Optimal Portfolio Liquidation with Limit Orders


Our model describes the intraday price and variance of the asset as a mean-reverting process. Through our choice of ansatz solution, we reduce the problem BTC from 5 to 3 dimensions. Furthermore, we show that our numerical scheme is stable and converges to the viscosity solution. We compare our model to the original model and a simple strategy, which pegs quotes at a fixed distance from the best bid and ask prices. Our back-test results show that the market making models significantly outperform the simple strategy concerning the certainty-equivalent value and inventory control.

Then the future stock prices are predicted as a nonlinear mapping of the combination of these components in an Inverse Fourier Transform fashion. Stay informed on the latest trending ML papers with code, research developments, libraries, methods, and datasets. If your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they’ll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication.

In the framework of the optimal trading strategy for high-frequency trading in a LOB, there have been many papers following early studies of Grossman and Miller and Ho and Stoll . 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. The literature on the optimal market making problem has been burgeoning since 2008 with the work of Avellaneda and Stoikov , inspiring Guilbaud and Pham to derive a model involving limit and market orders with optimal stochastic spreads. Bayraktar and Ludkovski have considered the optimal liquidation problem where they model the order arrivals with intensities depending on the liquidation price.


This consideration makes rb and ra reasonable reference prices around which to construct the market maker’s spread. Avellaneda and Stoikov define rb and ra, however, for a passive agent with no orders in the limit order book. In practice, as Avellaneda and Stoikov did in their original paper, when an agent is running and placing orders both rb and ra ra are approximated by the average of the two, r . Table12 obtained from all simulations illustrates that the traders using the Model c have relatively higher return but also relatively a higher standard deviation comparing to other models.

Hence, the optimal spreads which maximize the supremums in the verification Eq. Of exists and is unique that should be guaranteed by the verification theorem so that this classical solution is the value function of the HJB equation and the spreads, defined by , are indeed the optimal ones. Overall, both Alpha-AS models obtain higher and more stable returns, as well as a better P&L-to-inventory profile than AS-Gen and the non-AS baseline models. That is, they achieve a better P&L profile with less exposure to market movements. Conversely, test days for which the Alpha-ASs did worse than Gen-AS on P&L-to-MAP in spite of performing better on Max DD are highlighted in red (Alpha-AS “worse”).

Dealing with the inventory risk: a solution to the market making problem

To start filling Alpha-AS memory replay buffer and training the model (Section 5.2). Therefore, by choosing a Skew value the Alpha-AS agent can shift the output price upwards or downwards by up to 10%. Mean decrease impurity , a feature-specific measure of the mean reduction of weighted impurity over all the nodes in the tree ensemble that partition the data samples according to the values of that feature . Market indicators, consisting of features describing the state of the environment. Thus, the DQN approximates a Q-learning function by outputting for each input state, s, a vector of Q-values, which is equivalent to checking the row for s in a Qs,a matrix to obtain the Q-value for each action from that state.

Our community is full of market makers and arbitrageurs who are willing to help each other make the best use of Hummingbot. You can join our Discord channel to talk about the hummingbot, strategies, liquidity mining, and anything else related to the cryptocurrency world and receive direct support from our team. So, as the trading session is getting closer to the end, order spreads will be smaller, and the reservation price position will be more “aggressive” on rebalancing the inventory.

Two Models on Limit Order Trading

Moreover, in practice the importance of being able to get out with back of queue orders is very important and is completely exogenous to the model. 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. The combination of the choice of one from among four available values for γ, with the choice of one among five values for the skew, consequently results in 20 possible actions for the agent to choose from, each being a distinct (γ, skew) pair. We chose a discrete action space for our experiment to apply RL to manipulate AS-related parameters, aiming keep the algorithm as simple and quickly trainable as possible. A continuous action space, as the one used to choose spread values in , may possibly perform better, but the algorithm would be more complex and the training time greater. In its beginner mode, the user will be asked to enter min and max spread limits, and it’s aversion to inventory risk scaled from 0 to 1 .

On the optimal quotes will have just the opposite effect of when k is employed. While we do not change the rest of the parameters in Table1 and we observe our expectations in solutions which can be tracked by Table8, in coherence with . While keeping the other parameters same as in the Table1, our above expectation matches with the solutions obtained and be seen Table7. In order to see the time evolution of the process for larger inventory bounds. For a fixed inventory level q and a representation of the asset volatility which are obtained from one simulation.


In that case, the original article is easy to find on a quick internet search, or you can find the original publication here. This article will explain the idea behind the classic paper released by Marco Avellaneda and Sasha Stoikov in 2008 and how we implemented it in Hummingbot.

If γ value is close to zero, the reservation price will be very close to the market mid-price. Therefore, the trader will have the same risk as if he was using the symmetrical price strategy. 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. 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. It sets a target of base asset balance in relation to a total asset allocation value .

For each subsequent avellaneda-stoikov modeleration 45 new individuals run through the data and then added to the cumulative population, retaining all the individuals from previous generations. The 10 generations thus yield a total of 450 individuals, ranked by their Sharpe ratio. Note that, since we retain all individuals from generation to generation, the highest Sharpe ratio the cumulative population never decreases in subsequent generations. 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. The selected action is then taken repeatedly, once every market tick, in the following 5-second window, at the end of which the reward (the Asymmetric Dampened P&L) obtained from this repeated execution of the action is computed. The data on which the metrics for our market features were calculated correspond to one full day of trading .


The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented. So, if we’re modeling more or less the same things, all good/accurate models tend to be analogous; they must correspond to one another if they each correspond to the same underlying physical phenomena. Comparison of values for Max DD and P&L-to-MAP between the Gen-AS model and the Alpha-AS models (αAS1 and αAS2).

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Where the 0 subscript denotes the best avellaneda-stoikov model price level on the ask and on the bid side, i.e., the price levels of the lowest ask and of the highest bid, respectively. The procedure, therefore, has two steps, which are applied at each time increment as follows. If you want to end the trading session with your entire inventory allocated to USDT, you set this value to 0.

The selection of https://www.beaxy.com/ based on these three metrics reduced their number from 112 to 22 . The features retained by each importance indicator are shown in Table 1. The target for the random forest classifier is simply the sign of the difference in mid-prices at the start and the end of each 5-second timestep. That is, classification is based on whether the mid-price went up or down in each timestep. The ranges of possible values of the features that are defined in relation to the market mid-price, are truncated to the interval [−1, 1] (i.e., if a value exceeds 1 in magnitude, it is set to 1 if it is positive or -1 if negative).

  • We introduce an expert deep-learning system for limit order book trading for markets in which the stock tick frequency is longer than or close to 0.5 s, such as the Chinese A-share market.
  • We were able to achieve some parallelisation by running five backtests simultaneously on different CPU cores.
  • Based on the market state and the agent’s private indicators (i.e., its latest inventory levels and rewards), a prediction neural network outputs an action to take.
  • To minimize inventory risk, prices should be skewed to favor the inventory to come back to its targeted ideal balance point.
  • That is, these agents decide the bid and ask prices of their orderbook quotes at each execution step.

Decentralized exchanges allow parties to participate in financial markets while retaining full custody of their funds. For instance, even after comments about reference formatting, some references have missing publications, years, issues, or even author names . Also, there seems to be a large number of arxiv or SSRN preprints listed for references which are actually published, either as working papers by some institutions or even in peer reviewed journals . Some of these will most likely be handled by the editorial team, but the extent of the errors is too large, evidently due to the revisions made by authors being mostly superficial. In general, the legibility of the paper is hardly improved, and the revisions in this regards were mostly superficial. The reviewer can point in the directions and give some examples but it is simply impossible to list all of the specific details, and it should be on the authors to check the manuscript in detail.

Ensure you ETH have enough quote and base tokens to place the bid and ask for orders. The strategy will not place any orders if you do not have sufficient balance on either side of the order. In expert mode, the user will need to directly define the algorithm’s basic parameters described in the foundation paper, and no recalculation of parameters will happen. A Hamilton Jacobi Bellman approach to optimal trade execution.

More advanced models have been developed with adverse selection effects and stronger market order dynamics, see for example the paper of Cartea et al. . Guéant et al. have extended and formalized the results of Avellaneda and Stoikov . Another extended market making model with inventory constraints has been provided by Fodra and Labadie who consider a general case of midprice by linear and exponential utility criteria and find closed-form solutions for the optimal spreads. Cartea and Jaimungal have proposed a solution to deal with the problem of including the market impact on the midprice and have worked on risk metrics for the high-frequency trading strategies they have developed.

Together, a) and b) result in a set of 2×10d contiguous buckets of width 10−d, ranging from −1 to 1, for each of the features defined in relative terms. Approximately 80% of their values lie in the interval [−0.1, 0.1], while roughly 10% lie outside the [−1, 1] interval. Values that are very large can have a disproportionately strong influence on the statistical normalisation of all values prior to being inputted to the neural networks.

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The two most important features for all three methods are the latest bid and ask quantities in the orderbook , followed closely by the bid and ask quantities immediately prior to the latest orderbook update and the latest best ask and bid prices . There is a general predominance of features corresponding to the latest orderbook movements (i.e., those denominated with low numerals, primarily 0 and 1). This may be a consequence of the markedly stochastic nature of market behaviour, which tends to limit the predictive power of any feature to proximate market movements. Nevertheless, the prices 4 and 8 orderbook movements prior the action setting instant also make fairly a strong appearance in the importance indicator lists , suggesting the existence of slightly longer-term predictive component that may be tapped into profitably.

Since the distance between two consecutive items cannot be either defined or presumed equal, this kind of variable cannot be analysed by either statistical methods defined on a metric space or parametric tests. Therefore, Likert-type variables cannot be used as segmentation variables of a traditional cluster analysis unless pre-transformed. In such context, fuzzy numbers have been suggested as a way to recode Likert-type variables. Fuzzy numbers are defined by a membership function whose form is usually determined by an expert. In practice, researchers usually define one membership function for each Likert-type scale, not considering the peculiar characteristics of neither questions nor respondents.

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