A stability result on optimal Skorokhod embedding

TL;DR

This paper investigates a stability property of an optimization problem related to the Skorokhod embedding, showing convergence as more vanilla option prices are incorporated, with implications for numerical methods in derivative pricing.

Contribution

It establishes a stability result and convergence analysis for a finite-option version of the optimal Skorokhod embedding problem, linking it to the classical problem.

Findings

Convergence of the finite-option problem to the classical Skorokhod embedding as options increase.

Duality and geometric characterization of the optimizers.

Quantitative convergence rate analysis under certain metrics.

Abstract

Motivated by the model- independent pricing of derivatives calibrated to the real market, we consider an optimization problem similar to the optimal Skorokhod embedding problem, where the embedded Brownian motion needs only to reproduce a finite number of prices of Vanilla options. We derive in this paper the corresponding dualities and the geometric characterization of optimizers. Then we show a stability result, i.e. when more and more Vanilla options are given, the optimization problem converges to an optimal Skorokhod embedding problem, which constitutes the basis of the numerical computation in practice. In addition, by means of different metrics on the space of probability measures, a convergence rate analysis is provided under suitable conditions.