Optimal Execution of Backstopped Block Trades

Abstract

In this thesis, we introduce and study a model for a broker who executes a client order and takes over its execution risk at some transition time. Such agreements between clients and brokers are often called backstopped trades. To minimize risk, it may be beneficial for the broker to trade on his/her own book, even before taking over the execution risk of the order. The broker is not allowed to trade in the same stock while executing on behalf of the client, but the broker may trade in a different, correlated stock. We formulate this question as a mean-variance optimization problem with two correlated stocks, also incorporating permanent and temporary market impacts. We consider this problems in three different cases. In the first case, we assume that the transition time is deterministic. We then manage to find an explicit formula for the optimal trading strategy and analyze it in a numerical example. In the second case, we consider a stochastic transition time, but restrict the analysis to deterministic strategies. Under this assumption, we can characterize the optimal trading strategy through an ordinary differential equation. Finally, in the most general case of a stochastic transition time with stochastic strategies, we derive the Hamilton-Jacobi-Bellman equation corresponding to the optimization problem and describe a numerical implementation to find an approximate solution.