Optimal execution in high-frequency trading with Bayesian learning

We consider optimal trading strategies in which traders submit bid and ask quotes to maximize the expected quadratic utility of total terminal wealth in a limit order book. The trader’s bid and ask quotes will be changed by the Poisson arrival of market orders. Meanwhile, the trader may update his estimate of other traders’ target sizes and directions by Bayesian learning. The solution of optimal execution in the limit order book is a two-step procedure. First, we model an inactive trading with no limit order in the market. The dealer simply holds dollars and shares of stocks until terminal time. Second, he calibrates his bid and ask quotes to the limit order book. The optimal solutions are given by dynamic programming and in fact they are globally optimal. We also give numerical simulation to the value function and optimal quotes at the last part of the article. •We propose a stochastic model to describe the optimal execution in high-frequency Trading.•Traders’ behaviors are described using Bayesian rules in the model.•The explicit solutions to the stochastic model can be deduced by HJB equations.•Our analysis gives the numerical solutions based on static and dynamic situation.