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# Transformation of the Black-Scholes PDE into the diffusion equation - shift of coordinate system

The aim of transforming the Black-Scholes PDE is of course to find a form where an relatively easy solution exists. Most of the steps seem to be straightforward - please use this reference: http://planetmath.org/encyclopedia/AnalyticSolutionOfBlackScholesPDE.html

...all but one, actually the last one where a convection-diffusion equation is being transformed into the basic diffusion equation.

[In the article you find it here: "Under the new coordinate system (z), we have the relations amongst vector fields ... leading to the following transformation of equation..."]

The u_x-term vanishes by some magic coordinate transformation. When you look at the derivatives they even seems wrong to me because they state that tau=s and then derive delta/delta tau = delta/delta s + some extra term (delta/delta y * -(r-1/2 sigma^2).

I simply don't get it: first how it works and second how they find that kind of transformation.