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Convergence of Sinkhorn's Algorithm for Entropic Martingale Optimal Transport Problem

Fan Chen, Giovanni Conforti, Zhenjie Ren, Xiaozhen Wang

TL;DR

This work studies Entropic Martingale Optimal Transport (EMOT) on $\mathbb{R}$, motivated by calibration of stochastic volatility models, and develops a rigorous Sinkhorn-based solver. It derives a dual formulation, proves uniform bounds and exponential convergence of the dual iterates to an optimal triple $(f^*,g^*,h^*)$, and shows that the corresponding transport $\pi(f^*,g^*,h^*)$ minimizes the EMOT objective with no primal-dual gap. The analysis accommodates martingale constraints alongside entropy regularization and does not assume the a priori absence of a primal-dual gap. Numerical experiments illustrate convergence behavior and demonstrate a practical one-period EMOT calibration workflow using Heston-based market data, highlighting the method’s stability and computational efficiency for finance applications.

Abstract

In this paper, we study the Entropic Martingale Optimal Transport (EMOT) problem on \mathbb{R}. The investigation of the EMOT problem arises in the calibration problem of the Stochastic Volatility Models, where martingale constraints reflect no-arbitrage pricing conditions under the risk-neutral measure, as originally proposed by Henry-Labordere. We first establish the dual formulation of the EMOT problem and prove that Sinkhorn's algorithm achieves an exponential convergence rate under mild conditions. Notably, our analysis does not presuppose the existence of optimal potentials and rigorously confirms the absence of a primal-dual gap. These results provide a theoretical foundation for solving EMOT via Sinkhorn's method and constructing the optimal distribution from dual coefficients.

Convergence of Sinkhorn's Algorithm for Entropic Martingale Optimal Transport Problem

TL;DR

This work studies Entropic Martingale Optimal Transport (EMOT) on , motivated by calibration of stochastic volatility models, and develops a rigorous Sinkhorn-based solver. It derives a dual formulation, proves uniform bounds and exponential convergence of the dual iterates to an optimal triple , and shows that the corresponding transport minimizes the EMOT objective with no primal-dual gap. The analysis accommodates martingale constraints alongside entropy regularization and does not assume the a priori absence of a primal-dual gap. Numerical experiments illustrate convergence behavior and demonstrate a practical one-period EMOT calibration workflow using Heston-based market data, highlighting the method’s stability and computational efficiency for finance applications.

Abstract

In this paper, we study the Entropic Martingale Optimal Transport (EMOT) problem on \mathbb{R}. The investigation of the EMOT problem arises in the calibration problem of the Stochastic Volatility Models, where martingale constraints reflect no-arbitrage pricing conditions under the risk-neutral measure, as originally proposed by Henry-Labordere. We first establish the dual formulation of the EMOT problem and prove that Sinkhorn's algorithm achieves an exponential convergence rate under mild conditions. Notably, our analysis does not presuppose the existence of optimal potentials and rigorously confirms the absence of a primal-dual gap. These results provide a theoretical foundation for solving EMOT via Sinkhorn's method and constructing the optimal distribution from dual coefficients.
Paper Structure (15 sections, 11 theorems, 136 equations, 8 figures, 2 tables, 1 algorithm)

This paper contains 15 sections, 11 theorems, 136 equations, 8 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notations
  3. Main Results
  4. Applications
  5. Numerical Test on Convergence Rate
  6. Coefficient of Entropy regularization
  7. Dependence on $\nu$
  8. Calibration of Stochastic Volatility Models via EMOT
  9. Background
  10. Numerical Implementation
  11. Problem Setting
  12. Numerical Results
  13. Proof of the Main Results
  14. Uniform bound for dual functions
  15. Exponential convergence

Key Result

Proposition 2.7

Under Assumptions assu:mu_nu, assu:lipschitz_c_conti and assu:measuable_init_bdd, the iteration steps $(\tilde{f}^{n+1}, \tilde{g}^{n+1})_{n\ge 1}$ by Sinkhorn's iterations are well-defined on $\mathbb R$ in eq:update_formula_fg, and $(\tilde{h}^{n+1})_{n\ge1}$ are well defined on $\mathcal{X}_0$ in

Figures (8)

  • Figure 1: Convergence rate for different coefficient of Entropy regulization.
  • Figure 2: Convergence rate for different level of convex order in $\nu$.
  • Figure 3: Histogram of stock prices at time $t_1$ and $t_2$ and volatility at time $t_1$
  • Figure 4: Lines: density function of probability measures $\mu,\nu$; Dotted lines: marginal distribution $X,Y$ of optimal transport matrix $\pi^\star$.
  • Figure 5: Martingale constraints: $\int_{\mathcal{Y},\mathcal{Z}} (y-x)d\hat{\pi}_{y,z|x}$ for different $x$.
  • ...and 3 more figures

Theorems & Definitions (33)

  • Remark 1.1
  • Remark 1.2
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.7
  • Proposition 2.8
  • Remark 2.9
  • Theorem 2.10
  • Theorem 2.11
  • Remark 2.12
  • ...and 23 more