Finance and Financial Management
Prices of currency options commonly deffer from the Black-Scholes formula along two dimensions: implied volatilities vary by strike price (volatility smiles) and maturity (implied volatility of at-the-money options increases, on average, with maturity). We account for both using Gram-Charlier expansions to approximate the conditional distribution of the logarithm of the price of the underlying security. In this setting, volatility is approximately a quadratic function of moneyness, a result we use to infer skewness and kurtosis from volatility smiles. Evidence suggests that both kurtosis in currency prices and biases in Black-Scholes option prices decline with maturity.
Backus, David; Foresi, Silverio; Li, Kai; and Wu, Liuren, "Accounting for Biases in Black-Scholes" (1997). CRIF Working Paper series. Paper 30.