Black-Scholes Equation in Laplace Transform Domain

Laplace transformation is one of the most popular methods of solution of diffusion equations in many areas of science and technology. It is much less used in financial engineering. One reason is obvious: it is not supposed to be a way to solve a Nobel Prize winning problem. Another one is technical: not many people know that all that they need to do is to make simple calculations in the Laplace domain. Three years before Black-Scholes formula, the famous (in other areas) Stehfest algorithm of numerical inversion of Laplace transforms was published (Stehfest, 1970). The performance of the numerical procedure is comparable with evaluation of cumulative density functions: your solution in Laplace domain will be calculated in 10-14 predefined points on a real axes to achieve 5-6 digits accuracy. It took more then 10 years to “discover” Stehfest algorithm in Hydrodynamics of porous media. In short time employing of the algorithm dramatically improved the quality of well testing analysis in the 80s