Optimal Trading Algorithms: Portfolio Transactions, Multiperiod Portfolio Selection, and Competitive Online Search

Abstract

This thesis deals with optimal algorithms for trading of financial securities. It is divided into four parts: risk-averse execution with market impact, Bayesian adaptive trading with price appreciation, multiperiod portfolio selection, and the generic online search problem k-search. Risk-averse execution with market impact. We consider the execution of portfolio transactions in a trading model with market impact. For an institutional investor, especially in equity markets, the size of his buy or sell order is often larger than the market can immediately supply or absorb, and his trading will move the price (market impact). His order must be worked across some period of time, exposing him to price volatility. The investor needs to find a trade-off between the market impact costs of rapid execution and the market risk of slow execution. In a mean-variance framework, an optimal execution strategy minimizes variance for a specified maximum level of expected cost, or conversely. In this setup, Almgren and Chriss (2000) give path-independent (also called static) execution algorithms: their trade-schedules are deterministic and do not modify the execution speed in response to price motions during trading. We show that the static execution strategies of Almgren and Chriss (2000) can be significantly improved by adaptive trading. We first illustrate this by constructing strategies that update exactly once during trading: at some intermediary time they may readjust in response to the stock price movement up to that moment. We show that such single-update strategies yield lower expected cost for the same level of variance than the static trajectories of Almgren and Chriss (2000), or lower variance for the same expected cost. Extending this first result, we then show how optimal dynamic strategies can be computed to any desired degree of precision with a suitable application of the dynamic programming principle. In this technique the control variables are not only the shares traded at each time step, but also the maximum expected cost for the remainder of the program; the value function is the variance of the remaining program. This technique reduces the determination of optimal dynamic strategies to a series of single-period convex constrained optimization problems. The resulting adaptive trading strategies are “aggressive-in-the-money”: they accelerate the execution when the price moves in the trader’s favor, spending parts of the trading gains to reduce risk. The relative improvement over static trade schedules is larger for large initial positions, expressed in terms of a new nondimensional parameter, the market power μ. For small portfolios, μ → 0, optimal adaptive trade schedules coincide with the static trade schedules of Almgren and Chriss (2000). Bayesian adaptive trading with price appreciation. This part deals with another major driving factor of transaction costs for institutional investors, namely price appreciation (price trend) during the time of a buy (or sell) program. An investor wants to buy (sell) a stock before other market participants trade the same direction and push up (respectively, down) the price.