Optimal High-Frequency Trading in a Pro-Rata Microstructure with Predictive Information

We propose a framework to study optimal trading policies in a one-tick pro-rata limit order book, as typically arises in short-term interest rate futures contracts. The high-frequency trader chooses to post either market orders or limit orders, which are represented respectively by impulse controls and regular controls. We discuss the con- sequences of the two main features of this microstructure: first, the limit orders are only partially executed, and therefore she has no control on the executed quantity. Second, the high frequency trader faces the overtrading risk, which is the risk of brutal varia- tions in her inventory. The consequences of this risk are investigated in the context of optimal liquidation. The optimal trading problem is studied by stochastic control and dynamic programming methods, and we provide the associated numerical resolution procedure and prove its convergence. We propose dimension reduction techniques in several cases of practical interest. We also detail a high frequency trading strategy in the case where a (predictive) directional information on the price is available. Each of the resulting strategies are illustrated by numerical tests