Rigorous accuracy analysis of multiscale asymptotic approximations for optimal trading with price impact

Establish rigorous error bounds and validity results for the multiscale asymptotic expansions of the value function and the corresponding optimal control in the infinite-horizon mean-variance optimal trading problem featuring predictable returns (modeled by an Ornstein–Uhlenbeck signal), instantaneous quadratic transaction costs, and linear transient price impact with exponential decay, under multiscale stochastic volatility driven by a fast factor and a slow factor. Specifically, justify the accuracy of the singular (fast) and regular (slow) perturbation expansions used to approximate solutions of the nonlinear Hamilton–Jacobi–Bellman equation in this setting, in the spirit of the accuracy analyses available for option pricing and the classical Merton portfolio problem.

Background

The paper derives formal multiscale asymptotic approximations for a nonlinear Hamilton–Jacobi–Bellman equation arising in an infinite-horizon mean-variance optimal trading problem with both instantaneous transaction costs and linear transient price impact, and with stochastic volatility driven by fast and slow factors. The authors obtain second-order corrections in the fast-scale case and first-order corrections in the slow-scale case, and demonstrate numerical improvements in profit and loss.

While rigorous accuracy analyses exist for similar multiscale asymptotic methods in option pricing and for the classical Merton portfolio problem, the present frictional optimal trading model involves a nonlinear HJB with price impact and stochastic volatility, for which a corresponding proof of accuracy is not provided. The authors explicitly state that establishing such a theoretical justification is left for future work.