Laplace Transform and finite difference methods for the Black-Scholes equation

In this paper we explore discrete monitored barrier options in the Black-Scholes framework. The discontinuity arising at each monitoring data requires a careful numerical method to avoid spurious oscillations when low volatility is assumed. Here a technique mixing the Laplace Transform and the finite difference method to solve Black-Scholes PDE is considered. Equivalence between the Post-Widder inversion formula joint with finite difference and the standard finite difference technique is proved. The mixed method is positivity-preserving, satisfies the discrete maximum principle according to financial meaning of the involved PDE and converges to the underlying solution. In presence of low volatility, equivalence between methods allows some physical interpretations