This distance between levels is defined as a percentage of the optimal spread calculated by the strategy. Given that optimal spreads tend to be tight, the level_distances values should be in general in tens or hundreds of percents. Now, as another extension of a stock price impact on optimal market making problem, we work on the problem that the stochastic volatility of the asset is affected by the arrival of market orders and perform this case on the optimal trading prices. The models in literature assume that the stock price is followed by mostly Brownian motion with constant volatility, and in the cases of stochastic volatility, they consider only a diffusive volatility. Picking up the volatility as diffusive is highly permanent but this dynamics can allow the increments to increase only by a sequence of normally distributions. To start, we set up a high-frequency trading model in order to gain from the expected profit by building trading strategies on limit buy and sell orders.

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. The performances of Sharpe ratios of each models indicates that the stock price models with stochastic volatility based on a quadratic utility function produces more attractive portfolios than the other models. It is demonstrated that the Model d has a Gaussian normal distribution while the others are positively skewed. 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.

## About this article

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. In Section 2, we introduce some basic concepts and describe the input LOB datasets. On the P&L-to-MAP ratio, Alpha-AS-1 was the best-performing model for 11 test days, with Alpha-AS-2 coming second on 9 of them, whereas Alpha-AS-2 was the best-performing model on P&L-to-MAP for 16 of the test days, with Alpha-AS-1 coming second on 14 of these. Here the single best-performing model was Alpha-AS-2, winning for 16 days and coming second on 10 (on 9 of which losing to Alpha-AS-1).

Extensive data recovery experiments are conducted on two real industrial processes to evaluate the proposed method in comparison with existing state-of-the-art restorers. The results show that the proposed methods can impute better with different missing rates and have strong competitiveness in practical application. Reinforcement learning algorithms have been shown to be well-suited for use in high frequency trading contexts [16, 24–26, 37, 45, 46], which require low latency in placing orders together with a dynamic logic that is able to adapt to a rapidly changing environment. 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. These models, therefore, must learn everything about the problem at hand, and the learning curve is steeper and slower to surmount than if relevant available knowledge were to be leveraged to guide them.

## High-frequency trading and market performance

We relied on random forests to filter state-defining features based on their importance according to three indicators. Various techniques are worth exploring in future work for this purpose, such as PCA, Autoencoders, Shapley values or Cluster Feature Importance . Other modifications to the neural network architectures presented here may prove advantageous. We mention neuroevolution to train the neural network using genetic algorithms and adversarial networks to improve the robustness of the market making algorithm. In the training phase we fit our two Alpha-AS models with data from a full day of trading .

To minimize inventory risk, prices should be skewed to favor the inventory to come back to its targeted ideal balance point. The closer the risk_factor is to zero, the more symmetrical will be orders will be created, and the Reservation price will be pretty much equal to the market mid price. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Using the exponential utility function and the results are provided for the following models.

## IEEE Transactions on Knowledge and Data Engineering

The theory encourages the study of optimizing activities in financial markets as it allows to accomplish the complex optimization problems involving constraints that are consistent with the price dynamics while managing the inventory risk. In order to detect the optimal quotes in the market, it is, therefore, necessary to solve the corresponding nonlinear Hamilton-Jacobi-Bellman equation for the optimal stochastic control problem. This is generally achieved by applying various root-finding algorithms that can handle the complexity and high-dimensionality of the equation.

- Data normalization for features and labeling for signals are required for classification.
- While the market maker wants to maximize her profit from the transactions over a finite time horizon, she also wants to keep her inventories under control and get rid of the remaining inventories at the final time T by the penalization terms.
- For the case of a quadratic utility function, we derive the optimal spreads for limit orders and observe their behaviors.

Machine learning approaches have been explored to obtain dynamic limit order placement strategies that attempt to adapt in real time to changing market conditions. As regards market making, the AS algorithm, or versions of it , have been used as benchmarks against which to measure the improved performance of the machine learning algorithms proposed, either working with simulated data or in backtests with real data. The literature on machine learning approaches to market making is extensive. In 2008, Avellaneda and Stoikov published a procedure to obtain bid and ask quotes for high-frequency market-making trading .

## Related Papers

Single feature importance , an out-of-sample estimator of the individual importance of each feature, that avoids the substitution effect found with MDI and MDA GAL . S′ is the state the MDP has transitioned to when taking action a from state s, to which it arrived at the previous iteration. R is the latest reward obtained from state s by taking action a. The procedure, therefore, has two steps, which are applied at each time increment as follows. The description below is a general approximation of this strategy.

For even moderately large numbers of states and avellaneda stoikovs, let alone when the state space is practically continuous , it becomes computationally prohibitive to maintain a Qs,a matrix and iteratively to get the values contained in it to converge to the optimal Q-value estimates. To overcome this problem, a deep Q-network approximates the Qs,a matrix using a deep neural network. The DQN computes an approximation of the Q-values as a function, Q(s, a, θ), of a parameter vector, θ, of tractable size. To train a DQN is to let it evolve the values of these internal parameters based on the agent’s experiences acting in its environment, so that the value function approximated with them maps the input state to Q-values that increasingly approach the optimal Q-values for that state. There are various methods to achieve this, a particularly common one being gradient descent. What is common to all the above approaches is their reliance on learning agents to place buy and sell orders directly.

Moreover, the spread can also be considered to be normally distributed due to its skewness and kurtosis values. Increment means that more buy market orders arrived and are filled by sell orders which causes larger spreads. Similar to the proof of Proposition2, the optimal spreads can be found by the first order optimality conditions.

3.2.1, we consider the case of the jumps in https://www.beaxy.com/ of the price. The paper is also equipped with an Appendix on how to use the method of finite differences for the numerical solution of the corresponding nonlinear differential equation. Likert-type scales are commonly used in both academia and industry to capture human feelings since they are user-friendly, easy-to-develop and easy-to administer.

That is, these agents decide the bid and ask prices of their orderbook quotes at each execution step. The main contribution we present in this paper resides in delegating the quoting to the mathematically optimal Avellaneda-Stoikov procedure. What our RL algorithm determines are, as we shall see shortly, the values of the main parameters of the AS model. It is then the latter that calculates the optimal bid and ask prices at each step.

### Should you hedge or should you wait? – Risk.net

Should you hedge or should you wait?.

Posted: Wed, 24 Aug 2022 07:00:00 GMT [source]