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In this paper, we obtain asymptotic formulas with error estimates for theimplied volatility associated with a European call pricing function. We showthat these formulas imply Lee's moment formulas for the implied volatility andthe tail-wing formulas due to Benaim and Friz. In addition, we analyzePareto-type tails of stock price distributions in uncorrelated Hull-White,Stein-Stein, and Heston models and find asymptotic formulas with errorestimates for call pricing functions in these models.

Asymptotic Formulas with Error Estimates for Call Pricing Functions and the Implied Volatility at Extreme Strikes