  This paper starts by defining the criteria where the early-exercise of an
American option is never optimal, under positive, or negative rates. It follows
with a short analysis of the various shapes of the exercise region under
negative interest rates. It then presents a new integral equation, which
establishes the option price, and the two early exercise boundaries, under
negative rates. It shows how to solve this new equation, through modifications
of the modern and efficient algorithm of Andersen and Lake, from the initial
guess of the two boundaries to more subtle changes required in their fixed
point method for stability. Finally, the performance and accuracy of the
resulting algorithm is assessed against a cutting edge finite difference method
implementation.
