Multi-asset optimal execution and statistical arbitrage strategies under Ornstein-Uhlenbeck dynamics

TL;DR

This paper develops a rigorous framework for multi-asset optimal execution and statistical arbitrage strategies using Ornstein-Uhlenbeck dynamics, employing stochastic control and matrix Riccati ODEs to derive solutions validated with market data.

Contribution

It introduces a novel approach to multi-asset optimal execution under Ornstein-Uhlenbeck dynamics, including existence and uniqueness proofs for the associated Riccati ODEs, and applies the results to real market data.

Findings

Validated strategies with foreign exchange and stock market data.

Demonstrated the effectiveness of the model in practical scenarios.

Provided theoretical guarantees for the solution's existence and uniqueness.

Abstract

In recent years, academics, regulators, and market practitioners have increasingly addressed liquidity issues. Amongst the numerous problems addressed, the optimal execution of large orders is probably the one that has attracted the most research works, mainly in the case of single-asset portfolios. In practice, however, optimal execution problems often involve large portfolios comprising numerous assets, and models should consequently account for risks at the portfolio level. In this paper, we address multi-asset optimal execution in a model where prices have multivariate Ornstein-Uhlenbeck dynamics and where the agent maximizes the expected (exponential) utility of her PnL. We use the tools of stochastic optimal control and simplify the initial multidimensional Hamilton-Jacobi-Bellman equation into a system of ordinary differential equations (ODEs) involving a Matrix Riccati ODE for which classical existence theorems do not apply. By using \textit{a priori} estimates obtained thanks to optimal control tools, we nevertheless prove an existence and uniqueness result for the latter ODE, and then deduce a verification theorem that provides a rigorous solution to the execution problem. Using examples based on data from the foreign exchange and stock markets, we eventually illustrate our results and discuss their implications for both optimal execution and statistical arbitrage.