Trading with propagators and constraints: applications to optimal execution and battery storage

Abstract: Motivated by optimal execution with stochastic signals, market impact and constraints in financial markets, and optimal storage management in commodity markets, we formulate and solve an optimal trading problem with a general propagator model under linear functional inequality constraints. The optimal control is given explicitly in terms of the corresponding Lagrange multipliers and their conditional expectations, as a solution to a linear stochastic Fredholm equation. We propose a stochastic version of the Uzawa algorithm on the dual problem to construct the stochastic Lagrange multipliers numerically via a stochastic projected gradient ascent, combined with a least-squares Monte Carlo regression step to approximate their conditional expectations. We illustrate our findings on two different practical applications with stochastic signals: (i) an optimal execution problem with an exponential or a power law decaying transient impact, with either a no-shorting' constraint in the presence of asell' signal, a no-buying' constraint in the presence of abuy' signal or a stochastic `stop-trading' constraint whenever the exogenous price drops below a specified reference level; (ii) a battery storage problem with instantaneous operating costs, seasonal signals and fixed constraints on both the charging power and the load capacity of the battery.