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Finance Series: investment-and-risk

Optimal trade execution under endogenous pressure to liquidate

Theory and numerical solutions

Author(s): Professor Ales Cerny - Cass Business School; Pavol Brunovský - Department of Applied Mathematics and Statistics, Comenius University Bratislava; Ján Komadel - Department of Applied Mathematics and Statistics, Comenius University Bratislava


  • New look at optimal liquidation when fast sale pushes price against the seller.
  • Negative price drift induces endogenous pressure to liquidate.
  • Liquidation time horizon not fixed but modeled by a stopping time.
  • Study of extremely singular ODE IVP with a proposal for a stable numerical scheme.
  • Confirms square-root law for per-share impact of a block order.

We study optimal liquidation of a trading position (so-called block order or meta-order) in a market where seller’s order flow exhibits a linear temporary price impact on the unaffected price (Kyle, 1985). We endogenise 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. This contrasts with much of the existing literature where the liquidation time horizon is assumed exogenous and fixed. We find that the optimal liquidation strategy is non-linear and consistent with the empirically observed square-root law which states that the average price impact per share is proportional to the square root of the size of the meta-order (Bershova & Rakhlin,2013; Farmer et al., 2013; Donier et al., 2015; Tóth et al., 2016).

The paper is now available to read in its entirety at the ScienceDirect website. Access is free until 1st December 2017. It will be published in the European Journal of Operational Research.

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