  We use a powerful extension of the classical method of heat potentials,
recently developed by the present author and his collaborators, to solve
several significant problems of financial mathematics. We consider the
following problems in detail: (A) calibrating the default boundary in the
structural default framework to a constant default intensity; (B) calculating
default probability for a representative bank in the mean-field framework; (C)
finding the hitting time probability density of an Ornstein-Uhlenbeck process.
Several other problems, including pricing American put options and finding
optimal mean-reverting trading strategies, are mentioned in passing. Besides,
two non-financial applications -- the supercooled Stefan problem and the
integrate-and-fire neuroscience problem -- are briefly discussed as well.
