An Optimal Transport approach to arbitrage correction: Application to volatility Stress-Tests

We present a method based on optimal transport to remove arbitrage opportunities within a finite set of option prices. The method is notably intended for regulatory stress-tests, which impose to apply important local distortions to implied volatility surfaces. The resulting stressed option prices are naturally associated to a family of signed marginal measures: we formulate the process of removing arbitrage as a projection onto the subset of martingale measures with respect to a Wasserstein metric in the space of signed measures. We show how this projection problem can be recast as an optimal transport problem; in view of the numerical solution, we apply an entropic regularization technique. For the regularized problem, we derive a strong duality formula, show convergence results as the regularization parameter approaches zero, and formulate a multi-constrained Sinkhorn algorithm, where each iteration involves, at worse, finding the root of an explicit scalar function. The convergence of this algorithm is also established. We compare our method with the existing approach by [Cohen, Reisinger and Wang, Appl.\ Math.\ Fin.\ 2020] across various scenarios and test cases.