Lob-based deep learning models for stock price trend prediction: a benchmark study

A stock exchange employs a matching engine for storing and matching the orders issued by the trading agents. This is achieved by updating the so-called Limit Order Book (LOB) data structure. Each security (tradable asset) has a LOB, recording all the outstanding bid and ask orders currently available on an exchange or a trading platform. The shape of the order book gives traders a simultaneous view of the market demand and supply.

There are three major types of orders. Market orders are executed immediately at the best available price. Limit orders, instead, include the specification of a desired target price: a limit sell [buy] order will be executed only when it is matched to a buy [sell] order whose price is greater [lower] than or equal to the target price. Finally, a cancel order removes a previously submitted limit order.

Figure 1 depicts an example of a LOB snapshot, characterized by buy orders (bid) and sell orders (ask) of different prices. A level, shown on the horizontal axis, represents the number of shares with the same price either on the bid or ask side. In the example of Fig. 1, there are three bid and three ask levels. The best bid is the price of the shares with the highest price on the buy side; analogously, the best ask is the price of the shares with the lowest price on the sell side. When the former exceeds or equals the latter, the corresponding limit ask and bid orders are executed. The LOB is updated with each event (order insertion/modification/cancellation) and can be sampled at regular time intervals.

In Huang and Stoll (1994), Tran et al. (2022), Pascual and Veredas (2003), Cao et al. (2008), Cao et al. (2009), Duong and Kalev (2014) it has been empirically demonstrated, using both linear and non-linear models, that the orders behind the best bid and ask prices have a significant impact to price discovery and contain information about short-term future price movements, supporting the hypothesis that leveraging deeper levels of the limit order book is essential for improving the performance of SPTP tasks. This is the main reason for not restricting the input to the best levels. Additionally, deep learning models are well suited to handle high-dimensional input.

We represent the evolution of a LOB as a time series \({\mathbb {L}}\), where each \({\mathbb {L}}(t) \in {\mathbb {R}}^{4L}\) is called a LOB record, for \(t=1, \ldots , N\), being N the number of LOB observations and L the number of levels. In particular,where \(P^{\texttt {ask}}(t), P^\texttt {bid}(t) \in {\mathbb {R}}^{L}\) represent the prices of levels 1 to L of the LOB, on the ask (\(s=\texttt {ask}\)) side and bid (\(s=\texttt {bid}\)) side, respectively, at time t. Analogously, \(V^\texttt {ask}(t), V^\texttt {bid}(t) \in {\mathbb {R}}^{L}\) represent the volumes. This means that for each t and every \(j\in \{1,\ldots ,L\}\) on the ask side, \(V^\texttt {ask}_{j}(t)\) shares can be sold at price \(P^\texttt {ask}_{j}(t)\). The mid-price m(t) of the stock at time t, is defined as the average value between the best bid and the best ask,On average, if most of the executed orders are on the ask [bid] side, the mid-price increases [decreases] accordingly.

We use a ternary classification for trends: U (“upward”) if the price trend is increasing; D (“downward”) for decreasing prices; and S (“stable”) for prices with negligible variations. Among all the possible single values, mid-prices provide the most reliable indication of the actual stock price for equity markets. Nevertheless, because of the market’s inherent fluctuations and shocks, they can exhibit highly volatile trends. For this reason, using a direct comparison of consecutive mid-prices, i.e., m(t) and \(m(t+1)\), for stock price labelling would result in a noisy labelled dataset. As a result, labelling strategies typically employ smoother mid-price functions instead of raw mid-prices. Such functions consider mid-prices over arbitrarily long time intervals, called horizons. Our experiments adopt the labelling proposed in Ntakaris et al. (2018) and repurposed in several other SOTA solutions we selected for benchmarking. The adopted labelling strategy compares the current mid-price to the average mid-prices \(a^+(k, t)\) in a future horizon of k time units, formally:The average mid-prices are used to define a static threshold \(\theta \in (0,1)\) that is used to identify an interval around the current mid-price and define the class of the trend at time t as follows:With this labelling, we beat the effect of mid-price fluctuations by considering their average over a desired horizon k and considering a trend to be stable when the average mid-price variations do not change significantly, thus avoiding over-fitting. We highlight that timestamp t can come either from a homogeneous or an event-based process. In our experiments, we consider the latter approach. Hence, the horizon k is expressed in the number of future events.

Given the time series of a LOB \({\mathbb {L}}\) and a temporal window \(T = [t-h, t]\), \(h\in {\mathbb {N}}\), we can extract market observations on T, \({\mathbb {M}}(T)\), by considering the sub-sequence of LOB observations starting from time \(t-h\) up to t. Fig. 2 gives a representation of a market observation \({\mathbb {M}}(T) \in {\mathbb {R}}^{h\times 4L}\). The market observation over the window \([t-h,t]\) is associated with the label computed through Eqs. 1 and 2 at time t. An SPTP predictor takes as an input a market observation and outputs a probability distribution over the trend classes U, D, and S.

Table 1 summarizes the most peculiar characteristics of the selected models. The temporal shape represents the length of the input market observation for the model. In the table, the features shape refers to the number of features used by the models to infer the trend in the original papers. In the Table, we also indicate whether the authors released the code, and if so, whether they have used PyTorch Paszke et al. (2019) or TensorFlow Abadi et al. (2016). This is relevant because to ensure consistency and compatibility within our proposed framework, based on PyTorch Lightning, we found it necessary to re-implement models for which the code was not available or was only available in Tensorflow. We made every effort to validate and verify the correctness of our re-implementations, including making our code publicly available, which facilitates collaboration and scrutiny from the research community. We remark that to improve the reproducibility of the results, it would be advisable for the research community to publish the code developed to carry out the experiments.

In High-Frequency Trading (HFT) and algorithmic trading in general, minimizing latency between model querying and order placement is of utmost importance Gomber and Haferkorn (2015). To explore this aspect, we analyzed the inference time in milliseconds of all models, based on the experiments reported in Sect. 6.3. As shown in Table 1, DEEPLOB, DEEPLOBAT, AXIALLOB, TRANSLOB, and ATNBoF had inference times in the order of milliseconds, potentially unsuitable for HFT applications compared to other models with shorter times. Finally, we have reported the number of trainable parameters for each model. A noteworthy observation is that the average number of parameters is very low compared to other classical deep learning fields, such as computer vision He et al. (2016) and natural language processing (Devlin et al. 2018; Brown et al. 2020). This leads us to conjecture that current systems are inadequate in effectively handling the complexity of LOB data, as we will verify in the rest of this paper.

To explore the possibility of achieving new SOTA performance by combining the predictions of all 15 models, we have implemented two ensemble methods: MAJORITY and METALOB.

The MAJORITY ensemble assigns the class label that appears most frequently among the predictions of the classifiers. To account for variations in the performance of individual classifiers, we incorporate a weighting scheme based on their F1-Scores. This ensures that predictions from higher-performing models carry more influence in the final decision.

The METALOB meta-classifier is implemented as a multilayer perceptron (MLP) with two fully connected layers. It is designed to learn how to effectively combine the outputs of the 15 DL models, which serve as the base classifiers to produce the final output. The input to the meta-classifier is a 1D tensor with a probability distribution over the trends (up, stationary, down) for each of the models, resulting in a tensor of \(3\cdot 15\) elements.

The most widely spread public LOB dataset is FI-2010 which is licensed under Creative Commons Attribution 4.0 International (CC BY 4.0) and was proposed in 2017 by Ntakaris et al. Ntakaris et al. (2018) with the objective of evaluating the performance of machine learning models on the SPTP task. The dataset consists of LOB data from five Finnish companies: Kesko Oyj, Outokumpu Oyj, Sampo, Rautaruukki, and Wärtsilä Oyj of the NASDAQ Nordic stock market. Data spans the time period between June 1st to June 14th, 2010, corresponding to 10 trading days (trading happens only on business days). About 4 million limit order messages are stored for ten levels of the LOB. The dataset has an event-based granularity, meaning that the time series records are not uniformly spaced in time. LOB observations are sampled at intervals of 10 events, resulting in a total of 394,337 events. This dataset has the intrinsic limitation of being already pre-processed (filtered, normalized, and labelled) so that the original LOB cannot be backtracked, thus hampering thorough experimentation. Additionally, the labelling method employed is found to be prone to instability, as demonstrated by Zhang et al. in Zhang et al. (2019).

The dataset provides the time series and the classes relative to five horizons \(k \in {\mathcal {K}} = \{1, 2, 3, 5, 10\}\) by leveraging the trend definitions described in Eq. 2. Such a labelling scheme is very sensitive to the threshold \(\theta\) regarding the resulting balancing between “upward”, “downward” and “stable” trends. Table 2 shows the class balancing for different horizons \(k\in {\mathcal {K}}\). The authors of the dataset employed a single threshold \(\theta = 0.002\) for all horizons, but it balances only the case of \(k = 5\). As it can be observed, the stationary class S is progressively less predominant in favour of the upward and downward classes. In our experimental campaign, the class imbalance is not addressed to guarantee a fair robustness evaluation since the considered works do not claim to have done so.

To test the generalizability of the models in a more realistic scenario, we used data extracted from LOBSTER [13], an online LOB data provider for order book data, which is publicly available for the research community with an annual fee. LOBSTER limit order books are reconstructed directly from NASDAQ traded stocks. To compare the performance of the algorithms in a wide range of scenarios, we have created a large LOB dataset, including several stocks and time periods. The chosen pool of stocks includes those from the top 50% more liquid stocks of NASDAQ. To construct a diversified evaluation scenario, we selected six stocks, namely: SoFi Technologies (SOFI), Netflix (NFLX), Cisco Systems (CSCO), Wing Stop (WING), Shoals Technologies Group (SHLS), and Landstar System (LSTR). The periods in consideration are July 2021 (2021-07-01 to 2021-07-15, 10 trading days) making up LOB-2021, and February 2022 (2022-02-01 to 2022-02-15, 10 trading days) making up LOB-2022. The selection of these two periods aimed to capture data from periods with different levels of market volatility. February 2022 exhibited higher volatility compared to July 2021, largely influenced by the Ukrainian crisis. This allows for an assessment of models across varying market conditions. Further details on stock selection and processing are provided in the following two paragraphs.

LOB time series are susceptible to distribution shift due to the dynamic nature of the stock markets. This implies that when used in production, the input data seen by the model may deviate from the data it was trained on Bennett and Clarkson (2022). Such shifts can lead to serious consequences that directly translate into misclassification and significant economic losses if the model is not effectively monitored. It is important to note that the primary focus of this work, and the considered studies, is on the trend classification task. Deploying these models into production demands additional effort, particularly in addressing challenges like distribution shift. Being able to observe the market in time, allows for new data collection facilitating continual model retraining, data weighting for past forgetting, and fine tuning. These methods should be adopted in conjugation with the usual techniques to prevent data overfitting, such as early stopping, data augmentation, and regularization.

For evaluating the robustness of the surveyed models, we used the hyperparameters reported in the original papers whenever they were available. However, we encountered cases where hyperparameters were not declared at all, such as in LSTM Tsantekidis et al. (2017a) and CNN1 Tsantekidis et al. (2017b), while in other cases, including CNNLSTM Tsantekidis et al. (2020), AXIALLOB Kisiel and Gorse (2022), ATNBOF Tran et al. (2022) and DAIN Passalis et al. (2019) only partial information was provided. To address these gaps, we performed a grid search exploring different values for the batch size, including \(\{16, 32, 64, 128, 256\}\) and the learning rate, including \(\{0.01, 0.001, 0.0001, 0.00001\}\).

Regarding the generalizability experiment, we found that the majority of models using the hyperparameters from the robustness experiment performed poorly on the LOB-2021/2022 datasets. So, we conducted a comprehensive hyperparameter search on horizon \(k=5\) (which is the most balanced) using a grid search approach for all 16 models. For this search, we maintained the same number of epochs and optimizer used in the robustness analysis while searching for batch size and learning rate using the same domains mentioned above. F1-Score maximization over the validation set was the chosen criterion for optimizing the hyperparameters. For a complete overview of the hyperparameters utilized in our experiments, refer to Table 5.

To test robustness and generalizability, we conducted our experiments for each model using five different seeds to mitigate the impact of random initialization of network weights and training dataset shuffling. The necessity to produce reliable results within reasonable time constraints led us to opt for a set of five seeds. The training process involved training the 15 models for each seed on each of the considered prediction horizons (\({\mathcal {K}} = \{1, 2, 3, 5, 10\}\)). On average, the training process for all the models took approximately 155 h for FI-2010 and 258 h for each LOB dataset, utilizing a cluster comprised of 8 GPUs (1 NVIDIA GeForce RTX 2060, 2 NVIDIA GeForce RTX 3070, and 5 NVIDIA Quadro RTX 6000).

In Table 6, we summarize the results of our experiments. Our choice of F1-Score as the evaluation metric was motivated by the following reasons: (1) it captures both precision and recall in a single value, (2) the datasets are not well balanced and F1-Score is robust to the class imbalance problem that affects the accuracy measure, (3) it is the only metric that is reported in every SOTA paper. The Table compares the claimed performance of each system (column F1 Claim) with those measured in the robustness (FI-2010) and generalizability (LOB-2021 and 2022) experiments. For each dataset, we show the average performance and the standard deviation achieved by each model in all the horizons, along with its rank.

To evaluate the robustness and the generalizability of the models, we compute the robustness and the generalizability scores, a value \(\le 100\) that is computed as \(100 - (|A| + S)\), where A and S are defined as follows. A is the average difference between the F1-Score reported in the original paper and the one that we observed in our experiments on FI-2010 for robustness, and on LOB-2021 and LOB-2022 for generalizability. S is the standard deviation of these differences. The score penalizes models that demonstrate higher variability in their performance by subtracting the standard deviation. The average and standard deviation were computed over the declared horizons for each model and considering all five seeds.

Table 6 clearly highlights the following:

In this section, we delve into additional experiments. In Sect. 6.4.1, we measure the impact of labeling parameters on the quality of the SPTP task. In Sect. 6.4.2, we go beyond deep learning models and explore how tree-based methods perform on the SPTP task using the same experimental setting presented in the previous section. In Sect. 6.4.3, we incorporate profit considerations through backtesting.

Our results lead to a crucial observation: on the LOBSTER dataset, SOTA DL models for LOB data exhibit low generalizability. We suggest that this phenomenon is due to two factors: the higher complexity of the LOBSTER dataset compared to the FI-2010 dataset and the overfitting of the best-performing models to the FI-2010 dataset, which lowers their performance on the LOBSTER dataset. In fact, in the original papers, all the considered models (except for DEEPLOB, DEEPLOBAT and DLA) were trained, validated, and tested only on FI-2010, which is a smaller dataset with less frequent and voluminous orders than those contained in LOBSTER-derived datasets. Our conjecture is supported by insightful findings reported in the existing literature: several works (e.g., Orimoloye et al. 2020; Ruff et al. 2021; Najafabadi et al. 2015) have shown that while some datasets exhibit simple patterns that can be effectively captured by a shallow model, others may require deeper architectures to model complex relationships.

Another key finding of this study is that the top models with the highest performance on both datasets employ attention mechanisms. This suggests that the attention technique enhances the extraction of informative features and the discovery of patterns in LOB data. However, in general, it appears that current models cannot cope with the complexity of financial forecasting with LOB data.

Financial trends can be influenced by both local and international political events, in fact, political actions and decisions can significantly impact economic conditions, market sentiment, and investor confidence Engle et al. (2013). These factors are not captured by LOB data alone. For this reason, we believe that price predictors may benefit from integrating LOB data with additional information, for example, sentiment analysis relying on social media and press data, representing an easily accessible source of exogenous factors impacting the market Ren et al. (2018). This is particularly true for mid- and long-term price trend prediction, whereas it might not hold for HFT strategies Bouchaud et al. (2018). We remark that micro and macroscopic market trends are fundamentally different, and the microscopic behaviour of the market is very much driven by HFT algorithms, making it almost exclusively dependent on financial movements rather than external factors. In this scenario, granular and raw LOBs may suffice to provide data for price trend prediction. Several works have used sentiment analysis for price trend prediction. The work in Jin et al. (2020) combines comments made by investors on the online platform StockTwits with Apple stock price time series and uses LSTM for stock closing price prediction. By creating a dataset composed of tweets and historical stock prices, the work in Xu and Cohen (2018) proposes a new architecture for stock price prediction. Similar approaches are used in Li et al. (2014), Nguyen et al. (2015), where the authors show that the performance of their predictors improves when their dataset is enriched with sentiment data. Despite the wide research in this field, to the best of our knowledge, no one has ever used LOB data together with sentiment analysis for price trend prediction. The existing literature suggests that this could be a valuable research direction.

Another weakness in dataset generation is the potential for training, validation, and test splits to have dissimilar distributions. This occurs due to the distinct characteristics of the historical periods covered by the stock time series. This can negatively affect the model’s ability to generalize effectively and make reliable predictions on unseen data. A last limitation regarding the dataset is the representation of the limit order book which has been shown to be sensitive to permutations by Wu et al. (2021). In Wu et al. (2022), the same authors proposed robust alternative representations.

As we discussed in Sections 2, 5 and 6.3, the choice of the threshold for class definition in Eq. 2 plays a crucial role in determining the trend associated to a market observation. We believe that current solutions present room for improvement. As discussed in Sect. 5, in FI-2010, the parameter \(\theta\) was chosen to obtain a balanced dataset in the number of classes for the horizon \(k=5\) (which is the mean value of the considered interval in the set \({\mathcal {K}}\)). Thus, \(\theta\) is not chosen in accordance with its financial implication but rather serves the purpose of improving model performance. We recall that the dataset is made of different stocks. With such a labelling system, with a fixed \(\theta\), stocks with low volatility become associated with stable trends, as their behaviour is overshadowed by stocks exhibiting higher volatility. Good practices that could be investigated are to use a weighted look-behind moving average to absorb data noise instead of mid-prices as in Eq. 2 or to define a dynamically adapting \(\theta\) which accounts for changing trends of a stock’s mid-price. Moreover, the labelling approach of Eq. 2, used by all surveyed models, fails to leverage important aspects available in LOB data, so another possible improvement is the definition and use of other insightful features that can be extrapolated from the LOB in addition to the mid-price. Such values could encapsulate other peculiar and informative features, such as stocks’ spread and volumes which directly influence stock volatility and so the returns.

In the context of stock prediction tasks, it is of utmost importance to go beyond standard statistical performance metrics such as accuracy and F1-Score and incorporate trading simulations to assess the practical value of algorithms. SPTP predictors’ ultimate measure of success lies in their ability to generate profits under real market conditions. It is essential to conduct trading simulations using real simulators that go beyond testing on historical data. Recent progress has been made in the context of reactive simulators (Coletta et al. 2022a, b; Mizuta 2016; Shi and Cartlidge 2023).

We acknowledge that our study is subject to some limitations, which should be considered when interpreting our findings. First, we conducted a grid hyperparameter search for the models which did not specify them. Since hyperparameter search is not exhaustive, our chosen best hyperparameters could potentially undermine the quality of the original systems. Secondly, due to computational resource limitations, we could not train the benchmarked models on LOB datasets spanning longer periods, e.g., years rather than weeks. We recognize that doing so could have led to different results.

We also highlight that using DL or, more generally, AI models for solving SPTP and exploiting them for trading can have a number of risks. Some of them are inherently technical. This is the case for data biases, that is, incomplete or unrepresentative data, which can cause predictive algorithms to favor groups that are better represented in the training data Boukherouaa et al. (2021). Lack of explainability is another risk that could expose organizations to vulnerabilities (such as biased data, unsuitable modeling techniques, or incorrect decision-making) and potentially undermine the trust in their robustness Silberg and Manyika (2019). AI models for trading are also vulnerable to cyber-attacks. Malicious users can exploit AI model vulnerabilities to evade detection and prompt the models to make the wrong decisions or to extract information by manipulating data at some stage of the model lifecycle Comiter (2019). Other risks have an ethical nature and can impact financial stability. One of them is inequality among investors. As training and predicting are expensive in terms of hardware equipment and energy, and because of the challenges in model interpretation and prediction, AI trading can lead to a concentration of information among those who can afford the required technology, exacerbating income inequality and asymmetry in the market, with an uncertain impact on financial stability (Robledo Costales 2023; Boukherouaa et al. 2021). Because of limited regulation of the AI trading systems, there is a lack of transparency, making it difficult to detect possible unfair strategies Robledo Costales (2023). Governing legal and regulatory framework regulators should welcome the advancements of AI in finance and undertake the necessary preparations to harness its potential advantages and address its associated risks.