On the Approximation of the SABR Model : A Probabilistic Approach

In this paper, we derive a probabilistic approximation for three different versions of the SABR model: Normal, Log-Normal and a displaced diffusion version for the general constant elastic of variance case. Specifically, we focus on capturing the terminal distribution of the underlying process (conditional on the terminal volatility) to arrive at the implied volatilities of the corresponding European options for all strikes and maturities. Our resulting method allows us to work with a variety of parameters which cover the long dated options and highly stress market condition. This is a different feature from other current approaches which rely on the assumption of very small total volatility and usually fail for longer than 10 years maturity or large Volvol