The main topic of the thesis is the optimal stopping theory and its application to certain areas of finance. The optimal stopping time is represented by a martingale approach on a discrete and finite time horizon. We apply the theory on a binomial market model to American options and equity-linked insurance products with guarantees. We construct a Snell envelope for the value process of an American option, which we use to define a condition for the minimum optimal exercise time for the option buyer. We also prove, that the optimal hedging strategy for the option seller is equal to the martingale part of the Doob decomposition of the same Snell envelope. Finally, we present the hedging strategy for equity-linked insurance products with guarantees, where we prove that hedging by buying a European put option at the optimal time represents a smaller and less variable loss for the insurance company than hedging without the use of derivatives or optimal stopping theory.
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