The Black-Scholes model assumes a normal distribution of returns. However, non-normal skewness and kurtosis are found to contribute significantly to the phenomenon of volatility smile.
Compared with the normal probability density function, the Gram-Charlier model allows for more flexibility because it uses skewness and kurtosis as input parameters. Montgomery Investment Technology provides Online Calculators for option pricing implementing the Gram-Charlier model.
Skewness and kurtosis can be calculated from historical stock prices. Also, the market prices of call and put European options can be used to identify the skewness and kurtosis. An adjusted volatility can then be derived and used as an input to Black-Scholes.
The Gram-Charlier estimates are of better quality when the distribution is significantly different with respect to the normal one. It is therefore recommended to check first for departures from normality. If there is a significant departure then we should resort to Gram-Charlier instead of Black-Scholes.
For more information about the model, see our working paper Derivative Valuation – Gram Charlier.
Excel template examples generated by our software:
Gram-Charlier_Matrix_Skewness_Kurtosis.pdf
Gram-Charlier_Validity.pdf