A new class of option-pricing model is discussed, motivated initially by the practical observation that contracts with embedded options are not always exercised immediately when an implicit barrier is breached; this may occur for a number of reasons, for example linked to behavior of the investor. Rather than using a conventional barrier, this option class model takes the first touching of the payoff function by the option value as the start of a waiting period before exercise. This presents itself as a free-boundary problem, similar to, but somewhat more complicated than, that found with the usual American option. It turns out that this gives insight into the dynamics of the American option itself, as the “Ameripean” delayed-exercise option model provides a fluid link between a European and an American option. It also prompts the development of an improved numerical technique, based on boundary-fitted coordinates, together with some useful asymptotic analyses (which give further insights into valuations).