Optimal trading without optimal control

Abstract

A hypothetical risk-neutral agent who trades to maximize the expected profit of the next trade will approximately exhibit long-term optimal behavior as long as this agent uses the vector p = ∇V (t, x) as effective microstructure alphas, where V is the Bellman value function for a smooth relaxation of the problem. Effective microstructure alphas are the steepest-ascent direction of V , equal to the generalized momenta in a dual Hamiltonian formulation. This simple heuristics has wide-ranging practical implications; indeed, most utility-maximization problems that require implementation via discrete limit-order-book markets can be treated by our method. * Bastien Baldacci gratefully acknowledge the financial support of the ERC Grant 679836 Staqamof and would like to thank Iuliia Manziuk (Ecole Polytechnique) for fruitful discussions.