High Frequency Trading in a Markov Renewal Model

Abstract

We study an optimal high frequency trading problem within a market micro-structure model aiming at a good compromise between accuracy and tractability. The stock price is modeled by a Markov Renewal Process (MRP), while market orders arrive in the limit order book via a point process correlated with the stock price, and taking into account the adverse selection risk. We apply stochastic control methods in this semi-Markov framework, and show how to reduce remarkably the complexity of the associated Hamilton-Jacobi-Bellman equation by suitable change of variables that exploits the specific symmetry of the problem. We then handle numerically the remaining part of the HJB equation, simplified into an integro-ordinary differential equation, by a bi-dimensional Euler scheme. Statistical procedures and numerical tests for computing the optimal limit order strategies illustrate our results.

Keywords: High frequency trading – Markov renewal process, Marked Cox process, adverse selection, integro-ordinary differential equation

JEL Classification: G10, C51