Optimal and Equilibrium Execution Strategies with Generalized Price Impact Masamitsu OHNISHI†‡ and Makoto SHIMOSHIMIZU† †Graduate School of Economics, Osaka University, Osaka, 560–0043, Japan ‡Center for Mathematical Modeling and Data Science, Osaka University, Osaka, 560–8531, Japan January 26, 2019 Abstract This paper examines the execution problems of large traders with generalized price im- pact models. Constructing two related models in a discrete–time setting, we solve these problems by applying the backward induction method of the dynamic programming. In the first problem, we formulate the expected utility maximization problem of a single large trader as a Markov decision process and derive an optimal execution strategy. Then, in the second model, we formulate the expected utility maximization problem of two large traders as a Markov game and derive an equilibrium execution strategy at a Markov perfect equi- librium. Both of these two models enable us to investigate how the execution strategies and trade performances of a large trader are affected by the existence of the other traders. More- over, we find that these optimal and equilibrium execution strategies become deterministic when the total execution volumes of non–large traders are deterministic. We also show, by some numerical examples, the comparative statics results with respect to several problem parameters. 1 Introduction In the security market analysis, there is a growing awareness among academic researchers or practitioners that some kind of institutional traders called ‘large trader’ cause the ‘price impact’ through their own trades. A life insurance company, trust company, or a company who manages pension fund exhibit the typical examples of such traders of great importance. Large traders recognize these price impacts as ‘liquidity risk’. They can reduce the liquidity risk by splitting their order into small size over the course of the trading epoch. Conversely, submitting the small pieces of order gradually may expose them to the price risk. Consequently, when large traders allocate large orders into (small) pieces, they have to pay attention to two distinct facets; the liquidity risk which arises owing to the large orders they submit and the price risk which corresponds to the price fluctuations in the future. 1 Electronic copy available at: https://ssrn.com/abstract=3323335