Gram-Charlier Distribution


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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)

Published By: The Econometric Society

[5] Jondeau, E. et al. (2007). Financial Modeling Under Non-Gaussian Distributions. Springer.

[6] Zapelov, A. Development of Model of Sea Surface Elevations Distributions Created by Wind Waves and Swell. In Physical and Mathematical Modeling of Earth and Environment Processes (2018) 4th International Scientific School for Young Scientists, Ishlinskii Institute for Problems in Mechanics of Russian Academy of Sciences