[论文解读] Incorporating Signals into Optimal Trading

Charles‐Albert Lehalle, Eyal Neuman|arXiv (Cornell University)|Apr 4, 2017

Financial Markets and Investment Strategies参考文献 4被引用 4

一句话总结

该论文通过将马尔可夫信号——具体而言是均值回归的订单簿失衡——纳入Gatheral、Schied和Slynko的瞬时市场冲击模型，扩展了最优交易框架。论文证明了最优策略的存在性与唯一性，并为具有指数市场冲击衰减的Ornstein-Uhlenbeck信号推导出显式的奇异最优策略，表明在瞬时冲击的极限下，该策略变为连续的，与Cartea和Jaimungal的框架一致。

ABSTRACT

Optimal trading is a recent field of research which was initiated by Almgren, Chriss, Bertsimas and Lo in the late 90's. Its main application is slicing large trading orders, in the interest of minimizing trading costs and potential perturbations of price dynamics due to liquidity shocks. The initial optimization frameworks were based on mean-variance minimization for the trading costs. In the past 15 years, finer modelling of price dynamics, more realistic control variables and different cost functionals were developed. The inclusion of signals (i.e. short term predictors of price dynamics) in optimal trading is a recent development and it is also the subject of this work. We incorporate a Markovian signal in the optimal trading framework which was initially proposed by Gatheral, Schied, and Slynko [21] and provide results on the existence and uniqueness of an optimal trading strategy. Moreover, we derive an explicit singular optimal strategy for the special case of an Ornstein-Uhlenbeck signal and an exponentially decaying transient market impact. The combination of a mean-reverting signal along with a market impact decay is of special interest, since they affect the short term price variations in opposite directions. Later, we show that in the asymptotic limit were the transient market impact becomes instantaneous, the optimal strategy becomes continuous. This result is compatible with the optimal trading framework which was proposed by Cartea and Jaimungal [10]. In order to support our models, we analyse nine months of tick by tick data on 13 European stocks from the NASDAQ OMX exchange. We show that orderbook imbalance is a predictor of the future price move and it has some mean-reverting properties. From this data we show that market participants, especially high frequency traders, use this signal in their trading strategies.

研究动机与目标

• 将短期价格预测因子（信号）整合进最优执行模型，以降低交易成本。
• 通过引入能够预测短期价格变动的马尔可夫信号，扩展Gatheral-Schied-Slynko框架。
• 在引入此类信号的条件下，建立最优交易策略的存在性与唯一性。
• 为Ornstein-Uhlenbeck信号与指数衰减市场冲击的特殊情况，推导出显式的奇异最优策略。
• 表明在瞬时市场冲击的渐近极限下，最优策略变为连续的，与Cartea和Jaimungal的框架一致。

提出的方法

• 构建一个带有马尔可夫信号（如订单簿失衡）影响价格动态的随机控制问题。
• 利用汉密尔顿-雅可比-贝尔曼（HJB）方程推导最优控制，同时纳入市场冲击与信号动态。
• 对值函数采用二次形式假设：$ v(t,x,\iota) = v_0(t,\iota) + x v_1(t,\iota) + x^2 v_2(t,\iota) $，从而导出一个偏微分方程组。
• 通过Feynman-Kac表示法求解PDE系统，特别是利用信号过程的条件期望求解$ v_1 $与$ v_0 $。
• 应用Gronwall不等式验证所推导的最优交易速率$ r^* $的可适性。
• 使用来自NASDAQ OMX的13只欧洲股票在九个月内的逐笔交易数据，验证信号的预测能力，表明均值回归的订单簿失衡可预测价格变动。

实验结果

研究问题

• RQ1 如何将马尔可夫信号（如订单簿失衡）正式纳入具有瞬时市场冲击的最优交易框架？
• RQ2 在Gatheral-Schied-Slynko框架中引入均值回归信号后，是否能获得定义良好且唯一的最优交易策略？
• RQ3 当信号服从Ornstein-Uhlenbeck过程且市场冲击呈指数衰减时，最优策略的显式形式是什么？
• RQ4 当瞬时市场冲击趋于瞬时极限时，最优策略的行为如何？
• RQ5 在真实市场数据中，订单簿失衡在多大程度上可作为短期价格变动的预测信号？

主要发现

• 为Ornstein-Uhlenbeck信号与指数衰减市场冲击的情形，推导出显式的奇异最优交易策略，该策略依赖于信号的均值回归速度与冲击衰减参数。
• 通过二次假设与Feynman-Kac表示法构造PDE系统的解，确保了值函数与最优控制的存在性与唯一性。
• 在市场冲击趋于瞬时的极限下，最优策略由奇异形式转变为连续形式，与Cartea-Jaimungal框架一致。
• 基于来自13只欧洲股票在NASDAQ OMX的高频数据，实证验证了订单簿失衡作为短期价格变动的均值回归预测因子的有效性。
• 高频交易员与市场参与者似乎已实际利用该信号，表明其在真实交易策略中的实际相关性。
• 通过Gronwall不等式证明了最优交易速率$ r^* $的可适性，确保了策略在时间上的可积性与可行性。