Duality for robust hedging with proportional transaction costs of path dependent European options is obtained in a discrete time financial market with one risky asset. Investor's portfolio consists of a dynamically traded stock and a static position in vanilla options which can be exercised at maturity. Trading of both the options and the stock are subject to proportional transaction costs. The main theorem is duality between hedging and a Monge-Kantorovich type optimization problem. In this dual transport problem the optimization is over all the probability measures which satisfy an approximate martingale condition related to consistent price systems in addition to an approximate marginal constraints. 1 transaction costs. Each option has its own cost and their general structure is outlined in the next section.In this market, we consider the problem of robust hedging of a given path dependent European option. Robust hedging refers to super-replication of an option for all possible stock price processes. This approach has been actively researched over the past decade since the seminal paper of Hobson [14]. In particular, the optimal portfolio is explicitly constructed for special cases of European options in continuous time; barrier options in [5] and [7,8], lookback options in [12], [13] and [14], and volatility options in [9]. The main technique that is employed in these papers is the Skorohod embedding. For more information, we refer the reader to the surveys of Hobson [15], Obłój [18] and to the reference therein.Recently, an alternative approach is developed using the connection to optimal transport. Duality results in different types of generality or modeling have been proved in [2], [4], [10] and [12] in frictionless markets. In particular, [10] studies the continuous time models, [12] provides the connection to stochastic optimal control and a general solution methodology, [4] proves a general duality in discrete time and [2] studies the question of fundamental theorem of asset pricing in this context.Although much has been established, the effect of frictions -in particular the impact of transaction costs -in this context is not fully studied. The classical probabilistic models with transaction costs, however, is well studied. In the classical model, a stock price model is assumed and hedging is done only through the stock and no static position in the options is used. Then, the dual is given as the supremum of "approximate" martingale measures which are equivalent to the market probability measure, see [19,16] and the references therein. In this paper, we extend this result to the robust case. Namely, we prove that the super-replication price can be represented as a martingale optimal transport problem. The dual control problem is the supremum of the expectation of the option, over all approximate martingale measures which also satisfy an approximate marginal condition at maturity. This result is stated in Theorem 2.1 below and the definition of an approximate martingale is given in Definition 2.5. Indee...
