Multi-Asset Execution and Statistical Arbitrage with Ornstein–Uhlenbeck Prices

Summary

The paper studies how to execute large orders across multiple assets when prices follow multivariate Ornstein–Uhlenbeck dynamics. It frames the trader’s objective as maximizing expected exponential utility of profit and loss, so the execution decision accounts for portfolio-level risk rather than treating each asset in isolation.

The authors use stochastic optimal control to reduce a multidimensional Hamilton–Jacobi–Bellman equation to ordinary differential equations, including a matrix Riccati equation. They derive existence and uniqueness for that equation using bounds from optimal control, then establish a verification result for the proposed solution. Examples using foreign exchange and stock market data illustrate the approach and its implications for execution and statistical arbitrage. The document’s abstract does not provide specific numerical results or implementation details, so it indicates the model’s theoretical foundation and application areas rather than enough information to assess live trading performance.

Key ideas

• The model optimizes execution across a portfolio of assets with multivariate Ornstein–Uhlenbeck prices.
• The objective is expected exponential utility of profit and loss.
• Stochastic control reduces the problem to ordinary differential equations, including a matrix Riccati equation.
• Optimal control bounds support existence and uniqueness results for the Riccati equation.
• Foreign exchange and stock market examples connect the framework to execution and statistical arbitrage.

Tags

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


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









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.

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