Asymptotics of Implied Volatility in Extreme Regimes

A financial option contract on an underlying asset is parameterized by two quantities: a strike price and an expiration date. In the "extreme" regimes where the strike is small/large, and/or the expiration date is small/large, we find high-order asymptotic approximations for the option's value, expressed either in dollars or in the language of implied volatility. The latter expressions have remarkable accuracy in some cases, even outside of the extreme regimes.