  The SABR model is a stochastic volatility model not admitting a closed form
solution. Hagan, Kumar, Leniewski and Woodward have obtained an approximate
solution by means of perturbative techniques. A more precise approximation was
found by Henry-Labord\`ere with the heat kernel expansion method. The latter
relies on deep and hard theorems from Riemannian geometry which are almost
totally unknown to the professionals of finance, who however are those
primarily interested in these results.
  The goal of this report is to fill this gap and to make these topics
understandable with a basic knowledge of calculus and linear algebra.
