The McCormick martingale optimal transport

Abstract: Martingale optimal transport (MOT) often yields broad price bounds for options, constraining their practical applicability. In this study, we extend MOT by incorporating causality constraints among assets, inspired by the nonanticipativity condition of stochastic processes. This, however, introduces a computationally challenging bilinear program. To tackle this issue, we propose McCormick relaxations to ease the bicausal formulation and refer to it as McCormick MOT. The primal attainment and strong duality of McCormick MOT are established under standard assumptions. Empirically, we apply McCormick MOT to basket and digital options. With natural bounds on probability masses, the average price reduction for basket options is approximately 1.08% to 3.90%. When tighter probability bounds are available, the reduction increases to 12.26%, compared to the classic MOT, which also incorporates tighter bounds. For most dates considered, there are basket options with suitable payoffs, where the price reduction exceeds 10.00%. For digital options, McCormick MOT results in an average price reduction of over 20.00%, with the best case exceeding 99.00%.