Optimal execution with nonlinear impact functions and trading-enhanced risk

We determine optimal trading strategies for liquidation of a large single-asset portfolio to minimize a combination of volatility risk and market impact costs. We take the market impact cost per share to be a power law function of the trading rate, with an arbitrary positive exponent. This includes, for example, the square-root law that has been proposed based on market microstructure theory. In analogy to the linear model, we define a “characteristic time” for optimal trading, which now depends on the initial portfolio size and decreases as execution proceeds. We also consider a model in which uncertainty of the realized price is increased by demanding rapid execution; we show that optimal trajectories are described by a “critical portfolio size” above which this effect is dominant and below which it may be neglected.