An Optimal Transport approach to arbitrage correction: application to Volatility Stress-Tests

Abstract

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 require applying significant local distortions to implied volatility surfaces, thereby introducing arbitrage. The resulting stressed option prices being associated with 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, to which we then 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 worst, finding the root of an explicit scalar function. The convergence of this algorithm is also established. We compare our method with the existing approach of [Cohen, Reisinger and Wang, Appl.\ Math.\ Fin.\ 2020] across various scenarios and test cases.