The SABR Model - Densities and MC

Different Approximation to SABR. Including Kienitz, Doust, Hagan, Obloj, Lesniewski, Kainth method

We consider the well known SABR model. We give formulae for implied vol, densities and Monte Carlo simulation. We also cover no-arbitrage densities for parameter sets where standard formulae break down.
We also cover the recent Doust method and the Kienitz method for density extrapolation.