[Paper Review] Bid--Ask Martingale Optimal Transport

Bryan Liang, Marcel Nutz+2 more|arXiv (Cornell University)|Mar 14, 2026

Stochastic processes and financial applications0 citations

TL;DR

This paper extends Martingale Optimal Transport (MOT) to incorporate bid–ask frictions for vanilla options, defining Bid–Ask MOT (BAMOT) with marginals constrained by bid and ask spreads in convex order. It proves strong duality, analyzes convergence to MOT as spreads vanish, and demonstrates practical implications with synthetic and real-data examples.

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.

Motivation & Objective

• Motivate robust pricing and hedging for illiquid derivatives under bid–ask frictions in vanilla options.
• Generalize MOT by replacing exact marginal calibration with bid–ask constrained marginals in convex order.
• Establish a strong duality between BAMOT primal and dual problems.
• Prove convergence of BAMOT to classical MOT as bid–ask spreads vanish and quantify convergence rates.
• Provide theoretical insights and numerical illustrations using synthetic and real-market data.

Proposed method

• Formulate BAMOT primal problem: maximize expected payoff over martingale measures with bid and ask marginals in convex order.
• Formulate BAMOT dual problem: minimize cost of a static convex hedge bought at the ask minus sold at the bid plus dynamic hedging, ensuring superhedging of the payoff.
• Establish strong duality (P(h)=D(h)) first for the single-maturity case via a Hahn–Banach argument, then extend to multi-maturity via a minimax approach and MOT duality.
• Show consistency with classical MOT as bid–ask spreads shrink, proving convergence of BAMOT value and optimal strategies.
• Introduce a bid–ask distance metric in the single-maturity case to quantify convergence rates for payoffs that are differences of convex functions and for upper semicontinuous payoffs.
• Apply the framework to analytical and numerical examples, including a closed-form digital option in one-sided markets and real SPX data.

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.

Experimental results

Research questions

• RQ1 How can MOT be adapted to accommodate bid and ask price frictions in vanilla options?
• RQ2 What is the dual representation and strong duality for BAMOT in single and multiple maturities?
• RQ3 How does BAMOT relate to classical MOT as bid–ask spreads vanish?
• RQ4 What are the convergence rates of BAMOT prices under bid–ask frictions for specific payoff classes?
• RQ5 Do BAMOT bounds differ significantly from mid-marginal MOT in practical pricing and hedging of illiquid derivatives?

Key findings

• BAMOT yields realistic price bounds for illiquid derivatives under bid–ask frictions.
• Strong duality holds: BAMOT primal value equals the dual superhedging cost for upper semicontinuous payoffs with linear growth.
• As spreads vanish, BAMOT converges to classical MOT, with a novel bid–ask distance guiding the convergence rate.
• The dual problem remains nontrivial even for vanilla payoffs, due to the need to optimally combine bought and sold convex hedges.
• Numerical experiments show that mid-marginal pricing can substantially underestimate superhedging prices under bid–ask frictions, and BAMOT can capture convergence behavior for risk-reversal and at-the-money digital options.

Figure 2 : Bid–ask spread ( $\mathdollar$ ) of S&P 500 Index (SPX) options as of 02-07-2025.