Optimal Order Scheduling for Deterministic Liquidity Patterns

We consider a broker who has to place a large order which consumes a sizable part of average daily trading volume. The broker's aim is thus to minimize execution costs he incurs from the adverse impact of his trades on market prices. In contrast to the previous literature (see, e. g., Obizhaeva and Wang [A. Obizhaeva and J. Wang, J. Financial Markets, 16 (2013), pp. 1-32] and Predoiu, Shaikhet, and Shreve [SIAM J. Financial Math., 2 (2011), pp. 183-212]), we allow the liquidity parameters of market depth and resilience to vary deterministically over the course of the trading period. The resulting singular optimal control problem is shown to be tractable by methods from convex analysis, and, under minimal assumptions, we construct an explicit solution to the scheduling problem in terms of some concave envelope of the resilience-adjusted market depth.