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

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6American contingent claims

So far, we have studied European contingent claims whose payoff is due at a fixed maturity date. In the case of American options, the buyer can claim the payoff at any time up to the expiration of the contract.

First, we take the point of view of the seller, whose aim is to hedge against all possible claims of the buyer. In Section 6.1, this problem is solved under the assumption of market completeness, using the Snell envelope of the contingent claim. The buyer tries to choose the best date for exercising the claim, contingent on the information available up to that time. Since future prices are usually unknown, a formulation of this problem will typically involve subjective preferences. If preferences are expressed ...

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