On short-time behavior of implied volatility in a market model with indexes

Abstract: 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 new phenomenon termed the quasi-blow-up phenomenon.