Optimal investment with transient price impact

Moritz Voß (TU Berlin)

with Peter Bank

We propose a variant of the limit order book model by Obizhaeva and Wang (2013) which allows for both selling and buying stock. Specifically, our price impact model determines bid- and ask-prices via a coupled system of controlled diffusions, allowing us to retain the possibility to specify market depth, tightness and resilience. We provide existence of an optimal solution to the problem of maximizing expected utility from terminal liquidation wealth at a finite planning horizon. In the specific case when market uncertainty is generated by an arithmetic Brownian motion with drift and the investor exhibits constant absolute risk aversion, we show that the resulting singular optimal control problem reduces to a deterministic optimal tracking problem of the frictionless constant Merton portfolio in the presence of convex liquidity costs. We construct the solution explicitly by methods from convex analysis and calculus of variations. As expected by previous studies in the literature, it is optimal to trade towards the frictionless Merton position taking into account the initial bid-ask spread as well as the optimal liquidation of the accrued position when approaching terminal time. It turns out that this leads to a surprisingly rich phenomenology of possible trajectories for the optimal share holdings.