Advanced Analytics for the SABR Model

Abstract

In this paper, we present advanced analytical formulas for SABR model option pricing. The first technical result consists of a new exact formula for the zero correlation case. This closed form is a simple 2D integration of elementary functions, particularly attractive for numerical implementation. The second result is an effective approximation of the general correlation case. We use a map to the zero correlation case having a nice behavior on strike edges. The map formulas are easily implemented and do not contain any numerical integration.

These formulas are important in volatility surface construction and CMS products replication because they provide correct behavior for far strikes and reduced approximation error. The latter is also helpful for dynamic SABR models.

Keywords: SABR, closed formula, map approximation, swaption price, CMS replication, volatility surface

JEL Classification: C1, C3, C5, C6