A sample-path approach to optimal position liquidation

We consider the problem of optimal position liquidation with the aim of maximizing the expected cash flow stream from the transaction in the presence of a temporary or permanent market impact. We use a stochastic programming approach to derive trading strategies that differentiate decisions with respect to different realizations of market conditions. The scenario set consists of a collection of sample paths representing possible future realizations of state variable processes (price of the security, trading volume etc.) At each time moment the set of paths is partitioned into several groups according to specified criteria, and each group is controlled by its own decision variable(s), which allows for adequate representation of uncertainties in market conditions and circumvents anticipativity in the solutions. In contrast to traditional dynamic programming approaches, the stochastic programming formulation admits incorporation of different types of constraints in the trading strategy, e.g. risk constraints, regulatory constraints, various decision-making policies etc. We consider the lawn-mower principle, which increases stability of the solution with respect to paths partitioning and saturation of the scenario pattern, but leads to non-convex optimization problems. It is shown that in the case of temporary market impact the optimal liquidation strategy with the lawn-mower principle can be approximated by a solution of convex or linear programming problems. Implemented as a linear programming problem, our approach is capable of handling large-scale instances and produces robust optimal solutions. A risk-averse trading strategy was constructed by incorporating risk constraints in the stochastic programming problem. We controlled the risk, associated with trading, using the Conditional Value-at-Risk measure. Numerical results and optimal trading patterns for different forms of market impact are presented.