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Stochastic Finance, 4th Edition by Alexander Schied, Hans Föllmer

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8Efficient hedging

In an incomplete financial market model, a contingent claim typically will not admit a perfect hedge. Superhedging provides a method for staying on the safe side, but the required cost is usually too high both from a theoretical and from a practical point of view. It is thus natural to relax the requirements.

As a first preliminary step, we consider strategies of quantile hedging which stay on the safe side with high probability. In other words, we maximize the probability for staying on the safe side under a given cost constraint. The main idea consists in reducing the construction of such strategies for a given claim H to a problem of superhedging for a modified claim , which is the solution to a static optimization problem ...

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