Application of the Gram-Charlier Approximation for Option Valuation

The Hermite polynomials form the basis of a Hilbert space and may be used to get an expansion of the probability density function. Usually, this series is called Gram-Charlier. For practical purposes, only the first few terms of this expansion are taken into consideration. The resulting truncated series may be viewed as the normal probability density function multiplied by a polynomial that accounts for the effects of departure from normality. The Gram-Charlier series uses the moments of the real distribution. The Edgeworth series is similar to Gram-Charlier but uses cumulants instead of moments. Although the series are equivalent, for computational purposes the Gram-Charlier series seems to perform better than the Edgeworth series (Johnson et al., 1994).