Optimal portfolio execution under time-varying liquidity constraints

In this article, we take an algorithmic approach to solve the problem of optimal execution under time-varying constraints on the depth of a limit order book (LOB). Our algorithms are within the resilience model proposed by Obizhaeva and Wang (2013) with a more realistic assumption on the order book depth; the amount of liquidity provided by an LOB market is finite at all times. For the simplest case where the order book depth stays at a fixed level for the entire trading horizon, we reduce the optimal execution problem into a one-dimensional root-finding problem which can be readily solved by standard numerical algorithms. When the depth of the order book is monotone in time, we apply the Karush-Kuhn-Tucker conditions to narrow down the set of candidate strategies. Then, we use a dichotomy-based search algorithm to pin down the optimal one. For the general case, we start from the optimal strategy subject to no liquidity constraints and iterate over execution strategy by sequentially adding more constraints to the problem in a specific fashion until primal feasibility is achieved. Numerical experiments indicate that our algorithms give comparable results to those of current existing convex optimization toolbox CVXOPT with significantly lower time complexity.