Entropically Regularized Martingale Optimal Transport with $L_1$ Relaxation

Martingale Optimal Transport (MOT) provides a framework for model-free derivative pricing, but the hard martingale constraint becomes statistically unstable under finite-sample estimation. We introduce a regularized formulation that replaces the exact martingale condition with an $L_1$ penalty on the martingale defect, combined with entropic smoothing of the transport plan. The resulting dual problem admits a natural interpretation in terms of leverage-constrained hedging strategies and is amenable to proximal gradient methods with $O(1/k)$ convergence. Our main theoretical contribution is a concentration analysis establishing that the intrinsic statistical fluctuation in the martingale constraint scales as $O(N^{-1/2})$, yielding a principled lower bound for the relaxation parameter. Experiments on synthetic data confirm this scaling and demonstrate that the relaxed formulation remains tractable as the constraint tightens.