A Bismut-Elworthy-Li Formula for Singular SDE's Driven by a Fractional Brownian Motion and Applications to Rough Volatility Modeling

TL;DR

This paper develops a Bismut-Elworthy-Li formula for singular SDEs driven by fractional Brownian motion with H<1/2, enabling analysis of financial models with rough volatility using Malliavin calculus.

Contribution

It introduces a novel Bismut-Elworthy-Li formula for singular SDEs driven by fractional Brownian motion, extending tools for rough volatility modeling.

Findings

Derived a Bismut-Elworthy-Li formula for singular SDEs with fractional Brownian motion.

Applied the formula to study price sensitivities in rough volatility models.

Utilized Malliavin calculus and local time variational calculus techniques.

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 multi-dimensional 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 price sensitivities 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".