On some potential applications of the heat equation with a repulsive point interaction to derivative pricing

In this note we first investigate in detail the “heat equation” with the free Laplacian replaced by the one with a repulsive point interaction centred at the origin in the case where the initial condition is given by any function proportional to e^x/2 χ(−∞,0](x). The solution is expressed in terms of the cumulative function of the normal distribution in view of its direct application to derivative pricing. In the second part of the paper, with reference to the quantum mechanical approach to option pricing proposed in the last decade, we use the results in order to solve explicitly the Black-Scholes equation with a perturbing term given by a point interaction of the type λ · δ(ln( s/E )), s being the price of the underlying asset and E the exercise price of the option.