Multi-Dimensional Martingales from Mutual Information

TL;DR

This paper introduces a unique method for constructing multi-dimensional martingales using Martingale Entropic Optimal Transport, addressing calibration issues in multi-asset markets and providing new theoretical insights.

Contribution

It proposes a novel, unique construction of multi-dimensional martingales via Martingale Entropic Optimal Transport, filling a gap in the theory for dimensions greater than one.

Findings

Provides a constructive proof of Strassen's classic result.

Demonstrates limitations of local correlation models in FX markets.

Introduces a new calibration method for multi-asset martingales.

Abstract

In the context of Risk Neutral Pricing theory, we consider the classic problem of calibrating a martingale over $\mathbb{R}^n$ to a finite number of marginals thereof, or more practically, to prices of an arbitrary finite set of (joint) European contingent claims. For $n=1$, one can rely on the work of Dupire, while for $n\geq 2$ an analogous natural unique construction seems to be lacking. We provide such a unique candidate as the result of pure Martingale Entropic Optimal Transport. As a byproduct, the latter allows us to obtain a constructive proof of a classic result of Strassen. Finally, and in contrast to the proposed approach, we prove a result that demonstrates how a certain class of local correlation models fails in general to calibrate to basket option prices, particularly in the foreign exchange market.