American Options with guarantee – A class of two-sided stopping problems

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 under weak conditions. If the driving process is a diffusion we use harmonic-functions techniques to obtain general results. For an explicit solution we derive two differential equations that characterize the optimal strategies. Furthermore we study the case of Lévy processes. An explicit solution is obtained for spectrally negative processes using scale function