American Options with Last Exit Times: A Free-Boundary Approach

We study the valuation of an American put option with a random time horizon given by the last exit time of the underlying asset from a fixed level. Since this random time is not a stopping time, the problem falls outside the classical optimal stopping framework. Using enlargement of filtrations and the associated Azéma supermartingale, we transform the problem into an equivalent optimal stopping problem with a semi-continuous, time-dependent gain function whose partial derivatives exhibit singular behaviour. The resulting formulation introduces significant analytical challenges, including the loss of smoothness of the optimal stopping boundary. We develop new arguments to characterise the continuation and stopping regions, establishing monotonicity of the free boundary under suitable conditions, and analyse the regularity of the value function. In particular, we derive nonlinear integral equations that uniquely characterise both the free-boundary and the value function. Our results extend the classical theory of American options to a class of problems with random horizons and provide a framework for incorporating default-type features modelled by last exit times.