Before revision:  As an intermediary result, we also obtain high-order expansions for at-the-money digital call option prices . The results obtained herein are remarkably different from those obtained in recent papers close-to-the-money option prices in that they aid in understanding how the behavior of implied volatility near expiry is affected by important model parameters , such as the leverage and vol vol parameters, that were not present in the aforementioned earlier results .
After revision:  Unlike recent results for "near-the-money" option prices and implied volatility, the results herein aid in understanding how the behavior of implied volatility near expiry is affected by important model parameters , such as the leverage and vol vol parameters, that were not present in the aforementioned earlier results .