Optimal execution under a generalized price impact model with Markovian exogenous orders in a continuous-time setting (Financial Modeling and Analysis)

Abstract

In this paper, we analyze a continuous-time analog of the optimal trade execution problem with generalized price impacts, which was recently discussed in [11] for a discrete-time setting. The market model considers transient price impacts of random trade execution volumes posed by small traders as well as a large trader. Our problem is formulated as a stochastic continuous control problem over a finite horizon of maximizing the expected utility from the final wealth of the large trader with Constant Absolute Risk Aversion (CARA) von Neumann-Morgenstern (vN-M) utility function. By examining the Hamilton-Jacobi-Bellman (HJB) equation, we characterize the optimal value function and optimal trade execution strategy, and conclude that the trade execution strategy is a time-dependent affine function of three state variables: the remained trade execution volume of the large trader and, so-called, the residual effects of past price impacts caused by both of the large trader and other small traders and the small traders' aggregate volume of orders itself. Further, the time-dependent coefficients could be derived from a solution of a system of ordinary differential equations (ODEs) with terminal conditions, which is numerically tractable.