Optimal liquidation in dark pools in discrete and continuous time

In recent years there has been an increasing interest in financial market models that account for the impact of large transactions on asset prices. In this thesis we consider an “illiquid market” with a risk-averse investor who has to liquidate a large portfolio within a finite time horizon [0, T ]. At any point in time, the investor has the option to trade at a traditional exchange (the “primary venue”) which yields price impact and to place orders in a so-called dark pool. The liquidity in dark pools is not openly displayed and dark pools do not contribute to the price formation process. Instead, orders are executed at the price of the primary venue if matching liquidity is available. Therefore, these orders have no price impact. However, in contrast to the primary venue, their execution is uncertain. The investor thus faces the trade-off between the direct costs resulting from the price impact of her orders at the primary venue and the indirect costs resulting from the execution uncertainty in the dark pool. This thesis considers stochastic control problems arising in portfolio liquidation problems using dark pools. At first we consider a discrete-time market model and a cost functional which incorporates both the expected price impact costs and the market risk of a large portfolio. For linear temporary price impact, this results in a linear-quadratic cost functional and we obtain the optimal trading strategy in the form of an explicit backward recursion. If a single asset position is to be liquidated, the investor trades a certain proportion of the position at the primary venue, with the remainder being placed in the dark pool. For multi asset liquidation this is generally not optimal, and the optimal strategy depends strongly on the correlation of the assets. We generalize this model for single asset liquidation and analyze adverse selection effects. This reduces the attractiveness of the dark pool and thus optimal dark pool orders are smaller; it can even be optimal to not use the dark pool at all. In the second part of the thesis we consider the counterparts of the described optimization problems in a continuous-time market model. In continuous time the liquidation constraint implies a singularity of the value function at the terminal time T . Via the HJB equation and a quadratic ansatz, we obtain a candidate for the value function of the linear-quadratic optimization problem without adverse selection. This candidate is the limit of a sequence of solutions of initial value problems for a matrix differential equation. Although the differential equation is not a Riccati equation, we are able to show that this limit actually exists by using an appropriate matrix inequality and a known comparison result for Riccati equations. Additionally, we thereby obtain upper and lower bounds of the solutions of the initial value problems, which enables us to prove a verification theorem. The complexity of the optimization problem with adverse selection stems form the non-linear-quadratic form of the cost functional. Via the HJB equation and extensive heuristic considerations we prove that the value function is a “quasi-polynomial” whose coefficients depend on the asset position in a non-trivial way. For single asset liquidation, all optimization problems are solved in closed form both in discrete and in continuous time. The solutions of the discrete-time optimization problems converge to the solutions of the continuous-time optimization problems under appropriate conditions.