DISCRETE-TIME OPTIMAL EXECUTION UNDER A GENERALIZED PRICE IMPACT MODEL WITH MARKOVIAN EXOGENOUS ORDERS

Abstract

This paper examines a discrete-time optimal trade execution problem with generalized price impact. We extend a model recently discussed, which considers price impacts of aggregate random trade orders posed by small traders as well as a large trader. In contrast that assumes aggregate trading volumes submitted by small traders are serially independent, this paper allows a Markovian dependence. Our new problem is formulated as a Markov decision process with state variables including the last small traders' aggregate orders. Over a finite horizon, the large trader with Constant Absolute Risk Aversion (CARA) von Neumann-Morgenstern (vN-M) utility function maximizes the expected utility from the final wealth. By applying the backward induction method of dynamic programming, we characterize the optimal value function and optimal trade execution strategy, and conclude that the execution strategy is a time-dependent affine function of three state variables. Moreover, numerical analysis prevails that the optimal execution strategy admits a `statistical arbitrage' via a round-trip trading, although our model considers a linear permanent price impact, which does not admit any price manipulation or arbitrage. The reason is that our model considers price impacts caused by small traders' orders with a Markovian dependence.