Optimal placement in a limit order book: an analytical approach
The paper under review is a valuable contribution to applied probability and operational research at their interface as expressed by (i) stochastic processes and (ii) optimization, with many possible applications in economics, finance and commodity markets, industrial and chemical engineering, etc. This study is theoretically very rigorous and practically highly relevant; based on it, future research can be established and real-world applications conducted, in the aforementioned areas of modern and emerging industries, but also in earth- and geosciences and in developmental sciences. In this article, the authors offer and investigate a problem on optimal placement in a limit order book. On the basis of a correlated random walk model with mean reversion for the best ask/bid price, the authors derive optimal placement strategies for both static and dynamic cases. Actually, in the static case, the optimal strategy just involves market order, best bid and second best bid. In the dynamic case, the optimal strategy is shown to be of a threshold type which hinges on remaining trading time, market momentum and price mean-reversion factor. Of central importance for the analysis is a generalized reflection principle on correlated random walks, enabling a significant dimension reduction. The four sections of this paper are these: 1 Introduction, namely, 1.1 Limit order book, 1.2 The optimal placement problem, 1.3 Relaxation to the optimal execution problem, 1.4 Relationship to the market-making problem, 1.5 Our (the authors') contributions, 1.6 Outline of the paper, 2 The model and preliminary analysis, namely, 2.1 The model, and 2.2 Preliminary analysis, 3 Optimal strategy for the optimal placement problem, namely, 3.1 The static case, 3.2. The dynamic case, and 4 Conclusions. These sections are followed by an Appendix: Proofs. In future, refinements and advances might be expected in analytic and stochastic theory and in numerical methods by the scientific community, initiated by this paper. These could be made in terms of stochastic bi- and multilevel programming, stochastic mixed-integer programming, stochastic robust optimization, stochastic optimal control under regime switches, and stochastic game theory. Such future progress could catalyze and support advances in economics and finance, neuroscience, neuro-economics and -finance, medicine, natural sciences and engineering, environmental and earth-sciences, as well as in sciences on population dynamics, societal complexity and sustainable development.
scientific article; zbMATH DE number 6692620
scientific article