Sensitivity Analysis with respect to a Stock Price Model with Rough Volatility via a Bismut-Elworthy-Li Formula for Singular SDEs

TL;DR

This paper establishes the existence of Malliavin differentiable solutions for singular SDEs driven by fractional Brownian motion, introduces a novel stock price model with rough, correlated volatility, and derives a Bismut-Elworthy-Li formula for it.

Contribution

It provides the first proof of solution regularity for singular fractional SDEs and develops a new stock model capturing roughness and regime switching effects.

Findings

Existence of unique Malliavin differentiable solutions for singular fractional SDEs.

Development of a stock price model with rough, correlated volatility.

Derivation of a Bismut-Elworthy-Li formula for the proposed model.

Abstract

In this paper, we show the existence of unique Malliavin differentiable solutions to SDE`s driven by a fractional Brownian motion with Hurst parameter H<1/2 and singular, unbounded drift vector fields, for which we also prove a stability result. Further, using the latter results, we propose a stock price model with rough and correlated volatility, which also allows for capturing regime switching effects. Finally, we also derive a Bismut-Elworthy-Li formula with respect to our stock price model for certain classes of vector fields.