Optimal execution with uncertain order fills in Almgren–Chriss framework

The classical price impact model of Almgren and Chriss is extended to incorporate the uncertainty of order fills. The extended model can be recast as alternatives to uncertain impact models and stochastic liquidity models. Optimal strategies are determined by maximizing the expected final profit and loss (P&L) and various P&L-risk tradeoffs including utility maximization. Closed form expressions for optimal strategies are obtained in linear cases. The results suggest a type of adaptive volume weighted average price, adaptive percentage of volume and adaptive Almgren–Chriss strategies. VWAP and classical Almgren–Chriss strategies are recovered as limiting cases with a different characteristic time scale of liquidation for the latter.