Optimal High Frequency Trading with limit and market orders

TL;DR

This paper develops a mathematical framework for optimal high-frequency market making in a limit order book, balancing limit and market orders to maximize short-term expected utility while managing inventory risk.

Contribution

It introduces a novel mixed control model for market making that incorporates regime switching, and provides calibration methods and computational tests on real data.

Findings

Optimal trading strategies depend on spread and inventory dynamics.

Inclusion of execution priority improves profitability.

Model effectively captures real market behaviors.

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