Price Impact and Optimal Transport

This project studies (a) regularized optimal transport and (b) price impact in financial markets. Fueled by computational advances, optimal transport has become ubiquitous in applications from machine learning to image processing and economics. In such applications, optimal transport is often regularized with an entropic or quadratic penalty. Entropic regularization is the most frequent choice as it greatly facilitates computation and enhances smoothness. On the other hand, quadratic regularization is chosen when sparse solutions are desired. Studied mostly in the machine learning literature, its mathematical foundations are much less developed. The project will provide those foundations and theoretical guarantees for sparsity. Price impact in financial markets refers to the fact that prices are displaced during the execution of institutional-size orders; for instance, large buy orders push prices up. However, prices revert back over time. This resilience is vital for optimizing transaction costs in practice, but not modeled in many academic studies. The project features a range of broader impact activities, including advising and mentoring a diverse group of postdocs, graduate and undergraduate students, organizing interdisciplinary scientific meetings and summer schools, and serving on editorial boards and professional societies. The first part of this project investigates entropically regularized optimal transport, where couplings are penalized by KL-divergence. Specifically, it studies the convergence of the optimal coupling as the regularization parameter tends to zero. The long-standing conjecture of entropic selection predicts that the optimal coupling converges to a certain solution of the unregularized optimal transport problem; that is, the limit selects a particular solution out of the possibly large set of optimal transports. The project aims to prove this in the most important setting, namely for Monge's distance cost. The second part of the project investigates quadratically regularized optimal transport, where couplings are penalized by the squared norm. It aims to analytically describe the empirically observed phenomenon of sparse support as well as the convergence for vanishing regularization, in both discrete and continuous settings. While quadratic regularization was mostly used in computational works so far, the project provides a robust mathematical toolbox for its study. The third part of the project investigates financial markets with transient price impact. The project studies predatory trading and liquidity provision, and in particular the regulatory issue of pre-hedging, in the presence of price impact and resilience. On the methodological side, it also establishes how to correctly formalize the Obizhaeva--Wang model (and related models) for non-cooperative games with price impact. Separately, the project studies the optimal execution of order flows for central risk books and market makers. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.