Studies on optimal trade execution

Abstract

This dissertation deals with the question of how to optimally execute orders for financial assets that are subject to transaction costs. We study the problem in a discrete--time model where the asset price processes of interest are subject to stochastic volatility and liquidity.

First, we consider the case for the execution of a single asset. We find predictable strategies that minimize the expectation, mean--variance and expected exponential of the implementation cost.

Second, we extend the single asset case to incorporate a dark pool where the orders can be crossed at the mid--price depending on the liquidity available. The orders submitted to the dark pool face execution uncertainty and are assumed to be subject to adverse selection risk. We find strategies that minimize the expectation and the expected exponential of the implementation shortfall and show that one can incur less costs by also making use of the dark pool.

Next chapter studies a multi asset setting in the presence of a dark pool. We find strategies that minimize the expectation and expected exponential of a cost functional that consists of the implementation shortfall and an aversion term that penalizes the orders crossed in the dark pool. In the expected exponential of the cost case, the dimensionality of the problem does not allow for the numerical computation of optimal strategies. Therefore, we first solve the expected exponential case for a second order Taylor approximation and then provide a framework via a duality argument which can be used to generate approximate strategies.

Lastly, we treat the case where the single asset execution problem exhibits ambiguity for the distribution of stochastic liquidity and volatility. We see the implementation cost as the sum of risk terms arising at each execution period. We consider the problem obtained from aggregating worst case expectations of these risk terms, by penalizing the distributions used with dynamic indicator, relative entropy and Gini indices. Next, we formulate the problem as the multi--prior first order certainty equivalent of the exponential cost and lastly we consider a second order certainty equivalence formulation.