Optimal Execution

Unwinding Stochastic Order Flow: When to Warehouse Trades ArXiv ID: 2310.14144 “View on arXiv” Authors: Unknown Abstract We study how to unwind stochastic order flow with minimal transaction costs. Stochastic order flow arises, e.g., in the central risk book (CRB), a centralized trading desk that aggregates order flows within a financial institution. The desk can warehouse in-flow orders, ideally netting them against subsequent opposite orders (internalization), or route them to the market (externalization) and incur costs related to price impact and bid-ask spread. We model and solve this problem for a general class of in-flow processes, enabling us to study in detail how in-flow characteristics affect optimal strategy and core trading metrics. Our model allows for an analytic solution in semi-closed form and is readily implementable numerically. Compared with a standard execution problem where the order size is known upfront, the unwind strategy exhibits an additive adjustment for projected future in-flows. Its sign depends on the autocorrelation of orders; only truth-telling (martingale) flow is unwound myopically. In addition to analytic results, we present extensive simulations for different use cases and regimes, and introduce new metrics of practical interest. ...

Macroscopic Market Making ArXiv ID: 2307.14129 “View on arXiv” Authors: Unknown Abstract We propose a macroscopic market making model à la Avellaneda-Stoikov, using continuous processes for orders instead of discrete point processes. The model intends to bridge the gap between market making and optimal execution problems, while shedding light on the influence of order flows on the optimal strategies. We demonstrate our model through three problems. The study provides a comprehensive analysis from Markovian to non-Markovian noises and from linear to non-linear intensity functions, encompassing both bounded and unbounded coefficients. Mathematically, the contribution lies in the existence and uniqueness of the optimal control, guaranteed by the well-posedness of the strong solution to the Hamilton-Jacobi-Bellman equation and the (non-)Lipschitz forward-backward stochastic differential equation. Finally, the model’s applications to price impact and optimal execution are discussed. ...

Optimal Execution Using Reinforcement Learning ArXiv ID: 2306.17178 “View on arXiv” Authors: Unknown Abstract This work is about optimal order execution, where a large order is split into several small orders to maximize the implementation shortfall. Based on the diversity of cryptocurrency exchanges, we attempt to extract cross-exchange signals by aligning data from multiple exchanges for the first time. Unlike most previous studies that focused on using single-exchange information, we discuss the impact of cross-exchange signals on the agent’s decision-making in the optimal execution problem. Experimental results show that cross-exchange signals can provide additional information for the optimal execution of cryptocurrency to facilitate the optimal execution process. ...

Optimal Portfolio Execution in a Regime-switching Market with Non-linear Impact Costs: Combining Dynamic Program and Neural Network ArXiv ID: 2306.08809 “View on arXiv” Authors: Unknown Abstract Optimal execution of a portfolio have been a challenging problem for institutional investors. Traders face the trade-off between average trading price and uncertainty, and traditional methods suffer from the curse of dimensionality. Here, we propose a four-step numerical framework for the optimal portfolio execution problem where multiple market regimes exist, with the underlying regime switching based on a Markov process. The market impact costs are modelled with a temporary part and a permanent part, where the former affects only the current trade while the latter persists. Our approach accepts impact cost functions in generic forms. First, we calculate the approximated orthogonal portfolios based on estimated impact cost functions; second, we employ dynamic program to learn the optimal selling schedule of each approximated orthogonal portfolio; third, weights of a neural network are pre-trained with the strategy suggested by previous step; last, we train the neural network to optimize on the original trading model. In our experiment of a 10-asset liquidation example with quadratic impact costs, the proposed combined method provides promising selling strategy for both CRRA (constant relative risk aversion) and mean-variance objectives. The running time is linear in the number of risky assets in the portfolio as well as in the number of trading periods. Possible improvements in running time are discussed for potential large-scale usages. ...

Optimal Market Making in the Chinese Stock Market: A Stochastic Control and Scenario Analysis ArXiv ID: 2306.02764 “View on arXiv” Authors: Unknown Abstract Market making plays a crucial role in providing liquidity and maintaining stability in financial markets, making it an essential component of well-functioning capital markets. Despite its importance, there is limited research on market making in the Chinese stock market, which is one of the largest and most rapidly growing markets globally. To address this gap, we employ an optimal market making framework with an exponential CARA-type (Constant Absolute Risk Aversion) utility function that accounts for various market conditions, such as price drift, volatility, and stamp duty, and is capable of describing 3 major risks (i.e., inventory, execution and adverse selection risks) in market making practice, and provide an in-depth quantitative and scenario analysis of market making in the Chinese stock market. Our numerical experiments explore the impact of volatility on the market maker’s inventory. Furthermore, we find that the stamp duty rate is a critical factor in market making, with a negative impact on both the profit of the market maker and the liquidity of the market. Additionally, our analysis emphasizes the significance of accurately estimating stock drift for managing inventory and adverse selection risks effectively and enhancing profit for the market maker. These findings offer valuable insights for both market makers and policymakers in the Chinese stock market and provide directions for further research in designing effective market making strategies and policies. ...

Optimal execution and speculation with trade signals ArXiv ID: 2306.00621 “View on arXiv” Authors: Unknown Abstract We propose a price impact model where changes in prices are purely driven by the order flow in the market. The stochastic price impact of market orders and the arrival rates of limit and market orders are functions of the market liquidity process which reflects the balance of the demand and supply of liquidity. Limit and market orders mutually excite each other so that liquidity is mean reverting. We use the theory of Meyer-$σ$-fields to introduce a short-term signal process from which a trader learns about imminent changes in order flow. Her trades impact the market through the same mechanism as other orders. With a novel version of Marcus-type SDEs we efficiently describe the intricate timing of market dynamics at moments when her orders concur with that of others. In this setting, we examine an optimal execution problem and derive the Hamilton–Jacobi–Bellman (HJB) equation for the value function of the trader. The HJB equation is solved numerically and we illustrate how the trader uses the signals to enhance the performance of execution problems and to execute speculative strategies. ...

Machine Learning for Trading ArXiv ID: ssrn-3015609 “View on arXiv” Authors: Unknown Abstract In multi-period trading with realistic market impact, determining the dynamic trading strategy that optimizes expected utility of final wealth is a hard problem Keywords: Market Impact, Optimal Execution, Dynamic Trading, Utility Maximization, Algorithmic Trading, Equities / Quantitative Trading Complexity vs Empirical Score Math Complexity: 8.5/10 Empirical Rigor: 3.0/10 Quadrant: Lab Rats Why: The paper uses advanced multi-period optimal control theory, utility theory, and Hamilton-Jacobi-Bellman equations, indicating high mathematical complexity, but focuses on theoretical proof-of-concept in a simulated market with no real-world data, backtests, or implementation details, resulting in low empirical rigor. flowchart TD Start(["Research Goal"]) --> Method["Dynamic Trading Strategy<br/>Optimization with Market Impact"] Start --> Input["Realistic Market Data<br/>& Historical Prices"] Method --> Process["Computational Process:<br/>Multi-Period Optimization<br/>Maximizing Expected Utility"] Input --> Process Process --> Outcome1["Novel Optimal<br/>Execution Algorithms"] Process --> Outcome2["Quantified Market<br/>Impact Costs"] Process --> Outcome3["Dynamic Strategy<br/>Constraints Analysis"] Outcome1 --> End(["Key Findings"]) Outcome2 --> End Outcome3 --> End