Optimal Liquidation Behaviour Analysis for Stochastic Linear and Nonlinear Systems of Self-Exciting Model with Decay

When the market environment changes, we extend the self-exciting price impact model and further analysis of investors’ liquidation behaviour. It is assumed that the model is accompanied by an exponential decay factor when the temporary impact and its coefficient are linear and nonlinear. Using the optimal control method, we obtain that the optimal liquidation behaviours satisfy the second-order nonlinear ODEs with variable coefficients in the case of linear and nonlinear temporary impact. Next, we solve the ODEs and get the form of the investors’ optimal liquidation behaviour in four cases. Furthermore, we prove the decreasing properties of the optimal liquidation behaviour under the linear temporary impact. Through numerical simulation, we further explain the influence of the changed parameters ρ , a , b , x , and α on the investors’ liquidation strategy X t in twelve scenarios. Some interesting properties have been found.

OPTIMAL LIQUIDATION TRAJECTORIES FOR THE ALMGREN–CHRISS MODEL

We consider an optimal liquidation problem with infinite horizon in the Almgren–Chriss framework, where the unaffected asset price follows a Lévy process. The temporary price impact is described by a general function that satisfies some reasonable conditions. We consider a market agent with constant absolute risk aversion, who wants to maximize the expected utility of the cash received from the sale of the agent’s assets, and show that this problem can be reduced to a deterministic optimization problem that we are able to solve explicitly. In order to compare our results with exponential Lévy models, which provide a very good statistical fit with observed asset price data for short time horizons, we derive the (linear) Lévy process approximation of such models. In particular we derive expressions for the Lévy process approximation of the exponential variance–gamma Lévy process, and study properties of the corresponding optimal liquidation strategy. We then provide a comparison of the liquidation trajectories for reasonable parameters between the Lévy 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 Lévy 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.