This paper is devoted to studying the difference between the fair strike of a volatility swap and the at-the-money implied volatility (ATMI) of a European call option. It is well-known that the difference between these two quantities converges to zero as the time to maturity decreases. In this paper, we make use of a Malliavin calculus approach to derive an exact expression for this difference. This
representation allows us to establish that the order of the convergence is different in the correlated and in the uncorrelated case, and that it depends on the behavior of the Malliavin derivative of the volatility process. In particular, we will see that for volatilities driven by a fractional Brownian motion, this order depends on the corresponding Hurst parameter H. Moreover, in the case H≧1/2, we develop a model-free approximation formula for the volatility swap, in terms of the ATMI and its skew.