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In a further step, we can investigate the role of the spread on these dynamics. In the case of a large spread observation, then priority can be achieved with limit orders, it is thus expected that the imbalance effect will be less pronounced (as found for example in Stoikov, ( 2017 ) but that it will interact with [MAT... |
[EQUATION] where [MATH] if the observed spread is larger than its mean, and [MATH] if it is smaller. Obviously, one could use the spread value in ticks for finer models, but we choose categorical variable for graphic illustration purposes. Note that the spread distribution is very stable, so that the mean can for examp... |
We now have four curves representing the effect of the imbalance on the next trade sign, depending on the last trade sign and the current spread. Empirical curves are noisier as the samples are split into subsamples to compute conditional probabilities. However, we observe as expected that a large spread flattens the i... |
Since we are exploring the role of several covariates, it is natural to apply the information criteria mentioned in Proposition 4.1 . Among other possible sequences [MATH] , we use the QAIC for [MATH] , the QCAIC for [MATH] , and the QBIC for [MATH] , all based on the QMLE [MATH] . As an illustration, Table (left panel... |
Several comments can be made about this illustration. First of all, it turns out that the simplest model depending only on the imbalance is never selected. Recall for example that the weighted mid-price, commonly used in microstructure as a proxy for a “future” price, depends only on the imbalance. Our observation plea... |
Our investigation on the influence of the imbalance signal on the intensities of bid and ask market orders can also illustrate the use of the penalized QMLE described in Section . In this example, we use the ratio model to try to decide whether traders use the imbalance as a trading signal on a given stock looking at t... |
[EQUATION] where [MATH] . Let [MATH] [MATH] be our ratio parameters, to be estimated by likelihood maximization. We estimate the model using both standard quasi-likelihood maximization, as well as a penalized estimation using the penarized QMLE [MATH] of Equations ( 29 ) and ( 30 ). We compute daily fits of the model a... |
It turns out that the parameters associated to the imbalance at level 1 is always very significant, but that the additional information given by the corrective terms [MATH] is not translated into significant [MATH] [MATH] . Penalization greatly helps reducing the confidence intervals of estimates. This results thus see... |
5.3 Spread and order flows at the best quotes Spread as a potential trading signal seems less investigated in the microstructure literature than other factors. One potential reason is that the spread of many highly traded stocks is almost always equal to one tick (so-called large tick stocks). This property has given b... |
We can therefore propose a version of our ratio model to investigate the influence of the spread on orders flows occurring at the best quotes. Let us consider a model of best quotes of the limit order book where market orders, limit orders and cancellations are submitted with intensities [MATH] [MATH] and [MATH] respec... |
[EQUATION] Letting [MATH] and [MATH] [MATH] , our intensities ratio model is easily estimated by maximizing the quasi-log likelihood. |
We estimate the model for each stock and each trading days, which gives after data cleaning 8052 different samples and associated model fits. As in the previous cases, we compute the intensities ratios to estimate the probabilities of each event given the observed spread. Figure plots the probabilities of market orders... |
It is satisfying to observe that the model provide good fits in a wide range of spread distribution, from a large tick stock (Figure , bottom left) where the spread is almost always equal to one tick, to the more interesting case of small tick stocks (e.g., Figure , top right), where the probability is correctly estima... |
5.4 Equilibrium behaviour with respect to the queues sizes We propose a final illustration of the ratio model to investigate the role of the observed queue size in determining the flows of limit orders and cancellations. Some empirical observations of the role of the queue size can be observed in Huang et al. , ( 2015 ... |
We investigate these insights by building one simple ratio model for limit orders and cancellations occurring inside the book, from level 2 to 10. Level 1 (best quote) is left aside as its dynamics also involves market orders. For each level [MATH] , let [MATH] and [MATH] be the counting processes of limits orders and ... |
[EQUATION] where [MATH] is a potentially random baseline intensity and [MATH] is the volume standing in the book at the level of submission in group [MATH] and time [MATH] [MATH] is expressed in integer multiple of the median trade size (see e.g., Huang et al. , ( 2015 ); Muni Toke & Yoshida, ( 2017 about this normaliz... |
As in the previous examples, we estimate the model for each stock and each month, and then compute the intensities ratios to estimate the probabilities of each event given the observed volume of the level of submission. Each monthly fit gives 9 fits, one for each level, but for the sake of readability, we plot only thr... |
It is interesting that the probabilities of occurrence of a limit order as a function of the observed volume show similar patterns across stocks. The model obviously recovers the expected equilibrium property that decreases the interest (hence the probability) of a limit order when liquidity is already here (i.e. [MATH... |
Empirical results : Prediction Empirical results from Section are in-sample analysis aimed at providing new descriptions of some dependencies observed on financial markets. We now turn to the possible use of the ratio model as a prediction tool. We consider the problem of the sign of the next trade, previously introduc... |
[EQUATION] where [MATH] is the baseline intensity and the kernel matrix [EQUATION] describes the self- and cross- excitation parts. |
A simple ratio model with only current observations of the state and not taking into account the history of the order flow (except for the last trade sign) can probably not compete with the Hawkes description. However, we can incorporate some history into covariates of the ratio model. One may for example consider the ... |
[EQUATION] where [MATH] counts the number of bid and ask market orders. These covariates add some self-exciting history into the ratio model. |
We can thus examine different methods to predict the trade sign, given that one trade is observed a given time. This is of course a theoretical exercise, as we predict the trade sign with all the information available just before its occurrence, not taking into account latency, information delays, reaction times, etc.,... |
Last : the trade sign is set to be the same as the last observed trade sign ; Imbalance : the trade sign is set to [MATH] if the imbalance observed before its submission is negative, [MATH] otherwise; |
Hawkes Full : at each time we compute the intensity [MATH] of the Hawkes process described at Equations ( 42 )-( 43 ), given the observed history, and set the sign to be [MATH] if [MATH] [MATH] otherwise. |
Hawkes NoCross : same as the previous method, except that we independently fit two Hawkes processes for the bid and ask market orders, i.e. we set [MATH] |
Ratio [MATH] : we use the probabilities given by the basic ratio model of Equation ( 36 ) and set the sign to be [MATH] if the probability of an ask market order given by the ratio model is lower than [MATH] , and [MATH] otherwise ; |
Ratio [MATH] : same as the previous method, but we use only the covariates [MATH] in the ratio model, instead of the covariates describing the state of the book ; |
Ratio [MATH] : same as the previous method, but using both [MATH] and the covariates [MATH] in the ratio model. We test these methods on the sample described at Section 5.1 . “Last” and “Imbalance” methods do no require any calibration. For the other methods, calibration is carried out on the trading day preceding the ... |
Results are illustrated in Figure For all stocks, the “Last” indicator correctly predicts more than [MATH] of the trade signs, e.g. less than one trade out of 5 has a sign different from the preceding trade. The “Imbalance” indicator is less performant, signing correctly about [MATH] of the trades. The model “Ratio [MA... |
We can analyse a bit further these performances. We now focus on the prediction of a sign change, i.e. we compute the performance using only the trades that have a sign different from the previous trade (roughly [MATH] of the sample, given the observation above). Results are averaged for each stock on Figure |
The performance of the “Imbalance” signal is roughly similar. The “Last” indicator has obviously a performance equal to [MATH] , as it never predicts a sign change. The ratio model without any history also performs poorly (around [MATH] ) in this specific subsample, since its only history relies on the last trade sign.... |
Conclusion We have presented a model based on ratios of Cox-type intensities sharing a common, possibly random, baseline intensity. Consistency and asymptotic normality of the estimators have been proved. Such a model may be very useful in cases where one tries to investigate the role of given covariates on point proce... |
Acknowledgements This work was in part supported by Japan Science and Technology Agency CREST JPMJCR14D7; Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research No. 17H01702 (Scientific Research), No. 26540011 (Challenging Exploratory Research); and by a Cooperative Research Program of the Ins... |
Appendix A Mathematical proofs A.1 Proof of Theorem 3.1 We have [EQUATION] and [EQUATION] Simple calculus yields [EQUATION] and [EQUATION] |
where [MATH] In other words, [EQUATION] For the closed convex hull [MATH] of [MATH] , let [EQUATION] Then [EQUATION] and the equality holds if and only if |
[EQUATION] Condition ( 53 ) implies that [EQUATION] due to [MATH] Therefore, [EQUATION] and hence [MATH] for all [MATH] due to Condition [A3] applied to [MATH] |
We see [MATH] Since [MATH] is positive definite and [MATH] for all [MATH] there exists a constant [MATH] such that [EQUATION] for all [MATH] |
Let [MATH] , where [MATH] The quasi-likelihood ratio random field is define by [EQUATION] We will use also [EQUATION] For this extension, we notice [MATH] and hence [MATH] is naturally extended to [MATH] The random field [MATH] will be used to obtain the so-called polynomial type large deviation inequality for [MATH] |
Define [MATH] by [EQUATION] with the naturally extended [MATH] Define a [MATH] -dimensional random variable [MATH] and a [MATH] random matrix [MATH] by |
[EQUATION] and [EQUATION] respectively. Let [MATH] for [MATH] Then [EQUATION] Take parameters [MATH] [MATH] [MATH] and [MATH] so that |
[EQUATION] where [MATH] We have Lemma A.1 Suppose that [MATH] and [MATH] are satisfied. Let [MATH] be any positive number. Then (i) |
[MATH] (ii) [MATH] (iii) [MATH] (iv) [MATH] Proof. Let [MATH] Let [MATH] for [MATH] Define [MATH] [MATH] by [EQUATION] Then [EQUATION] |
where [EQUATION] We have [EQUATION] and [EQUATION] for some constant [MATH] depending on [MATH] By the Burkholder-Davis-Gundy inequality, we obtain |
[EQUATION] for any [MATH] and any measurable function [MATH] of at most polynomial growth. By Equation ( 65 ) and induction, we obtain (i). |
Set [EQUATION] and [EQUATION] where [EQUATION] Then [EQUATION] Similarly to the proof of (i), we obtain [EQUATION] for every [MATH] Moreover, applying Theorem 6.3 of Rio, ( 2017 under [MATH] and [MATH] we obtain |
[EQUATION] for every [MATH] . Now Sobolev’s embedding inequality in [MATH] for [MATH] gives (ii). In a similar fashion, it is possible to prove (iii). The proof of (iv) is also similar to that of (ii), and rather simpler. |
Lemma A.2 Suppose that [MATH] [MATH] are satisfied. Then (i) For any [MATH] [EQUATION] where [MATH] (ii) [MATH] admits a locally asymptotically normal representation |
[EQUATION] with [MATH] as [MATH] for every [MATH] and [MATH] as [MATH] Proof. We will verify the conditions of Theorem 3 (c) of Yoshida, ( 2011 for the naturally extended random field |
[MATH] over [MATH] Condition [MATH] therein holds according to Lemma A.1 (iii) and (iv). Condition [MATH] therein is satisfied with [MATH] under Equation ( 63 ). Lemma A.1 (i) and (ii) ensures Condition [MATH] therein. Condition [MATH] therein is obvious and Condition [MATH] therein is ( 56 ). Therefore, by Theorem 3 o... |
[EQUATION] which gives (i). The term [MATH] is defined by ( 77 ) for [MATH] For each [MATH] , for sufficiently large [MATH] [MATH] admits the representation |
[EQUATION] Then Lemma A.1 (iii) and (iv) verify the convergence [MATH] as [MATH] For [MATH] and [MATH] [EQUATION] Since it is assumed that [MATH] [MATH] ) do not have common jumps, |
[EQUATION] Under [MATH] and [MATH] , the term on the right-hand side converges in probability to [MATH] as [MATH] The conditional type Lindeberg condition is easily verified by dividing the range [MATH] of the integral ( 65 ) into |
[MATH] subintervals, and as a result, we see [MATH] , which concludes the proof of (ii). It is possible to extend [MATH] to [MATH] so that the extension has a compact support and |
[EQUATION] We will denote this extended random field by the same [MATH] Then [MATH] is a random variable taking values in the Banach space |
[MATH] equipped with sup-norm. On some probability space, we prepare a random field [EQUATION] where [MATH] By Lemma A.2 (ii), we have a finite-dimensional convergence |
[EQUATION] as [MATH] For [MATH] and [MATH] , define [MATH] by [EQUATION] for large [MATH] , where [MATH] Then by Lemma A.1 (iii) and the definition of [MATH] we have |
[EQUATION] for every [MATH] and [MATH] According to e.g. Theorem 4 of Yoshida, ( 2011 we obtain ( 13 ) for [MATH] since Conditions [MATH] and [MATH] therein are ensured by ( 86 ) and by ( 84 ), respectively, and Condition [MATH] is trivial now. |
The properties ( 86 ) and ( 84 ) give functional convergence [EQUATION] as [MATH] for each [MATH] Moreover, Lemma A.2 already provided the PLD inequality for [MATH] Thus e.g. Theorem 10 of Yoshida, ( 2011 |
proves ( 13 ) for [MATH] once the estimate [EQUATION] is established. However, ( 88 ) can be verified e.g. with Lemma 2 of Yoshida, ( 2011 This completes the proof of Theorem 3.1 |
A.2 Proof of Proposition 4.1 Suppose that [MATH] . Then [EQUATION] by Lemma A.1 (ii) and ( 56 ) applied to the sub-models [MATH] and [MATH] in place of [MATH] . Then (i) holds since [MATH] |
Next, suppose that [MATH] and [MATH] . Obviously, [MATH] . Denote by [MATH] the matrix consisting of the elements of [MATH] with indices in [MATH] . Then [MATH] implies [MATH] . Thus, we can obtain the same results as in Theorem 3.1 for [MATH] . In particular, |
[EQUATION] for both QMLE and QBE for the model [MATH] since [MATH] , where [MATH] . This is also valid for the model [MATH] with |
[MATH] Therefore, [EQUATION] as [MATH] , which completes the proof. Appendix B List of stocks Table lists all the stocks investigated in the paper. |
# Source: arxiv 1805.08550 # Title: Anticipating cryptocurrency prices using machine learning # Sections: all # Downloaded: 2026-03-03T04:48:07.894172+00:00 |
Anticipating cryptocurrency prices using machine learning Abstract Machine learning and AI-assisted trading have attracted growing interest for the past few years. Here, we use this approach to test the hypothesis that the inefficiency of the cryptocurrency market can be exploited to generate abnormal profits. We analy... |
Introduction The popularity of cryptocurrencies has skyrocketed in 2017 due to several consecutive months of super-exponential growth of their market capitalisation |
, which peaked at more than [MATH] billions in Jan. 2018. Today, there are more than [MATH] actively traded cryptocurrencies. Between [MATH] and [MATH] millions of private as well as institutional investors are in the different transaction networks, according to a recent survey |
, and access to the market has become easier over time. Major cryptocurrencies can be bought using fiat currency in a number of online exchanges (e.g., Binance |
, Upbit , Kraken , etc) and then be used in their turn to buy less popular cryptocurrencies. The volume of daily exchanges is currently superior to [MATH] billions. Since 2017, over [MATH] hedge funds specialised in cryptocurrencies have emerged and bitcoin futures have been launched to address institutional demand for... |
The market is diverse and provides investors with many different products. Just to mention a few, Bitcoin was expressly designed as a medium of exchange |
; Dash offers improved services on top of Bitcoin’s feature set, including instantaneous and private transactions ; Ethereum is a public, blockchain-based distributed computing platform featuring smart contract (scripting) functionality, and Ether is a cryptocurrency whose blockchain is |
; Ripple is a real-time gross settlement system (RTGS), currency exchange and remittance network Ripple , and IOTA is focused on providing secure communications and payments between agents on the Internet of Things |
The emergence of a self-organised market of virtual currencies and/or assets whose value is generated primarily by social consensus |
has naturally attracted interest from the scientific community . Recent results have shown that the long-term properties of the cryptocurrency marked have remained stable between 2013 and 2017 and are compatible with a scenario in which investors simply sample the market and allocate their money according to the crypto... |
. While this is true on average, various studies have focused on the analysis and forecasting of price fluctuations, using mostly traditional approaches for financial markets analysis and prediction |
The success of machine learning techniques for stock markets prediction , suggests that these methods could be effective also in predicting cryptocurrencies prices. However, the application of machine learning algorithms to the cryptocurrency market has been limited so far to the analysis of Bitcoin prices, using rando... |
, Bayesian neural network , long short-term memory neural network and other algorithms . These studies were able to anticipate, to different degrees, the price fluctuations of Bitcoin, and revealed that best results were achieved by neural network based algorithms. Deep reinforcement learning was showed to beat the uni... |
in predicting the prices of [MATH] cryptocurrencies over one year period Other attempts to use machine learning to predict the prices of cryptocurrencies other than Bitcoin come from non-academic sources |
. Most of these analyses focused on a limited number of currencies and did not provide benchmark comparisons for their results. Here, we test the performance of three models in predicting daily cryptocurrency price for 1,681 currencies. Two of the models are based on gradient boosting decision trees |
and one is based on long short-term memory (LSTM) recurrent neural networks . In all cases, we build investment portfolios based on the predictions and we compare their performance in terms of return on investment. We find that all of the three models perform better than a baseline ‘simple moving average’ model |
where a currency’s price is predicted as the average price across the preceding days, and that the method based on long short-term memory recurrent neural networks systematically yields the best return on investment. |
The article is structured as follows: In section Materials and Methods we describe the data (see Data description and pre-processing ), the metrics characterizing cryptocurrencies that are used along the paper (see Metrics ), the forecasting algorithms (see Forecasting algorithms ), and the evaluation metrics (see Eval... |
Materials and Methods Data description and pre-processing Cryptocurrency data was extracted from the website Coin Market Cap , collecting daily data from [MATH] exchange markets platforms starting in the period between November 11, 2015 and April 24, 2018. The dataset contains the daily price in U.S. dollars, the marke... |
Fig. shows the number of currencies with trading volume larger than [MATH] over time, for different values of [MATH] . In the following sections, we consider that only currencies with daily trading volume higher than [MATH] USD can be traded at any given day. |
The website lists cryptocurrencies traded on public exchange markets that have existed for more than 30 days and for which an API as well as a public URL showing the total mined supply are available. Information on the market capitalization of cryptocurrencies that are not traded in the 6 hours preceding the weekly rel... |
Metrics Cryptocurrencies are characterised over time by several metrics, namely Price, The exchange rate, determined by supply and demand dynamics. |
Market capitalization, The product of the circulating supply and the price. Market share, The market capitalization of a currency normalized by the total market capitalization. |
Rank, The rank of currency based on its market capitalization. Volume, Coins traded in the last 24 hours. Age, Lifetime of the currency in days. |
The profitability of a currency [MATH] over time can be quantified through the return on investment (ROI), measuring the return of an investment made at day [MATH] relative to the cost |
. The index [MATH] rolls across days and it is included between [MATH] and [MATH] , with [MATH] January 1, 2016, and [MATH] April 24, 2018. Since we are interested in the short-term performance, we consider the return on investment after [MATH] day defined as |
[EQUATION] In Fig. , we show the evolution of the [MATH] over time for Bitcoin (orange line) and on average for currencies whose volume is larger than [MATH] USD at [MATH] (blue line). In both case, the average return on investment over the period considered is larger than [MATH] , reflecting the overall growth of the ... |
Forecasting algorithms We test and compare three supervised methods for short-term price forecasting. The first two methods rely on XGboost |
, an open-source scalable machine learning system for tree boosting used in a number of winning Kaggle solutions (17/29 in 2015) |
. The third method is based on the long short-term memory (LSTM) algorithm for recurrent neural networks that have demonstrated to achieve state-of-the-art results in time-series forecasting |
Method 1: The first method considers one single regression model to describe the change in price of all currencies (see Fig. ). The model is an ensemble of regression trees built by the XGboost algorithm. The features of the model are characteristics of a currency between time [MATH] and [MATH] and the target is the RO... |
). The features for the regression are built across the window between [MATH] and [MATH] included (see Fig. ). Specifically, we consider the average, the standard deviation, the median, the last value and the trend (e.g. the difference between last and first value) of the properties listed above. In the training phase,... |
Method 2: Also the second method relies on XGboost, but now the algorithm is used to build a different regression model for each currency [MATH] (see Fig. |
). The features of the model for currency [MATH] are the characteristics of all the currencies in the dataset between [MATH] and [MATH] included and the target is the ROI of [MATH] at day [MATH] (i.e., now the algorithm learns to predict the price of the currency [MATH] based on the features of all the currencies in th... |
Method 3: The third method is based on Long Short Term Memory networks, a special kind of Recurrent Neural Networks, capable of learning long-term dependencies. As for Method 2, we build a different model for each currency. Each model predicts the ROI of a given currency at day [MATH] based on the values of the ROI of ... |
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