Solving the Optimal Trading Trajectory Problem Using a Quantum Annealer

TL;DR

This paper demonstrates how a quantum annealer can be used to solve a complex multi-period portfolio optimization problem, incorporating realistic market impact costs without matrix inversion, and discusses scalability and current limitations.

Contribution

It presents a novel formulation of a multi-period portfolio optimization problem suitable for quantum annealing, including encoding schemes and scalability considerations.

Findings

High success rates in numerical examples

Formulation includes transaction costs and market impact

Current technology limits problem size

Abstract

We solve a multi-period portfolio optimization problem using D-Wave Systems' quantum annealer. We derive a formulation of the problem, discuss several possible integer encoding schemes, and present numerical examples that show high success rates. The formulation incorporates transaction costs (including permanent and temporary market impact), and, significantly, the solution does not require the inversion of a covariance matrix. The discrete multi-period portfolio optimization problem we solve is significantly harder than the continuous variable problem. We present insight into how results may be improved using suitable software enhancements, and why current quantum annealing technology limits the size of problem that can be successfully solved today. The formulation presented is specifically designed to be scalable, with the expectation that as quantum annealing technology improves, larger problems will be solvable using the same techniques.