We study a portfolio optimization problem in a market which is under the threat of crashes. At random times, the investor receives a warning that a crash in the risky asset might occur. We construct a strategy which renders the investor indifferent about an immediate crash of maximum size and no...

Summary We introduce a class of optimal stopping problems in which the gain is at least a fraction of the initial value. From a financial point of view this structure can be seen as a guarantee for the holder of an American option. It turns out that the optimal strategies are of two-sided type...