However, adding secure points to a WANET can be costly in terms of price and time, so minimizing the number of secure points is of utmost importance. Graph theory provides a great foundation to tackle the emerging problems in WANETs. A vertex cover is a set of vertices where every edge is incident to at least one vertex. The minimum weighted connected VC problem can be defined as finding the VC of connected nodes having the minimum total weight. MWCVC is a very suitable infrastructure for energy-efficient link monitoring and virtual backbone formation.

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. Therefore, the corresponding HJB equation can be obtained by applying the stochastic control approach.

4 Comparison with existing models

Double DQN is a deep RL approach, more specifically deep Q-learning, that relies on two neural networks, as we shall see shortly (in Section 4.1.7). In this paper we present a double DQN applied to the market-making decision process. The RL agents (Alpha-AS) developed to use the Avellaneda-Stoikov equations to determine their actions are described in Section 4.1. An agent that simply applies the Avellaneda-Stoikov procedure with fixed parameters (Gen-AS), and the genetic algorithm to obtain said parameters, are presented in Section 4.2. In this section, we compare the existing optimal market making models based on the stock price impacts with the models that we introduce in the previous sections.

This is obtained from the algorithm’s P&L, discounting the losses from speculative positions. The Asymmetric dampened P&L penalizes speculative positions, as speculative profits are not added while losses are discounted. The procedure, therefore, has two steps, which are applied at each time increment as follows.

High-Frequency Trading Meets Online Learning

These formulas prescribe the AS strategy for placing limit orders. The rationale behind the strategy is, in Avellaneda and Stoikov’s words, to perform a ‘balancing act between the dealer’s personal risk considerations and the market environment’ [ibid.]. In electronic markets, any trader can become a market maker who provides the liquidity to the markets in Limit Order Books ; and market makers are allowed to submit the orders on both buy and sell sides of the market by the trading mechanisms. Deciding for the best bid and ask prices that a market maker sets up is a hard and complex problem in many aspects due to the fact that the problem should be tackled as a combined problem of the modeling the asset price dynamics and the optimal spreads. Fortunately, the stochastic control theory helps to handle such kind of optimization problem by seeking an optimal strategy in order to maximize the trader’s objective function and to face a dyadic problem for the high-frequency trading. 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.

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.

High frequency trading and the new market makers

It can then start exploiting this knowledge to apply an action selection policy that takes it closer to achieving its reward maximization goal. However, I do not see any specification of bounds for this reservation price and therefore I think there is no guarantee that ask prices computed by the market-maker will be higher or bid prices will be lower than the current price of the process. To put it simply, as the trading session is nearing the end, the reservation price will approach the market mid-price, reducing the risk of holding the inventory too far from the desired target. Allows your bid and ask order prices to be adjusted based on the current top bid and ask prices in the market. This parameter will be the limit time for this “trading cycle”. We call trading cycles the interval of time where spreads start the widest possible and end up the smallest.

• 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.
• Mann-Whitney tests comparing the four daily performance indicator values (Sharpe, Sortino, Max DD and P&L-to-MAP) obtained for the Gen-AS model with the corresponding values obtained for the other models, over the 30 test days.
• 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.
• Both Alpha-AS models performed better than the rest on 19 days.
• The DQN computes an approximation of the Q-values as a function, Q(s, a, θ), of a parameter vector, θ, of tractable size.

In this paper, we investigated the high-frequency trading strategies for a market maker using a mean-reverting stochastic volatility models that involve the influence of both arrival and filled market orders of the underlying asset. First, we design a model with variable utilities where the effects of the jumps corresponding to the orders are introduced in returns of the asset and generate optimal bid and ask prices for trading. Then, we develop another, but novel, approach considering an underlying asset model with jumps in stochastic volatility. Such an extension allows one to fit the implied volatility smile better in practice.

Nevertheless, it is still interesting to note that AS-avellaneda & stoikov performs much better on this indicator than on the others, relative to the Alpha-AS models. This means that, provided its parameter values describe the market environment closely enough, the pure AS model is guaranteed to output the bid and ask prices that minimise inventory risk, and any deviation from this strategy will entail a greater risk. Throughout a full day of trading, it is more likely than within shorter time frames that there will be intervals at which the market is indeed closely matched by the AS formula parameters.

The agent can also skew the bid and ask prices output by the Avellaneda-Stoikov procedure, tweaking them and, by so doing, potentially counteract the limitations of a static Avellaneda-Stoikov model by reacting to local market conditions. The agent learns to adapt its risk aversion and skew its bid and ask prices under varying market behaviour through reinforcement learning using two variants (Alpha-AS-1 and Alpha-AS-2) of a double DQN architecture. The central notion is that, by relying on a procedure developed to minimise inventory risk (the Avellaneda-Stoikov procedure) by way of prior knowledge, the RL agent can learn more quickly and effectively.

The greater inventory risk taken by the Alpha-AS models during such intervals can be punished with greater losses. Conversely, the gains may also be greater, a benefit which is indeed reflected unequivocally in the results obtained for the P&L-to-MAP performance indicator. The latter is an important feature for market maker algorithms. Indeed, this result is particularly noteworthy as the Avellaneda-Stoikov method sets as its goal precisely to minimize the inventory risk. Nevertheless, the flexibility that the Alpha-AS models are given to move and stretch the bid and ask price spread entails that the Alpha-AS models can, and sometimes do, operate locally with higher risk. Overall performance is more meaningfully obtained from the other indicators (Sharpe, Sortino and P&L-to-MAP), which show that, at the end of the day, the Alpha-AS models’ strategy pays off.

Furthermore, we design an effective optimization algorithm based on alternating direction minimization to solve the model of OMBG. Extensive experiments performed on four frequently-used benchmark multi-view datasets illustrate the superiority of OMBG which is compared with some state-of-the-art clustering baselines. Wireless ad hoc networks are infrastructureless networks BTC and are used in various applications such as habitat monitoring, military surveillance, and disaster relief.

To improve stability, a stores its experiences in a replay buffer, in terms of the value function given by Eq , where now the Q-value estimates are not stored in a matrix but obtained as the outputs of the neural network, given the current state as its input. The DQN then learns periodically, with batches of random samples drawn from the replay buffer, thus covering more of the state space, which accelerates the learning while diminishing the influence of single or of correlated experiences on the learning process. These successes with games have attracted attention from other areas, including finance and algorithmic trading.

Using the exponential https://www.beaxy.com/ and the results are provided for the following models. Increment means that more buy market orders arrived and are filled by sell orders which causes larger spreads. For a fixed inventory level q and a representation of the asset volatility which are obtained from one simulation.