  This paper investigates short-term behaviors of implied volatility of
derivatives written on indexes in equity markets when the index processes are
constructed by using a ranking procedure. Even in simple market settings where
stock prices follow geometric Brownian motion dynamics, the ranking mechanism
can produce the observed term structure of at-the-money (ATM) implied
volatility skew for equity indexes. Our proposed models showcase the ability to
reconcile two seemingly contradictory features found in empirical data from
equity markets: the long memory of volatilities and the power law of ATM skews.
Furthermore, the models allow for the capture of a novel phenomenon termed the
quasi-blow-up phenomenon.
