Asymptotics of Implied Volatility to Arbitrary Order

In a unified model-free framework that includes long-expiry, short-expiry, extreme-strike, and jointly-varying strike-expiry regimes, we find asymptotic implied volatility and implied variance formulas in terms of L, with rigorous error estimates of order 1/L to any given power, where L denotes the absolute log of an option price that approaches zero. Our results therefore sharpen, to arbitrarily-high order of accuracy, the model-free asymptotics of implied volatility in extreme regimes. We then apply these general formulas to particular examples: Levy and Heston