The aim of this contribution is to develop a numerical method
based on the Crank-Nicolson scheme for time discretization and
the F-transform to solve Black-Scholes partial differential
equations (PDEs). The F-transform is an efficient method for
the approximation of (multivariate) functions, where a
transformation of a space of locally square integrable functions
into a simple vector space of F-transform components is used.
We show how the PDE after the application of the Crank-Nicolson
scheme for time discretization can be transformed into a simple
vector space of F-transform components in order to find a
solution in terms of the F-transform components. An approximate
solution of the PDE is derived by the inverse F-transform.