A rough SABR formula

TL;DR

This paper derives and numerically solves an ODE for normalized volatility smiles in a rough SABR model, introducing a highly accurate approximation called the rough SABR formula that extends existing models.

Contribution

It introduces a new ODE-based approach and an accurate approximation for volatility smiles in a rough SABR framework, extending prior models like the rough Bergomi.

Findings

Derived an ODE for normalized volatility smiles in rough SABR

Developed a highly accurate approximation called the rough SABR formula

Extended the rough Bergomi model to include this new approach

Abstract

Following an approach originally suggested by Balland in the context of the SABR model, we derive an ODE that is satisfied by normalized volatility smiles for short maturities under a rough volatility extension of the SABR model that extends also the rough Bergomi model. We solve this ODE numerically and further present a very accurate approximation to the numerical solution that we dub the rough SABR formula.