PDF Optimal High-Frequency Market Making


Once the cycle is reset, spreads will start again, being the widest possible. This parameter, denoted by the letter gamma, is related to the aggressiveness when setting the spreads to achieve the inventory target. It is directly proportional to the asymmetry between the bid and ask spread. The Avellaneda Market Making Strategy is designed to scale inventory and keep it at a specific target that a user defines it with.

avellaneda & stoikovimum drawdown registers the largest loss of portfolio value registered between any two points of a full day of trading. Similarly, on the Sortino ratio, one or the other of the two Alpha-AS models performed better, that is, obtained better negative risk-adjusted returns, than all the baseline models on 25 (12+13) of the 30 days. Again, on 9 of the 12 days for which Alpha-AS-1 had the best Sharpe ratio, Alpha-AS-2 had the second best; and for 10 of the 13 test days for which after Alpha-AS-2 obtained the best Sortino ratio, Alpha-AS-1 performed second best. Both Alpha-AS models performed better than the rest on 19 days.

Simplified Avellaneda-Stoikov Market Making

Meanwhile, AS-, again the best of the rest, won on Sortino on only 3 test days. The mean and the median of the Sortino ratio were better for both Alpha-AS models than for the Gen-AS model , and for the latter it was significantly better than for the two non-AS baselines. Thus, the Alpha-AS models came 1st and 2nd on 20 out of the 30 test days (67%). The btc-usd data for 7th December 2020 was used to obtain the feature importance values with the MDI, MDA and SFI metrics, to select the most important features to use as input to the Alpha-AS neural network model. The data for the first use of the genetic algorithm was the full day of trading on 8th December 2020. Our algorithm works through 10 generations of instances of the AS model, which we will refer to as individuals, each with a different chromosomal makeup .

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The more specific context of market making has its own peculiarities. DRL has been used generally to determine the actions of placing bid and ask quotes directly [23–26], that is, to decide when to place a buy or sell order and at what price, without relying on the AS model. Spooner proposed a RL system in which the agent could choose from a set of 10 spread sizes on the buy and the sell side, with the asymmetric dampened P&L as the reward function (instead of the plain P&L). Combining a deep Q-network (see Section 4.1.7) with a convolutional neural network , Juchli achieved improved performance over previous benchmarks.

Appendix: Numerical Solution of the Optimal Stochastic Control Problem

If published, this will include your full peer review and any attached files. The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception . The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. Participant privacy or use of data from a third party—those must be specified. In view of the referees’ feedback and my own reading of your paper, I invite you to address all issues noted below.

  • Although 2 reviewers consider that the manuscript is suitable of publication in its current stand, one of the reviewers still show some concerns that need to be addressed before to deserve this manuscript for publication.
  • In the literature, reinforcement learning approaches to market making typically employ models that act directly on the agent’s order prices, without taking advantage of knowledge we may have of market behaviour or indeed findings in market-making theory.
  • Figures in parenthesis are the number of days the Alpha-AS model in question was second best only to the other Alpha-AS model (and therefore would have computed another overall ‘win’ had it competed alone against the baseline and AS-Gen models).
  • For more developments in optimal market making literature, we refer the reader to Guéant , Ahuja et al. , Cartea et al. , Guéant and Lehalle , Nyström and Guéant et al. .
  • Meanwhile, AS-Gen, again the best of the rest, won on Sortino on only 3 test days.

It is demonstrated that the Model d has a Gaussian normal distribution while the others are positively skewed. Low-rank approximation algorithms aim to utilize convex nuclear norm constraint of linear matrices to recover ill-conditioned entries caused by multi-sampling rates, sensor drop-out. However, LINK these existing algorithms are often limited in solving high-dimensionality and rank minimization relaxation. In this paper, a robust kernel factorization embedding graph regularization method is developed to statically impute missing measurements. Specifically, the implicit high-dimensional feature space of ill-conditioned data is factorized by kernel sparse dictionary.

A value close to 1 will indicate that you don’t want to take too much https://www.beaxy.com/ risk, and hummingbot will “push” the reservation price more to reach the inventory target. This potential weakness of the analytical AS approach notwithstanding, we believe the theoretical optimality of its output approximations is not to be undervalued. On the contrary, we find value in using it as a starting point from which to diverge dynamically, taking into account the most recent market behaviour. With the above definition of our Alpha-AS agent and its orderbook environment, states, actions and rewards, we can now revisit the reinforcement learning model introduced in Section (4.1.2) and specify the Alpha-AS RL model.

  • One way to improve the performance of an AS model is by tweaking the values of its constants to fit more closely the trading environment in which it is operating.
  • Avellaneda and Stoikov have revised the study of Ho and Stoll building a practical model that considers a single dealer trading a single stock facing with a stochastic demand modeled by a continuous time Poisson process.
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  • PhD Thesis, The London School of Economics and Political Sciences.
  • The successive orders generated by this procedure maximize the expected exponential utility of the trader’s profit and loss (P&L) profile at a future time, T , for a given level of agent inventory risk aversion.
  • But this kind of approach, depending on the market situation, might lead to market maker inventory skewing in one direction, putting the trader in a wrong position as the asset value moves against him.

The prediction DQN receives as input the state-defining features, with their values normalised, and it outputs a value between 0 and 1 for each action. The DQN has two hidden layers, each with 104 neurons, all applying a ReLu activation function. An ε-greedy policy is followed to determine the action to take during the next 5-second window, choosing between exploration , with probability ε, and exploitation , with probability 1-ε.

With these values, the AS model will determine the next reservation price and spread to use for the following orders. In other words, we do not entrust the entire order placement decision process to the RL algorithm, learning through blind trial and error. Rather, taking inspiration from Teleña , we mediate the order placement decisions through the AS model (our “avatar”, taking the term from ), leveraging its ability to provide quotes that maximize profit in the ideal case.

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By our numerical results, we deduce that the jump effects and comparative statistics metrics provide us with the information for the traders to gain expected profits. For instance, the model given by has a considerable Sharpe ratio and inventory management with a lower standard deviation comparing to the symmetric strategy. Besides, we further quantify the effects of a variety of parameters in models on the bid and ask spreads and observe that the trader follows different strategies on positive and negative inventory levels, separately. The strategy derived by the model , for instance, illustrates that when time is approaching to the terminal horizon, the optimal spreads converge to a fixed, constant value. Furthermore, in case of the jumps in volatility, it is observed that a higher profit can be obtained but with a larger standard deviation.

Again, the probability of selecting a specific individual for parenthood is proportional to the Sharpe ratio it has achieved. A weighted average of the values of the two parents’ genes is then computed. Private indicators, consisting of features describing the state of the agent. We model the market-agent interplay as a Markov Decision Process with initially unknown state transition probabilities and rewards.

However, tree outputs may be unreliable in presence of scarce data. The imprecise Dirichlet model provides workaround, by replacing point probability estimates with interval-valued ones. This paper investigates a new tree aggregation method based on the theory of belief functions to combine such probability intervals, resulting in a cautious random forest classifier.


This parameter denoted in the letter eta is related to the aggressiveness when setting the order amount to achieve the inventory target. It is inversely proportional to the asymmetry between the bid and ask order amount. Table11 which is obtained from all simulations depicts the results of these two strategies. We can see that when the jumps occur in volatility, it causes not only larger profits but also larger standard deviation of the profit and loss function. This is a small inventory-risk aversion value but is enough to force the inventory process to revert to zero at the end of the trading. With the same assumptions and quadratic utility function as in Case 1 in Sect.

how much is 1 eth

At each training step the parameters of the prediction DQN are updated using gradient descent. An early stopping strategy is followed on 25% of the training sets to avoid overfitting. The architecture of the target DQN is identical to that of the prediction DQN, the parameters of the former being copied from the latter every 8 hours. Balancing exploration and exploitation NEAR advantageously is a central challenge in RL. Γd is a discount factor (γd∈) by which future expected rewards are given less weight in the current Q-value than the latest observed reward.

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