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 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.

💡 Research Summary

This paper addresses a practical problem faced by financial institutions during regulatory stress‑testing: the introduction of large, localized distortions to implied volatility surfaces often creates static arbitrage opportunities in the resulting option price data. Such arbitrage invalidates the fundamental theorem of asset pricing, making the stressed data unsuitable for downstream risk‑metrics such as VaR or for pricing exotic derivatives. The authors propose a novel framework that formulates arbitrage removal as a projection problem in the space of signed measures, leveraging optimal transport (OT) theory and entropic regularization.

The methodology proceeds in several logical steps. First, the authors adapt the construction of Cousot (who provided a non‑rectangular grid test for no‑arbitrage) to the case where the observed option prices contain arbitrage. By augmenting each maturity with two artificial strikes and defining a piecewise‑affine call‑price function π_i(k) as the lower boundary of the convex hull of observed (strike, price) points, they obtain a signed discrete measure μ̃_i for each maturity. When arbitrage is present, μ̃_i may have negative mass, but it still satisfies a linear relationship with the normalized option prices.

Second, the paper introduces a Wasserstein‑type distance between signed measures, building on recent work on unbalanced OT (Piccoli, Ambrosio, etc.). This distance allows the authors to quantify how far a given signed measure is from the set of martingale measures (probability measures that satisfy the martingale condition across maturities). The arbitrage‑free target is defined as the martingale measure ν that minimizes this distance to the stressed signed measure μ̃.

Third, to make the high‑dimensional projection computationally tractable, the authors add an entropy penalty to the transport cost, yielding an entropic OT (EOT) problem. They prove strong duality for the regularized problem, establish existence of optimal dual potentials, and show that as the regularization parameter ε → 0 the solution of the EOT problem converges to the original Wasserstein projection. These theoretical results are formalized in a series of lemmas and theorems (Section 5), providing a rigorous foundation for the numerical scheme.

Fourth, the paper develops a multi‑constrained Sinkhorn algorithm tailored to the specific constraints of the problem. Classical Sinkhorn iterations enforce only two marginal constraints (row and column sums). Here, additional constraints are required: (i) the martingale condition linking successive maturities, and (ii) the total mass consistency of the signed measure. The algorithm alternates between scaling updates for the row potentials (which enforce the martingale constraints) and column updates (which enforce marginal consistency). Each row update reduces to solving a scalar equation φ_i(λ)=0, where φ_i is a monotone function derived from the exponential kernel; the root can be found efficiently by bisection or Newton methods. The authors prove convergence of the iterative scheme by interpreting it as a block‑coordinate descent on a strictly convex Bregman divergence.

The numerical section validates the approach on three fronts. (1) Synthetic data with deliberately injected arbitrage across five maturities and twenty strikes demonstrates that the OT‑Sinkhorn method achieves a lower mean absolute error (MAE) and faster runtime than the L1‑minimization approach of Cohen, Reisinger, and Wang (2020). (2) Real market data (S&P 500 options) are subjected to a 30 % volatility stress; the proposed method restores a smooth, arbitrage‑free smile while preserving the shape of the stress, and the corrected prices can be directly used in Monte‑Carlo VaR calculations. (3) A high‑dimensional test with ten maturities and fifty strikes shows that the algorithm scales roughly linearly with the number of grid points, confirming its suitability for realistic regulatory scenarios.

In conclusion, the authors contribute a mathematically rigorous and computationally efficient solution to the arbitrage‑correction problem in stressed volatility surfaces. By recasting the issue as a Wasserstein projection of a signed measure onto the martingale set, regularizing with entropy, and solving via a customized multi‑constraint Sinkhorn algorithm, they overcome the limitations of previous linear‑programming based methods. The paper also opens several avenues for future research, including extensions to continuous (infinite‑dimensional) volatility surfaces, multi‑asset settings with joint martingale constraints, and integration into real‑time stress‑testing pipelines.