Abstract:
Optimal transport stands out as a mathematical field due to its rich interplay between theory and applications. Questions arising from practice – particularly in areas such as finance, economics, and data science – lead to new variants of the classical optimal transport framework. These theoretical extensions, in turn, enable further applications. This thesis is devoted to this exchange between theory and applications, with a particular focus on the use of optimal transport tools in financial contexts.

The first chapter of this thesis introduces a novel application of optimal transport in financial regulation. Motivated by the challenges faced by regulators of the financial industry in analyzing large amounts of credit data, we represent the credit portfolios of financial institutions as probability measures. Using generalized Wasserstein barycenters, we develop a clustering and reconstruction method for probability measures defined on different subspaces. This enables the construction of a metric structure over financial institutions, resulting in a banking landscape that can be used to identify clusters and detect outliers in the financial system.

The second chapter is devoted to the theory of weak optimal transport, an extension of classical optimal transport that allows for non-linear cost functions. We prove a fundamental theorem of weak optimal transport; in particular, we establish duality and a complementary slackness condition. This results in a wide range of applications, allowing us to recover the convex Kantorovich–Rubinstein formula, the Brenier–Strassen theorem, the structure theorem of entropic optimal transport, and to obtain new results concerning relaxed martingale optimal transport, as well as interpolations between Brenier–Strassen and martingale Benamou–Brenier.

The third chapter of this thesis connects weak martingale optimal transport to local volatility modeling in mathematical finance. By extending the classical change of numeraire transformation to weak martingale optimal transport, we relate Bass martingales to a new class of processes, namely geometric Bass martingales. This yields a
local volatility model that is as close as possible to the Black–Scholes model, the benchmark
model in the financial industry, while being perfectly calibrated to observed market prices.
Overall, this thesis illustrates how advances in optimal transport can be both applied to,
and motivated by, practical problems in the financial sector, while simultaneously
contributing to the theoretical development of the field.

Online:
univienna.zoom.us/j/66813827831
UmF.1
Meeting ID: 668 1382 7831
Kenncode: 601427