Frictional martingale optimal transport and robust hedging

TL;DR

This work develops a robust, model-free framework for martingale optimal transport under state-dependent trading frictions in discrete time. By introducing a convex friction cost $f^{(a,b)}_t(x,v)$ and a frictional Spence–Mirrlees condition, it uncovers a frictional left-monotone geometry where, at each step, transport is bi-atomic outside a no-trade band and mass on the diagonal prevails inside the band; end-points satisfy an equal-slope condition balancing continuation value and trading costs. The authors establish stability and a vanishing-friction limit to the frictionless left-curtain, provide a dynamic-programming extension to the multi-marginal setting, and prove strong duality between the primal frictional MOT and its dual. These results yield a rigorous foundation for model-independent pricing and robust hedging of path-dependent derivatives in the presence of liquidity costs, with explicit band geometries and tractable off-band displacements that inform both theory and potential numerical schemes.

Abstract

We study the martingale optimal transport problem with state-dependent trading frictions and develop a geometric and duality framework extending from the one time-step to the multi-marginal setting. Building on the left-monotone structure of frictionless MOT (Beiglböck and Juillet, Ann. Probab., 2016; Henry-Labordère and Touzi, Finance Stoch., 2016; Beiglböck et al., Ann. Probab., 2017), we introduce a convex frictional cost combining proportional bid-ask spreads and quadratic liquidity impacts. The framework extends the martingale Spence-Mirrlees condition to nonlinear frictions and establishes a frictional monotonicity principle. At each time step, the joint distribution between consecutive asset prices exhibits a bi-atomic, monotone geometry: conditional on the current price, the next price lies on one of two monotone branches representing upward and downward rebalancing. A no-transaction region, or trade band, arises where maintaining the position is optimal, while outside the band, transitions follow two monotone graphs whose endpoints satisfy an equal-slope condition balancing continuation value and marginal trading cost. The framework extends dynamically via a recursive identity, ensuring stability and convergence to the frictionless left-curtain limit, and applies to model-independent pricing and robust hedging of path-dependent derivatives.

Figures (1)

• Figure 1: Schematic illustration of the no–trade region and optimal rebalancing under frictions. The orange line shows the price process $S_t$ evolving within the state–dependent trade band $B_t$ defined by \ref{['eq:band_def_repeat']}. While $S_t$ remains inside $B_t$, the optimal martingale coupling coincides with the identity—no rebalancing occurs and the portfolio position is held fixed. When $S_t$ exits the band, a trade is triggered (blue markers), corresponding to an update along the monotone graphs $T_d^{(t)}$ and $T_u^{(t)}$ described in \ref{['thm:fric_monotone']}. The band width reflects the local liquidity or impact cost parameters $(\alpha_t,\beta_t)$. Widening $\alpha_t$ increases the inaction region, whereas larger $\beta_t$ reduces off–band displacements. Financially, $B_t$ captures the threshold behaviour of market makers: small price fluctuations are absorbed without trading, while sufficiently large moves trigger optimal rebalancing.

Theorems & Definitions (44)

• Definition 2.1: Admissible Payoff
• Lemma 2.1: Conjugate of \ref{['eq:friction_function']}
• proof
• Definition 2.2: Frictional Superhedging Price
• Theorem 2.1: Strong Duality with State-Dependent Frictions
• Remark 2.1: Subhedging with frictions
• Definition 3.1: Martingale competitors
• Definition 3.2: Rectangle MSM inequality
• Definition 3.3: Irreducible components of $\Delta F_t$
• Theorem 3.1: Frictional left–curtain geometry and bi–atomic kernel