Optimal Execution under Time-Inhomogeneous Price Impact and Volatility

We consider the problem of optimally liquidating a large position in a security within a specified period of time, to maximize a weighted sum of the expected value and variance of the profit from liquidating the position. To exploit predictable seasonal components in liquidity and volatility, we incorporate time-inhomogeneous linear price impact slopes and volatility into the formulation. We show that the optimal liquidation trajectory can be found by solving a Fredholm integral equation of the second type, or a related two-point boundary value problem (TPBVP) for a linear homogeneous secondorder ordinary differential equation (when the price impact slopes vary smoothly). We also show how the liquidation trajectory design method can be extended to liquidation under drifts and basket trading that accounts for cross-price impacts in liquidating a basket of assets. As in the driftless single security case, the optimal liquidation strategies can be found via solving associated linear integral equations. We illustrate the optimal liquidation trajectory design method with an example.