Frictional martingale optimal transport and robust hedging

Abstract: We study martingale optimal transport (MOT) with state-dependent trading frictions and develop a geometric-duality framework that extends from one time-step to the multi-marginal setting. Building on the frictionless left-monotone structure (Beiglb\"ock and Juillet, Ann. Probab. 2016; Henry-Labord`ere and Touzi, Finance Stoch. 2016; Beiglb\"ock et al, Ann. Probab. 2017), we penalize trade increments by a convex cost and prove a frictional monotonicity principle. Optimal couplings admit bi-atomic disintegrations on active components, while a trade band emerges as a no-transaction region where the identity coupling is optimal. The band is characterized in dual variables by a subgradient condition at the origin; off the band, transport moves along two monotone graphs whose endpoints solve an equal-slope system balancing continuation values and marginal trading costs. We establish strong duality with attainment for state-dependent frictions, a dynamic programming identity that aggregates the one time-step problem, and stability of optimal couplings and endpoints under perturbations of marginals and friction. In the vanishing-friction limit the transport maps converge to the frictionless left-curtain coupling. For linear-quadratic frictions, motivated by spread and transient impact (Almgren and Chriss, J. Risk 2001; Obizhaeva and Wang, J. Financ. Mark. 2013), we obtain explicit off-band displacements and comparative statics in liquidity parameters. Applications include model-independent pricing/superhedging of lookback, barrier, and Asian options. Overall, the results link robust superhedging duality (Beiglb\"ock et al, Finance Stoch. 2013; Dolinsky and Soner, Finance Stoch. 2014) with the fine geometry of frictional MOT, providing a unified basis for analysis, stability, and computation in multi-marginal robust pricing.