A Bismut–Elworthy–Li formula for singular SDEs driven by a fractional Brownian motion and applications to rough volatility modeling

Abstract

In this paper we derive a Bismut-Elworthy-Li type formula with respect to strong solutions to singular stochastic differential equations (SDE's) with additive noise given by a multidimensional fractional Brownian motion with Hurst parameter H <1/2. Singular here means that the drift vector field of such equations is allowed to be merely bounded and integrable. As an application we use this representation formula for the study of the δ price sensitivity of financial claims based on a stock price model with stochastic volatility, whose dynamics is described by means of fractional Brownian motion driven SDE's. Our approach for obtaining these results is based on Malliavin calculus and arguments of a recently developed local time variational calculus.