The SABR model is a stochastic volatility model not admitting a closed form
solution. Hagan, Kumar, Leniewski and Woodward have given an approximate
solution by means of perturbative techniques. A more precise approximation was
obtained by Henry-Labord\`ere using the heat kernel expansion method. The
latter relies on deep and hard theorems from Riemannian geometry which are
almost totally unknown to people working in finance, who however are those
primarily interested in these results.