Weak Transport for Non-Convex Costs and Model-independence in a Fixed-Income Market

Abstract

A. We consider a model-independent pricing problem in a fixed-income market and show that it leads to a weak optimal transport problem as introduced by Gozlan et al. We use this to characterize the extremal models for the pricing of caplets on the spot rate and to establish a first robust super-replication result that is applicable to fixed-income markets.Notably, the weak transport problem exhibits a cost function which is non-convex and thus not covered by the standard assumptions of the theory. In an independent section, we establish that weak transport problems for general costs can be reduced to equivalent problems that do satisfy the convexity assumption, extending the scope of weak transport theory. This part could be of its own interest independent of our financial application, and is accessible to readers who are not familiar with mathematical finance notations.

Algorithms for Weak Optimal Transport with an Application to Economics

“…The production of a firm type x ∈ X depends non-linearly on the distribution π x ∈ P(Y) of its employees' types. 1 Barycentric WOT problem We will say that the WOT problem ( 4) is barycentric when F(x, p) only depends on the barycenter of p, that is when F(x, p) = F x, y dp(y) for some function F : X × conv(Y) → R, where conv(Y) is the convex hull of Y. 2 In the economic context, the barycentric specification is valid if the production of a firm depends on the distribution of its employees' types, p ∈ P(Y), only through their aggregate skills, y dp(y).…”

“…Over the past few years, optimal transport (OT) has gained importance in the machine learning community as a useful tool to analyze data, with applications to various domains such as graphics [37,8], imaging [34,15], generative models [5,35], biology [25,36], NLP [24,3], finance [7,22,1] or economics [21,20,30].…”