Probability Distribution in the SABR Model of Stochastic Volatility (original) (raw)

We study the SABR model of stochastic volatility . This model is essentially an extension of the local volatility model [6], , in which a suitable volatility parameter is assumed to be stochastic. The SABR model admits a large variety of shapes of volatility smiles, and it performs remarkably well in the swaptions and caps / floors markets. We refine the results of [10] by constructing an accurate and efficient asymptotic form of the probability distribution of forwards. Furthermore, we discuss the impact of boundary conditions at zero forward on the volatility smile. Our analysis is based on a WKB type expansion for the heat kernel of a perturbed Laplace-Beltrami operator on a suitable hyperbolic Riemannian manifold.