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The Black-Scholes-Merton model is a mathematical model for the dynamics of afinancial market that includes derivative investment instruments, and itsformula provides a theoretical price estimate of European-style options. Themodel's fundamental idea is to eliminate risk by hedging the option bypurchasing and selling the underlying asset in a specific way, that is, toreplicate the payoff of the option with a portfolio (which continuously tradesthe underlying) whose value at each time can be verified. One of the mostcrucial, yet restrictive, assumptions for this task is that the market followsa geometric Brownian motion, which has been relaxed and generalized in variousways.  The concept of robust finance revolves around developing models that accountfor uncertainties and variations in financial markets. Martingale OptimalTransport, which is an adaptation of the Optimal Transport theory to the robustfinancial framework, is one of the most prominent directions. In this paper, weconsider market models with arbitrarily many underlying assets whose values areobserved over arbitrarily many time periods, and demonstrates the existence ofa portfolio sub- or super-hedging a general path-dependent derivative securityin terms of trading European options and underlyings, as well as the portfolioreplicating the derivative payoff when the market model yields the extremalprice of the derivative given marginal distributions of the underlyings. Inmathematical terms, this paper resolves the question of dual attainment for themulti-period vectorial martingale optimal transport problem.

Replication of financial derivatives under extreme market models given marginals