Optimal Trade Execution Under Endogenous Pressure to Liquidate: Theory and Numerical Solutions

Abstract

We study optimal liquidation of a trading position (so-called block order or metaorder) in a market with a linear temporary price impact (Kyle, 1985). We endogenize the pressure to liquidate by introducing a downward drift in the unaffected asset price while simultaneously ruling out short sales. In this setting the liquidation time horizon becomes a stopping time determined endogenously, as part of the optimal strategy. We find that the optimal liquidation strategy is consistent with the squareroot law which states that the average price impact per share is proportional to the square root of the size of the meta-order (Bershova and Rakhlin, 2013;Farmer et al., 2013;Donier et al., 2015;Tóth et al., 2016).Mathematically, the Hamilton-Jacobi-Bellman equation of our optimization leads to a severely singular and numerically unstable ordinary differential equation initial value problem. We provide careful analysis of related singular mixed boundary value problems and devise a numerically stable computation strategy by re-introducing time dimension into an otherwise time-homogeneous task.