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

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.

On the Analysis of a Generalised Rough Ait-Sahalia Interest Rate Model

“…For instance, the authors in [5] employ the fractional Brownian motion with H < 1 2 to model the 'rough' volatility process of asset prices and derive a representation of the sensitivity parameter delta for option prices. Similarly, the authors in [6] also consider an asset price model in connection with the sensitivity analysis of option prices whose correlated 'rough' volatility dynamics is described by means of a SDE driven by a fractional Brownian motion with H < 1 2 . The reader may consult [4,15] for the coverage of properties and financial applications of the fractional Brownian motion with H < 1 2 .…”