[Paper Review] Incorporating Signals into Optimal Trading

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

Financial Markets and Investment Strategies4 references4 citations

TL;DR

This paper extends optimal trading frameworks by incorporating Markovian signals—specifically mean-reverting order book imbalances—into the transient market impact model of Gatheral, Schied, and Slynko. It proves existence and uniqueness of an optimal strategy and derives an explicit singular optimal strategy for an Ornstein-Uhlenbeck signal with exponential market impact decay, showing that in the limit of instantaneous impact, the strategy becomes continuous, aligning with Cartea and Jaimungal's framework.

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.

Motivation & Objective

• To integrate short-term price predictors (signals) into optimal execution models to reduce transaction costs.
• To extend the Gatheral-Schied-Slynko framework by incorporating a Markovian signal that predicts short-term price movements.
• To establish existence and uniqueness of an optimal trading strategy under the inclusion of such signals.
• To derive an explicit singular optimal strategy for the special case of an Ornstein-Uhlenbeck signal and exponentially decaying market impact.
• To show that in the asymptotic limit of instantaneous market impact, the optimal strategy becomes continuous, consistent with Cartea and Jaimungal's framework.

Proposed method

• Formulates a stochastic control problem with a Markovian signal (e.g., order book imbalance) affecting price dynamics.
• Uses the Hamilton-Jacobi-Bellman (HJB) equation to derive the optimal control, incorporating both market impact and signal dynamics.
• Employs a quadratic ansatz for the value function: $ v(t,x,\iota) = v_0(t,\iota) + x v_1(t,\iota) + x^2 v_2(t,\iota) $, leading to a system of PDEs.
• Solves the PDE system via Feynman-Kac representation, particularly for $ v_1 $ and $ v_0 $, using conditional expectations of the signal process.
• Applies Gronwall's inequality to verify the admissibility of the derived optimal trading rate $ r^* $.
• Validates the signal's predictive power using nine months of tick-by-tick data from 13 European stocks on NASDAQ OMX, showing mean-reverting order book imbalance predicts price moves.

Experimental results

Research questions

• RQ1 How can a Markovian signal, such as order book imbalance, be formally incorporated into optimal trading frameworks with transient market impact?
• RQ2 Does the inclusion of a mean-reverting signal lead to a well-defined and unique optimal trading strategy under the Gatheral-Schied-Slynko framework?
• RQ3 What is the explicit form of the optimal strategy when the signal follows an Ornstein-Uhlenbeck process and market impact decays exponentially?
• RQ4 How does the optimal strategy behave in the limit as transient market impact becomes instantaneous?
• RQ5 To what extent does order book imbalance serve as a predictive signal for short-term price movements in real market data?

Key findings

• An explicit singular optimal trading strategy is derived for the case of an Ornstein-Uhlenbeck signal and exponentially decaying market impact, with the strategy depending on the signal's mean-reversion and impact decay parameters.
• The solution to the PDE system is constructed using a quadratic ansatz and Feynman-Kac representation, ensuring existence and uniqueness of the value function and optimal control.
• In the limit where market impact becomes instantaneous, the optimal strategy transitions from singular to continuous, aligning with the Cartea-Jaimungal framework.
• The order book imbalance is empirically validated as a mean-reverting predictor of short-term price moves in high-frequency data from 13 European stocks on NASDAQ OMX.
• High-frequency traders and market participants appear to exploit this signal, indicating its practical relevance in real trading strategies.
• The optimal trading rate $ r^* $ is shown to be admissible via Gronwall’s inequality, ensuring integrability and feasibility of the strategy over time.