  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.
