Paper: arXiv 2406.07523

Authors: Mathias Beiglböck, Gudmund Pammer, Lorenz Riess

Abstract

Change of numeraire is a classical tool in mathematical finance. Campi-Laachir-Martini established its applicability to martingale optimal transport. We note that the results of Campi-Laachir-Martini extend to the case of weak martingale transport. We apply this to shadow couplings, continuous time martingale transport problems in the framework of Huesmann-Trevisan and in particular to establish the correspondence between stretched Brownian motion with its geometric counterpart. Note: We emphasize that we learned about the geometric stretched Brownian motion gSBM (defined in PDE terms) in a presentation of Loeper \cite{Lo23} before our work on this topic started. We noticed that a change of numeraire transformation in the spirit of \cite{CaLaMa14} allows for an alternative viewpoint in the weak optimal transport framework. We make our work public following the publication of Backhoff-Loeper-Obloj’s work \cite{BaLoOb24} on arxiv.org. The article \cite{BaLoOb24} derives gSBM using PDE techniques as well as through an independent probabilistic approach which is close to the one we give in the present article.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is a highly theoretical mathematical finance work that extends the Change of Numeraire (CN) technique to weak martingale optimal transport and geometric stretched Brownian motion. While mathematically rigorous and advanced in its proofs, it contains virtually no empirical evaluation or backtesting on real market data, focusing instead on theoretical formulations and numerical simulation schemes.

Research Flowchart

  flowchart TD
    A[Research Goal: Extend Change of Numeraire to Weak Martingale Transport WMT] --> B[Inputs: Campi-Laachir-Martini Framework & Loeper's gSBM Concept]
    B --> C[Methodology: Apply Change of Numeraire Transformation to WMT]
    C --> D[Computational Analysis: Evaluate Shadow Couplings & Huesmann-Trevisan Continuous Time WMT]
    D --> E[Outcome 1: Alternative WMT Viewpoint for Geometric Stretched Brownian Motion gSBM]
    D --> F[Outcome 2: Established Correspondence Between Stretched Brownian Motion & gSBM]

Cited References (2)

  1. Optimal Couplings on Wiener Space and An Extension of Talagrand’s Transport Inequality (2022) — link
  2. On the quadratic variation process of a continuous Martingale (1983) — link

0 of 2 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.