This talk deals with a research question that I have been working on for several years, but for which there is still no definitive answer. It addresses two of the fundamental problems of financial mathematics and stochastic processes: the distribution of the first hitting time on a curve of diffusion processes and the valuation of American options. Although the underlying theory is discussed in every introductory textbook, both the mathematical and applied treatment is challenging even in unrealistically simplified situations.

We discuss a non-standard approach to treating these questions. In the case of American options in the Black-Scholes model, the idea is as follows: We call a given American option representable if there exists a European claim which dominates the American payoff at any time and such that the values of the two options coincide in the continuation region of the American option. This concept has interesting implications from a probabilistic, analytical, financial, and numerical point of view. Relying, inter alia, on convex duality, we make a first step towards verifying representability of American options. This opens the door to a very efficient method for treating American options in both theory and practice. Similar ideas can be applied to the first hitting problem.

Based on joint work with Jan Kallsen, Matthias Lenga, Oskar Hallmann and Maike Klein.