  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.
