The duality between the robust (or equivalently, model independent) hedging of path dependent European options and a martingale optimal transport problem is proved. The financial market is modeled through a risky asset whose price is only assumed to be a continuous function of time. The hedging problem is to construct a minimal super-hedging portfolio that consists of dynamically trading the underlying risky asset and a static position of vanilla options which can be exercised at the given,

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... d maturity. The dual is a Monge-Kantorovich type martingale transport problem of maximizing the expected value of the option over all martingale measures that have a given marginal at maturity. In addition to duality, a family of simple, piecewise constant super-replication portfolios that asymptotically achieve the minimal super-replication cost is constructed. 123 392 Y. Dolinsky, H. M. Soner 123 394 Y. Dolinsky, H. M. Soner duality result together with the results of [21] implies that these two approachesnamely, robust hedging through the path-wise definition of this paper and the quasisure definition of [14, 38, 39] yield the same value. This is proved in Sect. 3 below. Our second result provides a class of portfolios which are managed on a finite number of random times and asymptotically achieve the minimal super-replication cost. This result may have practical implications allowing us to numerically investigate the corresponding discrete hedges, but we relegate this to a future study. Robust hedging has been an active research area over the past decade. The initial paper of Hobson [22] studies the case of the lookback option. The connection to the Skorokhod embedding is also made in this paper and an explicit solution for the minimal super-replication cost is obtained. This approach is further developed by Brown, Hobson and Rogers [7], Cox and Obłój [11, 13] and in several other papers, [23] [24] [25] [26] [27] [28] . We refer the reader to the excellent survey of Hobson [23] for robust hedging and to Obłój [34] for the Skorokhod embedding problem. In particular, the recent paper by Cox and Wang [13] provides a discussion of various constructions of Root's solution of the Skorokhod embedding. A similar modeling approach is applied to volatility options by Carr and Lee [9]. In a recent paper, Davis, Obłój and Raval [16] considers the variance swaps in a market with finitely many put options. In particular, in [16] the class of admissible portfolios is enlarged and numerical evidence is obtained by analyzing the S&P500 index options data. Furthermore, [16] contains a duality result in a simpler setting, using the classical Karlin-Isii duality in semi-infinite linear programming. As already mentioned above, the dual approach is used by Galichon, Henry-Labordère and Touzi [21] and Henry-Labordère and Touzi [32] as well. In these papers, the duality provides a connection to stochastic optimal control which can then be used to compute the solution in a more systematic manner. The proof of the main results is done in four steps. The first step is to reduce the problem to bounded claims. The second step is to represent the original robust hedging problem as a limit of robust hedging problems which live on a sequence of countable spaces. For these type of problems, robust hedging is the same as classical hedging, under the right choice of a probability measure. Thus we can apply the classical duality results for super-hedging of European options on a given probability space. The third step is to use the discrete structure and apply a standard min-max theorem (similar to the one used in [6]). The last step is to analyze the limit of the obtained prices in the discrete time markets. We combine methods from arbitrage-free pricing and limit theorems for stochastic processes. The paper is organized as follows. The main results are formulated in the next section. In Sect. 3, the connection between the quasi sure approach and ours is proved. The two sections that follow are devoted to the proof of one inequality which implies the main results. The final section discusses a possible extension. Preliminaries and main results The financial market consists of a savings account which is normalized to unity B t ≡ 1 by discounting and of a risky asset S t , t ∈ [0, T ], where T < ∞ is the maturity date. Let s := S 0 > 0 be the initial stock price and without loss of generality, we set s = 1.