Optimal liquidation trajectories for the Almgren-Chriss model with Levy processes

We consider an optimal liquidation problem with infinite horizon in the Almgren-Chriss framework, where the unaffected asset price follows a Levy process. The temporary price impact is described by a general function which satisfies some reasonable conditions. We consider an investor with constant absolute risk aversion, who wants to maximise the expected utility of the cash received from the sale of his assets, and show that this problem can be reduced to a deterministic optimisation problem which we are able to solve explicitly. In order to compare our results with exponential Levy models, which provides a very good statistical fit with observed asset price data for short time horizons, we derive the (linear) Levy process approximation of such models. In particular we derive expressions for the Levy process approximation of the exponential Variance-Gamma Levy process, and study properties of the corresponding optimal liquidation strategy. We then provide a comparison of the liquidation trajectories for reasonable parameters between the Levy process model and the classical Almgren-Chriss model. In particular, we obtain an explicit expression for the connection between the temporary impact function for the Levy model and the temporary impact function for the Brownian motion model (the classical Almgren-Chriss model), for which the optimal liquidation trajectories for the two models coincide.