Gram-Charlier Distribution

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 [1] 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 [2]:

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 [3] 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 [4]. Gallant and Tauchen [5] 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 [6].

References

[1] Haight, F. (1958). Index to the Distributions of Mathematical Statistics. National Bureau of Standards Report.

[2] Jondeau, E. & Rockinger, M. (1999). ESTIMATING GRAM-CHARLIER EXPANSIONS WITH POSITIVITY CONSTRAINTS. Les Notes d’Études et de Recherche. Bank of France. Online: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.867.1681&rep=rep1&type=pdf

[3] Quensel, C.-E. (1945). Studies of the logarithmic normal curve, Skandinavisk Aktuari[1]etidskrift, 28, 141-153.

[4] Gallant and Tauchen (1989). Seminonparametric Estimation of Conditionally Constrained Heterogeneous Processes: Asset Pricing Applications. Econometrica

Vol. 57, No. 5 (Sep., 1989), pp. 1091-1120 (30 pages)