A Remark on the Heat Equation with a Point Perturbation, the Feynman–Kac Formula with Local Time and Derivative Pricing

We discuss the probabilistic representation of the solutions of the heat equation perturbed by a repulsive point interaction in terms of a perturbation of Brownian motion, via a Feynman–Kac formula involving a local time functional. An application to option pricing is given, interpolating between the extreme cases of classical Black–Scholes options and knockouts having the barrier situated exactly at the exercise price.