In a discrete-time market, we study the problem of model-independent superhedging of exotic options under portfolio constraints. The superhedging portfolio consists of {\it   static positions in liquidly traded vanilla  options, and a dynamic trading strategy , subject to certain constraints, on the risky asset. By the theory of Monge-Kantorovich  optimal transport, we establish a  superhedging duality, which admits a natural connection to convex risk measures. With the aid of this duality, we derive a model-independent version of the fundamental theorem of asset pricing under portfolio constraints . It is worth noting that our method covers a large class of Delta constraints as well as Gamma constraint.
In a discrete-time market, we study model-independent superhedging , while the semi-static superhedging portfolio consists of {\it three parts: static positions in liquidly traded vanilla calls, static positions in other tradable, yet possibly less liquid, exotic options, and a dynamic trading strategy in risky assets under certain constraints. By considering the limit order book of each tradable exotic option and employing the Monge-Kantorovich theory of optimal transport, we establish a general superhedging duality, which admits a natural connection to convex risk measures. With the aid of this duality, we derive a model-independent version of the fundamental theorem of asset pricing . The notion "finite optimal arbitrage profit", weaker than no-arbitrage, is also introduced . It is worth noting that our method covers a large class of Delta constraints as well as Gamma constraint.
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