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, fixed 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.

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...

We prove limit theorems for the super-replication cost of European options in a Binomial model with friction. The examples covered are markets with proportional transaction costs and the illiquid markets. The dual representation for the superreplication cost in these models are obtained and used to prove the limit theorems. In particular, the existence of the liquidity premium for the continuous time limit of the model proposed in [6] is proved. Hence, this paper extends the previous convergence result of [13] to the general non-Markovian case. Moreover, the special case of small transaction costs yields, in the continuous limit, the G-expectation of Peng as earlier proved by Kusuoka in [14].

Abstract. The dual representation of the martingale optimal transport problem in the Skorokhod space of multi dimensional cádlág processes is proved. The dual is a minimisation problem with constraints involving stochastic integrals and is similar to the Kantorovich dual of the standard optimal transport problem. The constraints are required to hold for very path in the Skorokhod space. This problem has the financial interpretation as the robust hedging of path dependent European options.

We study the problems of efficient hedging of game (Israeli) options when the initial capital in the portfolio is less than the fair option price. In this case a perfect hedging is impossible and one can only try to minimise the risk (which can be defined in different ways) of having not enough funds in the portfolio to pay the required amount at the excercise time. We solve the minimization problems and find via dynamical programming appropriate efficient hedging strategies for discrete time game options in multinomial markets. The approach and some of the results are new also for standard American options.

We show that the shortfall risk of binomial approximations of game (Israeli) options converges to the shortfall risk in the corresponding Black--Scholes market considering Lipschitz continuous path-dependent payoffs for both discrete- and continuous-time cases. These results are new also for usual American style options. The paper continues and extends the study of Kifer [Ann. Appl. Probab. 16 (2006) 984--1033] where estimates for binomial approximations of prices of game options were obtained. Our arguments rely, in particular, on strong invariance principle type approximations via the Skorokhod embedding, estimates from Kifer [Ann. Appl. Probab. 16 (2006) 984--1033] and the existence of optimal shortfall hedging in the discrete time established by Dolinsky and Kifer [Stochastics 79 (2007) 169--195].Comment: Published in at http://dx.doi.org/10.1214/07-AAP503 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

We introduce a setup of model uncertainty in discrete time. In this setup we derive dual expressions for the super-replication prices of game options with upper semicontinuous payoffs. We show that the super-replication price is equal to the supremum over a special (non dominated) set of martingale measures, of the corresponding Dynkin games values. This type of results is also new for American options.

We study super-replication of contingent claims in an illiquid market with model uncertainty. Illiquidity is captured by nonlinear transaction costs in discrete time and model uncertainty arises as our only assumption on stock price returns is that they are in a range specified by fixed volatility bounds. We provide a dual characterization of super-replication prices as a supremum of penalized expectations for the contingent claim's payoff. We also describe the scaling limit of this dual representation when the number of trading periods increases to infinity. Hence, this paper complements the results in [11] and [19] for the case of model uncertainty.