e.g., follows the Markov property).
Renewal processes instead, introduce a basic history dependence. In our case,
for modeling LOB dynamics, the most appropriate choice is the Hawkes process,
thanks to the characteristics shown in the following section.

###  4.2 Hawkes process

Hawkes processes are a point processes subclass. The Hawkes choice depends on
the fact that they can capture the LOB self-excitation phenomena, see Lu and
Abergel (2018). When new orders are placed, the market hype increases and the
probability of having new orders placed increases as well. In Hawkes, the
intensity function is defined (according to Hawkes (1971)) as:

$\lambda(t)=\mu+\sum_{k}^{k(t_{k}<t)}\alpha\cdot e^{-\beta\cdot(t-t_{k})},$
(2)

where $\mu$ is the baseline, (e.g., the event rate in the absence of past
events), $\alpha$ is the weight associated with each event, $\beta$ is the
decay rate and controls the exponential decay over time. The collection
$\theta=[\mu,\alpha,\beta]$ constitutes the parameter vector that must be
identified during Hawkes training. $\theta^{*}=[\mu^{*},\alpha^{*},\beta^{*}]$
is identified through maximum likelihood estimation (MLE) on a time window
$T^{H}_{train}$ of historical data. MLE identifies $\theta^{*}$ maximizing the
joint probability of the observed data, e.g., the likelihood function
$\mathcal{L}_{n}(\theta)$,

$\theta^{*}=\underset{\theta\in\Theta}{\operatorname{arg\;max}}\,\mathcal{L}_{n}(\theta;\mathbf{y}),$
(3)

where y is the collection of observed data samples and $\Theta$ is the
parameter space. For additional details on MLE see Myung (2003). Given the
current evolution of $\lambda$ up to the current time $t$, we predict the
timing of the next event $\hat{t}_{k+1}$ as follows. A random value $x$ is
extracted from an exponential distribution with mean $\frac{1}{\lambda(t)}$.
$x$ is added to the current time instant $t$ to predict the next event time.

$x\sim\mathrm{Exp}(\lambda(t,\theta)),\text{ with
}\mathbb{E}[x]=\frac{1}{\lambda(t,\theta)},$ (4) $\hat{t}_{k+1}=t+x.$ (5)

We repeat this operation every second, updating the training set in a moving
window fashion. The forecast must occur within the forecast time window
$\Delta T$. If more than one event prediction occurs within $\Delta T$, only
the first prediction is saved. Figure 4 shows how the intensity function
identified by the Hawkes model captures the self-excitation phenomenon. As the
number of interactions with the LOB increases, the intensity function
increases and new interactions will occur with higher probability. Viceversa,
as less interactions happen, new interactions will occur less probably. In the
next section, we introduce the second component of the return sign prediction
algorithm: the COE model.

Figure 4: Intensity function self-excitation.

###  4.3 Continuous Output Error Model

Following Chen et al. (2013), we employ the simplified refined instrumental
variable method for continuous-time models (SRIVC) algorithm to predict future
return sign. We assume the existence of a Continuous Output Error (COE) model
that connects BI and future returns as follows:

$R(t)=\frac{B_{0}(p)}{A_{0}(p)}\cdot BI(t)+e(t),$ (6)

where $p$ is the differential operator $p^{i}F(t)=\frac{d^{i}F(t)}{dt^{i}}$,
$A_{0}(p)=p^{n_{a}}+a_{1}p^{n_{a}-1}+\ldots+a_{n_{a}}$,
$B_{0}(p)=b_{0}p^{n_{b}}+b_{1}p^{n_{b}-1}+\ldots+b_{n_{b}}$ subject to
$n_{a}=2\geq n_{b}=1$, and $e(t)$ is the random gaussian noise. The COE model
parameters $\theta_{COE}^{*}$ are estimated as:

$\theta_{COE}^{*}=\operatorname{argmin}_{\theta_{COE}}\sum_{k=1}^{N}\left\|\zeta(t_{k})\left(R^{n}(t_{k})-\varphi^{T}(t_{k})\theta_{COE}\right)\right\|_{2}^{2}$
(7)

where $\theta_{COE}=[a_{1}...a_{n_{a}},b_{0},b_{1}...b_{n_{b}}]^{T}$ and
$\varphi^{T}(t_{k})=[-R^{n_{a}-1}(t_{k}),...,-R(t_{k}),BI^{n_{b}}(t_{k}),...,BI(t_{k})]$
is the instrumental variable. This model, having available the values of the
base imbalance and the LOB event time vector (assumed known in Busetto and
Formentin (2023) and predicted by Hawkes in this work), predicts the return
sign.

###  4.4 Hawkes-based COE Return Prediction

The complete algorithm is obtained by merging the output of the point process,
the Hawkes-predicted next event times (see Equations (4) and (5)), with the
COE model in Equation (6). To predict the return value (and therefore its
sign) $\hat{R}_{k+1}$, we do not only need the predicted next event time
$\hat{t}_{k+1}$ but also the most recent $BI_{k}$, at the time when prediction
is computed. The reference return value used to compare the predicted return
is the nearest in terms of absolute relative temporal distance:
$R^{ref}_{k+1}\rightarrow\|t_{k+1}-\hat{t}_{k+1}\|$. Remark: before using the
identified intensity function to predict the next LOB events, a warm-up period
$T_{warm}$ is needed to let the intensity function see simulated events and
prepare for the prediction step with value not necessarily equal to the
baseline $\mu$. In algorithm 1 we show the procedure at each iteration. For
sake of compactness, for notation please refer to Table 1.

Algorithm 1 Hawkes-based COE Return Prediction

1:Initialize $t_{0}$.

2:Collect Hawkes training set $[t_{0}-T^{H}_{train}\cdot 60,t_{0}]$.

3:Identify Hawkes model $\theta^{*}$ with (3).

4:for t = 0,…,$T_{sim}-1$ do

5: Compute $\lambda(t_{0}+t,\theta^{*})$ with (2).

6: Extract $x$ from $\mathrm{Exp}(\frac{1}{\lambda(t_{0}+t,\theta^{*})})$.

7: Predict the next event time $\hat{t}_{k+1}=t_{0}+t+x$.

8: Update the event index $k=k+1$.

9:end for

10:Collect COE training set $[t_{0}-T^{COE}_{train}\cdot 60,t_{0}]$

11:Identify COE model $\theta_{COE}^{*}$ with (7).

12:for k = 1,…,$K$ do

13: Compute $BI_{k}$.

14: Compute the predicted return $\hat{R}_{k+1}$ with (6).

15:end for

##  5 Numerical Simulations and Results

In this section, we show the performances of the proposed architecture on $50$
$2$-minutes long validation scenarios, extracted from the available dataset.
We compare its performance against that of $3$ benchmark algorithms:

  * •

Oracle: perfect knowledge of the next event time (ideal benchmark for return
sign).

  * •

Naive: next event occurs exactly $1$ s from the instant at which we are making
the prediction. This coincides with the minimum resolution of the system as
the dataset shows that events are at least a part of $1$ second. See Table 3.

  * •

Moving Average (MA): predicts the next event time given the average over the
previous $W=60$ seconds.

Each event time prediction algorithm is integrated with the same COE model for
comparison. It’s worth noting that the Oracle algorithm, when combined with
the COE model, replicates the process performed in Busetto and Formentin
(2023), but with our dataset. In essence, it assumes knowledge of the timing
of the next event.

Settings |
---|---
$t_{k}-t_{k-1}\leq 2.2s$ | average maximum time distance
$t_{k}-t_{k-1}\geq 1s$ | minimum time distance
$R_{k}\neq 0$ | non-zero return events
Hyperparameters $\Gamma$ |
$T^{H}_{train}=20\,\,min$ | Hawkes training time
$T^{COE}_{train}=50\,\,min$ | COE training time
$T_{warm}=2.5\,\,min$ | Hawkes warm up time
$\Delta T=5\,\,s$ | Hawkes forecast window
$T_{sim}=2\,\,min$ | simulation time
$i=8$ | LOB depth
Table 3: Simulation settings - Hyperparameters $\Gamma$.

Table 3 collects the simulation settings and hyperparameters $\Gamma$. The
most restrictive setting requirement is on the average maximum time distance
between two consecutive events. This highlights the need of having temporal
horizons where a reasonable number of events occurs, captured by the average.
Each hyperparameter is obtained as result of a sensitivity analysis minimizing
the average absolute difference between the actual and the predicted event
times, on the validation set: May, 3, 2019. The sensitivity analysis procedure
can be described compactly through the following equation:

$\Gamma=\arg\min{\frac{\sum^{K_{v}}_{k=1}{\|t_{k}-\hat{t}_{k}(\Gamma)}\|}{K_{v}}}$
(8)

where $\Gamma$ is the hyperparameter set, and $K_{v}$ is the number of events
in the validation set. The COE model’s hyperparameters, instead, were tuned on
the validation set to optimize return prediction accuracy.

###  5.1 Return sign accuracy

Given the classes (TP-true positive, TN-true negative, FP-false positive, FN-
false negative) where positive corresponds to $sign(\hat{R}_{k+1})>0$ and
negative corresponds to $sign(\hat{R}_{k+1})<0$, we show the different
algorithms accuracy, defined as:

$Accuracy=\frac{TP+TN}{TP+TN+FP+FN}$ (9)

in Figure 5. Oracle algorithm is the best, as expected, as it has exact
knowledge of the next event’s time. Following that, Hawkes outperforms the
Naive and MA due to its machine learning-based prediction method for the next
event time.

Figure 5: Return sign accuracy on $50$ validation scenarios.

###  5.2 Trading Simulation

In this section we apply the proposed algorithm and the benchmark strategies
in a trading simulation framework. In this way we show that, for the same COE
model, the profit magnitude depends on the quality with which the next event
time instant is predicted. The trading strategy is summarized as follows: if
the predicted sign is positive (negative), the automatic trading algorithm
will purchase (short-sell) USDT for a value of 10,000 $. If the actual sign
matches the predicted one, this will generate a profit; otherwise, if the
prediction is incorrect, we will incur in a loss. No transaction costs are
considered in this exercise.

Figure 6 shows the total profit [$] obtained in 50 validation scenarios (the
same used to evaluate accuracy), computed as the sum of all the profits minus
all the losses occurred during the trading exercise. We observe how the same
quality ranking highlighted in Figure 5 is maintained. Oracle statistically
obtains the greatest profit, with a higher median value and a smaller range of
variation between the maximum and minimum quantiles. Hawkes shows satisfactory
performances, approaching ideal performances—next, the MA and Naive
benchmarking strategies.

Figure 6: Total profit on $50$ validation scenarios.

Figure 7 shows one case of the cumulative profit time series obtained with the
different trading algorithms. Initially, all the strategies fluctuate around
zero profit. Then, there is a drastic downward change in the MidPrice $P_{k}$
(shown in the lower panel), followed by a more volatile period. This allows us
to achieve a higher profit with the most accurate strategies.

Figure 7: Upper panel: Profit time series in an example scenario. Lower panel:
MidPrice $P_{k}$ fluctuations.

##  6 Limitations, Conclusions, and Future Work

The proposed algorithm overcomes the initial assumption of knowing, a priori,
the actual time of the new incoming event on the LOB. This is possible thanks
to a point process model, Hawkes, that can be trained on the original, not
uniformly sampled LOB dataset. We show that the methodology outperforms
benchmark alternatives in return sign prediction accuracy and, consequently,
the profit on 50 real validation scenarios. This work also confirms the value
of the base imbalance as a candidate regressor, as it is used effectively on
cryptocurrencies LOB data, showing a high correlation with the returns with
data sources different from the stock market on which it was developed.
However, the proposed architecture finds some limitations that can be improved
in the next steps. In particular, we stick to remove zero return events,
possibly losing information content. Additionally, comparing the predicted
return with the closest in terms of absolute time might be an approximation,
and different choices could be made (e.g., taking the previous or the
following event time). It’s also worth noting that this procedure cannot be
applied to all the data in the dataset. In fact, our sample has specific
constraints to meet (as outlined in Table 3). In the end, transaction fees are
not included in the profit final balance sheet. Nonetheless, the results are
promising and will lead to extensive future work, such as a vast validation of
cryptocurrency data and the comparison with other alternatives to point
process models.

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