Entropy in Optimal Transport and Finance

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This project investigates entropic penalties in two contexts: optimal transport and finance. In optimal transport, entropic regularization is an approximation enabling fast and robust computations in data-rich settings such as machine learning or image processing. In finance, entropic penalties yield a flexible and systematic method to calibrate a benchmark security model to market data on option prices. The research advances a vast array of applications in technology and science where entropic methods are used and leads to a transfer of knowledge between optimal transport, mathematical finance and probability theory. A diverse group of postdocs, graduate and undergraduate students is trained as part of the project.The first part of this project investigates entropically regularized optimal transport (EOT). Optimal transport provides a natural way to lift a distance or cost from a base space to its space of probability measures, hence has become ubiquitous in applications where data sets or statistical distributions are compared. Entropic regularization allows for Sinkhorn's algorithm and is the most important method for approximate computation is high-dimensional settings. The project develops a novel geometric method to study the convergence of EOT to its unregularized counterpart as the regularization parameter decreases. Moreover, it studies the stability of EOT with respect to the marginal distributions, as well as the so-called Schrödinger potentials that solve an associated dual problem. The second part of this proposal studies the calibration of option pricing models in finance. Starting with a reference model, such as a standard stochastic volatility model, and option price data given by the market, a calibrated model is chosen by minimizing the relative entropy with respect to the reference among all martingales fitting the data. This nonparametric approach is data-driven and flexible while retaining desirable qualities of the reference model. The calibrated model is an analogue to the classical Schrödinger bridge process but incorporates an additional martingale constraint to avoid dynamic arbitrages. The project investigates this model from financial and probabilistic perspectives.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.