David vs Goliath (You against the Markets), A Dynamic Programming Approach to Separate the Impact and Timing of Trading Costs

Abstract

A trader's conundrum is whether (and how much) to trade during a given interval or wait for the next interval when the price momentum is more favorable to his direction of trading. We develop a fundamentally different stochastic dynamic programming model of trading costs based on the Bellman principle of optimality. Built on a strong theoretical foundation, this model can provide insights to market participants by splitting the overall move of the security price during the duration of an order into the Market Impact (price move caused by their actions) and Market Timing (price move caused by everyone else) components. Plugging different distributions of prices and volumes into this framework can help traders decide when to bear higher Market Impact by trading more in the hope of offsetting the cost of trading at a higher price later. We derive formulations of this model under different laws of motion of the security prices. We start with a benchmark scenario and extend this to include multiple sources of uncertainty, liquidity constraints due to volume curve shifts and relate trading costs to the spread. We develop a numerical framework that can be used to obtain optimal executions under any law of motion of prices and demonstrate the tremendous practical applicability of our theoretical methodology including the powerful numerical techniques to implement them. This decomposition of trading costs into Market Impact and Market Timing allows us to deduce the zero sum game nature of trading costs. It holds numerous lessons for dealing with complex systems, especially in the social sciences, wherein reducing the complexity by splitting the many sources of uncertainty can lead to better insights in the decision process.