Optimal Pair–Trade Execution with Generalized Cross–Impact

We examine a discrete–time optimal pair–trade execution problem with generalized cross–impact. This research is an extension of Fukasawa et al. (2020b), which considers the price impact of aggregate random orders posed by small traders with a Markovian dependence. We focus on how a risk–averse large trader optimally executes two correlated assets to maximize his/her expected utility from the terminal wealth over a finite horizon. A Markov decision process modeling constitutes the basis for the formulation of the optimal pair–trade execution problem. Then, under some regularity conditions, the backward induction method of dynamic programming enables us to derive the optimal pair–trade execution strategy and its associated optimal value function. The trading orders of each risky asset posed by small traders do affect the optimal execution volume of both risky assets. Moreover, numerical results with simulation experiments show that the cross–impact affects the optimal execution strategy and a round–trip trade exists for the large trader to utilize a ‘statistical’ arbitrage and to increase his/her expected utility under our model setting of cross–impact.