Market Making

Logarithmic regret in the ergodic Avellaneda-Stoikov market making model ArXiv ID: 2409.02025 “View on arXiv” Authors: Unknown Abstract We analyse the regret arising from learning the price sensitivity parameter $κ$ of liquidity takers in the ergodic version of the Avellaneda-Stoikov market making model. We show that a learning algorithm based on a maximum-likelihood estimator for the parameter achieves the regret upper bound of order $\ln^2 T$ in expectation. To obtain the result we need two key ingredients. The first is the twice differentiability of the ergodic constant under the misspecified parameter in the Hamilton-Jacobi-Bellman (HJB) equation with respect to $κ$, which leads to a second–order performance gap. The second is the learning rate of the regularised maximum-likelihood estimator which is obtained from concentration inequalities for Bernoulli signals. Numerical experiments confirm the convergence and the robustness of the proposed algorithm. ...

Consistent time travel for realistic interactions with historical data: reinforcement learning for market making ArXiv ID: 2408.02322 “View on arXiv” Authors: Unknown Abstract Reinforcement learning works best when the impact of the agent’s actions on its environment can be perfectly simulated or fully appraised from available data. Some systems are however both hard to simulate and very sensitive to small perturbations. An additional difficulty arises when a RL agent is trained offline to be part of a multi-agent system using only anonymous data, which makes it impossible to infer the state of each agent, thus to use data directly. Typical examples are competitive systems without agent-resolved data such as financial markets. We introduce consistent data time travel for offline RL as a remedy for these problems: instead of using historical data in a sequential way, we argue that one needs to perform time travel in historical data, i.e., to adjust the time index so that both the past state and the influence of the RL agent’s action on the system coincide with real data. This both alleviates the need to resort to imperfect models and consistently accounts for both the immediate and long-term reactions of the system when using anonymous historical data. We apply this idea to market making in limit order books, a notoriously difficult task for RL; it turns out that the gain of the agent is significantly higher with data time travel than with naive sequential data, which suggests that the difficulty of this task for RL may have been overestimated. ...

The Negative Drift of a Limit Order Fill ArXiv ID: 2407.16527 “View on arXiv” Authors: Unknown Abstract Market making refers to a form of trading in financial markets characterized by passive orders which add liquidity to limit order books. Market makers are important for the proper functioning of financial markets worldwide. Given the importance, financial mathematics has endeavored to derive optimal strategies for placing limit orders in this context. This paper identifies a key discrepancy between popular model assumptions and the realities of real markets, specifically regarding the dynamics around limit order fills. Traditionally, market making models rely on an assumption of low-cost random fills, when in reality we observe a high-cost non-random fill behavior. Namely, limit order fills are caused by and coincide with adverse price movements, which create a drag on the market maker’s profit and loss. We refer to this phenomenon as “the negative drift” associated with limit order fills. We describe a discrete market model and prove theoretically that the negative drift exists. We also provide a detailed empirical simulation using one of the most traded financial instruments in the world, the 10 Year US Treasury Bond futures, which also confirms its existence. To our knowledge, this is the first paper to describe and prove this phenomenon in such detail. ...

Explainable AI in Request-for-Quote ArXiv ID: 2407.15038 “View on arXiv” Authors: Unknown Abstract In the contemporary financial landscape, accurately predicting the probability of filling a Request-For-Quote (RFQ) is crucial for improving market efficiency for less liquid asset classes. This paper explores the application of explainable AI (XAI) models to forecast the likelihood of RFQ fulfillment. By leveraging advanced algorithms including Logistic Regression, Random Forest, XGBoost and Bayesian Neural Tree, we are able to improve the accuracy of RFQ fill rate predictions and generate the most efficient quote price for market makers. XAI serves as a robust and transparent tool for market participants to navigate the complexities of RFQs with greater precision. ...

Macroscopic Market Making Games via Multidimensional Decoupling Field ArXiv ID: 2406.05662 “View on arXiv” Authors: Unknown Abstract Building on the macroscopic market making framework as a control problem, this paper investigates its extension to stochastic games. In the context of price competition, each agent is benchmarked against the best quote offered by the others. We begin with the linear case. While constructing the solution directly, the \textit{“ordering property”} and the dimension reduction in the equilibrium are revealed. For the non-linear case, we extend the decoupling approach by introducing a multidimensional \textit{“characteristic equation”} to analyse the well-posedness of the forward-backward stochastic differential equations. Properties of the coefficients in this characteristic equation are derived using tools from non-smooth analysis. Several new well-posedness results are presented. ...

Loss-Versus-Fair: Efficiency of Dutch Auctions on Blockchains ArXiv ID: 2406.00113 “View on arXiv” Authors: Unknown Abstract Milionis et al.(2023) studied the rate at which automated market makers leak value to arbitrageurs when block times are discrete and follow a Poisson process, and where the risky asset price follows a geometric Brownian motion. We extend their model to analyze another popular mechanism in decentralized finance for onchain trading: Dutch auctions. We compute the expected losses that a seller incurs to arbitrageurs and expected time-to-fill for Dutch auctions as a function of starting price, volatility, decay rate, and average interblock time. We also extend the analysis to gradual Dutch auctions, a variation on Dutch auctions for selling tokens over time at a continuous rate. We use these models to explore the tradeoff between speed of execution and quality of execution, which could help inform practitioners in setting parameters for starting price and decay rate on Dutch auctions, or help platform designers determine performance parameters like block times. ...

A Tick-by-Tick Solution for Concentrated Liquidity Provisioning ArXiv ID: 2405.18728 “View on arXiv” Authors: Unknown Abstract Automated market makers with concentrated liquidity capabilities are programmable at the tick level. The maximization of earned fees, plus depreciated reserves, is a convex optimization problem whose vector solution gives the best provision of liquidity at each tick under a given set of parameter estimates for swap volume and price volatility. Surprisingly, early results show that concentrating liquidity around the current price is usually not the best strategy. ...

Trading Volume Maximization with Online Learning ArXiv ID: 2405.13102 “View on arXiv” Authors: Unknown Abstract We explore brokerage between traders in an online learning framework. At any round $t$, two traders meet to exchange an asset, provided the exchange is mutually beneficial. The broker proposes a trading price, and each trader tries to sell their asset or buy the asset from the other party, depending on whether the price is higher or lower than their private valuations. A trade happens if one trader is willing to sell and the other is willing to buy at the proposed price. Previous work provided guidance to a broker aiming at enhancing traders’ total earnings by maximizing the gain from trade, defined as the sum of the traders’ net utilities after each interaction. In contrast, we investigate how the broker should behave to maximize the trading volume, i.e., the total number of trades. We model the traders’ valuations as an i.i.d. process with an unknown distribution. If the traders’ valuations are revealed after each interaction (full-feedback), and the traders’ valuations cumulative distribution function (cdf) is continuous, we provide an algorithm achieving logarithmic regret and show its optimality up to constant factors. If only their willingness to sell or buy at the proposed price is revealed after each interaction ($2$-bit feedback), we provide an algorithm achieving poly-logarithmic regret when the traders’ valuations cdf is Lipschitz and show that this rate is near-optimal. We complement our results by analyzing the implications of dropping the regularity assumptions on the unknown traders’ valuations cdf. If we drop the continuous cdf assumption, the regret rate degrades to $Θ(\sqrt{“T”})$ in the full-feedback case, where $T$ is the time horizon. If we drop the Lipschitz cdf assumption, learning becomes impossible in the $2$-bit feedback case. ...

Adaptive Optimal Market Making Strategies with Inventory Liquidation Cos ArXiv ID: 2405.11444 “View on arXiv” Authors: Unknown Abstract A novel high-frequency market-making approach in discrete time is proposed that admits closed-form solutions. By taking advantage of demand functions that are linear in the quoted bid and ask spreads with random coefficients, we model the variability of the partial filling of limit orders posted in a limit order book (LOB). As a result, we uncover new patterns as to how the demand’s randomness affects the optimal placement strategy. We also allow the price process to follow general dynamics without any Brownian or martingale assumption as is commonly adopted in the literature. The most important feature of our optimal placement strategy is that it can react or adapt to the behavior of market orders online. Using LOB data, we train our model and reproduce the anticipated final profit and loss of the optimal strategy on a given testing date using the actual flow of orders in the LOB. Our adaptive optimal strategies outperform the non-adaptive strategy and those that quote limit orders at a fixed distance from the midprice. ...

Price-Aware Automated Market Makers: Models Beyond Brownian Prices and Static Liquidity ArXiv ID: 2405.03496 “View on arXiv” Authors: Unknown Abstract In this paper, we introduce a suite of models for price-aware automated market making platforms willing to optimize their quotes. These models incorporate advanced price dynamics, including stochastic volatility, jumps, and microstructural price models based on Hawkes processes. Additionally, we address the variability in demand from liquidity takers through models that employ either Hawkes or Markov-modulated Poisson processes. Each model is analyzed with particular emphasis placed on the complexity of the numerical methods required to compute optimal quotes. ...