My name is Julia Ackermann and I am a postdoc in the Applied and Computational Mathematics (ACM) group at the University of Wuppertal, Germany. In this blog post I give a brief introduction to optimal trade execution and my research in this area.
The task
When a large institution needs to buy (or sell) a significant amount of shares, it can lead to unfavorable consequences in an illiquid market: there might not be enough shares available at the current price level, so the institution has to purchase the remaining shares at higher price levels. Hence, the institution’s trading activity adversely impacts the price and incurs so-called execution costs.
However, the institution might be able to extend their trading until a small fixed time horizon T. In this case, it is usually better (in terms of smaller execution costs) to split the large order into several smaller ones that are executed consecutively. The question then arises: how should the trading activities be scheduled to achieve minimal execution costs while reaching the buying or selling goal?
Models
Mathematically, buying or selling a certain amount of shares by time T while minimizing execution costs can be modeled as an optimal control problem. Pioneering works in optimal trade execution that set up and analyze relevant control problems include Bertsimas & Lo [1], Almgren & Chriss [2], and Obizhaeva & Wang [3]. If you are interested in playing with a numerical tool for solving such a control problem and computing optimal trade execution strategies (following the model in Chen et al. [4]), the Jupyter Notebook [5] might be of interest.
Models for optimal trade execution can differ in several regards. For example, trading can be allowed only at a finite number of fixed time points (discrete time), or trading can be possible during the whole trading interval from 0 to the terminal time T (continuous time). This is closely linked to the set of possible trading strategies one considers. Further, one needs to specify how trading according to a certain strategy affects the price and the execution costs.
To motivate their choice for the price behavior and the execution costs, Obizhaeva and Wang [3] based their trade execution model on a simplified version of a limit order book (LOB). An LOB stores the orders of the market participants. For a real-world example of an electronic LOB, visit a suitable stock exchange’s website [6].
A glimpse into our work
Inspired by the seminal work [3], researchers have extended or modified this model to better capture features such as more general shapes, variation during the day, or stochastic behavior of real-world LOBs. Specifically, we studied optimal trade execution in an LOB-model where both order book depth and resilience are allowed to evolve stochastically over time in a discrete-time setting [7], and in a continuous-time setting [8], [9]. Additionally, in [8] and [9], we considered trading strategies that go beyond the ones that had typically been used. The challenge was thus not only the stochastic nature of the LOB-parameters but also extending the control problem to more general classes of trading strategies. In [9], we established that under appropriate assumptions, the problem can be continuously extended, ensuring that a unique optimal trading strategy exists, and we derived formulas for the optimal strategy and its execution costs in terms of the solution to a backward stochastic differential equation.
When there is more than one asset
Often, a large institution needs to buy or sell in more than one asset. It is then crucial to take potential cross-impact between these assets into account. For instance, consider one asset of a certain company and a second asset of another company in the same sector. Trading in the first asset will then likely also affect the price of the second asset. If the institution handles the optimal trade execution tasks for these two assets separately, it can lead to sub-optimal overall trading behavior and unnecessarily high execution costs. This scenario illustrates the need to explicitly study optimal trade execution in a portfolio of assets, which is a topic of my ongoing research.
References
- [1] D. Bertsimas, A. W. Lo (1998): Optimal control of execution costs. Journal of Financial Markets 1(1), 1-50.
- [2] R. Almgren, N. Chriss (2001): Optimal execution of portfolio transactions. Journal of Risk 3, 5-40.
- [3] A. A. Obizhaeva, J. Wang (2013): Optimal trading strategy and supply/demand dynamics. Journal of Financial Markets 16, 1-32.
- [4] T. Chen, M. Ludkovski, M. Voss (2023): On parametric optimal execution and machine learning surrogates. Quantitative Finance 24(1), 15-34.
- [5] https://github.com/moritz-voss/Parametric_Optimal_Execution_ML
- [6] https://www.boerse-frankfurt.de/equities/open-xetra-orderbook/DE0008469008
- [7] J. Ackermann, T. Kruse, M. Urusov (2021): Optimal trade execution in an order book model with stochastic liquidity parameters. SIAM Journal on Financial Mathematics 12, 788-822.
- [8] J. Ackermann, T. Kruse, M. Urusov (2021): Cadlag semimartingale strategies for optimal trade execution in stochastic order book models. Finance and Stochastics 25, 757-810.
- [9] J. Ackermann, T. Kruse, M. Urusov (2024): Reducing Obizhaeva-Wang-type trade execution problems to LQ stochastic control problems. Finance and Stochastics 28, 813-863.