Optimal High Frequency Trading with limit and market orders (1106.5040v1)

Abstract: We propose a framework for studying optimal market making policies in a limit order book (LOB). The bid-ask spread of the LOB is modelled by a Markov chain with finite values, multiple of the tick size, and subordinated by the Poisson process of the tick-time clock. We consider a small agent who continuously submits limit buy/sell orders and submits market orders at discrete dates. The objective of the market maker is to maximize her expected utility from revenue over a short term horizon by a tradeoff between limit and market orders, while controlling her inventory position. This is formulated as a mixed regime switching regular/ impulse control problem that we characterize in terms of quasi-variational system by dynamic programming methods. In the case of a mean-variance criterion with martingale reference price or when the asset price follows a Levy process and with exponential utility criterion, the dynamic programming system can be reduced to a system of simple equations involving only the inventory and spread variables. Calibration procedures are derived for estimating the transition matrix and intensity parameters for the spread and for Cox processes modelling the execution of limit orders. Several computational tests are performed both on simulated and real data, and illustrate the impact and profit when considering execution priority in limit orders and market orders

• The paper introduces a mixed regime-switching control model that integrates limit and market orders to optimize high-frequency trading outcomes.
• It applies dynamic programming and stochastic processes, including Markov, Poisson, and Cox models, to mitigate execution and inventory risks.
• Calibration on synthetic and real data demonstrates that strategic spread management significantly boosts market maker profitability.

Optimal High-Frequency Trading with Limit and Market Orders

The research paper by Fabien Guilbaud and Huyền Pham presents a sophisticated framework for optimizing market-making policies within the construct of a limit order book (LOB). It addresses the challenges faced by market makers in high-frequency trading environments through a mathematical model that integrates both limit and market orders.

Model Overview

The authors propose a model where the bid-ask spread in the LOB is represented by a Markov chain affected by random events in tick-time, modeled as a Poisson process. Market participants, particularly small agents or market makers, can place limit orders for buying or selling at the best quotes. These orders face execution risk, as fulfiLLMent relies on counterpart market orders that follow Cox processes influenced by the spread and the market maker’s pricing strategy. Additionally, to mitigate inventory risk from holding volatile positions, market makers may use immediate market orders as a tactical element, despite incurring higher execution costs due to crossing the spread.

Objective and Methodology

The primary goal for the market maker is to maximize revenue within a short-term trading horizon while controlling inventory risk through a balance of limit and market orders. This objective is mathematically formulated as a mixed regime-switching control problem, devised using dynamic programming techniques. Remarkably, the researchers transform this complex problem into a system of simpler equations, focusing on inventory and spread variables. This transformation becomes feasible especially when considering a mean-variance criterion under a martingale reference price, or in scenarios where asset prices follow a Levy process, employing exponential utility functions.

Calibration and Numerical Results

A notable strength of the paper lies in its detailed calibration procedures, which are crucial for estimating transition matrices and intensities for the spread and limit order executions. These estimations are reliably performed both on synthetic and real market data, underlining the practical applicability of the proposed strategies. Numerical analyses reveal measurable impacts on market maker profitability when prioritizing order execution through strategic spread management.

Implications and Future Research

The implications of this research extend across theoretical and practical domains. Practically, the model enhances the understanding of inventory management and order execution strategies in algorithmic trading, which could potentially influence the development of advanced trading systems. Theoretically, it expands computational methods for solving stochastic control problems associated with financial markets.

Future research could further explore the integration of AI-driven prediction models to refine trading strategies and risk assessments. Additionally, extending this framework to multi-asset or multi-market environments could enrich the model's robustness, providing deeper insights into the dynamics of global financial markets.

Overall, Guilbaud and Pham’s work provides valuable contributions to the field of financial engineering, particularly in optimizing high-frequency trading strategies within the complex architecture of a limit order book.