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

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 $H \lt 1/2$. “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 $\delta$ 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 SDEs.

Our approach for obtaining these results is based on Malliavin calculus and arguments of a recently developed “local time variational calculus”.

Keywords

Bismut–Elworthy–Li formula, singular SDEs, fractional Brownian motion, Malliavin calculus, stochastic flows, stochastic volatility

2010 Mathematics Subject Classification

49N60, 60H10, 91G80

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The original title of this paper, as published Dec. 11, 2020 in print and online, was “Bismut–Elworthy–Li formula, singular SDEs, fractional Brownian motion, Malliavin calculus, stochastic flows, stochastic volatility”. The title of the online paper was revised on February 22, 2021. The printed publication remains unchanged.

Received 17 March 2019

Accepted 12 April 2020

Published 11 December 2020