Empirical martingale projections via the adapted Wasserstein distance

TL;DR

This paper introduces the smoothed empirical martingale projection distance (SE-MPD), a new metric for projecting empirical measures onto martingale couplings, with applications in testing arbitrage in neural network models for asset pricing.

Contribution

It provides an explicit characterization of SE-MPD, proves invariance under smoothing, and develops a consistent hypothesis test for arbitrage detection in financial models.

Findings

SE-MPD converges at a parametric rate under certain conditions.

The space of martingale couplings is invariant under smoothing.

The hypothesis test effectively detects arbitrage opportunities.

Abstract

Given a collection of multidimensional pairs $\{(X_i,Y_i):1 \leq i\leq n\}$, we study the problem of projecting the associated suitably smoothed empirical measure onto the space of martingale couplings (i.e. distributions satisfying $\mathbb{E}[Y|X]=X$) using the adapted Wasserstein distance. We call the resulting distance the smoothed empirical martingale projection distance (SE-MPD), for which we obtain an explicit characterization. We also show that the space of martingale couplings remains invariant under the smoothing operation. We study the asymptotic limit of the SE-MPD, which converges at a parametric rate as the sample size increases if the pairs are either i.i.d. or satisfy appropriate mixing assumptions. Additional finite-sample results are also investigated. Using these results, we introduce a novel consistent martingale coupling hypothesis test, which we apply to test the existence of arbitrage opportunities in recently introduced neural network-based generative models for asset pricing calibration.