On Short‐Term Behavior of Implied Volatility for Index Options

This paper investigates short‐term behavior of implied volatility of derivatives written on a market index when the index is constructed using a ranking procedure. Even when stock prices follow geometric Brownian motion dynamics, the ranking mechanism can lead to the observed term structure of at‐the‐money (ATM) implied volatility skew for index options. Our proposed models showcase the ability to reconcile two seemingly contradictory features found in empirical data from equity markets: the long memory of volatility and the power law decay of the ATM skew for long maturities. Furthermore, the models allow for the capture of a new phenomenon termed the quasi‐blow‐up phenomenon.