The approximation of a lognormal-pdf

The EE (lEE) scheme we derive in section (3.1) (section (4.1)) is exact in the sense that the Taylor expansion is valid for infinity terms. Hence, the generalized EE is an equivalent expression for the pdf (cdf). Now, perform- 

Given the mean /ij, and the standard deviation of a normal-distributed random variable y we obtain the pdf of a lognormal-distributed random variable X by 

Together with the one-to-one mapping between the cumulants and the moments (3.2) we can compute the cumulants of a lognormal-distributed random variable x as follows 

The summation over the set km satishes the nonnegarive integer equation 

It is well known that the lognormal pdf is more close-to-normal for low volatilities Gy. Hence, following the results of the last section we expect that the EE performs more accurately for lower volatilities Gy and means /iy representing a more close-to-normal pdf. This is also confirmed in figure (3.5) for the approximation of a lognormal pdf with O), = 0.05 and jiy = 0. Running an EE up to the critical order term ° Me =5 leads to a permanent decrease in the approximation error. Overall, the absolute error remains very 

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