Discrete-Time Optimal Execution Under A Generalized Price Impact Model With Markovian Exogenous Orders

This paper examines a discrete-time optimal execution problem with generalized price impact. Our main objective is to investigate the effect of price impact caused by aggregate random trade orders posed by small traders on the optimal execution strategy when orders of the small traders have a Markovian dependence. Our problem is formulated as a Markov decision process with state variables which include the last small traders’ aggregate orders. Over a finite horizon, a 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 execution strategy and optimal value function and conclude that the optimal 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. The result differs from the previous prevailing one that a linear permanent price impact model precludes any price manipulation or arbitrage. Thus, considering a price impact caused by small traders’ orders with a Markovian dependence is significant.