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Bid--Ask Martingale Optimal Transport

Bryan Liang, Marcel Nutz, Shunan Sheng, Valentin Tissot-Daguette

Abstract

Martingale Optimal Transport (MOT) provides a framework for robust pricing and hedging of illiquid derivatives. Classical MOT enforces exact calibration of model marginals to the mid-prices of vanilla options. Motivated by the industry practice of fitting bid and ask marginals to vanilla prices, we introduce a relaxation of MOT in which model-implied volatilities are only required to lie within observed bid--ask spreads; equivalently, model marginals lie between the bid and ask marginals in convex order. The resulting Bid--Ask MOT (BAMOT) yields realistic price bounds for illiquid derivatives and, via strong duality, can be interpreted as the superhedging price when short and long positions in vanilla options are priced at the bid and ask, respectively. We further establish convergence of BAMOT to classical MOT as bid--ask spreads vanish, and quantify the convergence rate using a novel distance intrinsically linked to bid--ask spreads. Finally, we support our findings with several synthetic and real-data examples.

Bid--Ask Martingale Optimal Transport

Abstract

Martingale Optimal Transport (MOT) provides a framework for robust pricing and hedging of illiquid derivatives. Classical MOT enforces exact calibration of model marginals to the mid-prices of vanilla options. Motivated by the industry practice of fitting bid and ask marginals to vanilla prices, we introduce a relaxation of MOT in which model-implied volatilities are only required to lie within observed bid--ask spreads; equivalently, model marginals lie between the bid and ask marginals in convex order. The resulting Bid--Ask MOT (BAMOT) yields realistic price bounds for illiquid derivatives and, via strong duality, can be interpreted as the superhedging price when short and long positions in vanilla options are priced at the bid and ask, respectively. We further establish convergence of BAMOT to classical MOT as bid--ask spreads vanish, and quantify the convergence rate using a novel distance intrinsically linked to bid--ask spreads. Finally, we support our findings with several synthetic and real-data examples.

Paper Structure

This paper contains 34 sections, 13 theorems, 115 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Bid--Ask Friction
  3. Primal Formulation
  4. Dual Formulation
  5. Contributions
  6. Related Literature
  7. Notation
  8. Organization
  9. Bid--Ask MOT
  10. Bid and Ask Marginals
  11. Primal (Measure) Problem
  12. Dual (Portfolio) Problem
  13. Duality
  14. Single Maturity
  15. Multiple Maturities
  16. ...and 19 more sections

Key Result

Lemma 3.1

We have $P(h)\le D(h)$ for all $h\in \textnormal{USC}_L(\mathbb{R}^N_+)$.

Figures (12)

  • Figure 1: Bid, model, and ask implied volatility skews for S&P 500 Index (SPX) options expiring on 03-21-2025, as of 02-27-2025.
  • Figure 2: Bid--ask spread ($\$$) of S&P 500 Index (SPX) options as of 02-07-2025.
  • Figure 3: Term structure of S&P 500 Index (SPX) call options struck at $K=5550$ as of 2025-02-27, and corresponding price envelopes.
  • Figure 4: Superhedging of a digital call in a one-sided market.
  • Figure 5: Digital call in a one-sided market. Illustrations of different optimal measures (left) and corresponding implied volatility skews (right).
  • ...and 7 more figures

Theorems & Definitions (34)

  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1: Weak Duality
  • proof
  • Theorem 3.2: Strong Duality, $N=1$
  • proof
  • Remark 3.3
  • Theorem 3.4: Strong Duality
  • proof
  • Proposition 4.1
  • ...and 24 more