Ornstein-Uhlenbeck动态下的多资产最优执行与统计套利策略
Multi-asset optimal execution and statistical arbitrage strategies under Ornstein-Uhlenbeck dynamics
摘要：
近年来，学术界、监管机构和市场从业者越来越多地关注流动性问题。在众多的问题中，大订单的最优执行可能是吸引了最多研究工作的问题，主要是在单一资产组合的情况下。然而，在实践中，最优执行问题通常涉及由众多资产组成的大型投资组合，因此模型应考虑投资组合层面的风险。在本文中，我们解决了多资产最优执行的价格有多变量Ornstein-Uhlenbeck动态模型和代理最大化的预期（指数）效用，她的PnL。利用随机最优控制的工具，将初始的多维哈密顿-雅可比-贝尔曼方程简化为一个包含Riccati矩阵的常微分方程组，经典的存在性定理不适用于该常微分方程组.通过使用\textit{a priori}估计得到的最优控制工具，我们仍然证明了后者ODE的存在性和唯一性结果，然后推导出一个验证定理，提供了一个严格的解决方案的执行问题。使用的例子的基础上，从外汇和股票市场的数据，我们最终说明我们的结果，并讨论其最优执行和统计套利的影响。
英文摘要：
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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