The Gram-Charlier distribution is a way to explicitly model departure from normality by using a series expansion around a normal distribution. It is more flexible than a normal distribution because it directly introduces a distribution’s kurtosis and skew as unknown parameters. The One practical application of the Gram-Charlier distribution is to model stock returns in option pricing.
Types of Gram-Charlier Distribution
The two-term Gram-Charlier distribution is defined by  as
Gram-Charlier’s Type A expansion has moments as inputs (up to order k). The expansion gives a probability distribution function (PDF) for a continuous random variable x. The type A distribution is defined by :
Where Hei is a Hermite polynomial defined as
The first six of which are:
- He0(z) = 1
- He1(z) = z
- He2(z) = z2 – 1
- He3(z) = z3 – 3z
- He4(z) = z4 – 6z2 + 3
- He5(z) = z5 – 10x3 + 15z
- He6(z) = z6 – 15z4 + 45z2 – 15
Quensel  presented a logarithmic Gram-Charlier distribution, where log X has a Gram-Charlier distribution.
Drawbacks of the Gram-Charlier Distribution
The Gram-Charlier distribution does have a couple of drawbacks. As it involves polynomial approximations, it can result in negative parameter values under certain conditions. In addition, there isn’t an easy and analytic characterization of a density which will take on only positive values . Gallant and Tauchen  suggested simply squaring the polynomial part of the series. However, that results in losing interpretation of parameters as moments. that A combined model may also be used to exclude negative values; where the range of random variables are small, the PDF is described by a truncated Gram-Charlier distribution. Outside of that area, the model is a normal distribution .
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 Quensel, C.-E. (1945). Studies of the logarithmic normal curve, Skandinavisk Aktuarietidskrift, 28, 141-153.
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Published By: The Econometric Society
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