American options with guarantee -- a class of two-sided stopping problems (Q2855514)

The paper deals with a class of two-sided stopping problems. If the payoff depends on the starting point then the classical stopping theory for one-dimensional processes does not work. The authors suggest in such a case to embed the optimal stopping problem to a two-dimensional one. One can look upon this situation as on the American options with guarantee where the guaranteed payoff is a fraction of the starting price. At first, the authors assume that the driving process is a diffusion. This assumption allows to apply the harmonic-functions techniques and to look for the explicit solution of two differential equations characterizing the optimal strategy. Then they study the case of Lévy processes where the explicit solution is obtained for spectrally negative processes applying scale functions.