An alternative perspective on the classical solution to an optimal stopping problem with a spectrally negative Lévy process

This study considers an optimal stopping problem for a real put option in which the underlying dynamics characterizing the payoff uncertainty are modeled as a spectrally negative Lévy process. It demonstrates that the solution, obtained using the conventional “value-matching” and “smooth-pasting” conditions, may not be optimal when the jumps are large in expectation and/or frequent. Specifically, at that threshold, the value of waiting can exceed the value of stopping. The objective of this study is to emphasize this potential limitation in the classical solution for optimal stopping problems of the described type, and to propose an alternative solution approach. It proves that our proposed solution is a viscosity sub-solution to the Hamilton–Jacobi–Bellman equation.