We consider a class of fractional stochastic volatility models (including the so-called rough Bergomi model), where the volatility is a superlinear function of a fractional Gaussian process. We show that the stock price is a true martingale if and only if the correlation $ρ$ between the driving Brownian motions of the stock and the volatility is nonpositive. We also show that for each $ρ<0$ and $m> \frac{1}{1-ρ^2}$, the $m$-th moment of the stock price is infinite at each positive time.