ABSTRACT
This chapter is devoted to the study of the short-time limit of the at-the-money implied (ATMI) volatility. Following and extending the work in Alos and Shiraya (2019), we deep in this short-time behaviour, and we study the relationship between the ATM implied volatility, the volatility swap, the variance swap, and the spot volatility. Our analysis is based on an adequate decom- position of the implied volatility in the uncorrelated case, plus a term due to the correlation. In particular, we see that the rate of convergence depends on the corresponding Hurst parameter H, being slower for rough volatilities than for classical volatility models. Using similar techniques, we can also study the difference between the volatility swap and the square root of a variance swap. Based on the above decompositions, we can deduce simple approximation formulas for the ATMI that recover some well-known results in the literature, and that can be applied to the case of fractional volatilities.
