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

TL;DR

The paper tackles the problem of removing arbitrage opportunities from stressed implied volatility surfaces by recasting option prices as a signed measure and projecting this measure onto the set of martingale measures under a Wasserstein-1 distance. It develops an entropically regularized, multi-constrained OT framework, proving strong duality and convergence as the regularization parameter vanishes, and introduces a scalable Sinkhorn-type algorithm to compute the corrected data. The method yields arbitrage-free martingale measures closest to the stressed data, with convergence guarantees and empirical validation on SPX data, including single- and multi-maturity cases. This approach provides regulators and practitioners with a robust, data-driven way to generate arbitrage-free risk scenarios for volatility stress-testing, potentially improving consistency and pricing for exotic instruments while maintaining computational feasibility.

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.

Figures (7)

• Figure 1: Sampled and stressed IVS of the SPX as of 02-09-2024 used to illustrate convergence results.
• Figure 2: Left: convergence of Algorithm \ref{['algo:multiconstrainedsinkhorn']} performed under the stressed conditions described by Figure \ref{['fig:data_numerical_convergence']}, as $n\rightarrow +\infty$, with $\varepsilon=1$. $\vert\vert \cdot \vert\vert_{F}$ stands for the Frobenius norm. Right: heatmap of the relative pointwise error between $\mathbf{M}_{n,R-1}$ and $\mathbf{M}_\varepsilon$, after the stopping criterion $\mathcal{E}_{\mathrm{tol}}$ was met ($n\approx 1000$).
• Figure 3: Empirical relation between $\mathcal{E}(n,R-1)$ and $\vert\vert \mathbf{M}_{n,R-1}-\mathbf{M}_\varepsilon \vert\vert_{F}$, along iterations of Algorithm \ref{['algo:multiconstrainedsinkhorn']}, in the toy example described by Figure \ref{['fig:data_numerical_convergence']}. The red line was obtained by linear regression in the log space.
• Figure 4: Evolution of the cost of $\mathbf{M}_\varepsilon$ when $\varepsilon$ approaches zero, in comparison with the optimal value of \ref{['optproblem:linearprogram']} solved in the situation described by Figure \ref{['fig:data_numerical_convergence']}.
• Figure 5: Left column: comparison of our method (in green) vs. cohenReisinger (in purple), for different types of stress-tests (in red), applied to observed SPX smile (in blue). Right column: illustration of the effect of regularization, for the same stress-tests.

Theorems & Definitions (60)

• Definition 2.1.1: Convex order and NDCO condition
• Remark 2.1.2
• Definition 2.2.1: Signed marginals from arbitrageable option prices
• Remark 2.2.2
• Definition 2.2.3: Piecewise-affine interpolation of prices
• Lemma 2.2.4: Lemma 3.1 in Cousot2004
• Remark 2.2.5
• Theorem 3.1.1: Kantorovich-Rubinstein duality formula, see villanitopics for $\alpha=1$
• Definition 3.2.1: Wasserstein distance between signed measures with unit mass
• Proposition 3.2.2