Multi-Dimensional Martingales from Mutual Information

TL;DR

The paper introduces Hessian martingales, the unique Markov martingales that minimize mutual information subject to a finite set of marginal or option-price constraints, thereby enabling risk-neutral calibration in multidimensional settings. It provides a constructive extension of Strassen's theorem to $\mathbb{R}^n$ via an entropic optimal transport framework, characterizing the optimal kernels as exponential families linked to a Legendre duality with a base measure, and showing these kernels yield the desired martingale couplings. It develops both sequential and global calibration theories for incomplete marginals, proving coercivity and existence results under convex-analytic conditions, and supplies a practical Monte Carlo calibration scheme for hedging basket options in FX with multiple currencies. Finally, it demonstrates a failure of a broad class of local-correlation models to fit arbitrage-free basket prices and outlines avenues for extending the framework to continuous-time dynamics and stochastic interest rates.

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.