I'm looking at pricing a very large deal and while the distribution is kind of "normal," there's quiet a bit of skew and kurtosis that isn't being considered when I use the normal Bachelier's future spread model, which assumes the spread is normally distributed.
The skew is: -0.5, kurtosis: 4.4
So likely, someone has modified the Bachelier formula to incorporate moment matching: mean, standard deviation, skew, and kurtosis, but I can't find a reference paper on the subject. This is a huge basket of 20 underlyings or so that is roughly approximated by Bachelier's, but it's a big enough deal that people aren't comfortable if I don't moment match the distribution. Of course they wouldn't ask me to plot the distribution if I had a lognormal spread model that completely misrepresented the spread dynamics... but it is what it is.