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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.

Much appreciated!

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  • $\begingroup$ Why don't you use a Bachelier model with stochastic volatility, or SABR with $\beta < 1$? $\endgroup$
    – user34971
    Apr 15, 2022 at 13:22
  • $\begingroup$ I think I found a paper integrating all the moments: Schaefer, M. P. 2002. “Pricing and Hedging European Options on Futures Spreads Using the Bachelier Spread Option Model.” Now just working through the paper... $\endgroup$
    – Matt
    Apr 15, 2022 at 15:43
  • $\begingroup$ Haven't read Schaefer's paper. But what you could also do is adjust the Bachelier normal assumption for skew and kurtosis using Gram-Charlier (a la Corrado-Su). $\endgroup$
    – user34971
    Apr 15, 2022 at 17:18
  • $\begingroup$ @FridoRolloos thank you that's much simpler than what I was looking at. Nothing easier than just adjusting a call and put price by a conversion factor for skew and kurtosis. And I don't really have to code much up: mathworks.com/matlabcentral/fileexchange/… $\endgroup$
    – Matt
    Apr 15, 2022 at 18:46
  • $\begingroup$ Welcome. You can't use the Black Scholes Corrado Su though, I think you'd need to derive a Bachelier Corrado Su to allow for negative prices (spreads) + skew + kurtosis. But I think that's doable. Good luck! $\endgroup$
    – user34971
    Apr 15, 2022 at 19:15

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