  This paper shows the relationship between the forward start volatility swap
price and the forward start zero vanna implied volatility of forward start
options in rough volatility models. It is shown that in the short
time-to-maturity limit the approximation error in the leading term of the
correlated case with $H\in(0,\frac12)$ does not depend on the time to forward
start date, but only on the difference between the maturity date and forward
start date and on the Hurst parameter $H$.
