  In this work we detail the application of a fast convolution algorithm
computing high dimensional integrals to the context of multiplicative noise
stochastic processes. The algorithm provides a numerical solution to the
problem of characterizing conditional probability density functions at
arbitrary time, and we applied it successfully to quadratic and piecewise
linear diffusion processes. The ability in reproducing statistical features of
financial return time series, such as thickness of the tails and scaling
properties, makes this processes appealing for option pricing. Since exact
analytical results are missing, we exploit the fast convolution as a numerical
method alternative to the Monte Carlo simulation both in objective and risk
neutral settings. In numerical sections we document how fast convolution
outperforms Monte Carlo both in velocity and efficiency terms.
