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

TL;DR

This paper introduces an optimal transport-based method to correct arbitrage in option prices, especially useful for regulatory stress-testing, by projecting onto martingale measures with convergence guarantees and an efficient algorithm.

Contribution

It develops a novel optimal transport framework with entropic regularization for arbitrage removal, including a new multi-constrained Sinkhorn algorithm with proven convergence.

Findings

The method effectively removes arbitrage in simulated scenarios.

It outperforms existing approaches in accuracy and computational efficiency.

Theoretical convergence results support practical applicability.

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.