When to Cross the Spread? Trading in Two-Sided Limit Order Books

In this paper the problem of optimal trading in illiquid markets is addressed when the deviations from a given stochastic target function describing, for instance, external aggregate client flow are penalized. Using techniques of singular stochastic control, we extend the results of [F. Naujokat and N. Westray, Math. Financ. Econ., 4 (2011), pp. 299--335] to a two-sided limit order market with temporary market impact and resilience, where the bid ask spread is now also controlled. In addition to using market orders, the trader can also submit orders to a dark pool. We first show existence and uniqueness of an optimal control. In a second step, a suitable version of the stochastic maximum principle is derived which yields a characterization of the optimal trading strategy in terms of a nonstandard coupled forward-backward stochastic differential equation (FBSDE). We show that the optimal control can be characterized via buy, sell, and no-trade regions. The new feature is that we now get a nondegenerate no-trade region, which implies that market orders are used only when the spread is small. This allows us to describe precisely when it is optimal to cross the bid ask spread, which is a fundamental problem of algorithmic trading. We also show that the controlled system can be described in terms of a reflected BSDE. As an application, we solve the portfolio liquidation problem with passive orders.