| Abstract: |
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 has the choice to trade via
market orders or limit orders, which are represented respectively by impulse
controls and regular controls. We model and discuss the consequences of the
two main features of this particular microstructure: first, the limit orders
sent by the high frequency trader are only partially executed, and therefore
she has no control on the executed quantity. For this purpose, cumulative
executed volumes are modelled by compound Poisson processes. Second, the high
frequency trader faces the overtrading risk, which is the risk of brutal
variations 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 progra\-mming methods, which lead to a
characterization of the value function in terms of an integro
quasi-variational inequality. We then provide the associated numerical
resolution procedure, and convergence of this computational scheme is proved.
Next, we examine several situations where we can on one hand simplify the
numerical procedure by reducing the number of state variables, and on the
other hand focus on specific cases of practical interest. We examine both a
market making problem and a best execution problem in the case where the
mid-price process is a martingale. We also detail a high frequency trading
strategy in the case where a (predictive) directional information on the
mid-price is available. Each of the resulting strategies are illustrated by
numerical tests. |