# Market Making under a Weakly Consistent Limit Order Book Model

## TL;DR

This paper introduces a novel market-making model tailored for high-frequency trading that respects the microstructure of limit order books, allowing for flexible order dynamics and ensuring price consistency.

## Contribution

It develops a flexible, microstructure-consistent market-making framework using impulse control techniques, extending classical models with more realistic order book features.

## Key findings

- Optimal trading strategies are numerically computed.
- Price inconsistencies can significantly overstate profits.
- Model calibration to ETF data demonstrates practical applicability.

## Abstract

We develop a new market-making model, from the ground up, which is tailored towards high-frequency trading under a limit order book (LOB), based on the well-known classification of order types in market microstructure. Our flexible framework allows arbitrary order volume, price jump, and bid-ask spread distributions as well as the use of market orders. It also honors the consistency of price movements upon arrivals of different order types. For example, it is apparent that prices should never go down on buy market orders. In addition, it respects the price-time priority of LOB. In contrast to the approach of regular control on diffusion as in the classical Avellaneda and Stoikov [1] market-making framework, we exploit the techniques of optimal switching and impulse control on marked point processes, which have proven to be very effective in modeling the order-book features. The Hamilton-Jacobi-Bellman quasi-variational inequality (HJBQVI) associated with the control problem can be solved numerically via finite-difference method. We illustrate our optimal trading strategy with a full numerical analysis, calibrated to the order-book statistics of a popular Exchanged-Traded Fund (ETF). Our simulation shows that the profit of market-making can be severely overstated under LOBs with inconsistent price movements.